Hans J.Fahr, Argelander Institute for Astronomy, University of Bonn, Auf dem
Huegel 71, 53121 Bonn Germany.

Hans J.Fahr, Argelander Institute for Astronomy, University of Bonn, Auf dem
Huegel 71, 53121 Bonn Germany.

Resumo: Portanto, um estudo sério deste problema certamente é e deve ser baseado em uma teoria bem fundamentada da expansão cósmica. Em um universo estático, a formação da estrutura segue as linhas que os astrônomos desenvolveram há muito tempo para o espaço estático (por exemplo, Jeans, 1909, 1929, ou mais tarde, ver, por exemplo, Fahr e Willerding, 1998). Os processos de formação de estrutura, é claro, são muito diferentes no universo em expansão, porque então a estrutura para
Por que colapsos de massas cósmicas deveriam acontecer em universos em expansão?

https://www.opastpublishers.com/open-access-articles/at-what-distance-from-us-and-when-starts-the-hubble-expansion.pdf

https://www.opastpublishers.com/open-access-articles/evolution-of-cosmic-structures-in-the-expanding-universe-could-not-one-have-known-it-all-before.pdf

Volume 5 | Issue 4 | 604
At what distance from us and when starts the Hubble expansion?
Research Article
Argelander Institut für Astronomie, Universität Bonn,
Auf dem Huegel 71, 53121 Bonn (Germany)
Hans J. Fahr
*
Corresponding author
Hans J.Fahr, Argelander Institute for Astronomy, University of Bonn, Auf dem
Huegel 71, 53121 Bonn Germany.
Submitted: 19 Oct 2022; Accepted: 01 Nov 2022; Published: 09 Nov 2022
Adv Theo Comp Phy, 2022
Citation:Fahr, H. J. (2022). At what distance from us and when starts the Hubble expansion? Adv Theo Comp Phy, 5(4), 604-607.
Advances in Theoretical & Computational Physics
ISSN: 2639-0108
Abstract
In this article we shall ask what may cause the uniformly distributed, cosmic matter to form singular local mass concentrations in a universe that has started, and according to the general belief never ended till now to expand since the event of the
Big-Bang. Though the so-called Big-Bang till now is a physically rather nebulous cosmic event, all modern cosmology is
centered around it and founded upon it. Our investigations here do show that in fact some forms of a cosmic Hubble expansion do allow for gravitationally driven local matter contractions – even though the universe as a whole is expanding to ever
and ever larger space volumes. For a universe undergoing an unaccelerated, “coasting” Hubble expansion we can show that
the forces connected with the centrifugal Hubble drifts are overcompensated by the centripetal forces of cosmic matter inside
critical local space volumes and thus do form mass concentrations up to Mega-solar masses as soon as the coasting phase of
the expansion has started. To the contrast, in an universe with an accelerated Hubble expansion which is nowadays favoured
by many astrophysicists structure formation is, however, stopped soon after the accelerated expansion has started. That may
serve as a criterion what form of the Hubble expansion in fact predominates in this actual universe.
Why and when in an expanding universe do distributed cosmic masses collapse?
It was Fred Hoyle who coined for the first time the stigmatic
concept of a “Big-Bang”- universe for the scientific community.
This was during a BBC interview in the year 1949. The denotation “Big-Bang” served furtheron as a paradigm for a universe
which originates from a singularity with a gigantic explosion
whose driver is unknown and it continues to expand since then.
But not at all Hoyle did so, because he was convinced by this
idea, rather to the opposite, because he wanted to blame his colleagues like Lémaître, de Sitter, Friedman and others for pushing, in his view, such an absurd scientifc idea. According to his
view the idea of a Big-Bang as origin of the universe was a sheer
nonsense. Nevertheless, however, this concept since these days
till now is indoctrinating the vision of the whole cosmologic science community and its modern cosmic concepts.
The question coming up from such a BB-paradigm necessarily
concerns the place where, if at all, this Big-Bang happened? And
where were we and all the rest of the universe at this event?
The answer is: We and everything else of this universe were
exactly at the same place where the BB happened, namely at
and within the same singularity. Somehow already the famous
Nikolaus Kusanus, the later Bishop of Brixen, at 1400pC. did
express it impressively with his visionary words: This world is
a creation whose center is everywhere, whose border, however,
is – nowhere! Though this perhaps is an intelligent paradigm for
the true nature of the universe, it nevertheless provocates the
fundamental question – if the universe started expanding with
the event of the BB, – why? and when? then after that – did it
stop to expand into larger and larger spaces – to instead locally
create material structures like stars, planetary systems and galaxies? And only these latter things we infact do see when looking
into the nearest and the farthest cosmic environments, while the
BB we do not see. Somehow the BB, however, must have had a
successor in form of the “BC”, the “Big-Collapse” or at least the
“LC”, the “Local collapse” from where local cosmic structures
originated.
What concerns the influence of the general Hubble expansion
on more local structures like e.g. the solar system there exists
already a long list of publications starting perhaps with the consideration of the problem of the “Einstein-Strauss vacuole” (Einstein and Strauss, 1945,1946) with the Einstein-Strauss radius as
that distance where a smooth transition between the Schwartzschild geometry of the local gravitational field into the global
Robertson-Walker geometry can be achieved. More recently this
concept has been specifically applied to the case of our solar
system and it has been shown there how the transition from the
local to the global spacetime geometry can be probed by radiotrackings of space probes like especially the NASA space probe
PIONEER-10 manifesting the spectacular Pioneer-10 radio
tracking anomaly ( Fahr and Siewert, 2007, 2008). But all these
studies do take the solar system and the global Hubble expansion
already as given facts, not asking how local mass structures can
originate in a globally expanding universe. To study this latter
point one rather has to pay attention to the following aspects:
Adv Theo Comp Phy, 2022 Volume 5 | Issue 4 | 605
Contraction of distributed matter in expanding universes
If matter is at rest with respect to an inertial rest frame, then
it must move with respect to that frame when a force is acting
upon it. If now somehow the cosmic inertial rest frame in fact
is a general-relativistic, dynamic rest frame – like the cosmic
Hubble-Lemaître rest frame – then a decoupling from the cosmic
expansion is only possible, if a counter-expansion force Kc
is
acting on the matter which is larger than the Hubble-Lemaître
force KHL i.e. if |KHL| ≤ |Kc
|. The force KHL is connected with the
general, differential cosmic Hubble drift vHL = H • D in a distance
D from the selected origin of the coordinate system and is given
by:
On the other hand, the counter-expansion force Kc may be immaginable as due to the gravitational attraction force of a central
mass Mc
at the origin of the selected coordinate system. This
mass Mc
is thought to be due to the accumulated cosmic, originally homogeneously distributed masses inside a sphere with
radius D. Hence one finds, with G denoting Newton‘s gravitational constant and ρ(R) = ρ0
• (R0
/R)
3
denoting the actual average cosmic mass density at the cosmic scale R
Contraction of distributed matter in expanding universes
If matter is at rest with respect to an inertial rest frame, then it must move with respect
to that frame when a force is acting upon it. If now somehow the cosmic inertial rest
frame in fact is a general-relativistic, dynamic rest frame – like the cosmic
Hubble-Lemaître rest frame – then a decoupling from the cosmic expansion is only
possible, if a counter-expansion force Kc is acting on the matter which is larger than the
Hubble-Lemaître force KHL i.e. if |KHL |  |Kc |. The force KHL is connected with the
general, differential cosmic Hubble drift vHL  H  D in a distance D from the selected
origin of the coordinate system and is given by:
KHL  m d
dt vHL  m d
dt H  D  mH  D  H  D   m R
R  R 2
R2 D  H  D 
On the other hand, the counter-expansion force Kc may be immaginable as due to the
gravitational attraction force of a central mass Mc at the origin of the selected coordinate
system. This mass Mc is thought to be due to the accumulated cosmic, originally
homogeneously distributed masses inside a sphere with radius D. Hence one finds , with
G denoting Newton‘s gravitational constant and R  0  R0/R3 denoting the actual
average cosmic mass density at the cosmic scale R
Kc  G mMc
D2   4
3 RD3 mG
D2   4
3 mG0   R0
R 3  D
Hereby the replacement of Mc was made by DcR  R  Mc/4/3R0
3o1/3, i.e. the
quantity D  o  R and R are strictly proportional to eachother. In order then to have the
Hubble expansion reversed into a local contraction one needs to have |Kc |  KHL, i.e.
4
3 G0R0/R3  D   R
R  R 2
R2 D  H  D
Accelerated Inertial Hubble frames:
Structure formation in an
expanding universe
Evolving mass center Centrifugal
Hubble forces
D
Centripetal
Newton forces
Figure 1: Schematic illustration of structure formation in a cosmic Hubble frame
To further study and analyse the meaning of this above relation, one needs to have a
look into the Hubble dynamics which determines the quantities R and R as functions of
R. These relations are multiform and have a large variety of possible solutions under
general cosmic conditions as recently again analysed in Fahr (2022). There it is shown
that the Hubble parameter H  HR is variable with the scale R of the universe in very
many different forms dependend on the relative contributions b , d, ,  of densities
of baryonic mass, dark matter mass, photons, and vacuum energy to the cosmic
Hereby the replacement of Mc
was made by Dc
(R) = R • [Mc
/(4π/3)R0
3
ρo]1/3, i.e. the quantity D = Δo • R and R are strictly proportional
to each other. In order then to have the Hubble expansion reversed into a local contraction one needs to have |Kc
| ≥ KHL, i.e.
On the other hand, the counter-expansion force Kc may be immaginable as due to the
gravitational attraction force of a central mass Mc at the origin of the selected coordinate
system. This mass Mc is thought to be due to the accumulated cosmic, originally
homogeneously distributed masses inside a sphere with radius D. Hence one finds , with
G denoting Newton‘s gravitational constant and R  0  R0/R3 denoting the actual
average cosmic mass density at the cosmic scale R
Kc  G mMc
D2   4
3 RD3 mG
D2   4
3 mG0   R0
R 3  D
Hereby the replacement of Mc was made by DcR  R  Mc/4/3R0
3o1/3, i.e. the
quantity D  o  R and R are strictly proportional to eachother. In order then to have the
Hubble expansion reversed into a local contraction one needs to have |Kc |  KHL, i.e.
4
3 G0R0/R3  D   R
R  R 2
R2 D  H  D
Accelerated Inertial Hubble frames:
Structure formation in an
expanding universe
Evolving mass center Centrifugal
Hubble forces
D
Centripetal
Newton forces
Figure 1: Schematic illustration of structure formation in a cosmic Hubble frame
To further study and analyse the meaning of this above relation, one needs to have a
look into the Hubble dynamics which determines the quantities R and R as functions of
R. These relations are multiform and have a large variety of possible solutions under
general cosmic conditions as recently again analysed in Fahr (2022). There it is shown
that the Hubble parameter H  HR is variable with the scale R of the universe in very
many different forms dependend on the relative contributions b , d, ,  of densities
of baryonic mass, dark matter mass, photons, and vacuum energy to the cosmic
Figure 1: Schematic illustration of structure formation in a cosmic Hubble frame
To further study and analyse the meaning of this above relation,
one needs to have a look into the Hubble dynamics which determines the quantities and as functions of R. These relations
are multiform and have a large variety of possible solutions under general cosmic conditions as recently again analysed in Fahr
(2022). There it is shown that the Hubble parameter H = H(R) is
variable with the scale R of the universe in very many different
forms dependend on the relative contributions ρb
, ρd
, ρv
, ρΛ of
densities of baryonic mass, dark matter mass, photons, and vacuum energy to the cosmic energy-momentum tensor.
Here in this article we do not play with all these possible options, instead we concentrate on one option mainly, namely the
one leading to a so-called “coasting” Hubble expansion with R ̈
= 0, describing the unaccelerated universe. This particular case
in fact always prevails, if vacuum energy density ρΛ dominates
over all the other contributions ρb
, ρd
, ρv
at later phases of the
cosmic expansion, since as shown in Fahr and Heyl (2021) and
Fahr (2022) ρΛ varies with the scale R like R-2. Under these prerequisites the Hubble parameter H(R) can be written in the form
(Fahr and Heyl, 2021):
energy-momentum tensor.
Here in this article we do not play with all these possible options, instead we
concentrate on one option mainly, namely the one leading to a so-called “coasting”
Hubble expansion with R  0, describing the unaccelerated universe. This particular
case in fact always prevails, if vacuum energy density  dominates over all the other
contributions b, d,  at later phases of the cosmic expansion, since as shown in Fahr
and Heyl (2021) and Fahr (2022)  varies with the scale R like R2. Under these
prerequisites the Hubble parameter HR can be written in the form (Fahr and Heyl,
2021):
H2R  R 2/R2  8G
3   8G
3 ,0  R0/R2
which allows to conclude that in this case one finds R 2  8G
3 ,0  R02  const, i.e.
the so-called “coasting expansion”.with R  0! and
HR  8G
3 ,0  R0/R  H0  R0/R
in which case one finds the above derived requirement given by the relation
4
3 G0R0/R3  D   R 2
R2 D  H  D  H0
2 R0
2
R2 D  H0
R0
R  D
which allows to conclude that in this case one finds R2
= 8πG/3 ρΛ,
0

• (R0
  )2
  = const, i.e. the so-called “coasting expansion”.with =
  0! and
  energy-momentum tensor.
  Here in this article we do not play with all these possible options, instead we
  concentrate on one option mainly, namely the one leading to a so-called “coasting”
  Hubble expansion with R  0, describing the unaccelerated universe. This particular
  case in fact always prevails, if vacuum energy density  dominates over all the other
  contributions b, d,  at later phases of the cosmic expansion, since as shown in Fahr
  and Heyl (2021) and Fahr (2022)  varies with the scale R like R2. Under these
  prerequisites the Hubble parameter HR can be written in the form (Fahr and Heyl,
  2021):
  H2R  R 2/R2  8G
  3   8G
  3 ,0  R0/R2
  which allows to conclude that in this case one finds R 2  8G
  3 ,0  R02  const, i.e.
  the so-called “coasting expansion”.with R  0! and
  HR  8G
  3 ,0  R0/R  H0  R0/R
  in which case one finds the above derived requirement given by the relation
  4
  3 G0R0/R3  D   R 2
  R2 D  H  D  H0
  2 R0
  2
  R2 D  H0
  R0
  R  D
  in which case one finds the above derived requirement given by the relation
  On the other hand, the counter-expansion force Kc may be immaginable as due to the
  gravitational attraction force of a central mass Mc at the origin of the selected coordinate
  system. This mass Mc is thought to be due to the accumulated cosmic, originally
  homogeneously distributed masses inside a sphere with radius D. Hence one finds , with
  G denoting Newton‘s gravitational constant and R  0  R0/R3 denoting the actual
  average cosmic mass density at the cosmic scale R
  Kc  G mMc
  D2   4
  3 RD3 mG
  D2   4
  3 mG0   R0
  R 3  D
  Hereby the replacement of Mc was made by DcR  R  Mc/4/3R0
  3o1/3, i.e. the
  quantity D  o  R and R are strictly proportional to eachother. In order then to have the
  Hubble expansion reversed into a local contraction one needs to have |Kc |  KHL, i.e.
  4
  3 G0R0/R3  D   R
  R  R 2
  R2 D  H  D
  Accelerated Inertial Hubble frames:
  Structure formation in an
  expanding universe
  Evolving mass center Centrifugal
  Hubble forces
  D
  Centripetal
  Newton forces
  Figure 1: Schematic illustration of structure formation in a cosmic Hubble frame
  To further study and analyse the meaning of this above relation, one needs to have a
  look into the Hubble dynamics which determines the quantities R and R as functions of
  R. These relations are multiform and have a large variety of possible solutions under
  general cosmic conditions as recently again analysed in Fahr (2022). There it is shown
  that the Hubble parameter H  HR is variable with the scale R of the universe in very
  many different forms dependend on the relative contributions b , d, ,  of densities
  of baryonic mass, dark matter mass, photons, and vacuum energy to the cosmic
  d, the counter-expansion force Kc may be immaginable as due to the
  on force of a central mass Mc at the origin of the selected coordinate
  Mc is thought to be due to the accumulated cosmic, originally
  tributed masses inside a sphere with radius D. Hence one finds , with
  s gravitational constant and R  0  R0/R3 denoting the actual
  ss density at the cosmic scale R
  G mMc
  D2   4
  3 RD3 mG
  D2   4
  3 mG0   R0
  R 3  D
  cement of Mc was made by DcR  R  Mc/4/3R0
  3o1/3, i.e. the
  and R are strictly proportional to eachother. In order then to have the
  eversed into a local contraction one needs to have |Kc |  KHL, i.e.
  4
  3 G0R0/R3  D   R
  R  R 2
  R2 D  H  D
  Accelerated Inertial Hubble frames:
  Structure formation in an
  expanding universe
  Evolving mass center Centrifugal
  Hubble forces
  D
  Centripetal
  Newton forces
  d, the counter-expansion force Kc may be immaginable as due to the
  on force of a central mass Mc at the origin of the selected coordinate
  Mc is thought to be due to the accumulated cosmic, originally
  tributed masses inside a sphere with radius D. Hence one finds , with
  s gravitational constant and R  0  R0/R3 denoting the actual
  ss density at the cosmic scale R
  G mMc
  D2   4
  3 RD3 mG
  D2   4
  3 mG0   R0
  R 3  D
  cement of Mc was made by DcR  R  Mc/4/3R0
  3o1/3, i.e. the
  and R are strictly proportional to eachother. In order then to have the
  eversed into a local contraction one needs to have |Kc |  KHL, i.e.
  4
  3 G0R0/R3  D   R
  R  R 2
  R2 D  H  D
  Accelerated Inertial Hubble frames:
  Structure formation in an
  expanding universe
  Evolving mass center Centrifugal
  Hubble forces
  D
  Centripetal
  Newton forces
  On the other hand, the counter-expansion force Kc may bgravitational attraction force of a central mass Mc at the origsystem. This mass Mc is thought to be due to the accumulahomogeneously distributed masses inside a sphere with radG denoting Newton‘s gravitational constant and R  0 average cosmic mass density at the cosmic scale R
  Kc  G mMc
  D2   4
  3 RD3 mG
  D2   4
  3 mHereby the replacement of Mc was made by DcR  R  quantity D  o  R and R are strictly proportional to eachothHubble expansion reversed into a local contraction one nee4
  3 G0R0/R3  D   R
  R  R 2
  R2 D Accelerated Inertial HubbleECentrifugal

Adv Theo Comp Phy, 2022 Volume 5 | Issue 4 | 606
3 in which case one finds the above derived requirement given by the relation
4
3 G0R0/R3  D   R 2
R2 D  H  D  H0
2 R0
2
R2 D  H0
R0
R  D
or :
4
3 G0R0/R3  H0
2 R0
2
R2   H0
R0
R  D
D
and using now the proportionality between D and R given by
DcR  R  Mc/4/3R0
3o1/3 one finds:
4
3 G0R0/R3  H0
2 R0
2
R2   H0
R0
R  R
R  0 !
That, interestingly enough, means for a coasting universe that this above requirement
is always fulfilled, as soon and as long as the coasting expansion prevails, meaning that
at all those times masses of all sizes Mc on the basis of the collapse of uniformly
distributed cosmic matter density 0 can be generated.
In contrast to the above, according to Einstein‘s introduction of the vacuum energy in
the form of the constant vacuum energy density  (Einstein‘s famous cosmologic
“constant”!) one obtains for the later phases of the cosmic expansion, derived from the
two Friedmann equations (see Goenner, 1996):
HR  8G
3   const
which means H  0! and R  R 8G
3  . This in contrast to the above relation thus
then implies
and using now the proportionality between D and R given by
4
3 G0R0/R3  D   R 2
R2 D  H  D  H0
2 R0
2
R2 D  H0
R0
R  D
or :
4
3 G0R0/R3  H0
2 R0
2
R2   H0
R0
R  D
D
and using now the proportionality between D and R given by
DcR  R  Mc/4/3R0
3o1/3 one finds:
4
3 G0R0/R3  H0
2 R0
2
R2   H0
R0
R  R
R  0 !
That, interestingly enough, means for a coasting universe that this above requirement
is always fulfilled, as soon and as long as the coasting expansion prevails, meaning that
at all those times masses of all sizes Mc on the basis of the collapse of uniformly
distributed cosmic matter density 0 can be generated.
In contrast to the above, according to Einstein‘s introduction of the vacuum energy in
the form of the constant vacuum energy density  (Einstein‘s famous cosmologic
“constant”!) one obtains for the later phases of the cosmic expansion, derived from the
two Friedmann equations (see Goenner, 1996):
HR  8G
3   const
which means H  0! and R  R 8G
3  . This in contrast to the above relation thus
then implies
That, interestingly enough, means for a coasting universe that
this above requirement is always fulfilled, as soon and as long as
the coasting expansion prevails, meaning that at all those times
masses of all sizes Mc on the basis of the collapse of uniformly
distributed cosmic matter density ρ0
can be generated.
In contrast to the above, according to Einstein‘s introduction of
the vacuum energy in the form of the constant vacuum energy
density Λ (Einstein‘s famous cosmologic “constant”!) one obtains for the later phases of the cosmic expansion, derived from
the two Friedmann equations (see Goenner, 1996):
4
3 G0R0/R3  D   R 2
R2 D  H  D  H0
2 R0
2
R2 D  H0
R0
R  D
or :
4
3 G0R0/R3  H0
2 R0
2
R2   H0
R0
R  D
D
and using now the proportionality between D and R given by
DcR  R  Mc/4/3R0
3o1/3 one finds:
4
3 G0R0/R3  H0
2 R0
2
R2   H0
R0
R  R
R  0 !
That, interestingly enough, means for a coasting universe that this above requirement
is always fulfilled, as soon and as long as the coasting expansion prevails, meaning that
at all those times masses of all sizes Mc on the basis of the collapse of uniformly
distributed cosmic matter density 0 can be generated.
In contrast to the above, according to Einstein‘s introduction of the vacuum energy in
the form of the constant vacuum energy density  (Einstein‘s famous cosmologic
“constant”!) one obtains for the later phases of the cosmic expansion, derived from the
two Friedmann equations (see Goenner, 1996):
HR  8G
3   const
which means H  0! and R  R 8G
3  . This in contrast to the above relation thus
then implies
which means H = 0! and R = This in contrast to the above relation thus then implies
4
3 G0R0/R3  H0  R
R  H0
2
meaning that as soon as the cosmic density R~R3 during the expansion of the
universe has fallen off too much, no mass contractions can happen anymore during all
the time of the ongoing expansion of the universe.
Conclusions
It may appear for cosmologists as one of the biggest enigmas that in an initially
homogeneous universe under the ongoing Hubble expansion local mass structures like
stars, stellar systems and systems of galaxies could have been formed. As we have,
however, shown in this article here, structure formation is possible even under conditions
of an expanding universe, though the form of the underlying expansion of the universe
must, however, be specific for that purpose; it namely must be an “unaccelerated”,
“coasting” expansion, while under famous astrophysicists of these decades the
accelerated expansion is strongly in favour. In order to explain the redshifts of galaxies
with the most distant SN-1a supernovae Perlmutter et al.(1998), Schmidt et al. (1998) or
Riess et al. (1998) have prefered an accelerated expansion of the universe, associated
with the action of a constant vacuum energy density  as initially proposed by Einstein
(1917). However, as we do show here, structure formation and build-up of solar systems
and galaxies is impeded as soon as the universe starts expanding in an accelerated
form, only as long as the expansion takes place in an unaccelerated, coasting form then
structure formation can continue to happen in the universe. And this is important, since
our solar system may live for about 108 years, but in a universe which is already about

8. 7  109 years old, such systems must be reborn, in order to be visible at our time
   period. Perhaps this can be used as a criterion which form of a Hubble expansion is
   characteristic for our actual present universe.
   meaning that as soon as the cosmic density ρ(R)~R-3 during the
   expansion of the universe has fallen off too much, no mass contractions can happen anymore during all the time of the ongoing
   expansion of the universe.
   Conclusions
   It may appear for cosmologists as one of the biggest enigmas
   that in an initially homogeneous universe under the ongoing
   Hubble expansion local mass structures like stars, stellar systems and systems of galaxies could have been formed. As we
   have, however, shown in this article here, structure formation
   is possible even under conditions of an expanding universe,
   though the form of the underlying expansion of the universe
   must, however, be specific for that purpose; it namely must be
   an “unaccelerated”, “coasting” expansion, while under famous
   astrophysicists of these decades the accelerated expansion is
   strongly in favour. In order to explain the redshifts of galaxies
   with the most distant SN-1a supernovae Perlmutter et al.(1998),
   Schmidt et al. (1998) or Riess et al. (1998) have prefered an
   accelerated expansion of the universe, associated with the action
   of a constant vacuum energy density ρΛ as initially proposed by
   Einstein (1917). However, as we do show here, structure formation and build-up of solar systems and galaxies is impeded
   as soon as the universe starts expanding in an accelerated form,
   only as long as the expansion takes place in an unaccelerated,
   coasting form then structure formation can continue to happen in
   the universe. And this is important, since our solar system may
   live for about 108
   years, but in a universe which is already about
9. 7 • 109
   years old, such systems must be reborn, in order to be
   visible at our time period. Perhaps this can be used as a criterion
   which form of a Hubble expansion is characteristic for our actual
   present universe.
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    HR 3 ,0 R0/R H0 R0/R
    in which case one finds the above derived requirement given by the relation
    4
    3 G0R0/R3  D   R 2
    R2 D  H  D  H0
    2 R0
    2
    R2 D  H0 R0
    R  D
    or :
    4
    3 G0R0/R3  H0
    2 R0
    2
    R2   H0 R0
    R  D
    D
    and using now the proportionality between D and R given by
    DcR  R  Mc/4/3R0
    3o1/3 one finds:
    4
    3 G0R0/R3  H0
    2 R0
    2
    R2   H0 R0
    R  R
    R  0 !
    That, interestingly enough, means for a coasting universe that this above requirement
    is always fulfilled, as soon and as long as the coasting expansion prevails, meaning that
    at all those times masses of all sizes Mc on the basis of the collapse of uniformly
    distributed cosmic matter density 0 can be generated.
    In contrast to the above, according to Einstein‘s introduction of the vacuum energy in
    the form of the constant vacuum energy density  (Einstein‘s famous cosmologic
    “constant”!) one obtains for the later phases of the cosmic expansion, derived from the
    two Friedmann equations (see Goenner, 1996):
    HR  8G
    3   const
    which means H  0! and R  R 8G
    3  . This in contrast to the above relation thus
    then implies
    Adv Theo Comp Phy, 2022 Volume 5 | Issue 4 | 607
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    Copyright: ©2022 Hans J.Fahr. This is an open-access article
    distributed under the terms of the Creative Commons Attribution
    License, which permits unrestricted use, distribution, and reproduction
    in any medium, provided the original author and source are credited.
    https://opastpublishers.com

Volume 5 | Issue 3 | Volume 5 | Issue 3 | 524
Evolution of cosmic structures in the expanding universe: Could not one have known
it all before?
Research Article
1
Argelander Institute for Astronomy, University of Bonn,
Auf dem Huegel 71, 53121 Bonn Germany
2
Deutsches Zentrum f¨ur Luft- und Raumfahrt (DLR),
K¨onigswinterer Strasse 522-524, 53227 Bonn (Germany)
Hans J.Fahr1* and M. Heyl2
*
Corresponding author
Hans J.Fahr, Argelander Institute for Astronomy, University of Bonn, Auf dem
Huegel 71, 53121 Bonn Germany.
Submitted: 28 Jul 2022; Accepted: 30 Jul 2022; Published: 05 Aug 2022
Adv Theo Comp Phy, 2022
Abstract
Most recent observations from the James Webb space telescope (JWST) have shown by highly resolved infrared
observations of highest sensitivity that structure formation in the universe into the forms of early galaxies has
already taken place at cosmic times less than 0.6 Gigayears after the Big-Bang. This is taken up with a big
surprise in the whole astronomic community, though, as it seems, it could have been predicted from simple
theoretical considerations. In this article, we are demonstrating that this result already would have clearly
come out from theoretical considerations of gravitational structure formation processes in the early expanding
universe just after the cosmic matter recombination period. While, however, it can be easily understood how
matter structures of the order of 108
solar masses could evolve in the cosmic meantime, it nevertheless remains
obscure, how galaxies of the type of the Milky way or more massive structures with 1011 or more solar masses
can have evolved up to the present cosmic days without some not yet specified collapse-accelerating processes.
Citation: Hans J.Fahr. (2022). Evolution of cosmic structures in the expanding universe: Could not one have known it all before?
Adv Theo Comp Phy, 5(3), 524-527.
Advances in Theoretical & Computational Physics
ISSN: 2639-0108
Collapse in Expanding Universes
In principle it is a problem hard to understand that matter may
be able to collapse into large local mass units, though in an expanding universe the initially widely and uniformly distributed
cosmic matter must be subject to the expansion into a permanently growing cosmic space with permanently decreasing cosmic mass densities. This only can be possible, if the structuring
collapse velocity is larger than the general expansion velocity.
The problem thus evidently is and must be connected with the
specific form of the actual expansion dynamics of the whole universe.
Therefore this study certainly is and must be based on the specific
form of the cosmic expansion of the universe. In a static universe
structure formation runs along the lines that astronomers have
developed since long ago for the static space [1, 2]. Processes
of structure formation of course are very much different in the
expanding universe, because then structure formation definitely
will depend on the specific form of the prevailing cosmic expansion (e.g. decelerated, accelerated or coasting expansion etc.).
To best explain the SN 1a luminosities Perlmutter et al. (1998),
Schmidt et al.(1998), or Riess et al. (1998) have preferred the
accelerated expansion of the universe connected with action of
a constant vacuum energy density [3-10], however, there are
attempts by Casado (2011) and Casado and Jou (2013) showing that a ”coasting non-accelerated universe” can equally well
explain these supernovae luminosities [11-12]. In our following
con- siderations we shall consider first here – mainly for mathematical reasons – the case of a “coasting expansion” [13-16],
which in fact can be expected to prevail, if the universe expands
under the form of thermodynamic and gravidynamic action of
vacuum pressure [17]. Alternative forms of a cosmic expansion
may be discussed at the end of this paper and lead to very interesting conclusions.
If then as our working basis such a ”coasting universe” can be
assumed to prevail, like given in the case when ρΛ~R−2 (ϱΛ denoting the mass density equivalent of the vacuum energy, R denoting the scale of the universe, see e.g. Fahr, 2022) and when
vacuum energy is the dominant ingredient to the cosmic mass
density ρΛ ≫ρb , ρd , ρν
, (indices b, d, ν standing for baryons, dark
matter, and photons, respectively) and to the relativistic energy-momentum tensor, then one unavoidably finds:

which in fact means and necessarily implies: a ”coasting expansion” of the universe! Then consequently, a Hubble parameter
must be expected falling off with the scale R like:
This means that the Hubble parameter in course of the coasting
cosmic expansion decreases like H~R−1!
Therefore this study certainly is and must be based on the specific form of the cosmic expastatic universe structure formation runs along the lines that astronomers have developed sispace [1, 2]. Processes of structure formation of course are very much different in the expthen structure formation definitely will depend on the specific form of the prevailingdecelerated, accelerated or coasting expansion etc.). To best explain the SN 1a luminositieSchmidt et al.(1998), or Riess et al. (1998) have preferred the accelerated expansion of thaction of a constant vacuum energy density [3-10], however, there are attempts by Casado ((2013) showing that a ”coasting non-accelerated universe” can equally well explain thes[11-12]. In our following con- siderations we shall consider first here – mainly for mathemat“coasting expansion” [13-16], which in fact can be expected to prevail, if the universe ethermodynamic and gravidynamics action of vacuum pressure [17]. Alternative forms of a discussed at the end of this paper and lead to very interesting conclusions.
If then as our working basis such a ”coasting universe” can be assumed to prevail, like give(ϱΛ denoting the mass density equivalent of the vacuum energy, R denoting the scale of th2022) and when vacuum energy is the dominant ingredient to the cosmic mass density ρΛ ≫standing for baryons, dark matter, and photons, respectively) and to the relativistic energone unavoidably finds:
which in fact means and necessarily implies: a ”coasting expansion” of the universe! Theparameter must be expected falling off with the scale R like:
This means that the Hubble parameter in course of the coasting cosmic expansion decreases Under these cosmic auspices one finds that the local free-fall time τff = (4πGϱ)
−1/2 of barJeans, 1929) is smaller than the expansion time τex = 1/H of that matter (i.e. so that mass stthe expanding universe!), as soon as:
Therefore this study certainly is and must be based on the specific form of the cosmic exstatic universe structure formation runs along the lines that astronomers have developedspace [1, 2]. Processes of structure formation of course are very much different in the then structure formation definitely will depend on the specific form of the prevaildecelerated, accelerated or coasting expansion etc.). To best explain the SN 1a luminosSchmidt et al.(1998), or Riess et al. (1998) have preferred the accelerated expansion ofaction of a constant vacuum energy density [3-10], however, there are attempts by Casad(2013) showing that a ”coasting non-accelerated universe” can equally well explain th[11-12]. In our following con- siderations we shall consider first here – mainly for mathem“coasting expansion” [13-16], which in fact can be expected to prevail, if the universethermodynamic and gravidynamics action of vacuum pressure [17]. Alternative forms ofdiscussed at the end of this paper and lead to very interesting conclusions.
If then as our working basis such a ”coasting universe” can be assumed to prevail, like g(ϱΛ denoting the mass density equivalent of the vacuum energy, R denoting the scale o2022) and when vacuum energy is the dominant ingredient to the cosmic mass density ρstanding for baryons, dark matter, and photons, respectively) and to the relativistic eneone unavoidably finds:
which in fact means and necessarily implies: a ”coasting expansion” of the universe! Tparameter must be expected falling off with the scale R like:
This means that the Hubble parameter in course of the coasting cosmic expansion decreasUnder these cosmic auspices one finds that the local free-fall time τff = (4πGϱ)
−1/2 of bJeans, 1929) is smaller than the expansion time τex = 1/H of that matter (i.e. so that massthe expanding universe!), as soon as:
(1)
(2)
Volume 5 | Issue 3 | 525
Under these cosmic auspices one finds that the local free-fall
time τff
= (4πGϱ)−1/2 of baryonic, cosmic matter (see Jeans, 1929)
is smaller than the expansion time τex = 1/H of that matter (i.e. so
that mass structures can grow even in the expanding universe!),
as soon as:
i,e, if actual free fall times are shorter than expansion times of
material structures. This means one would need to have the following relation fulfilled:
or:
This implies that the critical scale Rc from which upwards a
progress of structuring despite of cosmic expansion can and will
occur is given by:
That means for world times with R(t) ≥ Rc
one thus cannot expect
to have any more a homogeneous cosmic matter distribution, but
a hierarchical mass structure in the universe like described by
Fahr and Heyl (2019) [18].
On the other hand – since matter can anyway not gravitationally
condense, before it has recombined to neutral atoms due to the
strong interactions of free electrons with the strongly coupling
el-mag. radiation fields (photon fields), one can therefore start
this consideration here with the time t
0
= t
r
of matter recombination, since before that time no irreversible condensations are
possible in the form of enduring, persisting structures. Hence
along this argumentation one might find this critical scale by:
This obviously says that structuring of cosmic matter can only
start when the scale of the world has increased to at least Rc
=
5.02 • Rr , i.e. to about five times the recombination scale Rr
!
The question now may pose itself concerning the critical mass
Mc
that is connected with such a selfstructuring mass unit Mc
(Rc
).
The answer must come from the usual knowledge of the collapse-critical mass unit Mc
given by a comparison of the free-fall
time τff and the sound time (pressure counterreaction time) given
by τs
= D/cs
(D being the radial dimension of the collaps-critical
mass unit Mc) and thus leading to the following request:
with P and γ denoting the baryonic gas pressure and the polytropic index of the gas. This then with Fahr and Heyl (2021)
leads to the following expression for Mc
= Mc
(R):
Assuming that pressure and density during the cosmic expansion conserve the gas entropy, i.e.P/ϱγ
= const, then leads to the
result:
where hereby the typical collapse mass Mc
(Rr
) at the recombination scale has been calculated by Fahr and Heyl [19] to be
Mc
(Rr
) = 105 Mօ
.
It is interesting now to ask at what cosmic times t ≥ t
r
the first
gravitationally bound mass structures of galactic type according
to the above considerations can be expected as existing in the
universe? This now can be answered with the following calculation:
Under the prerequisites which we have discussed before this
leads to the following result:
where τ0
= 1/H0
denotes the present age of the universe (say 13.7
Gi- gayears!). The recombination scale Rr hereby was estimated with the redshift zr ≃ 103
of the CMB (Cosmic Microwave
Background, see Bennet et al., 2003) [20] through R0 / Rr = 1 +
zr ≃1000. That implies that already at a time of ∆t = 4 ∙ 10−3∙ τ0
after the recombination point matter could start creating gravitationally bound, selfsustained, collapsed structures! So far this
result at least seems to be out of any conflict with the most recent James-Webb ST observations stating that already at times
of 13.1 Giga years before our present time galaxies and stellar structures appear to have been present in the early universe
which were very much similar to our present day galaxies.
The only remaining problem from our theoretical explanations
presented here in this paper are the predicted typical masses of
present day galaxies com- pared to their realistic values which
are about a factor of 1000 larger. While present day galaxies –
like typically the one of our Milky way – have typical masses of
about 1011 solar masses, our above theoretical predictions for the
present-day collaps masses would rather give us
parameter must be expected falling off with the scale R like:
This means that the Hubble parameter in course of the coasting cosmic expansion decreases like H ˜ R−1
!
Under these cosmic auspices one finds that the local free-fall time τff = (4πGϱ)
−1/2 of baryonic, cosmic matter (see
Jeans, 1929) is smaller than the expansion time τex = 1/H of that matter (i.e. so that mass structures can grow even in
the expanding universe!), as soon as:
i,e, if actual free fall times are shorter than expansion times of materialstructures. This means one would
need to have the following relation fulfilled:
parameter must be expected falling off with the scale R like:
This means that the Hubble parameter in course of the coasting cosmic expansion decreases like H ˜ R−1
!
Under these cosmic auspices one finds that the local free-fall time τff = (4πGϱ)
−1/2 of baryonic, cosmic matter (see
Jeans, 1929) is smaller than the expansion time τex = 1/H of that matter (i.e. so that mass structures can grow even in
the expanding universe!), as soon as:
i,e, if actual free fall times are shorter than expansion times of materialstructures. This means one would
need to have the following relation fulfilled:
or:
This implies that the critical scale Rc from which upwards a progress of structuring despite of cosmic expansion can
and will occur is given by:
That means for world times with R(t) ≥ Rc one thus cannot expect to have any more a homogeneous cosmic matter
distribution, but a hierarchical mass structure in the universe like described by Fahr and Heyl (2019) [18].
On the other hand – since matter can anyway not gravitationally condense, before it has recombined to neutral atoms
due to the strong interactions of free electrons with the strongly coupling el-mag. radiation fields (photon fields), one
can therefore start this consideration here with the time t0 = tr of matter recombination, since before that time no
irreversible condensations are possible in the form of enduring, persisting structures. Hence along this argumentation
one might find this critical scale by:
This obviously says that structuring of cosmic matter can only start when the scale of the world has increased to at
least Rc = 5.02 · Rr, i.e. to about five times the recombination scale Rr!
The question now may pose itself concerning the critical mass Mc that is connected with such a selfstructuring mass
unit Mc(Rc). The answer must come from the usual knowledge of the collapse-critical mass unit Mc given by a
comparison of the free-fall time τff and the sound time (pressure counterreaction time) given by τs = D/cs (D being the
radial dimension of the collaps-critical mass unit Mc) and thus leading to the following request:
with P and γ denoting the baryonic gas pressure and the polytropic index of the gas. This then with Fahr and Heyl
(2021) leads to the following expression for Mc = Mc(R):
or:
This implies that the critical scale Rc from which upwards a progress of structuring despite of cosmic expansion can
and will occur is given by:
That means for world times with R(t) ≥ Rc one thus cannot expect to have any more a homogeneous cosmic matter
distribution, but a hierarchical mass structure in the universe like described by Fahr and Heyl (2019) [18].
On the other hand – since matter can anyway not gravitationally condense, before it has recombined to neutral atoms
due to the strong interactions of free electrons with the strongly coupling el-mag. radiation fields (photon fields), one
can therefore start this consideration here with the time t0 = tr of matter recombination, since before that time no
irreversible condensations are possible in the form of enduring, persisting structures. Hence along this argumentation
one might find this critical scale by:
This obviously says that structuring of cosmic matter can only start when the scale of the world has increased to at
least Rc = 5.02 · Rr, i.e. to about five times the recombination scale Rr!
The question now may pose itself concerning the critical mass Mc that is connected with such a selfstructuring mass
unit Mc(Rc). The answer must come from the usual knowledge of the collapse-critical mass unit Mc given by a
comparison of the free-fall time τff and the sound time (pressure counterreaction time) given by τs = D/cs (D being the
radial dimension of the collaps-critical mass unit Mc) and thus leading to the following request:
with P and γ denoting the baryonic gas pressure and the polytropic index of the gas. This then with Fahr and Heyl
(2021) leads to the following expression for Mc = Mc(R):
or:
This implies that the critical scale Rc from which upwards a progress of structuring despite of cosmic expansion can
and will occur is given by:
That means for world times with R(t) ≥ Rc one thus cannot expect to have any more a homogeneous cosmic matter
distribution, but a hierarchical mass structure in the universe like described by Fahr and Heyl (2019) [18].
On the other hand – since matter can anyway not gravitationally condense, before it has recombined to neutral atoms
due to the strong interactions of free electrons with the strongly coupling el-mag. radiation fields (photon fields), one
can therefore start this consideration here with the time t0 = tr of matter recombination, since before that time no
irreversible condensations are possible in the form of enduring, persisting structures. Hence along this argumentation
one might find this critical scale by:
This obviously says that structuring of cosmic matter can only start when the scale of the world has increased to at
least Rc = 5.02 · Rr, i.e. to about five times the recombination scale Rr!
The question now may pose itself concerning the critical mass Mc that is connected with such a selfstructuring mass
unit Mc(Rc). The answer must come from the usual knowledge of the collapse-critical mass unit Mc given by a
comparison of the free-fall time τff and the sound time (pressure counterreaction time) given by τs = D/cs (D being the
radial dimension of the collaps-critical mass unit Mc) and thus leading to the following request:
with P and γ denoting the baryonic gas pressure and the polytropic index of the gas. This then with Fahr and Heyl
(2021) leads to the following expression for Mc = Mc(R):
Adv Theo Comp Phy, 2022
or:
This implies that the critical scale Rc from which upwards a progress of structuring despite of cosmic expansion can
and will occur is given by:
That means for world times with R(t) ≥ Rc one thus cannot expect to have any more a homogeneous cosmic matter
distribution, but a hierarchical mass structure in the universe like described by Fahr and Heyl (2019) [18].
On the other hand – since matter can anyway not gravitationally condense, before it has recombined to neutral atoms
due to the strong interactions of free electrons with the strongly coupling el-mag. radiation fields (photon fields), one
can therefore start this consideration here with the time t0 = tr of matter recombination, since before that time no
irreversible condensations are possible in the form of enduring, persisting structures. Hence along this argumentation
one might find this critical scale by:
This obviously says that structuring of cosmic matter can only start when the scale of the world has increased to at
least Rc = 5.02 · Rr, i.e. to about five times the recombination scale Rr!
The question now may pose itself concerning the critical mass Mc that is connected with such a selfstructuring mass
unit Mc(Rc). The answer must come from the usual knowledge of the collapse-critical mass unit Mc given by a
comparison of the free-fall time τff and the sound time (pressure counterreaction time) given by τs = D/cs (D being the
radial dimension of the collaps-critical mass unit Mc) and thus leading to the following request:
with P and γ denoting the baryonic gas pressure and the polytropic index of the gas. This then with Fahr and Heyl
(2021) leads to the following expression for Mc = Mc(R):
with P and γ denoting the baryonic gas pressure and the polytropic index of the gas(2021) leads to the following expression for Mc = Mc(R):
Assuming that pressure and density during the cosmic expansion conserve the gas ento the result:
where hereby the typical collapse mass Mc(Rr) at the recombination scale has been cabe Mc(Rr) = 105
Mօ.
It is interesting now to ask at what cosmic times t ≥ tr the first gravitationally boundaccording to the above considerations can be expected as existing in the universe? Thfollowing calculation:
Under the prerequisites which we have discussed before this leads to the following rewhere τ0 = 1/H0 denotes the present age of the universe (say 13.7 Gi- gayears!). Thwas estimated with the redshift zr ≃ 103 of the CMB (Cosmic Microwave Backgrouthrough R0/Rr = 1 + zr ≃1000. That implies that already at a time of ∆t = 4 ∙ 10−3
∙ matter could start creating gravitationally bound, selfsustained, collapsed structures! be out of any conflict with the most recent James-Webb ST observations stating thyears before our present time galaxies and stellar structures appear to have been prewere very much similar to our present day galaxies.
The only remaining problem from our theoretical explanations presented here in thmasses of present day galaxies com- pared to their realistic values which are aboupresent day galaxies – like typically the one of our Milky way – have typical massesabove theoretical predictions for the present-day collaps masses would rather give us Obviously a further mass growth of more than three orders of magnitude would stbetter explanation. One idea for an ongoing mass growth is connected with the procof mass units collapsed before that time. This idea we shall briefly sketch here belowAssuming that pressure and density during the cosmic expansion conserve the gas entropy, i.e.P/ϱγ = const, then leadto the result:
where hereby the typical collapse mass Mc(Rr) at the recombination scale has been calculated by Fahr and Heyl [19] tbe Mc(Rr) = 105
Mօ.
It is interesting now to ask at what cosmic times t ≥ tr the first gravitationally bound mass structures of galactic typaccording to the above considerations can be expected as existing in the universe? This now can be answered with thfollowing calculation:
Under the prerequisites which we have discussed before this leads to the following result:
where τ0 = 1/H0 denotes the present age of the universe (say 13.7 Gi- gayears!). The recombination scale Rr herebwas estimated with the redshift zr ≃ 103 of the CMB (Cosmic Microwave Background, see Bennet et al., 2003) [20through R0/Rr = 1 + zr ≃1000. That implies that already at a time of ∆t = 4 ∙ 10−3
∙ τ0 after the recombination poinmatter could start creating gravitationally bound, selfsustained, collapsed structures! So far this result at least seems tbe out of any conflict with the most recent James-Webb ST observations stating that already at times of 13.1 Gigyears before our present time galaxies and stellar structures appear to have been present in the early universe whicwere very much similar to our present day galaxies.
The only remaining problem from our theoretical explanations presented here in this paper are the predicted typicamasses of present day galaxies com- pared to their realistic values which are about a factor of 1000 larger. Whilpresent day galaxies – like typically the one of our Milky way – have typical masses of about 1011 solar masses, ouabove theoretical predictions for the present-day collaps masses would rather give us
Obviously a further mass growth of more than three orders of magnitude would still be left over for an upcominbetter explanation. One idea for an ongoing mass growth is connected with the process of a cumulative mass growtof mass units collapsed before that time. This idea we shall briefly sketch here below.
Assuming that pressure and density during the cosmic expansion conserve the gas entropyto the result:
where hereby the typical collapse mass Mc(Rr) at the recombination scale has been calculatbe Mc(Rr) = 105
Mօ.
It is interesting now to ask at what cosmic times t ≥ tr the first gravitationally bound masaccording to the above considerations can be expected as existing in the universe? This nofollowing calculation:
Under the prerequisites which we have discussed before this leads to the following result:
where τ0 = 1/H0 denotes the present age of the universe (say 13.7 Gi- gayears!). The recwas estimated with the redshift zr ≃ 103 of the CMB (Cosmic Microwave Background, sthrough R0/Rr = 1 + zr ≃1000. That implies that already at a time of ∆t = 4 ∙ 10−3
∙ τ0 afmatter could start creating gravitationally bound, selfsustained, collapsed structures! So farbe out of any conflict with the most recent James-Webb ST observations stating that alryears before our present time galaxies and stellar structures appear to have been present were very much similar to our present day galaxies.
The only remaining problem from our theoretical explanations presented here in this papmasses of present day galaxies com- pared to their realistic values which are about a fapresent day galaxies – like typically the one of our Milky way – have typical masses of ababove theoretical predictions for the present-day collaps masses would rather give us
Obviously a further mass growth of more than three orders of magnitude would still bebetter explanation. One idea for an ongoing mass growth is connected with the process ofof mass units collapsed before that time. This idea we shall briefly sketch here below.
Assuming that pressure and density during the cosmic expansion conserve the gas entropto the result:
where hereby the typical collapse mass Mc(Rr) at the recombination scale has been calculbe Mc(Rr) = 105
Mօ.
It is interesting now to ask at what cosmic times t ≥ tr the first gravitationally bound maaccording to the above considerations can be expected as existing in the universe? This nfollowing calculation:
Under the prerequisites which we have discussed before this leads to the following resultwhere τ0 = 1/H0 denotes the present age of the universe (say 13.7 Gi- gayears!). The rewas estimated with the redshift zr ≃ 103 of the CMB (Cosmic Microwave Background,through R0/Rr = 1 + zr ≃1000. That implies that already at a time of ∆t = 4 ∙ 10−3
∙ τ0 matter could start creating gravitationally bound, selfsustained, collapsed structures! So fbe out of any conflict with the most recent James-Webb ST observations stating that ayears before our present time galaxies and stellar structures appear to have been presenwere very much similar to our present day galaxies.
The only remaining problem from our theoretical explanations presented here in this pmasses of present day galaxies com- pared to their realistic values which are about a present day galaxies – like typically the one of our Milky way – have typical masses of above theoretical predictions for the present-day collaps masses would rather give us
Obviously a further mass growth of more than three orders of magnitude would still bbetter explanation. One idea for an ongoing mass growth is connected with the process of mass units collapsed before that time. This idea we shall briefly sketch here below.
Assuming that pressure and density during the cosmic expansion conserve the gas entrto the result:
where hereby the typical collapse mass Mc(Rr) at the recombination scale has been calcbe Mc(Rr) = 105
Mօ.
It is interesting now to ask at what cosmic times t ≥ tr the first gravitationally bound maccording to the above considerations can be expected as existing in the universe? Thisfollowing calculation:
Under the prerequisites which we have discussed before this leads to the following resuwhere τ0 = 1/H0 denotes the present age of the universe (say 13.7 Gi- gayears!). The was estimated with the redshift zr ≃ 103 of the CMB (Cosmic Microwave Backgrounthrough R0/Rr = 1 + zr ≃1000. That implies that already at a time of ∆t = 4 ∙ 10−3
∙ τmatter could start creating gravitationally bound, selfsustained, collapsed structures! Sobe out of any conflict with the most recent James-Webb ST observations stating thatyears before our present time galaxies and stellar structures appear to have been preswere very much similar to our present day galaxies.
The only remaining problem from our theoretical explanations presented here in this masses of present day galaxies com- pared to their realistic values which are about present day galaxies – like typically the one of our Milky way – have typical masses oabove theoretical predictions for the present-day collaps masses would rather give us
Obviously a further mass growth of more than three orders of magnitude would stilbetter explanation. One idea for an ongoing mass growth is connected with the procesof mass units collapsed before that time. This idea we shall briefly sketch here below.
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
Adv Theo Comp Phy, 2022 Volume 5 | Issue 3 | 526
Obviously a further mass growth of more than three orders of
magnitude would still be left over for an upcoming better explanation. One idea for an ongoing mass growth is connected with
the process of a cumulative mass growth of mass units collapsed
before that time. This idea we shall briefly sketch here below.
The Idea of a Cumulative Mass Growth
Let us start from a homogeneous universe under Hubble expansion that has, as discussed above, started to produce the first generation of massive collapse centers with masses of the order of
Mc ≃105
Mօ
. Assuming furthermore a symmetric production of
collapse centers in this homogeneous universe one could then
assume that these collapse centers conserve the general Hubble
expansion dynamics. This would allow to assume that two of
such neighboring centers of masses Mc,A and Mc,B in a radial distance of twice the collapse radius D = D(Mc
) of such objects A
and B would have a mutual, relative Hubble migration velocity
of:
One can now compare the relative Hubble energy Ekin = (1/2)
Mc
V2
AB of these two objects A and B with the gravitational binding energy Ebind = GMc
2
/2D between these two mass centers and
can study their absolute magnitudes investigating:
or leading to:
Studying then the situation when first in the universe collapse
structures can be expected, i.e. at R = Rc
= 5Rr
, will lead to:
In view of Rc = 5Rr and we
then obtain:
which finally with τ0
= 1/H0
= 13.7 Gigayears means:
or:
That finally expresses the fact that the left side (i.e. kinetic energy Ekin,AB) is much greater than the right side (i.e. binding energy
Ebind,AB). This indicates that the two centers A and B of collapsed
masses would be essentially free to continue their Hubble dynamics, – i.e. no! further accumulation of collapsed materials
would occur. The latter to the contrast would, however, occur, if
the binding energy would turn out to be larger than the kinetic
energy, because in this case the two mass clusters would produce
one new gravitationally bound system decoupling from the free
Hubble expansion, and, in view of the other equivalent systems
in the neighborhood, would induce a multi-cluster collaps system.
As it looks so far, however, accumulation of cosmic masses
beyond a cluster mass of Mc ≃105 Mօ
until the present age of
the universe remains unexplained by our present theory. When
JWST in its highly resolving infrared observations can really
find indications for the existence of systems of clustered masses
with 1011Mօ
with an age of 13.1 Gigayears, – then this! is really
an exciting message.
On the other hand perhaps, taking serious what Perlmutter et al.
(1999), Schmidt et al. (1999), and Riess et al. (1999) are claiming, namely that they are seeing in the SN1a luminosities of the
most distant galaxies already clear indications of an accelerated
expansion of the universe would express the fact that already at
that early times [21-24], all the more at all later cosmic times, the
Hubble constant would have been given by:
meaning that the expansion times would have stayed
constant all the way since that time till today and later, while
the free-fall times are increasing permanently
since that time like
which would mean that since that early time no collapse
could have happened anymore, and especially young galaxies
could not at all be understood in this context of the universe.
Does that mean: H = HΛ = const can be ruled out?, while
. might appear as the better, since
more valid approach?
References

1. Jeans, J. (1929). Astronomy and cosmogony. CUP Archive.
2. Fahr, H. J., & Willerding, E. (1998). Die Entstehung von
   Sonnensystemen: eine Einführung in das Problem der Planetenentstehung. Spektrum, Akad. Verlag.
3. Perlmutter, S., Aldering, G., Goldhaber, G., Knop, R. A.,
   Nugent, P., Castro, P. G., … & Supernova Cosmology Project. (1999). Measurements of Ω and Λ from 42 high-redshift
   supernovae. The Astrophysical Journal, 517(2), 565.
4. Schmidt, B. P., Suntzeff, N. B., Phillips, M. M., Schommer,
   R. A., Clocchiatti, A., Kirshner, R. P., … & Ciardullo, R.
   (1998). The high-Z supernova search: measuring cosmic
   deceleration and global curvature of the universe using type
   Ia supernovae. The Astrophysical Journal, 507(1), 46.
5. Riess, A. G., Filippenko, A. V., Challis, P., Clocchiatti, A.,
   Diercks, A., Garnavich, P. M., … & Tonry, J. (1998). Observational evidence from supernovae for an accelerating
   ≃
   The Idea of a Cumulative Mass Growth
   Let us start from a homogeneous universe under Hubble expansion that has, as discussed above, started to
   produce the first generation of massive collapse centers with masses of the order of Mc ≃105
   Mօ. Assuming
   furthermore a symmetric production of collapse centers in this homogeneous universe one could then assume that
   these collapse centers conserve the general Hubble expansion dynamics. This would allow to assume that
   two of such neighboring centers of masses Mc,A and Mc,B in a radial distance of twice the collapse radius
   D = D(Mc) of such objects A and B would have a mutual, relative Hubble migrationvelocity of:
   One can now compare the relative Hubble energy Ekin = (1/2)
   of these two objects A and B with the
   gravitational binding energy Ebind = G
   /2D between these two mass centers and can study their absolute
   magnitudes investigating:
   or leading to:
   Studying then the situation when first in the universe collapse structurescan be expected, i.e. at R = Rc
   = 5Rr, will lead to:
   In view of Rc = 5Rr and we then obtain:
   which finally with τ0 = 1/H0 = 13.7 Gigayears means:
   or:
   ≃
   The Idea of a Cumulative Mass Growth
   Let us start from a homogeneous universe under Hubble expansion that has, as discussed above, started to
   produce the first generation of massive collapse centers with masses of the order of Mc ≃105
   Mօ. Assuming
   furthermore a symmetric production of collapse centers in this homogeneous universe one could then assume that
   these collapse centers conserve the general Hubble expansion dynamics. This would allow to assume that
   two of such neighboring centers of masses Mc,A and Mc,B in a radial distance of twice the collapse radius
   D = D(Mc) of such objects A and B would have a mutual, relative Hubble migrationvelocity of:
   One can now compare the relative Hubble energy Ekin = (1/2)
   of these two objects A and B with the
   gravitational binding energy Ebind = G
   /2D between these two mass centers and can study their absolute
   magnitudes investigating:
   or leading to:
   Studying then the situation when first in the universe collapse structurescan be expected, i.e. at R = Rc
   = 5Rr, will lead to:
   In view of Rc = 5Rr and we then obtain:
   which finally with τ0 = 1/H0 = 13.7 Gigayears means:
   or:
   ≃
   The Idea of a Cumulative Mass Growth
   Let us start from a homogeneous universe under Hubble expansion that has, as discussed above, started to
   produce the first generation of massive collapse centers with masses of the order of Mc ≃105
   Mօ. Assuming
   furthermore a symmetric production of collapse centers in this homogeneous universe one could then assume that
   these collapse centers conserve the general Hubble expansion dynamics. This would allow to assume that
   two of such neighboring centers of masses Mc,A and Mc,B in a radial distance of twice the collapse radius
   D = D(Mc) of such objects A and B would have a mutual, relative Hubble migrationvelocity of:
   One can now compare the relative Hubble energy Ekin = (1/2)
   of these two objects A and B with the
   gravitational binding energy Ebind = G
   /2D between these two mass centers and can study their absolute
   magnitudes investigating:
   or leading to:
   Studying then the situation when first in the universe collapse structurescan be expected, i.e. at R = Rc
   = 5Rr, will lead to:
   In view of Rc = 5Rr and we then obtain:
   which finally with τ0 = 1/H0 = 13.7 Gigayears means:
   or:
   ≃
   The Idea of a Cumulative Mass Growth
   Let us start from a homogeneous universe under Hubble expansion that has, as discussed above, started to
   produce the first generation of massive collapse centers with masses of the order of Mc ≃105
   Mօ. Assuming
   furthermore a symmetric production of collapse centers in this homogeneous universe one could then assume that
   these collapse centers conserve the general Hubble expansion dynamics. This would allow to assume that
   two of such neighboring centers of masses Mc,A and Mc,B in a radial distance of twice the collapse radius
   D = D(Mc) of such objects A and B would have a mutual, relative Hubble migrationvelocity of:
   One can now compare the relative Hubble energy Ekin = (1/2)
   of these two objects A and B with the
   gravitational binding energy Ebind = G
   /2D between these two mass centers and can study their absolute
   magnitudes investigating:
   or leading to:
   Studying then the situation when first in the universe collapse structurescan be expected, i.e. at R = Rc
   = 5Rr, will lead to:
   In view of Rc = 5Rr and we then obtain:
   which finally with τ0 = 1/H0 = 13.7 Gigayears means:
   or:
   ≃
   The Idea of a Cumulative Mass Growth
   Let us start from a homogeneous universe under Hubble expansion that has, as discussed above, started to
   produce the first generation of massive collapse centers with masses of the order of Mc ≃105
   Mօ. Assuming
   furthermore a symmetric production of collapse centers in this homogeneous universe one could then assume that
   these collapse centers conserve the general Hubble expansion dynamics. This would allow to assume that
   two of such neighboring centers of masses Mc,A and Mc,B in a radial distance of twice the collapse radius
   D = D(Mc) of such objects A and B would have a mutual, relative Hubble migrationvelocity of:
   One can now compare the relative Hubble energy Ekin = (1/2)
   of these two objects A and B with the
   gravitational binding energy Ebind = G
   /2D between these two mass centers and can study their absolute
   magnitudes investigating:
   or leading to:
   Studying then the situation when first in the universe collapse structurescan be expected, i.e. at R = Rc
   = 5Rr, will lead to:
   In view of Rc = 5Rr and we then obtain:
   which finally with τ0 = 1/H0 = 13.7 Gigayears means:
   or:
   ≃
   The Idea of a Cumulative Mass Growth
   Let us start from a homogeneous universe under Hubble expansion that has, as discussed above, started to
   produce the first generation of massive collapse centers with masses of the order of Mc ≃105
   Mօ. Assuming
   furthermore a symmetric production of collapse centers in this homogeneous universe one could then assume that
   these collapse centers conserve the general Hubble expansion dynamics. This would allow to assume that
   two of such neighboring centers of masses Mc,A and Mc,B in a radial distance of twice the collapse radius
   D = D(Mc) of such objects A and B would have a mutual, relative Hubble migrationvelocity of:
   One can now compare the relative Hubble energy Ekin = (1/2)
   of these two objects A and B with the
   gravitational binding energy Ebind = G
   /2D between these two mass centers and can study their absolute
   magnitudes investigating:
   or leading to:
   Studying then the situation when first in the universe collapse structurescan be expected, i.e. at R = Rc
   = 5Rr, will lead to:
   In view of Rc = 5Rr and we then obtain:
   which finally with τ0 = 1/H0 = 13.7 Gigayears means:
   or:
   ≃
   The Idea of a Cumulative Mass Growth
   Let us start from a homogeneous universe under Hubble expansion that has, as discussed above, started to
   produce the first generation of massive collapse centers with masses of the order of Mc ≃105
   Mօ. Assuming
   furthermore a symmetric production of collapse centers in this homogeneous universe one could then assume that
   these collapse centers conserve the general Hubble expansion dynamics. This would allow to assume that
   two of such neighboring centers of masses Mc,A and Mc,B in a radial distance of twice the collapse radius
   D = D(Mc) of such objects A and B would have a mutual, relative Hubble migrationvelocity of:
   One can now compare the relative Hubble energy Ekin = (1/2)
   of these two objects A and B with the
   gravitational binding energy Ebind = G
   /2D between these two mass centers and can study their absolute
   magnitudes investigating:
   or leading to:
   Studying then the situation when first in the universe collapse structurescan be expected, i.e. at R = Rc
   = 5Rr, will lead to:
   In view of Rc = 5Rr and we then obtain:
   which finally with τ0 = 1/H0 = 13.7 Gigayears means:
   or:
   That finally expresses the fact that the left side (i.e. kinetic energy Ekin,AB)is much greater than the right
   side (i.e. binding energy Ebind,AB). This indicates that the two centers A and B of collapsed masses would be
   essentially free to continue their Hubble dynamics, – i.e. no! further accumulation of collapsed materials would
   occur. The latter to the contrast would, however, occur, if the binding energy would turn out to be larger than
   the kinetic energy, because in this case the two mass clusters would produce one new gravitationally bound
   system decoupling from the free Hubble expansion, and, in view of the other equivalent systems in the
   neighborhood, would induce a multi-cluster collaps system.
   As it looks so far, however, accumulation of cosmic masses beyond a clustermass of Mc ≃105
   Mօ until the
   presentageoftheuniverseremainsunexplainedbyourpresenttheoryWhenJWSTinitshighlyresolvingThat finally expresses the fact that the left side (i.e. kinetic energy Ekin,AB)is much greater than the rightside (i.e. binding energy Ebind,AB). This indicates that the two centers A and B of collapsed masses would beessentially free to continue their Hubble dynamics, – i.e. no! further accumulation of collapsed materials wouldoccur. The latter to the contrast would, however, occur, if the binding energy would turn out to be larger thanthe kinetic energy, because in this case the two mass clusters would produce one new gravitationally bound
   system decoupling from the free Hubble expansion, and, in view of the other equivalent systems in theneighborhood, would induce a multi-cluster collaps system.
   As it looks so far, however, accumulation of cosmic masses beyond a clustermass of Mc ≃105
   Mօ until the
   present age of the universe remains unexplained by our present theory. When JWST in its highly resolving
   infrared observationscan really find indications for the existence of systems of clustered masses with 1011Mօ
   with an age of 13.1 Gigayears, – then this! is really an exciting message.
   On the other hand perhaps, taking serious what Perlmutter et al. (1999), Schmidt et al. (1999), and
   Riess et al. (1999) are claiming, namely that theyare seeing in the SN1a luminosities of the most distant
   galaxies already clearindications of an accelerated expansion of the universe would express the fact that
   already at that early times [21-24], all the more at all later cosmic times, the Hubble constant would
   have been given by:
   meaning that the expansion times would have stayed constant all the way since that time
   till today and later, while the free-fall times are increasing permanently since that time
   like
   which would mean that since that early time no collapse could have happenedanymore, and especially younggalaxies could not at all be understood in this context of the universe. Does that mean: H = HΛ = constcan be ruled out?, while might appear as the better, since more valid
   approach?
   That finally expresses the fact that the left side (i.e. kinetic energy Ekin,AB)is much greside (i.e. binding energy Ebind,AB). This indicates that the two centers A and B of collapsedessentially free to continue their Hubble dynamics, – i.e. no! further accumulation of collapseoccur. The latter to the contrast would, however, occur, if the binding energy would turn outhe kinetic energy, because in this case the two mass clusters would produce one new grasystem decoupling from the free Hubble expansion, and, in view of the other equivaleneighborhood, would induce a multi-cluster collaps system.
   As it looks so far, however, accumulation of cosmic masses beyond a clustermass of Mpresent age of the universe remains unexplained by our present theory. When JWST in itinfrared observationscan really find indications for the existence of systems of clustered mwith an age of 13.1 Gigayears, – then this! is really an exciting message.
   On the other hand perhaps, taking serious what Perlmutter et al. (1999), Schmidt eRiess et al. (1999) are claiming, namely that theyare seeing in the SN1a luminosities ogalaxies already clearindications of an accelerated expansion of the universe would exalready at that early times [21-24], all the more at all later cosmic times, the Hubbhave been given by:
   meaning that the expansion times would have stayed constant all the watill today and later, while the free-fall times are increasing permanentlike
   which would mean that since that early time no collapse could have happenedanymore, andgalaxies could not at all be understood in this context of the universe. Does that mean: can be ruled out?, while might appear as the better,approach?
   That finally expresses the fact that the left side (i.e. kinetic energy Ekin,AB)is much greater than the right
   side (i.e. binding energy Ebind,AB). This indicates that the two centers A and B of collapsed masses would be
   essentially free to continue their Hubble dynamics, – i.e. no! further accumulation of collapsed materials would
   occur. The latter to the contrast would, however, occur, if the binding energy would turn out to be larger than
   the kinetic energy, because in this case the two mass clusters would produce one new gravitationally bound
   system decoupling from the free Hubble expansion, and, in view of the other equivalent systems in the
   neighborhood, would induce a multi-cluster collaps system.
   As it looks so far, however, accumulation of cosmic masses beyond a clustermass of Mc ≃105
   Mօ until the
   present age of the universe remains unexplained by our present theory. When JWST in its highly resolving
   infrared observationscan really find indications for the existence of systems of clustered masses with 1011Mօ
   with an age of 13.1 Gigayears, – then this! is really an exciting message.
   On the other hand perhaps, taking serious what Perlmutter et al. (1999), Schmidt et al. (1999), and
   Riess et al. (1999) are claiming, namely that theyare seeing in the SN1a luminosities of the most distant
   galaxies already clearindications of an accelerated expansion of the universe would express the fact that
   already at that early times [21-24], all the more at all later cosmic times, the Hubble constant would
   have been given by:
   meaning that the expansion times would have stayed constant all the way since that time
   till today and later, while the free-fall times are increasing permanently since that time
   like
   which would mean that since that early time no collapse could have happenedanymore, and especially young
   galaxies could not at all be understood in this context of the universe. Does that mean: H = HΛ = const
   can be ruled out?, while might appear as the better, since more valid
   approach?
   That finally expresses the fact that the left side (i.e. kinetic energy Ekin,AB)is much greater than the rigside (i.e. binding energy Ebind,AB). This indicates that the two centers A and B of collapsed masses would essentially free to continue their Hubble dynamics, – i.e. no! further accumulation of collapsed materials wouoccur. The latter to the contrast would, however, occur, if the binding energy would turn out to be larger ththe kinetic energy, because in this case the two mass clusters would produce one new gravitationally bousystem decoupling from the free Hubble expansion, and, in view of the other equivalent systems in neighborhood, would induce a multi-cluster collaps system.
   As it looks so far, however, accumulation of cosmic masses beyond a clustermass of Mc ≃105
   Mօ until present age of the universe remains unexplained by our present theory. When JWST in its highly resolviinfrared observationscan really find indications for the existence of systems of clustered masses with 1011with an age of 13.1 Gigayears, – then this! is really an exciting message.
   On the other hand perhaps, taking serious what Perlmutter et al. (1999), Schmidt et al. (1999), aRiess et al. (1999) are claiming, namely that theyare seeing in the SN1a luminosities of the most distagalaxies already clearindications of an accelerated expansion of the universe would express the fact thalready at that early times [21-24], all the more at all later cosmic times, the Hubble constant wouhave been given by:
   meaning that the expansion times would have stayed constant all the way since that titill today and later, while the free-fall times are increasing permanently since that tilike
   which would mean that since that early time no collapse could have happenedanymore, and especially yougalaxies could not at all be understood in this context of the universe. Does that mean: H = HΛ = cocan be ruled out?, while might appear as the better, since more vaapproach?
   That finally expresses the fact that the left side (i.e. kinetic energy Ekin,AB)is much greater than the right
   side (i.e. binding energy Ebind,AB). This indicates that the two centers A and B of collapsed masses would be
   essentially free to continue their Hubble dynamics, – i.e. no! further accumulation of collapsed materials would
   occur. The latter to the contrast would, however, occur, if the binding energy would turn out to be larger than
   the kinetic energy, because in this case the two mass clusters would produce one new gravitationally bound
   system decoupling from the free Hubble expansion, and, in view of the other equivalent systems in the
   neighborhood, would induce a multi-cluster collaps system.
   As it looks so far, however, accumulation of cosmic masses beyond a clustermass of Mc ≃105
   Mօ until the
   present age of the universe remains unexplained by our present theory. When JWST in its highly resolving
   infrared observationscan really find indications for the existence of systems of clustered masses with 1011Mօ
   with an age of 13.1 Gigayears, – then this! is really an exciting message.
   On the other hand perhaps, taking serious what Perlmutter et al. (1999), Schmidt et al. (1999), and
   Riess et al. (1999) are claiming, namely that theyare seeing in the SN1a luminosities of the most distant
   galaxies already clearindications of an accelerated expansion of the universe would express the fact that
   already at that early times [21-24], all the more at all later cosmic times, the Hubble constant would
   have been given by:
   meaning that the expansion times would have stayed constant all the way since that time
   till today and later, while the free-fall times are increasing permanently since that time
   like
   which would mean that since that early time no collapse could have happenedanymore, and especially young
   galaxies could not at all be understood in this context of the universe. Does that mean: H = HΛ = const
   can be ruled out?, while might appear as the better, since more valid
   approach?
   (14)
   (15)
   (16)
   (17)
   (18)
   (19)
   (20)
   (21)
   (22)
   Volume 5 | Issue 3 | 527
   Copyright: ©2022 Hans J.Fahr. This is an open-access article
   distributed under the terms of the Creative Commons Attribution
   License, which permits unrestricted use, distribution, and reproduction
   in any medium, provided the original author and source are credited.
   Adv Theo Comp Phy, 2022
   universe and a cosmological constant. The Astronomical
   Journal, 116(3), 1009.
6. Einstein, A. (1917). Kosmologische Betrachtungen zur
   allgemeinen Relativitätstheorie, Sitzungsberichte der
   Königlich Preußischen Akademie der Wissenschaften (Berlin), Seite 142-152.
7. Peebles, P. J. E., & Ratra, B. (2003). The cosmological constant and dark energy. Reviews of modern physics, 75(2),
   559-599.
8. Fahr, H. J. (2004). The cosmology of empty space: How
   heavy is the vacuum? What we know enforces our belief. SCHRIFTENREIHE-WITTGENSTEIN GESELLSCHAFT, 33, 339-352.
9. Kragh, H.S., and Overduin, J.M. (2014). The weight of the
   vacuum, Springer Briefs in Physics.
10. Heyl, M., Fahr, H. J., & Siewert, M. (2014). The thermodynamics of a gravitating vacuum. arXiv preprint arXiv:1412.3667.
11. Casado, J., & Jou, D. (2013). Steady Flow cosmological
    model. Astrophysics and Space Science, 344(2), 513-520.
12. Casado, J. (2020). Linear expansion models vs. standard
    cosmologies: a critical and historical overview. Astrophysics and Space Science, 365(1), 1-14.
13. Kolb, E. W. (1989). A coasting cosmology. The Astrophysical Journal, 344, 543-550.
14. Gehlaut, S., Kumar, P., & Lohiya, D. (2003). A Concordant” Freely Coasting Cosmology”. arXiv preprint astro-ph/0306448.
15. Dev, A., Sethi, M., & Lohiya, D. (2001). Linear coasting in
    cosmology and SNe Ia. Physics Letters B, 504(3), 207-212.
16. Fahr, H. J., & Heyl, M. (2020). A universe with a constant
    expansion rate. Physics & Astronomy International Journal,
    4(4), 156-163.
17. Hans J.Fahr. (2022). How much could gravitational binding energy act as hidden cosmic vacuum energy? Adv Theo
    Comp Phy, 5(2), 449-457.
18. Fahr, H. J., & Heyl, M. (2019). Stellar matter distribution
    with scale-invariant hierarchical structuring. Physics & Astronomy International Journal, 3(4), 146-159.
19. Fahr, H. J., & Heyl, M. (2021). Structure Formation after
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    Phy, 4(3), 253-258.
20. Bennett, C. L., Hill, R. S., Hinshaw, G., Nolta, M. R.,
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    Supplement Series, 148(1), 97-117.
21. Fahr, H. J. (2021). The thermodynamics of cosmic gases in
    expanding universes based on Vlasow-theoretical grounds,
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22. Fahr, H. J. (2021). The baryon distribution function in the
    expanding universe after the recombination era, Phys. § Astron. Internat. Journal, 5(2), 37-41.
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24. Weinberg, S. (1989). The cosmological constant problem.
    Reviews of modern physics, 61(1), 1.
    https://opastpublishers.com

Volume 5 | Issue 3 | Volume 5 | Issue 3 | 524
Evolution of cosmic structures in the expanding universe: Could not one have known
it all before?
Research Article
1
Argelander Institute for Astronomy, University of Bonn,
Auf dem Huegel 71, 53121 Bonn Germany
2
Deutsches Zentrum f¨ur Luft- und Raumfahrt (DLR),
K¨onigswinterer Strasse 522-524, 53227 Bonn (Germany)
Hans J.Fahr1* and M. Heyl2
*
Corresponding author
Hans J.Fahr, Argelander Institute for Astronomy, University of Bonn, Auf dem
Huegel 71, 53121 Bonn Germany.
Submitted: 28 Jul 2022; Accepted: 30 Jul 2022; Published: 05 Aug 2022
Adv Theo Comp Phy, 2022
Abstract
Most recent observations from the James Webb space telescope (JWST) have shown by highly resolved infrared
observations of highest sensitivity that structure formation in the universe into the forms of early galaxies has
already taken place at cosmic times less than 0.6 Gigayears after the Big-Bang. This is taken up with a big
surprise in the whole astronomic community, though, as it seems, it could have been predicted from simple
theoretical considerations. In this article, we are demonstrating that this result already would have clearly
come out from theoretical considerations of gravitational structure formation processes in the early expanding
universe just after the cosmic matter recombination period. While, however, it can be easily understood how
matter structures of the order of 108
solar masses could evolve in the cosmic meantime, it nevertheless remains
obscure, how galaxies of the type of the Milky way or more massive structures with 1011 or more solar masses
can have evolved up to the present cosmic days without some not yet specified collapse-accelerating processes.
Citation: Hans J.Fahr. (2022). Evolution of cosmic structures in the expanding universe: Could not one have known it all before?
Adv Theo Comp Phy, 5(3), 524-527.
Advances in Theoretical & Computational Physics
ISSN: 2639-0108
Collapse in Expanding Universes
In principle it is a problem hard to understand that matter may
be able to collapse into large local mass units, though in an expanding universe the initially widely and uniformly distributed
cosmic matter must be subject to the expansion into a permanently growing cosmic space with permanently decreasing cosmic mass densities. This only can be possible, if the structuring
collapse velocity is larger than the general expansion velocity.
The problem thus evidently is and must be connected with the
specific form of the actual expansion dynamics of the whole universe.
Therefore this study certainly is and must be based on the specific
form of the cosmic expansion of the universe. In a static universe
structure formation runs along the lines that astronomers have
developed since long ago for the static space [1, 2]. Processes
of structure formation of course are very much different in the
expanding universe, because then structure formation definitely
will depend on the specific form of the prevailing cosmic expansion (e.g. decelerated, accelerated or coasting expansion etc.).
To best explain the SN 1a luminosities Perlmutter et al. (1998),
Schmidt et al.(1998), or Riess et al. (1998) have preferred the
accelerated expansion of the universe connected with action of
a constant vacuum energy density [3-10], however, there are
attempts by Casado (2011) and Casado and Jou (2013) showing that a ”coasting non-accelerated universe” can equally well
explain these supernovae luminosities [11-12]. In our following
con- siderations we shall consider first here – mainly for mathematical reasons – the case of a “coasting expansion” [13-16],
which in fact can be expected to prevail, if the universe expands
under the form of thermodynamic and gravidynamic action of
vacuum pressure [17]. Alternative forms of a cosmic expansion
may be discussed at the end of this paper and lead to very interesting conclusions.
If then as our working basis such a ”coasting universe” can be
assumed to prevail, like given in the case when ρΛ~R−2 (ϱΛ denoting the mass density equivalent of the vacuum energy, R denoting the scale of the universe, see e.g. Fahr, 2022) and when
vacuum energy is the dominant ingredient to the cosmic mass
density ρΛ ≫ρb , ρd , ρν
, (indices b, d, ν standing for baryons, dark
matter, and photons, respectively) and to the relativistic energy-momentum tensor, then one unavoidably finds:

which in fact means and necessarily implies: a ”coasting expansion” of the universe! Then consequently, a Hubble parameter
must be expected falling off with the scale R like:
This means that the Hubble parameter in course of the coasting
cosmic expansion decreases like H~R−1!
Therefore this study certainly is and must be based on the specific form of the cosmic expastatic universe structure formation runs along the lines that astronomers have developed sispace [1, 2]. Processes of structure formation of course are very much different in the expthen structure formation definitely will depend on the specific form of the prevailingdecelerated, accelerated or coasting expansion etc.). To best explain the SN 1a luminositieSchmidt et al.(1998), or Riess et al. (1998) have preferred the accelerated expansion of thaction of a constant vacuum energy density [3-10], however, there are attempts by Casado ((2013) showing that a ”coasting non-accelerated universe” can equally well explain thes[11-12]. In our following con- siderations we shall consider first here – mainly for mathemat“coasting expansion” [13-16], which in fact can be expected to prevail, if the universe ethermodynamic and gravidynamics action of vacuum pressure [17]. Alternative forms of a discussed at the end of this paper and lead to very interesting conclusions.
If then as our working basis such a ”coasting universe” can be assumed to prevail, like give(ϱΛ denoting the mass density equivalent of the vacuum energy, R denoting the scale of th2022) and when vacuum energy is the dominant ingredient to the cosmic mass density ρΛ ≫standing for baryons, dark matter, and photons, respectively) and to the relativistic energone unavoidably finds:
which in fact means and necessarily implies: a ”coasting expansion” of the universe! Theparameter must be expected falling off with the scale R like:
This means that the Hubble parameter in course of the coasting cosmic expansion decreases Under these cosmic auspices one finds that the local free-fall time τff = (4πGϱ)
−1/2 of barJeans, 1929) is smaller than the expansion time τex = 1/H of that matter (i.e. so that mass stthe expanding universe!), as soon as:
Therefore this study certainly is and must be based on the specific form of the cosmic exstatic universe structure formation runs along the lines that astronomers have developedspace [1, 2]. Processes of structure formation of course are very much different in the then structure formation definitely will depend on the specific form of the prevaildecelerated, accelerated or coasting expansion etc.). To best explain the SN 1a luminosSchmidt et al.(1998), or Riess et al. (1998) have preferred the accelerated expansion ofaction of a constant vacuum energy density [3-10], however, there are attempts by Casad(2013) showing that a ”coasting non-accelerated universe” can equally well explain th[11-12]. In our following con- siderations we shall consider first here – mainly for mathem“coasting expansion” [13-16], which in fact can be expected to prevail, if the universethermodynamic and gravidynamics action of vacuum pressure [17]. Alternative forms ofdiscussed at the end of this paper and lead to very interesting conclusions.
If then as our working basis such a ”coasting universe” can be assumed to prevail, like g(ϱΛ denoting the mass density equivalent of the vacuum energy, R denoting the scale o2022) and when vacuum energy is the dominant ingredient to the cosmic mass density ρstanding for baryons, dark matter, and photons, respectively) and to the relativistic eneone unavoidably finds:
which in fact means and necessarily implies: a ”coasting expansion” of the universe! Tparameter must be expected falling off with the scale R like:
This means that the Hubble parameter in course of the coasting cosmic expansion decreasUnder these cosmic auspices one finds that the local free-fall time τff = (4πGϱ)
−1/2 of bJeans, 1929) is smaller than the expansion time τex = 1/H of that matter (i.e. so that massthe expanding universe!), as soon as:
(1)
(2)
Volume 5 | Issue 3 | 525
Under these cosmic auspices one finds that the local free-fall
time τff
= (4πGϱ)−1/2 of baryonic, cosmic matter (see Jeans, 1929)
is smaller than the expansion time τex = 1/H of that matter (i.e. so
that mass structures can grow even in the expanding universe!),
as soon as:
i,e, if actual free fall times are shorter than expansion times of
material structures. This means one would need to have the following relation fulfilled:
or:
This implies that the critical scale Rc from which upwards a
progress of structuring despite of cosmic expansion can and will
occur is given by:
That means for world times with R(t) ≥ Rc
one thus cannot expect
to have any more a homogeneous cosmic matter distribution, but
a hierarchical mass structure in the universe like described by
Fahr and Heyl (2019) [18].
On the other hand – since matter can anyway not gravitationally
condense, before it has recombined to neutral atoms due to the
strong interactions of free electrons with the strongly coupling
el-mag. radiation fields (photon fields), one can therefore start
this consideration here with the time t
0
= t
r
of matter recombination, since before that time no irreversible condensations are
possible in the form of enduring, persisting structures. Hence
along this argumentation one might find this critical scale by:
This obviously says that structuring of cosmic matter can only
start when the scale of the world has increased to at least Rc
=
5.02 • Rr , i.e. to about five times the recombination scale Rr
!
The question now may pose itself concerning the critical mass
Mc
that is connected with such a selfstructuring mass unit Mc
(Rc
).
The answer must come from the usual knowledge of the collapse-critical mass unit Mc
given by a comparison of the free-fall
time τff and the sound time (pressure counterreaction time) given
by τs
= D/cs
(D being the radial dimension of the collaps-critical
mass unit Mc) and thus leading to the following request:
with P and γ denoting the baryonic gas pressure and the polytropic index of the gas. This then with Fahr and Heyl (2021)
leads to the following expression for Mc
= Mc
(R):
Assuming that pressure and density during the cosmic expansion conserve the gas entropy, i.e.P/ϱγ
= const, then leads to the
result:
where hereby the typical collapse mass Mc
(Rr
) at the recombination scale has been calculated by Fahr and Heyl [19] to be
Mc
(Rr
) = 105 Mօ
.
It is interesting now to ask at what cosmic times t ≥ t
r
the first
gravitationally bound mass structures of galactic type according
to the above considerations can be expected as existing in the
universe? This now can be answered with the following calculation:
Under the prerequisites which we have discussed before this
leads to the following result:
where τ0
= 1/H0
denotes the present age of the universe (say 13.7
Gi- gayears!). The recombination scale Rr hereby was estimated with the redshift zr ≃ 103
of the CMB (Cosmic Microwave
Background, see Bennet et al., 2003) [20] through R0 / Rr = 1 +
zr ≃1000. That implies that already at a time of ∆t = 4 ∙ 10−3∙ τ0
after the recombination point matter could start creating gravitationally bound, selfsustained, collapsed structures! So far this
result at least seems to be out of any conflict with the most recent James-Webb ST observations stating that already at times
of 13.1 Giga years before our present time galaxies and stellar structures appear to have been present in the early universe
which were very much similar to our present day galaxies.
The only remaining problem from our theoretical explanations
presented here in this paper are the predicted typical masses of
present day galaxies com- pared to their realistic values which
are about a factor of 1000 larger. While present day galaxies –
like typically the one of our Milky way – have typical masses of
about 1011 solar masses, our above theoretical predictions for the
present-day collaps masses would rather give us
parameter must be expected falling off with the scale R like:
This means that the Hubble parameter in course of the coasting cosmic expansion decreases like H ˜ R−1
!
Under these cosmic auspices one finds that the local free-fall time τff = (4πGϱ)
−1/2 of baryonic, cosmic matter (see
Jeans, 1929) is smaller than the expansion time τex = 1/H of that matter (i.e. so that mass structures can grow even in
the expanding universe!), as soon as:
i,e, if actual free fall times are shorter than expansion times of materialstructures. This means one would
need to have the following relation fulfilled:
parameter must be expected falling off with the scale R like:
This means that the Hubble parameter in course of the coasting cosmic expansion decreases like H ˜ R−1
!
Under these cosmic auspices one finds that the local free-fall time τff = (4πGϱ)
−1/2 of baryonic, cosmic matter (see
Jeans, 1929) is smaller than the expansion time τex = 1/H of that matter (i.e. so that mass structures can grow even in
the expanding universe!), as soon as:
i,e, if actual free fall times are shorter than expansion times of materialstructures. This means one would
need to have the following relation fulfilled:
or:
This implies that the critical scale Rc from which upwards a progress of structuring despite of cosmic expansion can
and will occur is given by:
That means for world times with R(t) ≥ Rc one thus cannot expect to have any more a homogeneous cosmic matter
distribution, but a hierarchical mass structure in the universe like described by Fahr and Heyl (2019) [18].
On the other hand – since matter can anyway not gravitationally condense, before it has recombined to neutral atoms
due to the strong interactions of free electrons with the strongly coupling el-mag. radiation fields (photon fields), one
can therefore start this consideration here with the time t0 = tr of matter recombination, since before that time no
irreversible condensations are possible in the form of enduring, persisting structures. Hence along this argumentation
one might find this critical scale by:
This obviously says that structuring of cosmic matter can only start when the scale of the world has increased to at
least Rc = 5.02 · Rr, i.e. to about five times the recombination scale Rr!
The question now may pose itself concerning the critical mass Mc that is connected with such a selfstructuring mass
unit Mc(Rc). The answer must come from the usual knowledge of the collapse-critical mass unit Mc given by a
comparison of the free-fall time τff and the sound time (pressure counterreaction time) given by τs = D/cs (D being the
radial dimension of the collaps-critical mass unit Mc) and thus leading to the following request:
with P and γ denoting the baryonic gas pressure and the polytropic index of the gas. This then with Fahr and Heyl
(2021) leads to the following expression for Mc = Mc(R):
or:
This implies that the critical scale Rc from which upwards a progress of structuring despite of cosmic expansion can
and will occur is given by:
That means for world times with R(t) ≥ Rc one thus cannot expect to have any more a homogeneous cosmic matter
distribution, but a hierarchical mass structure in the universe like described by Fahr and Heyl (2019) [18].
On the other hand – since matter can anyway not gravitationally condense, before it has recombined to neutral atoms
due to the strong interactions of free electrons with the strongly coupling el-mag. radiation fields (photon fields), one
can therefore start this consideration here with the time t0 = tr of matter recombination, since before that time no
irreversible condensations are possible in the form of enduring, persisting structures. Hence along this argumentation
one might find this critical scale by:
This obviously says that structuring of cosmic matter can only start when the scale of the world has increased to at
least Rc = 5.02 · Rr, i.e. to about five times the recombination scale Rr!
The question now may pose itself concerning the critical mass Mc that is connected with such a selfstructuring mass
unit Mc(Rc). The answer must come from the usual knowledge of the collapse-critical mass unit Mc given by a
comparison of the free-fall time τff and the sound time (pressure counterreaction time) given by τs = D/cs (D being the
radial dimension of the collaps-critical mass unit Mc) and thus leading to the following request:
with P and γ denoting the baryonic gas pressure and the polytropic index of the gas. This then with Fahr and Heyl
(2021) leads to the following expression for Mc = Mc(R):
or:
This implies that the critical scale Rc from which upwards a progress of structuring despite of cosmic expansion can
and will occur is given by:
That means for world times with R(t) ≥ Rc one thus cannot expect to have any more a homogeneous cosmic matter
distribution, but a hierarchical mass structure in the universe like described by Fahr and Heyl (2019) [18].
On the other hand – since matter can anyway not gravitationally condense, before it has recombined to neutral atoms
due to the strong interactions of free electrons with the strongly coupling el-mag. radiation fields (photon fields), one
can therefore start this consideration here with the time t0 = tr of matter recombination, since before that time no
irreversible condensations are possible in the form of enduring, persisting structures. Hence along this argumentation
one might find this critical scale by:
This obviously says that structuring of cosmic matter can only start when the scale of the world has increased to at
least Rc = 5.02 · Rr, i.e. to about five times the recombination scale Rr!
The question now may pose itself concerning the critical mass Mc that is connected with such a selfstructuring mass
unit Mc(Rc). The answer must come from the usual knowledge of the collapse-critical mass unit Mc given by a
comparison of the free-fall time τff and the sound time (pressure counterreaction time) given by τs = D/cs (D being the
radial dimension of the collaps-critical mass unit Mc) and thus leading to the following request:
with P and γ denoting the baryonic gas pressure and the polytropic index of the gas. This then with Fahr and Heyl
(2021) leads to the following expression for Mc = Mc(R):
Adv Theo Comp Phy, 2022
or:
This implies that the critical scale Rc from which upwards a progress of structuring despite of cosmic expansion can
and will occur is given by:
That means for world times with R(t) ≥ Rc one thus cannot expect to have any more a homogeneous cosmic matter
distribution, but a hierarchical mass structure in the universe like described by Fahr and Heyl (2019) [18].
On the other hand – since matter can anyway not gravitationally condense, before it has recombined to neutral atoms
due to the strong interactions of free electrons with the strongly coupling el-mag. radiation fields (photon fields), one
can therefore start this consideration here with the time t0 = tr of matter recombination, since before that time no
irreversible condensations are possible in the form of enduring, persisting structures. Hence along this argumentation
one might find this critical scale by:
This obviously says that structuring of cosmic matter can only start when the scale of the world has increased to at
least Rc = 5.02 · Rr, i.e. to about five times the recombination scale Rr!
The question now may pose itself concerning the critical mass Mc that is connected with such a selfstructuring mass
unit Mc(Rc). The answer must come from the usual knowledge of the collapse-critical mass unit Mc given by a
comparison of the free-fall time τff and the sound time (pressure counterreaction time) given by τs = D/cs (D being the
radial dimension of the collaps-critical mass unit Mc) and thus leading to the following request:
with P and γ denoting the baryonic gas pressure and the polytropic index of the gas. This then with Fahr and Heyl
(2021) leads to the following expression for Mc = Mc(R):
with P and γ denoting the baryonic gas pressure and the polytropic index of the gas(2021) leads to the following expression for Mc = Mc(R):
Assuming that pressure and density during the cosmic expansion conserve the gas ento the result:
where hereby the typical collapse mass Mc(Rr) at the recombination scale has been cabe Mc(Rr) = 105
Mօ.
It is interesting now to ask at what cosmic times t ≥ tr the first gravitationally boundaccording to the above considerations can be expected as existing in the universe? Thfollowing calculation:
Under the prerequisites which we have discussed before this leads to the following rewhere τ0 = 1/H0 denotes the present age of the universe (say 13.7 Gi- gayears!). Thwas estimated with the redshift zr ≃ 103 of the CMB (Cosmic Microwave Backgrouthrough R0/Rr = 1 + zr ≃1000. That implies that already at a time of ∆t = 4 ∙ 10−3
∙ matter could start creating gravitationally bound, selfsustained, collapsed structures! be out of any conflict with the most recent James-Webb ST observations stating thyears before our present time galaxies and stellar structures appear to have been prewere very much similar to our present day galaxies.
The only remaining problem from our theoretical explanations presented here in thmasses of present day galaxies com- pared to their realistic values which are aboupresent day galaxies – like typically the one of our Milky way – have typical massesabove theoretical predictions for the present-day collaps masses would rather give us Obviously a further mass growth of more than three orders of magnitude would stbetter explanation. One idea for an ongoing mass growth is connected with the procof mass units collapsed before that time. This idea we shall briefly sketch here belowAssuming that pressure and density during the cosmic expansion conserve the gas entropy, i.e.P/ϱγ = const, then leadto the result:
where hereby the typical collapse mass Mc(Rr) at the recombination scale has been calculated by Fahr and Heyl [19] tbe Mc(Rr) = 105
Mօ.
It is interesting now to ask at what cosmic times t ≥ tr the first gravitationally bound mass structures of galactic typaccording to the above considerations can be expected as existing in the universe? This now can be answered with thfollowing calculation:
Under the prerequisites which we have discussed before this leads to the following result:
where τ0 = 1/H0 denotes the present age of the universe (say 13.7 Gi- gayears!). The recombination scale Rr herebwas estimated with the redshift zr ≃ 103 of the CMB (Cosmic Microwave Background, see Bennet et al., 2003) [20through R0/Rr = 1 + zr ≃1000. That implies that already at a time of ∆t = 4 ∙ 10−3
∙ τ0 after the recombination poinmatter could start creating gravitationally bound, selfsustained, collapsed structures! So far this result at least seems tbe out of any conflict with the most recent James-Webb ST observations stating that already at times of 13.1 Gigyears before our present time galaxies and stellar structures appear to have been present in the early universe whicwere very much similar to our present day galaxies.
The only remaining problem from our theoretical explanations presented here in this paper are the predicted typicamasses of present day galaxies com- pared to their realistic values which are about a factor of 1000 larger. Whilpresent day galaxies – like typically the one of our Milky way – have typical masses of about 1011 solar masses, ouabove theoretical predictions for the present-day collaps masses would rather give us
Obviously a further mass growth of more than three orders of magnitude would still be left over for an upcominbetter explanation. One idea for an ongoing mass growth is connected with the process of a cumulative mass growtof mass units collapsed before that time. This idea we shall briefly sketch here below.
Assuming that pressure and density during the cosmic expansion conserve the gas entropyto the result:
where hereby the typical collapse mass Mc(Rr) at the recombination scale has been calculatbe Mc(Rr) = 105
Mօ.
It is interesting now to ask at what cosmic times t ≥ tr the first gravitationally bound masaccording to the above considerations can be expected as existing in the universe? This nofollowing calculation:
Under the prerequisites which we have discussed before this leads to the following result:
where τ0 = 1/H0 denotes the present age of the universe (say 13.7 Gi- gayears!). The recwas estimated with the redshift zr ≃ 103 of the CMB (Cosmic Microwave Background, sthrough R0/Rr = 1 + zr ≃1000. That implies that already at a time of ∆t = 4 ∙ 10−3
∙ τ0 afmatter could start creating gravitationally bound, selfsustained, collapsed structures! So farbe out of any conflict with the most recent James-Webb ST observations stating that alryears before our present time galaxies and stellar structures appear to have been present were very much similar to our present day galaxies.
The only remaining problem from our theoretical explanations presented here in this papmasses of present day galaxies com- pared to their realistic values which are about a fapresent day galaxies – like typically the one of our Milky way – have typical masses of ababove theoretical predictions for the present-day collaps masses would rather give us
Obviously a further mass growth of more than three orders of magnitude would still bebetter explanation. One idea for an ongoing mass growth is connected with the process ofof mass units collapsed before that time. This idea we shall briefly sketch here below.
Assuming that pressure and density during the cosmic expansion conserve the gas entropto the result:
where hereby the typical collapse mass Mc(Rr) at the recombination scale has been calculbe Mc(Rr) = 105
Mօ.
It is interesting now to ask at what cosmic times t ≥ tr the first gravitationally bound maaccording to the above considerations can be expected as existing in the universe? This nfollowing calculation:
Under the prerequisites which we have discussed before this leads to the following resultwhere τ0 = 1/H0 denotes the present age of the universe (say 13.7 Gi- gayears!). The rewas estimated with the redshift zr ≃ 103 of the CMB (Cosmic Microwave Background,through R0/Rr = 1 + zr ≃1000. That implies that already at a time of ∆t = 4 ∙ 10−3
∙ τ0 matter could start creating gravitationally bound, selfsustained, collapsed structures! So fbe out of any conflict with the most recent James-Webb ST observations stating that ayears before our present time galaxies and stellar structures appear to have been presenwere very much similar to our present day galaxies.
The only remaining problem from our theoretical explanations presented here in this pmasses of present day galaxies com- pared to their realistic values which are about a present day galaxies – like typically the one of our Milky way – have typical masses of above theoretical predictions for the present-day collaps masses would rather give us
Obviously a further mass growth of more than three orders of magnitude would still bbetter explanation. One idea for an ongoing mass growth is connected with the process of mass units collapsed before that time. This idea we shall briefly sketch here below.
Assuming that pressure and density during the cosmic expansion conserve the gas entrto the result:
where hereby the typical collapse mass Mc(Rr) at the recombination scale has been calcbe Mc(Rr) = 105
Mօ.
It is interesting now to ask at what cosmic times t ≥ tr the first gravitationally bound maccording to the above considerations can be expected as existing in the universe? Thisfollowing calculation:
Under the prerequisites which we have discussed before this leads to the following resuwhere τ0 = 1/H0 denotes the present age of the universe (say 13.7 Gi- gayears!). The was estimated with the redshift zr ≃ 103 of the CMB (Cosmic Microwave Backgrounthrough R0/Rr = 1 + zr ≃1000. That implies that already at a time of ∆t = 4 ∙ 10−3
∙ τmatter could start creating gravitationally bound, selfsustained, collapsed structures! Sobe out of any conflict with the most recent James-Webb ST observations stating thatyears before our present time galaxies and stellar structures appear to have been preswere very much similar to our present day galaxies.
The only remaining problem from our theoretical explanations presented here in this masses of present day galaxies com- pared to their realistic values which are about present day galaxies – like typically the one of our Milky way – have typical masses oabove theoretical predictions for the present-day collaps masses would rather give us
Obviously a further mass growth of more than three orders of magnitude would stilbetter explanation. One idea for an ongoing mass growth is connected with the procesof mass units collapsed before that time. This idea we shall briefly sketch here below.
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
Adv Theo Comp Phy, 2022 Volume 5 | Issue 3 | 526
Obviously a further mass growth of more than three orders of
magnitude would still be left over for an upcoming better explanation. One idea for an ongoing mass growth is connected with
the process of a cumulative mass growth of mass units collapsed
before that time. This idea we shall briefly sketch here below.
The Idea of a Cumulative Mass Growth
Let us start from a homogeneous universe under Hubble expansion that has, as discussed above, started to produce the first generation of massive collapse centers with masses of the order of
Mc ≃105
Mօ
. Assuming furthermore a symmetric production of
collapse centers in this homogeneous universe one could then
assume that these collapse centers conserve the general Hubble
expansion dynamics. This would allow to assume that two of
such neighboring centers of masses Mc,A and Mc,B in a radial distance of twice the collapse radius D = D(Mc
) of such objects A
and B would have a mutual, relative Hubble migration velocity
of:
One can now compare the relative Hubble energy Ekin = (1/2)
Mc
V2
AB of these two objects A and B with the gravitational binding energy Ebind = GMc
2
/2D between these two mass centers and
can study their absolute magnitudes investigating:
or leading to:
Studying then the situation when first in the universe collapse
structures can be expected, i.e. at R = Rc
= 5Rr
, will lead to:
In view of Rc = 5Rr and we
then obtain:
which finally with τ0
= 1/H0
= 13.7 Gigayears means:
or:
That finally expresses the fact that the left side (i.e. kinetic energy Ekin,AB) is much greater than the right side (i.e. binding energy
Ebind,AB). This indicates that the two centers A and B of collapsed
masses would be essentially free to continue their Hubble dynamics, – i.e. no! further accumulation of collapsed materials
would occur. The latter to the contrast would, however, occur, if
the binding energy would turn out to be larger than the kinetic
energy, because in this case the two mass clusters would produce
one new gravitationally bound system decoupling from the free
Hubble expansion, and, in view of the other equivalent systems
in the neighborhood, would induce a multi-cluster collaps system.
As it looks so far, however, accumulation of cosmic masses
beyond a cluster mass of Mc ≃105 Mօ
until the present age of
the universe remains unexplained by our present theory. When
JWST in its highly resolving infrared observations can really
find indications for the existence of systems of clustered masses
with 1011Mօ
with an age of 13.1 Gigayears, – then this! is really
an exciting message.
On the other hand perhaps, taking serious what Perlmutter et al.
(1999), Schmidt et al. (1999), and Riess et al. (1999) are claiming, namely that they are seeing in the SN1a luminosities of the
most distant galaxies already clear indications of an accelerated
expansion of the universe would express the fact that already at
that early times [21-24], all the more at all later cosmic times, the
Hubble constant would have been given by:
meaning that the expansion times would have stayed
constant all the way since that time till today and later, while
the free-fall times are increasing permanently
since that time like
which would mean that since that early time no collapse
could have happened anymore, and especially young galaxies
could not at all be understood in this context of the universe.
Does that mean: H = HΛ = const can be ruled out?, while
. might appear as the better, since
more valid approach?
References

1. Jeans, J. (1929). Astronomy and cosmogony. CUP Archive.
2. Fahr, H. J., & Willerding, E. (1998). Die Entstehung von
   Sonnensystemen: eine Einführung in das Problem der Planetenentstehung. Spektrum, Akad. Verlag.
3. Perlmutter, S., Aldering, G., Goldhaber, G., Knop, R. A.,
   Nugent, P., Castro, P. G., … & Supernova Cosmology Project. (1999). Measurements of Ω and Λ from 42 high-redshift
   supernovae. The Astrophysical Journal, 517(2), 565.
4. Schmidt, B. P., Suntzeff, N. B., Phillips, M. M., Schommer,
   R. A., Clocchiatti, A., Kirshner, R. P., … & Ciardullo, R.
   (1998). The high-Z supernova search: measuring cosmic
   deceleration and global curvature of the universe using type
   Ia supernovae. The Astrophysical Journal, 507(1), 46.
5. Riess, A. G., Filippenko, A. V., Challis, P., Clocchiatti, A.,
   Diercks, A., Garnavich, P. M., … & Tonry, J. (1998). Observational evidence from supernovae for an accelerating
   ≃
   The Idea of a Cumulative Mass Growth
   Let us start from a homogeneous universe under Hubble expansion that has, as discussed above, started to
   produce the first generation of massive collapse centers with masses of the order of Mc ≃105
   Mօ. Assuming
   furthermore a symmetric production of collapse centers in this homogeneous universe one could then assume that
   these collapse centers conserve the general Hubble expansion dynamics. This would allow to assume that
   two of such neighboring centers of masses Mc,A and Mc,B in a radial distance of twice the collapse radius
   D = D(Mc) of such objects A and B would have a mutual, relative Hubble migrationvelocity of:
   One can now compare the relative Hubble energy Ekin = (1/2)
   of these two objects A and B with the
   gravitational binding energy Ebind = G
   /2D between these two mass centers and can study their absolute
   magnitudes investigating:
   or leading to:
   Studying then the situation when first in the universe collapse structurescan be expected, i.e. at R = Rc
   = 5Rr, will lead to:
   In view of Rc = 5Rr and we then obtain:
   which finally with τ0 = 1/H0 = 13.7 Gigayears means:
   or:
   ≃
   The Idea of a Cumulative Mass Growth
   Let us start from a homogeneous universe under Hubble expansion that has, as discussed above, started to
   produce the first generation of massive collapse centers with masses of the order of Mc ≃105
   Mօ. Assuming
   furthermore a symmetric production of collapse centers in this homogeneous universe one could then assume that
   these collapse centers conserve the general Hubble expansion dynamics. This would allow to assume that
   two of such neighboring centers of masses Mc,A and Mc,B in a radial distance of twice the collapse radius
   D = D(Mc) of such objects A and B would have a mutual, relative Hubble migrationvelocity of:
   One can now compare the relative Hubble energy Ekin = (1/2)
   of these two objects A and B with the
   gravitational binding energy Ebind = G
   /2D between these two mass centers and can study their absolute
   magnitudes investigating:
   or leading to:
   Studying then the situation when first in the universe collapse structurescan be expected, i.e. at R = Rc
   = 5Rr, will lead to:
   In view of Rc = 5Rr and we then obtain:
   which finally with τ0 = 1/H0 = 13.7 Gigayears means:
   or:
   ≃
   The Idea of a Cumulative Mass Growth
   Let us start from a homogeneous universe under Hubble expansion that has, as discussed above, started to
   produce the first generation of massive collapse centers with masses of the order of Mc ≃105
   Mօ. Assuming
   furthermore a symmetric production of collapse centers in this homogeneous universe one could then assume that
   these collapse centers conserve the general Hubble expansion dynamics. This would allow to assume that
   two of such neighboring centers of masses Mc,A and Mc,B in a radial distance of twice the collapse radius
   D = D(Mc) of such objects A and B would have a mutual, relative Hubble migrationvelocity of:
   One can now compare the relative Hubble energy Ekin = (1/2)
   of these two objects A and B with the
   gravitational binding energy Ebind = G
   /2D between these two mass centers and can study their absolute
   magnitudes investigating:
   or leading to:
   Studying then the situation when first in the universe collapse structurescan be expected, i.e. at R = Rc
   = 5Rr, will lead to:
   In view of Rc = 5Rr and we then obtain:
   which finally with τ0 = 1/H0 = 13.7 Gigayears means:
   or:
   ≃
   The Idea of a Cumulative Mass Growth
   Let us start from a homogeneous universe under Hubble expansion that has, as discussed above, started to
   produce the first generation of massive collapse centers with masses of the order of Mc ≃105
   Mօ. Assuming
   furthermore a symmetric production of collapse centers in this homogeneous universe one could then assume that
   these collapse centers conserve the general Hubble expansion dynamics. This would allow to assume that
   two of such neighboring centers of masses Mc,A and Mc,B in a radial distance of twice the collapse radius
   D = D(Mc) of such objects A and B would have a mutual, relative Hubble migrationvelocity of:
   One can now compare the relative Hubble energy Ekin = (1/2)
   of these two objects A and B with the
   gravitational binding energy Ebind = G
   /2D between these two mass centers and can study their absolute
   magnitudes investigating:
   or leading to:
   Studying then the situation when first in the universe collapse structurescan be expected, i.e. at R = Rc
   = 5Rr, will lead to:
   In view of Rc = 5Rr and we then obtain:
   which finally with τ0 = 1/H0 = 13.7 Gigayears means:
   or:
   ≃
   The Idea of a Cumulative Mass Growth
   Let us start from a homogeneous universe under Hubble expansion that has, as discussed above, started to
   produce the first generation of massive collapse centers with masses of the order of Mc ≃105
   Mօ. Assuming
   furthermore a symmetric production of collapse centers in this homogeneous universe one could then assume that
   these collapse centers conserve the general Hubble expansion dynamics. This would allow to assume that
   two of such neighboring centers of masses Mc,A and Mc,B in a radial distance of twice the collapse radius
   D = D(Mc) of such objects A and B would have a mutual, relative Hubble migrationvelocity of:
   One can now compare the relative Hubble energy Ekin = (1/2)
   of these two objects A and B with the
   gravitational binding energy Ebind = G
   /2D between these two mass centers and can study their absolute
   magnitudes investigating:
   or leading to:
   Studying then the situation when first in the universe collapse structurescan be expected, i.e. at R = Rc
   = 5Rr, will lead to:
   In view of Rc = 5Rr and we then obtain:
   which finally with τ0 = 1/H0 = 13.7 Gigayears means:
   or:
   ≃
   The Idea of a Cumulative Mass Growth
   Let us start from a homogeneous universe under Hubble expansion that has, as discussed above, started to
   produce the first generation of massive collapse centers with masses of the order of Mc ≃105
   Mօ. Assuming
   furthermore a symmetric production of collapse centers in this homogeneous universe one could then assume that
   these collapse centers conserve the general Hubble expansion dynamics. This would allow to assume that
   two of such neighboring centers of masses Mc,A and Mc,B in a radial distance of twice the collapse radius
   D = D(Mc) of such objects A and B would have a mutual, relative Hubble migrationvelocity of:
   One can now compare the relative Hubble energy Ekin = (1/2)
   of these two objects A and B with the
   gravitational binding energy Ebind = G
   /2D between these two mass centers and can study their absolute
   magnitudes investigating:
   or leading to:
   Studying then the situation when first in the universe collapse structurescan be expected, i.e. at R = Rc
   = 5Rr, will lead to:
   In view of Rc = 5Rr and we then obtain:
   which finally with τ0 = 1/H0 = 13.7 Gigayears means:
   or:
   ≃
   The Idea of a Cumulative Mass Growth
   Let us start from a homogeneous universe under Hubble expansion that has, as discussed above, started to
   produce the first generation of massive collapse centers with masses of the order of Mc ≃105
   Mօ. Assuming
   furthermore a symmetric production of collapse centers in this homogeneous universe one could then assume that
   these collapse centers conserve the general Hubble expansion dynamics. This would allow to assume that
   two of such neighboring centers of masses Mc,A and Mc,B in a radial distance of twice the collapse radius
   D = D(Mc) of such objects A and B would have a mutual, relative Hubble migrationvelocity of:
   One can now compare the relative Hubble energy Ekin = (1/2)
   of these two objects A and B with the
   gravitational binding energy Ebind = G
   /2D between these two mass centers and can study their absolute
   magnitudes investigating:
   or leading to:
   Studying then the situation when first in the universe collapse structurescan be expected, i.e. at R = Rc
   = 5Rr, will lead to:
   In view of Rc = 5Rr and we then obtain:
   which finally with τ0 = 1/H0 = 13.7 Gigayears means:
   or:
   That finally expresses the fact that the left side (i.e. kinetic energy Ekin,AB)is much greater than the right
   side (i.e. binding energy Ebind,AB). This indicates that the two centers A and B of collapsed masses would be
   essentially free to continue their Hubble dynamics, – i.e. no! further accumulation of collapsed materials would
   occur. The latter to the contrast would, however, occur, if the binding energy would turn out to be larger than
   the kinetic energy, because in this case the two mass clusters would produce one new gravitationally bound
   system decoupling from the free Hubble expansion, and, in view of the other equivalent systems in the
   neighborhood, would induce a multi-cluster collaps system.
   As it looks so far, however, accumulation of cosmic masses beyond a clustermass of Mc ≃105
   Mօ until the
   presentageoftheuniverseremainsunexplainedbyourpresenttheoryWhenJWSTinitshighlyresolvingThat finally expresses the fact that the left side (i.e. kinetic energy Ekin,AB)is much greater than the rightside (i.e. binding energy Ebind,AB). This indicates that the two centers A and B of collapsed masses would beessentially free to continue their Hubble dynamics, – i.e. no! further accumulation of collapsed materials wouldoccur. The latter to the contrast would, however, occur, if the binding energy would turn out to be larger thanthe kinetic energy, because in this case the two mass clusters would produce one new gravitationally bound
   system decoupling from the free Hubble expansion, and, in view of the other equivalent systems in theneighborhood, would induce a multi-cluster collaps system.
   As it looks so far, however, accumulation of cosmic masses beyond a clustermass of Mc ≃105
   Mօ until the
   present age of the universe remains unexplained by our present theory. When JWST in its highly resolving
   infrared observationscan really find indications for the existence of systems of clustered masses with 1011Mօ
   with an age of 13.1 Gigayears, – then this! is really an exciting message.
   On the other hand perhaps, taking serious what Perlmutter et al. (1999), Schmidt et al. (1999), and
   Riess et al. (1999) are claiming, namely that theyare seeing in the SN1a luminosities of the most distant
   galaxies already clearindications of an accelerated expansion of the universe would express the fact that
   already at that early times [21-24], all the more at all later cosmic times, the Hubble constant would
   have been given by:
   meaning that the expansion times would have stayed constant all the way since that time
   till today and later, while the free-fall times are increasing permanently since that time
   like
   which would mean that since that early time no collapse could have happenedanymore, and especially younggalaxies could not at all be understood in this context of the universe. Does that mean: H = HΛ = constcan be ruled out?, while might appear as the better, since more valid
   approach?
   That finally expresses the fact that the left side (i.e. kinetic energy Ekin,AB)is much greside (i.e. binding energy Ebind,AB). This indicates that the two centers A and B of collapsedessentially free to continue their Hubble dynamics, – i.e. no! further accumulation of collapseoccur. The latter to the contrast would, however, occur, if the binding energy would turn outhe kinetic energy, because in this case the two mass clusters would produce one new grasystem decoupling from the free Hubble expansion, and, in view of the other equivaleneighborhood, would induce a multi-cluster collaps system.
   As it looks so far, however, accumulation of cosmic masses beyond a clustermass of Mpresent age of the universe remains unexplained by our present theory. When JWST in itinfrared observationscan really find indications for the existence of systems of clustered mwith an age of 13.1 Gigayears, – then this! is really an exciting message.
   On the other hand perhaps, taking serious what Perlmutter et al. (1999), Schmidt eRiess et al. (1999) are claiming, namely that theyare seeing in the SN1a luminosities ogalaxies already clearindications of an accelerated expansion of the universe would exalready at that early times [21-24], all the more at all later cosmic times, the Hubbhave been given by:
   meaning that the expansion times would have stayed constant all the watill today and later, while the free-fall times are increasing permanentlike
   which would mean that since that early time no collapse could have happenedanymore, andgalaxies could not at all be understood in this context of the universe. Does that mean: can be ruled out?, while might appear as the better,approach?
   That finally expresses the fact that the left side (i.e. kinetic energy Ekin,AB)is much greater than the right
   side (i.e. binding energy Ebind,AB). This indicates that the two centers A and B of collapsed masses would be
   essentially free to continue their Hubble dynamics, – i.e. no! further accumulation of collapsed materials would
   occur. The latter to the contrast would, however, occur, if the binding energy would turn out to be larger than
   the kinetic energy, because in this case the two mass clusters would produce one new gravitationally bound
   system decoupling from the free Hubble expansion, and, in view of the other equivalent systems in the
   neighborhood, would induce a multi-cluster collaps system.
   As it looks so far, however, accumulation of cosmic masses beyond a clustermass of Mc ≃105
   Mօ until the
   present age of the universe remains unexplained by our present theory. When JWST in its highly resolving
   infrared observationscan really find indications for the existence of systems of clustered masses with 1011Mօ
   with an age of 13.1 Gigayears, – then this! is really an exciting message.
   On the other hand perhaps, taking serious what Perlmutter et al. (1999), Schmidt et al. (1999), and
   Riess et al. (1999) are claiming, namely that theyare seeing in the SN1a luminosities of the most distant
   galaxies already clearindications of an accelerated expansion of the universe would express the fact that
   already at that early times [21-24], all the more at all later cosmic times, the Hubble constant would
   have been given by:
   meaning that the expansion times would have stayed constant all the way since that time
   till today and later, while the free-fall times are increasing permanently since that time
   like
   which would mean that since that early time no collapse could have happenedanymore, and especially young
   galaxies could not at all be understood in this context of the universe. Does that mean: H = HΛ = const
   can be ruled out?, while might appear as the better, since more valid
   approach?
   That finally expresses the fact that the left side (i.e. kinetic energy Ekin,AB)is much greater than the rigside (i.e. binding energy Ebind,AB). This indicates that the two centers A and B of collapsed masses would essentially free to continue their Hubble dynamics, – i.e. no! further accumulation of collapsed materials wouoccur. The latter to the contrast would, however, occur, if the binding energy would turn out to be larger ththe kinetic energy, because in this case the two mass clusters would produce one new gravitationally bousystem decoupling from the free Hubble expansion, and, in view of the other equivalent systems in neighborhood, would induce a multi-cluster collaps system.
   As it looks so far, however, accumulation of cosmic masses beyond a clustermass of Mc ≃105
   Mօ until present age of the universe remains unexplained by our present theory. When JWST in its highly resolviinfrared observationscan really find indications for the existence of systems of clustered masses with 1011with an age of 13.1 Gigayears, – then this! is really an exciting message.
   On the other hand perhaps, taking serious what Perlmutter et al. (1999), Schmidt et al. (1999), aRiess et al. (1999) are claiming, namely that theyare seeing in the SN1a luminosities of the most distagalaxies already clearindications of an accelerated expansion of the universe would express the fact thalready at that early times [21-24], all the more at all later cosmic times, the Hubble constant wouhave been given by:
   meaning that the expansion times would have stayed constant all the way since that titill today and later, while the free-fall times are increasing permanently since that tilike
   which would mean that since that early time no collapse could have happenedanymore, and especially yougalaxies could not at all be understood in this context of the universe. Does that mean: H = HΛ = cocan be ruled out?, while might appear as the better, since more vaapproach?
   That finally expresses the fact that the left side (i.e. kinetic energy Ekin,AB)is much greater than the right
   side (i.e. binding energy Ebind,AB). This indicates that the two centers A and B of collapsed masses would be
   essentially free to continue their Hubble dynamics, – i.e. no! further accumulation of collapsed materials would
   occur. The latter to the contrast would, however, occur, if the binding energy would turn out to be larger than
   the kinetic energy, because in this case the two mass clusters would produce one new gravitationally bound
   system decoupling from the free Hubble expansion, and, in view of the other equivalent systems in the
   neighborhood, would induce a multi-cluster collaps system.
   As it looks so far, however, accumulation of cosmic masses beyond a clustermass of Mc ≃105
   Mօ until the
   present age of the universe remains unexplained by our present theory. When JWST in its highly resolving
   infrared observationscan really find indications for the existence of systems of clustered masses with 1011Mօ
   with an age of 13.1 Gigayears, – then this! is really an exciting message.
   On the other hand perhaps, taking serious what Perlmutter et al. (1999), Schmidt et al. (1999), and
   Riess et al. (1999) are claiming, namely that theyare seeing in the SN1a luminosities of the most distant
   galaxies already clearindications of an accelerated expansion of the universe would express the fact that
   already at that early times [21-24], all the more at all later cosmic times, the Hubble constant would
   have been given by:
   meaning that the expansion times would have stayed constant all the way since that time
   till today and later, while the free-fall times are increasing permanently since that time
   like
   which would mean that since that early time no collapse could have happenedanymore, and especially young
   galaxies could not at all be understood in this context of the universe. Does that mean: H = HΛ = const
   can be ruled out?, while might appear as the better, since more valid
   approach?
   (14)
   (15)
   (16)
   (17)
   (18)
   (19)
   (20)
   (21)
   (22)
   Volume 5 | Issue 3 | 527
   Copyright: ©2022 Hans J.Fahr. This is an open-access article
   distributed under the terms of the Creative Commons Attribution
   License, which permits unrestricted use, distribution, and reproduction
   in any medium, provided the original author and source are credited.
   Adv Theo Comp Phy, 2022
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21. Fahr, H. J. (2021). The thermodynamics of cosmic gases in
    expanding universes based on Vlasow-theoretical grounds,
    Adv Theo Comp Phy, 4(2), 129-133.
22. Fahr, H. J. (2021). The baryon distribution function in the
    expanding universe after the recombination era, Phys. § Astron. Internat. Journal, 5(2), 37-41.
23. Perlmutter, S. (2003). Supernovae, dark energy, and the accelerating universe. Physics today, 56(4), 53-62.
24. Weinberg, S. (1989). The cosmological constant problem.
    Reviews of modern physics, 61(1), 1.
    https://opastpublishers.com

Adv

Teo

Comp

Phy,

2022

dominant

ingredient

to

the

cosmic

mass

density

ρ

Λ

≫

ρ

b

,

ρ

d

,

ρ

v

(indices

b,

d,

v

standing

for

baryons,

dark

matter,

and

photons,

respectively)

and

to

the

relativistic

energy-momentum

tensor,

then

one

unavoidably

fnds:

which

in

fact

because

of

R

=

0

means

and

necessarily

implies:

a

“coasting

expansion”

of

the

universe!

Then

consequently

a

Hub-

ble

parameter

must

be

expected

that

falls

of

with

the

cosmic

scale

R

like:

The

interesting

point

thus

is

that

the

Hubble

parameter

in

course

of

the

coasting

cosmic

expansion

permanently

decreases

like

-1

period

τ

ex

=

1/H(R),

permanently

grows

proportional

to

R!

Creation

of

gravitational

collapse

centers

during

the

cosmic

expansion

have

formed

over

the

epochs

of

cosmic

expansion,

in

order

to

clarify

whether

or

not

solar

systems

over

the

cosmic

epochs

have

had

diferent

parameters

and

consequently

have

looked

difer-

ent.

And

let

us

start

assuming

a

specifc

cosmic

expansion

state

characterized

by

the

actual

cosmic

scale

R

=

R

0

and

the

pre-

0

this

epoch.

Let

us

further

assume

that

in

this

cosmic

phase

by

a

locally

induced

gravitational

collapse

process

a

mass

center

with

a

central

mass

M,

just

equal

to

the

solar

mass

M0,

is

formed

from

all

the

matter

originally

uniformly

distributed

inside

the

origi-

nating

vacuole

of

a

linear

dimension

D

=

D(R),

i.e.

we

obtain

the

following

request:

This

makes

evident

that

the

actual

linear

dimension

D

=

D(R)

forming

one

solar

mass

unit

M

=

M0

in

the

expanding

universe

is

given

by:

which

tells

us

that

the

characteristic

“solar

mass”-

collaps

di-

mension

D0(R)

is

just

proportional

to

the

cosmic

scale

R,

i.e.

tacitly

been

assumed

that

the

universe

has

a

Euclidean

geometry

with

a

curvature

parameter

of

k

=

0.

Now

let

us

further

assume

for

reasons

given

in

detail

by

Fahr

a

coasting

expansion

of

the

universe

with

the

property

as

ex-

plained

that

the

Hubble

constant

is

given

by

H(R)

=

R/R

=

H0

.

center

of

a

sphere

with

radius

D(R)

might

mean

that

any

piece

of

matter

m

at

the

outer

surface

of

this

sphere

now

is

attracted

in

Newton‘s

sense

by

the

gravitational

feld

of

the

central

mass

M0,

dynamics

leading

to

its

diferential

Hubble

drift

of

v

H

=

D(R)

.

Looking

now

A)

for

the

kinetic

energy

E

kin

with

respect

to

the

bind

this

mass

m

to

the

central

mass

M0

one

fnds:

A)

and

B)

then

conclude

that

over

all

the

periods

of

the

whole

cosmic

ex-

pansion,

i.e.

over

all

cosmic

eons,

the

same

Kepler

problem

(i.e.

motion

of

a

planet

around

the

Sun)

would

be

appearing,

“if!”

the

ratio

ϵ

of

kinetic

over

binding

energy

would

have

turned

out

from

this

consideration

as

a

constant,

i.e.

if

one

would

fnd

ϵ

=

As

one

can

see

the

ratio

ϵ

obtained

above

turns

out

to

be

a

linear

function

of

the

scale

R

of

the

universe

meaning

that

the

Kepler

problem

all

the

time

in

the

universe

would

change

its

character

with

the

cosmic

scale

R,

making

for

instance

the

“Kepler

pen-

dulum”

(with

the

specifc

acceleration

g(R)

=

G

.

M0/D2(R)

at

a

distance

D(R)

from

a

solar

mass

M0)

something

like

“a

cosmic

clock”

with

a

cosmic

oscillation

period

of

i.e.

delivering

a

real

“linear”

cosmic

clock

τ(R)~R

with

G

=

G(R)

=

G0

.

(R/R0).

The

more

interesting

point,

however,

fnally

is

that

this

above

ratio

ϵ

would

in

fact

be!

A

cosmologic

constant

if

the

Newton

gravitational

coupling

coefcient

G

seen

over

the

cosmic

eons

would

not

be

a

constant,

but

instead

would

scale

with

R

according

to

the

formula

G

=

G(R)

=

G0

.

(R/R0)!

Then

simply

not

be

seen

as

a

casual

event

that

happened

just

at

New-

ton‘s

epoch

(~1660nC.),

but

as

an

event

with

a

deep,

fundamen-

Kepler‘s

laws

would

attain

the

rank

of

“cosmologically

relevant

laws”!!!

#

#

4

3

DR

3

R



4

3

DR

3

R

0



R

0

R



3



M



that

the

actual

linear

dimension

D

D

R

form

D



R



R



M



4

3

R

0

3



0



1/3

characteristic

“solar

mass”-

collaps

dimensio

E

kin



1

2

m

DRHR

2

E

bind



GmM



ould

find

const!,

instead

of

what

one

in

fact

numerically

obtains:





1

2

m

DRHR

2

GmM

DR



1

2

DR

2



HR

2

GM

DR



DR

3

HR

2

2GM





R

3



M



4

3

R

0

3



0

H

0

2

R

0

/R

2

2GM





R



H

0

2

4

3

R

0



0



2G

see

the

ratio



obt

ained

above

t

urns

out

to

be

a

linear

f

unction

of

R



2

LR/gR



2

DR/gR



2

D

3

R/GM





2

R

3



M



4

3

R

0

3



0

/GM





2R

R

GM



ring

a

real

“linear”

cosmic

clock

R

~R

with

G

G

R

G

R/R

constant



0



R

0



H

0

2

4

3

R

0



0



2G

0

Volume

5

|

Issue

3

|

598

..

.

Para melhor explicar as luminosidades SN 1a Perlmutter et al. (1998), Schmidt et al.(1998), ou Riess et al. (1998) preferiram uma expansão acelerada do universo conectada com a ação de uma densidade de
energia de vácuo constante (Einstein, 1917, Peebles e Ratra, 2003, Fahr,
2004, Kragh e Overduin, 2014, Fahr e Heyl, 2014), no entanto, há
tentativas mais recentes de Casa do (2011) e Casado e Jou (2013)
mostrando que um “universo costeiro e não acelerado” pode explicar
igualmente bem essas luminosidades de supernovas. Em nossas
considerações a seguir, consideraremos primeiro aqui – principalmente
por razões matemáticas simplificadas – o caso de uma “expansão por
inércia” (ver, por exemplo, Kolb, 1989, Gehlaut et al., 2003, Dev et al.,
2001, Fahr e Heyl, 2020). De fato, deve-se esperar que este caso
prevaleça, caso o universo se expanda sob a forma de ação termodinâmica
e gravodinâmica da pressão do vácuo, como mostra Fahr (2022).
Se então, como nossa base de trabalho, tal “universo costeiro” pode ser
assumido, como dado no caso em que ÿÿ~R-2 (ÿÿ denotando a densidade
de massa equivalente da energia do vácuo, R denotando a escala do
universo, veja por exemplo Fahr, 2022) e quando a energia do vácuo nas
fases posteriores da expansão cósmica se tornou o
A formação definitivamente dependerá da forma específica da expansão
cósmica prevalecente (por exemplo, expansão desacelerada, acelerada
Em princípio, é um problema dificilmente compreensível que a matéria ou por inércia, etc.).
cósmica com, – como geralmente se supõe -, uma distribuição inicialmente
perfeitamente uniforme no espaço possa ter sofrido formação de estrutura
por meio de colapsos locais induzidos gravitacionalmente em grandes
unidades de massa locais? Em um universo em expansão, a matéria
cósmica inicialmente ampla e uniformemente distribuída deve estar sujeita
apenas à expansão para um espaço cósmico em crescimento permanente
conectado com densidades de massa cósmica permanentemente
decrescentes. O oposto só pode ser possível, se a velocidade de um
processo de estruturação local induzido gravitacionalmente for maior que
a velocidade de expansão geral. O problema, portanto, evidentemente
está e deve estar ligado à forma específica da atual dinâmica de
expansão de todo o universo. Para dizer isso em palavras simples: se o
universo está se expandindo muito rápido, isso não deve permitir a
formação de nenhuma estrutura (consulte Fahr e Heyl, 2022)!
ISSN: 2639-0108
Artigo de Pesquisa
1Argelander Institut für Astronomie, Universität Bonn, *
,
Hans J. Fahr1* e M. Heyl2
Autor correspondente Hans
J.Fahr, Instituto Argelander de Astronomia, Universidade de Bonn, Auf dem
Huegel 71, 53121 Bonn Germany.
Resumo
As observações mais recentes do telescópio espacial James Webb (JWST) obviamente mostram, por meio de observações infravermelhas
altamente resolvidas de alta sensibilidade, que a formação estruturada no universo em formas de galáxias primitivas, regiões formadoras
de estrelas e sistemas planetários já ocorreu em tempos cósmicos menos de meio Gigaano após o Big-Bang. Isso é considerado uma
grande surpresa por toda a comunidade astronômica, embora, no entanto, pudesse ser previsível a partir de algumas considerações
teóricas básicas relativas à estrutura básica do universo e suas leis de formas invariantes. Uma pergunta um pouco irônica talvez pudesse
ser feita: Isaac Newton, ao ser derrubado por uma maçã que caiu sobre ele de sua macieira próxima, teria inventado a mesma lei
gravitacional, quando isso teria acontecido um Megaano antes, ou um Megaano depois, O tempo histórico real de Newton? Em outras
palavras, as leis de Kepler, derivadas com a ajuda da lei de Newton, refletiriam as mudanças nos tempos cósmicos? E se sim, – como eles
fariam isso? Neste artigo concluímos que de fato o pêndulo de Newton representaria um relógio cósmico, a menos que a constante
gravitacional de Newton varie com a escala do universo.
Citação:Fahr, HJ, Heyl, M. (2022). Gravidade variável em um universo em expansão: o problema de Kepler com relevância cósmica! Adv Theo
Comp Phy, 5(3), 597-600.
Volume 5 | Edição 3 | 597
Enviado: 09 de setembro de 2022; Aceito: 14 de setembro de 2022; Publicado: 20 de setembro de 2022
Adv Theo Comp Phy, 2022
Königswinterer Strasse 522-524, 53227 Bonn (Alemanha)
2Deutsches Zentrum für Luft- und Raumfahrt (DLR),
Avanços em Física Teórica e Computacional
Gravidade variável em um universo em expansão: o problema de Kepler com relevâncMachine Translated by Google
4ÿ
3
2
4ÿ
3
2
H0
H0
2
4ÿ
H0
2
4ÿ
3
4ÿ
3
Mÿ
4ÿ
3
2
4ÿ
3
H0
Mÿ
2
4ÿ
3
Mÿ
H0
4ÿ
..
.
#
#
Adv Theo Comp Phy, 2022
R0 3ÿ0 ÿ1/3
4ÿ
3
4ÿ 3
Mÿ
Ebind ÿ
Mÿ ÿ1/3
2
DÿRÿ
1
DÿRÿ
GMÿ
GmMÿ
GMÿ
R0ÿ0
GMÿ
4ÿ
3
R0ÿ0
R0ÿ0
DÿRÿ
1
1 2
GmMÿ
2
1 2
4ÿ
3
3 R0ÿ0
1 2
R0 3ÿ0
R0ÿ0
GmMÿ
Mÿ
4ÿ
3
Mÿ
DÿRÿ
R0ÿ0
1 2
DÿRÿ3ÿÿR0ÿÿ ÿ3 ÿ Mÿ pêndulo” (com a aceleração específica gÿRÿ ÿ G ÿ Mÿ/D2ÿRÿ a uma distância DÿRÿ d3 4ÿ R0
o universo em expansão é dado por:
aqui sob quais condições estrelas como o nosso Sol podem ter se formado ao longo
do m ÿ ÿDÿRÿ ÿ HÿRÿÿ2
DÿRÿ
rt assumindo um estado de expansão cósmica específico caracterizado pelo real
4ÿ R0
DÿRÿ
GmMÿ
4ÿ
1 m ÿ ÿDÿRÿ ÿ HÿRÿÿ2 2
GmMÿ
DÿRÿ3ÿÿRÿ ÿ 3
4ÿ 3
O ponto mais interessante, no entanto, finalmente, é que essa razão fornecendo acima um ÿ seria relógio de fato! cósmico nosque “linear” a característica real ÿÿRÿ~R solar”- “massa com colapsa G ÿ a dimensão DÿÿRÿ é apenas ou seja,
desta época. Vamos supor ainda que nesta fase cósmica por um processo de
colapso localmente cional um centro de massa com uma massa
central M um processo de colapso tacional um centro de massa com uma massa central M
A)
DÿRÿ , apenas igual
DÿRÿ ÿ HÿRÿ em relação ao centro de massa.
DÿÿRÿ ÿ R ÿ ÿ
DÿRÿ3ÿÿRÿ ÿ ÿ3 ÿ Mÿ 3
Newton pelo campo gravitacional da massa DÿRÿ central ÿ HÿRÿ Mÿ em n ÿÿ~Rÿ2 relação ao centro de massa atraído no sentido de Olhando denotando agora a densidade A) para a de energia (ÿÿ massa cinética equivalente Ekin em da energia do vácuo, mesmo cnto ver, tempo por está exemplo, sujeita Fahr, à dinâmica 2022) e quando geral de a expansão energia relação do por à vácuo massa inércia ao levando a t , ver, por exemplo, Fahr, 2022) e quando a energia do
massa m à massa central vácuo de Hubble Mÿ no campo diferencial de vH ÿgravitacional Desvio energia de ligação Eligação desta
DÿÿRÿ ÿ R ÿ ÿ
verdade, esta afirmação As leis de Kepler atingiriam Rÿtÿ de a “sentido massa o do nível universo central de Newton M0, e, portanto, deriva cosmologicamente gravitacional de no vH tempo ÿ DÿRÿ cósmico relevante ÿ HÿRÿ t . pelo em Na relação campo mas artigo, talvez deva aparecer como um altamente , Ekin m de em todas relação ao semelhantes as centro leis ao massivas”!!! centro de massa. de pedidos Mach de massa, Por A) ´ que para outro ase o B) massas lado, energia para no lembrar inerciais cinética final deste a ao centro ao inércia mesmo de A) massa, tempo para a e está energia B) para sujeito cinética partículas à expansão Ekin elementares em geral relação por não natureza”, suposição massa são desta constelação portanto, Newton genuinamente massa m à no mas provocativa massa depende do tempo m também à espaço massa central fixadas cósmico da de escala cósmico como central Mÿ que como t . encontra- energia a l Na Mÿ constante energia que “constantes verdade, encontra- os de se: hospeda , de ligação variar gravitacional esta ligação se: pré- escala outro com afirmação Conclusões Ebind selecionadas e Eligar lado, ssim Volume a no desta w Por final 5 om | da Edição a deste artigo, 3 | 598 talvez R = Rÿtÿ deva do aparecer universo e, ideia, como lembra veja: um pedidos Mach, altamente semelhantes 1883, R Thirring, ÿ Rÿtÿ do e 1918, universo MachSciama, ´ ian (para que 1953, as esta massas inerciais m de todos os massivos suposição provocativa de que a constante gravitacional de Newton
pré- depende da escala Barbour e Pfister, 1995, Fahr, elementares 2012). as não partículas são genuinamente fixadas como “constantes suporte lembra e, publicado inerciais selecionadas portanto, as surpreendentemente solicitações m por deno Thirring todos tempo da natureza”, os semelhantes cósmico (1918), R massivos bom mas que t . para Na de tmbém varia ÿ verdade, Mach a Rt ideia do R acordo de universo de o vja: = esta que Rÿtÿ Mach espaço Mach as com afirmação do para massas foi universo a cósmico 1883 constelação esta Um Thirrin idea que 1918 os hospeda Sciama e, 1953 portanto, com a escala,
1
R
R0 3ÿ0
Então o sonho atingindo- de Newton o induzido pela queda da maçã ga uma unidade constante, de massa mas escalariam de uma massa com solar R de acordo Mÿ no gravitacional seriam centro com uma a de fórmula fosse esfera constante, uma de leis Newton com relevantes” se raio mas o Gcoeficiente com visto em vez R ao R0/ de disso longo Rÿ. acordo de acoplamento éons escalariam dos com cósmicos éons a não cósmicos não seriam G = G(R) = G0 . (R/ R0)! Então (~1660nC.), mas como fórmula um evento com uma verdade induzido cosmológica pela queda da profunda maçã atingindo- e fundamental o com de uma duração unidade GÿRÿ massa G de ÿ ÿ de G0 uma ÿ ÿ R/ massa R0ÿ! Então solar Mÿ o sonho no centro de Newton de uma atingindo- esfera o com poderia raio ser O válido, sonho de – e, Newton todos os induzido mais, pela as leis queda de Kepler da maçã poderiam Conclusões atingiriam que simplesmente leis aconteceu relevantes” !!! o nível ser apenas de vistas “cosmologicamente na como época um de evento Nwton. não casual
expansão ic, a fim de esclarecer se há ou não sistemas solares
sobre a expansão mic, a fim de esclarecer se há ou não sistemas solares sobre o
,
3
2
R
R
,
2
2
0 ÿ
,
“
2
R0
3ÿ0 muda seu caráter com a escala cósmica R
2G0
“‘
ÿ
ÿ ÿ
ÿH0
ÿ
ÿ
Como se pode ver a razão ÿ obtida acima acaba por ser uma função linear da escala
isto é, se alguém encontrasse ÿ ÿ const!, em vez do que de fato obtém numericamente: 2G
ÿ ÿR0/ Rÿ2 Mÿ, é R formado do universo de toda significando a matéria que originalmente o problema DÿRÿ distribuída de Kepler ss o uniformemente tempo todo no dentro uiverso da função seria da
escala R do universo significando que o Kepler ass Mÿ, é formado de toda a matéria originalmente distribuída uniformemente dentro do problema todo o tempo no universo
mudaria D2(R) a uma seu caráter distância R3 D(R) ÿ ÿ vacúolo de uma massa de uma solar dimensão M0) algo linear como D “um ÿ DÿRÿ , relógio com cósmico” a escala com cósmica exemplo, uma oscilação R, fazendo o “Kepler por pen dulum” (com a aceleração específica g(R) = G . M0/ cósmica período de DÿRÿ3ÿÿR0ÿÿ massa solar Mÿÿ algo
como “um relógio cósmico” com um período de oscilação cósmica de 2ÿ R3ÿ muda seu caráter com
a escala cósmica R , fazendo por exemplo o “Kepler ÿ /GMÿ ÿ 2ÿR GMÿ
R0 3ÿ0 evidente que a dimensão linear real D ÿ DÿRÿ formando um
pêndulo de massa solar” (com a aceleração específica gÿRÿ ÿ G ÿ Mÿ/D2ÿRÿ cósmico” ÿ G0 ÿ a ÿR/ t uma com R0ÿ .distância ÿ um massa período slar DÿRÿ de M de oscilação ÿÿ aÿÿRÿ algo como cósmica, ÿ 2ÿ LÿRÿ/ “um ou “linear” relógio gÿRÿ seja, ÿreal 2ÿ fornecendo DÿRÿ/ÿRÿ~R gÿRÿ um com ÿ relógio 2ÿ G D3ÿRÿ/ ÿ GÿRÿ cósmico GMÿ
ÿ
ÿ
DÿRÿ3HÿRÿ2
G ÿ Mÿ/D2ÿRÿ a uma distância DÿRÿ de a Como se pode ver a relação ÿ obtida acima acaba por ser uma função linear da 2GMÿ 2G
ÿH0
os períodos de toda a expansão cósmica, ou seja, ao longo de todos os éons cósmicos, o mesmo problema de
Kepler (isto é, o movimento de um planeta ao redor do Sol) estaria aparecendo, “se!” a razão ÿ da energia
cinética sobre a energia de ligação resultaria dessa consideração como uma constante, com G denotando a
constante gravitacional de Newton. Pode-se então concluir que acima de tudo, ou seja, se encontrarmos ÿ ÿ const!, em vez do que de fato se obtém numericamente: os períodos de toda a expansão cósmicconcluir que ao Pode-se então como éons uma cósmicos, constante, longo o mesmo os de períodos toda Kepler, a energia de ou toda seja, cinética a expansão se alguém sobre cósmica, ligação encontrasse 2GMÿ ou seja, ÿ resultaria consideração ÿ const! , ao longo dessa ao de todos os invés do que
de fato se obtém numericamente: estaria aparecendo problema (ou seja, movimento de um planeta
ao redor do Sol), “se!” a razão ÿ de ÿ ÿR0/Rÿ2 ÿ DÿRÿ3HÿRÿ2 2GMÿ
ÿ ÿDÿRÿ2 ÿ R HÿRÿ2 ÿ
, fazendo por exemplo o “Kepler
ÿ ÿR0/Rÿ2
ÿ
ÿ ÿ
DÿRÿ/gÿRÿ ÿ 2ÿ D3ÿRÿ/GMÿ ÿ
R do universo, o que significa que o problema de Kepler o tempo todo no universo
constante cosmológica Mÿ R 2ÿ R3ÿ ÿ/GMÿ ÿ 2ÿR ou seja, “linear” “linear” DÿÿRÿ é real real apenas ÿ(R)~R ÿÿRÿ~R R0 com ÿ com ÿ G ÿ = G G0 . (R) fornecendo cósmico ÿ GÿRÿ (R/ R0 R0). 3ÿ0 ÿ G0 s um ÿ que ÿR/ relógio a R0ÿ . dimensão característica cósmico ou seja, fornecendo “massa solar”- um relógio colapsa
ÿ
mudar 2ÿ LÿRÿ/ seu gÿRÿ caráter ÿ 2ÿcom a escala cósmica R , fazendo por exemplo o “Kepler ÿÿRÿ ÿ
mensão escalaria D0(R) com é R apenas de acordo proporcional com a fórmula à escala umenta cósmica linearmente R, ou seja, com foi assumido a escala do GÿRÿ da uma do escala universo. que geometria universo . ÿ o G0 universo cósmica ÿ ÿR/ 2G0 Por euclidiana R0ÿ . que meio tem R, nos ou a escala eja, diz com que aumenta cósmica a a não característica sendo linearmente R, ou uma seja, “massa constante, aumenta com solar”- a escala linermente mas colapsa em dovez universo. di constante com disso a A escala cosmológicaConstante deste foi assumido cosmológica tacitamente ÿ0 = que o universo tem uma geometria euclidiana com um metro de k = 0.
GMÿ
Conclusões R ÿ
ÿ
ÿ0 ÿ
a energia cinética sobre a ligação teria resultado desta consideração como uma consÿ
R3 ÿ ÿ
R
GMÿ
ou 2GMÿ seja, 2G obtemos solar ignifica R do Mÿÿ universo, a que seguinte algo como problema o que massa “um de relógio Kepler cósmico” o tempo com tdo um noperíodo universode oscilação cósmica de
R
ÿ0 ÿ
ÿ
ÿ
2GMÿ
G ÿ GÿRÿ ÿ G0 ÿ ÿR/R0ÿ! Em seguida, o sonho de Newton induzido pela queda da maçã atingindo-o foi tacitamente assumido que o universo tem uma geometria euclidiana 2G0 ameter de k = 0. rther assuma por razões dadas em detalhes por Fahr e Heyl (2022), mas H0 também R0 ÿ ÿ poderia simplesmente não ser visto como um evento casual
que gravitacional aconteceu de apenas Newton na G época visto ao de longo Newton dos com éons um cósmicos parâmetro assumiria de curvatura detalhes ainda de k por mais = 0. Fahr se por o e razões coeficiente Heyl dadas de em acoplamento inércia do universo com o (~1660nC.), mas como um evento com uma verdade cosmológica profunda (2022), mas e também g deste artigo, para ter uma expansão por constante, as leis de fundamental universo Kepler mas escalaria atingiriam com de avalidade, duração com o nível R de não – de e, acordo seriaainda ” comprovado com umamais, afórmula cosmologicamente deste artigo, dada ter que inclinação uma por a constante HÿRÿ expansão G ÿ visto Rÿ /de por sobre R Hubble ÿ inércia H0 G ÿ ÿ ÿR0/ é GÿRÿ do Rÿ. ÿ se G0 o ÿ campo ÿR/R0ÿ! gravitacional de Newton coeficiente de
simplesmente não poderia ser visto como um evento casual que aconteceu apenas na época de Newtonÿ
ÿ ÿ
ÿDÿRÿ2 ÿ HÿRÿ2
pêndulo” (com a aceleração específica gÿRÿ ÿ R ÿ ÿ ÿ
ÿ
Como se pode ver a razão ÿ obtida acima acaba por ser uma função linear da escala
ÿÿRÿ R0 3ÿ0 ÿ2ÿ O ponto LÿRÿ/ 2ÿ ÿ 2ÿ R3ÿ mais gÿRÿ R interessante, ÿ 2ÿ DÿRÿ/gÿRÿ noentanto, ÿ 2ÿ D3ÿRÿ/ finalmente, GMÿ ÿ ÿ/é GMÿ que
razão acima ÿ seria de fato! esta
,
,
(ou constante seja, movimento gravitacional deum de Newton. planeta ao redor do Sol) ÿDÿRÿ2 G denotando estaria ÿ HÿRÿ2 aparecendo, a DÿRÿ3HÿRÿ2 “se!” a razão ÿ de com
A)
Agora, vamos assumir ainda mais por razões dadas em
detalhes por Fahr e Heyl (2022), mas também no início deste
artigo, para ter uma expansão por inércia do universo com a
propriedade como ex an que qualquer pedaço de matéria m na superfície externa dada pelo (R0/ R) . campo por desta superfície visto (~1660nC.), Em H(R) seguida, como gravitacional esfera = R/externa um R mas fica = produzindo evento H0 . como claro da esta significa casual massa agora um uma esfera evento que que que central unidade agora aconteceu a qualquer com constante Mÿ, de é uma sentido massa mas apenas pedaço de verdade simplesmente Hubble de por em ua de wton New matéria é massa não slar m na pode M0 ser nocosmológica profunda e fundamental de suportar o sentido de Ewton pelo campo gravitacional (~1660nC.), mas da como massa um central evento Mÿ, com mas um no centro outro profundo mão no e sujeita final fundamental deste ao general deva esfera artigo de uma aparecer dinâmica com talvez raio como D(R) de expansão pode uma significar época costeira de que alta levando qualquer tonelada à peça sua validade

• das ainda e, ainda leis mais, desvio relevantes mais, da matéria de as leis da m escala” !!! de na Kepler superfície atingiriam sujeito externa à dinâmica o desta nível esfera de geral “suposição agora de provocativa levando expansão é atraída cosmologicamente à sua de por em verdade que inércia um a constante cosmológica gravitacional total de validade de Newton duradoura, depende – e, massa. vH ÿ DÿRÿ ÿ Hÿ Rÿ em relação ao centro de
  que de fato por causa de R = 0 significa e implica necessariamente: a e:
  período A de criação de centros de colapso gravitacional durante o condições proporcional estrelas tempo como cósmico a R! o Expansão nosso ÿex ÿ Sol 1/Vamos HÿRÿ, podem cresce perguntar ter se formado permanentemente aqui em durante que as
  épocas
  de expansão cósmica, a fim de esclarecer se os sistemas
  solares durante as épocas cósmicas se formaram ou não. a expansão cósmica
  tinha parâmetros diferentes e, consequentemente, parecia diferente. A expansão
  diferentes um estado específico e, cósmica vamos consequentemente, começar de tinha expansão parâmetros assumindo cósmica parecia que diferente. teve parâmetros ent. E
  pareceram caracterizado massa cósmica diferentes. pela homogênea escala diferentes caracterizado HÿRÿÿ2 t assumindo cósmica variando ÿ = e, ÿ consequentemente, (R0) pelo real densidade um = R real ÿ0 = estado R0 de e ÿ de o R0 de pré e expansão a R3 densidade ÿ ÿ ÿH0 cósmica m de ÿÿDÿRÿ massaespecífico ÿ
  cósmica homogênea predominante R0 3ÿ0 nesta época. Vamos supor ainda que nesta massa predominante fase com cósmica m ÿ processo ÿDÿRÿ por um ÿ de HÿRÿÿ2 R colapso ÿ R0 desta e a gravitacional densidade época. localmente de massa cósmica induzido um homogênea centro de
  Como se pode ver, a razão ÿ obtida acima acaba por ser um linear R ÿ ÿ ÿ 2GMÿ
  se fato! o Uma coeficiente O longo constante ponto dos demais acoplamento cosmológica éons interessante, cósmicos gravitacional no seria. entanto, R0 ÿ finalmente, ÿ de razão Newton ÿ seria é G que visto de isso ao
  Suponhamos ainda que nesta fase cósmica por um localmente dentro linear por seja, D do = vacúolo toda DÿRÿ , obtemos igual a matéria a originário uma o seguinte massa originalmente de uma central pedido: dimensão distribuída M, vacúolo exatamente linear de uniformemente D dimensão igual = D( R), à massa ou seja, solar obtemos , M0, é formado ou
  O ponto interessante então é que o parâmetro de Hubble no
  curso B) da expansão cósmica por inércia diminui
  permanentemente como ponto ting assim é que o parâmetro de Hubble no
  curso da inércia H~R-1, e consequentemente o inverso dele, o tempo de expansão Hubble como permanentemente H~Rÿ1, O proporcional durante ponto e consequentemente o importante, período como a R! sion H~Rÿ1, de portanto, inércia diminui e consequentemente ÿex o permanentemente inverso é = que 1/ H(R), o dele, parâmetro cresce sion o permanentemente diminui de tempo ÿex ÿ inverso dele, período de
  e
  com G denotando a constante gravitacional de Newton. Pode-se então
  concluir
  Ebind ÿ que ao longo de todos os períodos de toda a expansão
  cósmica, ou seja, ao longo de todos os éons cósmicos, o mesmo
  problema de Kepler (ou seja, com G denotando a constante gravitacional de Newton.
  Pode-se então concluir que o movimento geral de um planeta em
  torno do Sun) estaria aparecendo, “se!” a razão ÿ da energia cinética
  sobre a energia de ligação resultaria dessa consideração como uma constante, ou seja, se alguém encontrasse ÿ = const!, em vez do que de numericamente: fato se obtém sob quais condições estrelas como o nosso Sol podem ter sIsso torna evidente que a dimensão linear real D = D(R)
  em expansão formando é evidente uma unidade que a de dimensão massa solar M = M0 no universo solar no universo em expansão é dada liner por: real é D dado ÿ DÿRÿ por:formando uma massa
  A escala do universo A
  escala do universo A
  expansão cósmica tornou-se o ingrediente dominante para a expansão cósmica
  tornou- densidade se o de ingrediente massa cósmica dominante ÿÿ ÿ ÿb , para ÿd , a ÿv dinâmica ÿÿ ÿ ÿb,ÿd,ÿÿ, cósmica (índices levando b,d,ÿ ao diferencial desvio representando de de Hubble vH = D(R) . ingrediente dominante para a relação ÿb,ÿd,ÿÿ, bárions, energia efetivamente) representando a ao de matéria relativística (índices centro ligação e aescura de b,d,ÿ tensor gravitacional massa tensor bárions, representando e relativístico Olhando H(R) energia- matéria Eligar em agora momento, relação escura energia- bárions, desta A) e ao para massa fótons, momento, então centro matéria a Olhando energia m ÿÿ um de à ÿ escura massa massa. a então cinética e um central (índices efetivamente) A) Ekin respectivamente) inds: Mÿ b, em d, v e para e ao tensor relativístico energia-momento, agora A) para a energia cinética Ekin em relação ao então inevitavelmente se esta massa m à massa central M0 encontra-se: e B) # # para Ekin a ÿenergia encontra: de ligação inds: gravitacional 1 centro de Ebind massa, de m ÿ ÿDÿRÿ ÿ HÿRÿÿ2 Ekin ÿ 2
  B)
  R
  Rÿ ÿ
  ÿ const ÿ
  const
  “expansão costeira” do universo! Então, conseqüentemente,
  deve-se esperar um parâmetro Hubble que cai com a escala
  cósmica R como: e: R0 B) ÿ H0 ÿ ÿ ÿ R0 R ÿ H0 ÿ ÿ ÿ R
  HÿRÿ ÿ
  dR
  dR
  dt dt
  Então, consequentemente, deve-se esperar um parâmetro de Hubble que caia Hubble e! Então, que caiaconsequentemente, deve-se esperar um parâmetro de
  amc acon o pressão de vácuo, conforme semeado y ar Então produzindo uma unidade de massa de uma massa solar Mÿ no centro de uma esfera ação amic de pressão de vácuo, geral de expansão por conforme mostrado por Fahr (2022). mesmo tempo está sujeito à dinâmicaDÿRÿ “universo externa densidade pode desta de designificar inércia” massa base de equivalente que pode trabalho inércia qulquer ser que assumido, esférica da leva pedaço nergia a tal sua como steiro” desvio “universo de do base matéria vácuo, dado de de pode Hubble trabalho, em m na ser superfície diferencial assumido, tal de como vH ÿdado em n ÿÿ~Rÿ2 (ÿÿ denotando a
  1/HÿRÿ, cresce permanentemente proporcional a R!
  #
  #
  Rÿ ÿ
  HÿRÿ ÿ
  t por causa de Rÿ ÿ 0 significa e necessariamente implica: uma “expansão
  por inércia” t por causa de Rÿ ÿ 0 significa e necessariamente implica: uma “expansão por inércia” e!
  Rÿ
  .
  escala ic R
  like : escala ic R like :
  Rÿ
  Machine Translated by Google
  RU expÿÿÿrÿ/2ÿr2dr RU expÿÿÿrÿ/2ÿr2dr
  RU expÿÿÿrÿ/2ÿr2dr RU expÿÿÿrÿ/2ÿr2dr RU expÿÿÿrÿ/2ÿr2dr
  RU expÿÿÿrÿ/2ÿr2dr
  ,
  c
  r
  Volume 5 | Edição 3 | 599
  Isso talvez possa dar algum suporte ao sempre esperado pela maioria dos cosmólogos.
  Exigindo agora cosmologia 4GMU 4GMU 4GMU e SNe
  Ia. Física Letras B, 504(3), 207-212. ! A igualdade entre os dois casos A) e B) obrigaria o valor Relativitätstheorie, X a ser igual Sitzungsberichte 3c2R 3c2R 3c2R der 6.t, Einstein, no entanto, A. (1917). parece Kosmologische realmente exigir Betrachtungen um zur implicação importante: Allgemeinen comportamento do tipo Machìan do t total, no entanto, parece realmente exigir um
  comportamento do tipo Machìan do t total, no entanto, parece realmente exigir exigir um comportamento do tipo Machìan do niverso proporcional total MU de 4GMU com ao niverso a Este escala pedido, MU R,pois cmno entanto, somente a escala parece R, quando pois realmente somente MU seria exigir quando proporcional um MU tipo KPAkademie MU proporcional seria Machìan ao com niverso a escala der a 3c2R Wissenschaften, R, pois de comportamento somente Phys.Math. quando da massa MU Klasse, seria do satisfeita universo numericamente. Mu com a escala a total ser 142-152. exaltação a poderia exaltação ser poderia satisfeita numericamente. a exaltação poderia ser satisfeita
  numericamente. R, pois somente quando Mu seria proporcional a R, o acima 7. Fahr, (2004). poderia A ser um cosmologia numericamente tipo machista do espaço de cumprido. comportamento vazio: HJ r Como mão é t, interessante no da entanto, relação total parece reconhecer realmente que não exigir
  precisaríamos de uma mão Machìan r é interessante reconhecer que não precisaríamos de uma mão Machìan r é interessante reconhecer que não precisaríamos seria do universo, proporcional niverse do universo, se de MU orequisito uma ao cm se lief. Machìan o a requisito escala a de massa euforia pesado R, já do pois discutido universo, já é somente discutido o vácuo? acima se quando O acima o que já da discutido massa sabemos puder MU ser requisito reforça numericamente nosso da massa be cumprido. Por outro lado, é interessante se dMachìan preferimos Mach. realizada. preferimos precisamos da A massa questão, reconhecer confiar A do de confiar questão, portanto, universo, um no que cumprimento comportamento no gostaríamos portanto, cumprimento se é se o 8.preferimos é de de Mach. SELLSCHAFT, confiar A questão, em 33, Mach, portanto, 339-352. não é se
  ,
  controverso e deve ser precisamente definido como “a massa simultânea de
  1 ÿ ÿ 1 ÿ ÿ 1 ÿ ÿ
  8ÿG 8ÿG
  8ÿG
  rc2 rc2 rc2 expÿÿÿrÿÿ é dada por; nt, visão preferencial de um
  universo flutuante com H ÿ H0 ÿ ÿR0/Rÿ e nt, visão preferencial de um universo flutuante com H ÿ H0 ÿ ÿR0/Rÿ e nt, visão preferencial de um universo
  flutuante com H ÿ H0 ÿ ÿR0/Rÿ e x2dx 8ÿG rc2 1 ÿ ÿHr/cÿ2
  expÿÿÿrÿÿ ÿ 1 ÿ ne, portanto, pode avaliar a
  expressão superior por (consulte Fahr e Heyl, 2006): ne, portanto, pode avaliar a expressão superior por (consulte Fahr e Heyl, 2006): ne, portanto, pode avaliar a expressão superior expressão por (ver Fahr e Heyl, 2006): Adv Theo Comp Phy, 2022
  MUÿtÿc2 ÿ 4ÿÿ0ÿtÿc2 ÿ MUÿtÿc2 ÿ 4ÿÿ0ÿtÿc2 ÿ
  a expÿÿÿrÿÿ nction é dada por; a expÿÿÿrÿÿ nction é dada por; a expÿÿÿrÿÿ nction é dada por;
  1 ÿ ÿHr/cÿ2 1 ÿ ÿHr/cÿ2 1 ÿ ÿHr/cÿ2
  MUÿtÿ ÿ MUÿtÿ ÿ MUÿtÿ ÿ
  1 ÿ ÿ 1 ÿ ÿ
  X ÿ KB X ÿ
  Journal, 5(2), 37-41. da seguinte forma: Não elaborada incluindo forma: equivalentes (1934), da por seguinte Tolman da 11. os seguinte Fahr, de HJ massa forma: dapor térmica qualquer conforme detector definição físico. já
  X ÿ
  expÿÿÿrÿÿ ÿ 1 ÿ expÿÿÿrÿÿ ÿ 1 ÿ expÿÿÿrÿÿ ÿ 1 ÿ
  Fahr, HJ (2021). A termodinâmica dos gases cósmicos em ontroversial massa linear de por Computational detector do trás acoplamento termal Tolman da da ontroversial massa Tolman detector termal quantidade simultânea físico . constante y nunca qualquer (1934), e De simultânea Physics, deve (1934), G físico. Não fato, – ” de de de mas 10. e de sr incluindo deve tector acordo para ontroversial acordo mas Não4(2), acoplamento Fahr, definida nunca de mencionar ser incluindo nunca 129-133. HJo com com físico. definida preferimos os de com equivalentes Advances acordo uma e uma deve de ention Não os Na precisão definição com Newton definição equivalentes verdade, incluindo com ser aceitar a Theoret. precisão verdade, definida como a de G uma já o ojá – massa conceito elaborada universos aumento os elaborada “a definição como” de com o equivalentes massa conceito massa do precisão por a termal em por simultânea já expansão elaborada por baseados quantidade Tolman y de como qualquer trás massa da de (1934), “a em por “MU” , do fundamentos ou seja, teóricos a massa de do Vlasow, y qualquer constante
  Isso talvez possa dar algum suporte para, em uma primeira da Terra visão no nível – suposições da água tratada do provocativas oceano em dois efeitos – da efeitos rotação das das forças da forças Terra centrífugas centrífugas no nível da da água da Terra rotação do em oceano rotação oceano centrífugas uma primeira da visão Terra em rotação no nível da água do no nível da água do oceano efeitos das forças variável fisicamente fisicamente a solidamente Terra G. idênticas st, em de uma e repouso escala B) idênticas idênticas em a primeira Terra contra- – A) G. com a visualizações – em Terra A) ) rotação visão um repouso a a Terra – Terra girando universo suposição – suposição ao girando girando fisicamente com redor. em solidamente um provocativa relação st, provocativa em em universo e relação idênticas relação B) às a Terra em visualizações – às de ao – – contra- de A) visualizações em uma universo a uma repouso Terra variável rotação em relação B) universo girando a com repouso, ao de Terra fisicamente redor. escala a em um uma solidamente em e repouso variável de com em escala contra- uma contagem G. rotação visualizações sólida ao redor. st, e B) Referências forças centrífugas efeitos demonstrar das forças com entrífugas abordagens universo do centrífugas da oceano Terraelativísticas er-girando em demonstrar demonstrar rotação ao gerais redor. no com com nível Como as abordagens abordagens forças da ele água pôde relativísticas relativísticas demonstrar com gerais gerais gen as as forças nestas duas centrífugas visões em fisicamente 1. Barbour , idênticas (1995). da relacionados JB, Terra & Pfister, – A A) em relativística a relação H. rotação pela (Eds.). seguinte primária aos es equação: equação: são de Mach relacionados es aborda es são são as relacionados pela forças seguinte centrífugas pela seguinte a Terra os redor. em equação: casos repouso Springer são ciclo: relacionados com Science do um balde universo & Business pela de Newton seguinte solidamente Media. à gravidade equação: em contra- st, quântica e B) rotação (Vol. 6). ao
  (2020). Modelos de expansão linear versus verso 4GMU
  padrão . Exigindo agora a identidade física entre os dois casos de cosmologias: uma visão crítica e histórica. A astrofisidade entre os casos importante: linear dois em A) B) com 5. casos R e Dev, forçaria B) denotando Implicação o forçaria ics A) A., and e Sethi, o B) valor Space forçaria a a importante: valor M., massa X & Science, a X Lohiya, ser o total igual valor a igualdade Implicação 365(1), e D. a X a 3c2R a (2001). escala er 1-14. A) a entre igualdade Deslizamento do e importante: B) Implicação universo. forçariam os dois entre Implicação o valor os dois X igual importante: casos a 1, A) e
  , , ,
  GG G pode-se assim avaliar a expressão superior por (ver Fahr e Heyl, 2006): como se um universo Machìan fosse solicitado com MU~RU, mas
  com o como se um universo Machìan fosse solicitado com MU~RU, mas com o como se um universo de Machìan fosse solicitado com MU~RU, mas com o
  X ÿ
  MUÿtÿc2 ÿ 4ÿÿ0ÿtÿc2 ÿ MUÿtÿc2 ÿ 4ÿÿ0ÿtÿc2 ÿ MUÿtÿc2 ÿ 4ÿÿ0ÿtÿc2 ÿ
  1 ÿ ÿ x2dx x2dx x2dx ÿ
  1ÿ
  X ÿ
  1 = 1 = 1 =
  MUÿtÿc2 ÿ 4ÿÿ0ÿtÿc2 ÿ
  4GMU 4GMU KA _
  (2021). A função de distribuição de bárions foi mostrada por Fahr e Heyl (2006) que a expressão
  de Tolman pode ser mostrada por Fahr e Heyl (2006) que a expressão de Tolman pode ser mostrada por Fahr e Heyl expressão (2006) que de aTolman pode ser Mu” , universo em expansão após a era da recombinação,
  Phys § De acordo com uma definição já elaborada por Tolman (1934), mas nunca ser definido com precisão como “a massa simultânea do universo” tron.
  Fahr, HJ (2012). “Was ist Trägheit?”, em “Wie gewiss ist r hand é interessante preferimos aumento aceitar reconhecer linear o aumento da que constante não linear pecisaríamos da de constante acoplamento de de um acoplamento de Machìan Newton preferimos G ? deNewton aceitar G aceitar – preferimos o o aumento linear da constante de acoplamento de Newton
  G – o requisito de G~R já discutido acima seria cumprido. unser Wissen?, Ed. H.Müller, Frank Verlag, & Ber Timme he mass of the universe, if o requisito já discutido
  acima de ie a massa do ie a massa do ie a massa do ention a verdade, o conceito por trás da quantidade “MU” ention a verdade, conceito o conceito portrás por trás da quantidade da quantidade “MU” “MU” A questão, ention portanto, a verdade, é, nós o preferimos confiar no prin
  lin de Mach, S.208-226. cumprido. A questão, portanto, é, nós preferimos confiar no princípio de Mach, ou preferimos aceitar o aumento linear do 9 de Newton.
  expÿÿÿrÿÿ ÿ 1 ÿ expÿÿÿrÿÿ ÿ 1 ÿ
  MUÿtÿ ÿ MUÿtÿ ÿ
  hora
  c2 c2 c2
  0 0
  0
  0 0
  ÿ2 ÿ2
  KA = X ÿ KB KA = X ÿ KB KA a: ano = X das ÿ KB observações demonstra do com Wilkinson abordagens Anisotropy relativísticas Probe (WMAP), gerais calculado ctor as forças X é como calculado centrífugas sendo como igual em igual 2. Bennet, a: ctor CI, X Halpern, é calculado M., Hinshaw, como igual G. a: et ctor al. (2003). X é calculado estão Primeiro, relacionados como onde o igual fator pela a: X es é seguinte equação: Astrophys. Journal Suplements, 148(1), 97-117.
  ÿ0 ÿ ÿ0 ÿ
  0 0 0
  r r r
  hora hora
  8ÿG 8ÿG
  rc2 rc2
  ÿ2 ÿ2 ÿ2
  ÿ0 ÿ ÿ0 ÿ ÿ0 ÿ
  ÿ0 ÿ
  r r
  0
  Hr Hr Hr
  c c c
  0 0 0
  c c
  Ru Ru Ru nt, visão preferida de um universo costeiro com H ÿ H0 ÿ ÿR0/Rÿ e

1. Casado, J., & Jou, D. (2013). Modelo cosmológico de fluxo constante 3c2R 3c2R 3c2R . Astrofísica e Ciência Espacial, 344(2), 513-520. ctor X é calculado para ser igual a: R denotando a massa total e a escala do universo. Exigindo agora R denotando a massa
   total e a escala do universo. Exigindo agora R denotando a massa
   total e a escala do universo. Requerendo agora com Mu e R denotando a massa total e a escala do uni 4. Casado, J.
   ou seja, a massa do universo”
   Mu ~ Ru, mas com a escala já discutida acima, depende da já discutida dependência da da ideia escala de Mach da constante foi publicado constante R0) Mu gravitacional de como isso fato seria G trazendo gravitacional, ÿ G0 um pela ÿ suporte ÿR/ de nce volta R0ÿ surpreendentemente já da isso um discutida constante universo traria de acima, gravitacional com fato uma dependência de bom volta massa para por um G a total = G0 constante0 (R/
   universo com uma massa total constante ma G ÿ G0 ÿ ÿR/R0ÿ isso traria de fato um
   universo dos cosmólogos. com ma Isso total talvez constante seja sempre Thirring esperado (1918) que pela comparou maioria os dos efeitos cosmólogos. da força centrífuga sempre esperados pela maioria
   é controverso e deve
   (2022). Quanto poderia ligação gravitacional, mas nunca
   demonstrado reconhecível por Fahr por qualquer e Heyl detector físico. Não inclusive foi Computacional, agindo como 5(2), energia 449-457. de a (2006) vácuo seguinte cósmica que forma: a expressão oculta? por Fahr Física de e Tolman Heyl (2006) pode que ser expressão a a energia de Tolman pode ser avaliada 12. Fahr, HJ, &
   Willerding, E. (1998). Die Entstehung von ated na seguinte forma: Sonnensystemen: eine Einführung in das Problem der Plan Spektrum, etenentstehung. Akad. Verlag.
   onde onde a função exp[ÿ(r)] é dada por; a função expÿÿÿrÿÿ função é dada expÿÿÿrÿÿ por; onde é a dada por;
2. Fahr, HJ e Heyl, M. (2006). Sobre a massa instantânea e
   a extensão de um universo em expansão. Astronomische
   Nachrichten: Notas Astronômicas, 327(7), 733-736.
3. Fahr, HJ, & Heyl, M. (2014). A termodinâmica de um
   SCHRIFTENREIHE-WITTGENSTEIN GE
   Conclusões
   Por outro lado, no final deste artigo, talvez deva aparecer x2dx x2dx como uma suposição altamente provocativa de que a constante gravitacional 1 ÿ ÿHr/cÿ2 1 ÿ ÿHr/cÿ2 de
   no tempo cósmico t. Na verdade, esta afirmação lembra: No Newton depende da escala R = R( t) do universo e assim
   presente, visão preferencial de um universo flutuante com H = H0 No presente, visão preferencial de um universo flutuante
   com H ÿ H0 ÿ ÿR0/Rÿ e No presente, visão preferencial de um universo flutuante universo e Heyl, 2006): constantes lecionadas”, mas sim pode- com também a se avaliar constelação avaliar variam a expressão a expressão do espaço com inerciais avaliar superior H cósmico superior a ÿ m expressão H0 de de ÿ pedidos c2 todas ÿR0/ por c2 (ver Rÿ superior Ru as emelhantes e massas Ru Fahr 0 (R0/ que por e R) Hey ÿHR/ os (ver e hospeda de ÿHR/ (HR/ c Fahr Mach ÿ2 c)2 cÿ2 ÿ e ´ c2 , e, Hey). ÿ ian ÿ c2 portanto, c2 , portanto, que pode- (ver portanto, as Fahr se massas com a pode- escala se
   um universo Machìan seria solicitado com R um = universo R(t) o qual do parece verso Machìan uni que G seria G (para solicitado esta ideia com ver: MU~RU, Mach, 1883, mas Thirring, sagacidade 1918, que Sciama, parece que como parece se um que universo de Machìan fosse solicitado com MU~RU, mas com 1953, Barbour e Pfister, 1995, Fahr, 2012).
   Machine Translated by Google
   Direitos autorais: ©2022 Hans J.Fahr. Este é um artigo de acesso
   aberto distribuído sob os termos da Creative Commons Attribution
   License, que permite uso, distribuição e reprodução irrestritos em
   qualquer meio, desde que o autor original e a fonte sejam
   Adv Theo Comp Phy, 2022 creditados. https://opastpublishers.com Volume 5 | Edição 3 | 600
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