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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    R2 D  H  D  H0
    2 R0
    2
    R2 D  H0 R0
    R  D
    or :
    4
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    2 R0
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    3o1/3 one finds:
    4
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    2 R0
    2
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    which means H  0! and R  R 8G
    3  . This in contrast to the above relation thus
    then implies
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