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Introduction
How does the initial matter uniformity disappear in an
expanding universe?
It is hardly understandable why cosmic matter with, - as generally
required by the cosmologic principle -, an initially perfectly
homogeneous distribution in space may at all have started at some
epoch in the past a process of forming local material substructures
like stars or galaxies. Such formations are generally understood as
driven by local, gravitationally induced collapse instabilities of
cosmic gases forming large local units of solar or Giga-solar masses.
In an expanding universe the uniformly distributed cosmic matter
should otherwise simply be subject to an ongoing redistribution into a
permanently growing cosmic space, accompanied by permanently and
unavoidably decreasing cosmic mass densities
3
00
() ( / )R RR
ρρ
= ⋅
. The opposite can only be expected, if the collapse period of a
gravitationally induced local structuring process is shorter than the
universal expansion period of the homogeneous matter distribution,
so that density structures can form and do grow decoupled from the
general cosmic expansion. The problem thus evidently is and must
be closely connected with the specic form of the actual expansion
dynamics of the whole universe, permitting matter to accumulate at
distinct places, even though the universe continues to expand.Given
an accelerated expansion of the universe, as is presently favoured
by several astrophysicists when trying to understand the redshifted
emissions of most distant galaxies1-3 it may be denitely harder to
understand these structure formation processes. Here in this article
we, however, mainly consider this problem on the basis of a “coasting
expansion of the universe” with a constant expansion velocity of its
scale
()R Rt=
with
R const=
and
0R=
. This latter form of the
cosmic expansion we do strongly favour in this article since it can be
based on solid scientic grounds.4,5
Why is the Hubble parameter a critical quantity?
Before the event of cosmic matter recombination anyway no
gravitationallly induced matter collapses were possible, because
then ionized matter - because of strong electron-photon couplings -
was repelled by the collapse-inherent increase of radiation pressure.
Thus the question arises, how much variation the Hubble parameter
()H Ht=
may have undergone since that cosmic time of matter
recombination when at rst in cosmic history matter accumulation or
condensation could have started? What in fact does one know at all
about the value of the Hubble parameter at earlier times in the cosmic
past, especially near and even before the point of recombination of
cosmic matter? To frankly confess the truth: Not very much, - and for
sure - nothing safe yet.
All about that is connected with the mainstream cosmic view which
cosmologists nowadays share concerning the state of the universe near
cosmic recombination time. One can only speculate about this point on
the basis of the Big-Bang cosmology, and perhaps question whether it
existed at all in the history of the universe, i.e. if at all cosmic matter
at some times in the past was in a fully ionized phase. The present
day value of the Hubble parameter with
70 / /
today
H km s Mpc=
. 6is
obtained from redshift observations of the more or less nearby galaxies
with redshifts
1z≤
, and not very much can be speculated from this
poor observational basis on the specic value of
()
rr
H Ht=
which
prevailed at the time of recombination
r
tt=
(i.e.
3
10Ζ
). If to the
contrary at least some fundamental theoretical prerequisites need to be
fullled, then at least the basis for estimations would be better.
For example: If the Hubble parameter
H
is predetermined for all
cosmic times by a constant vacuum energy density
Λ
, at present time
as well as back all the time till the recombination time
r
tt=
, then it
can be shown (see Fahr, 2021a) , that the Hubble parameter would
have been constant all over this time period from the recombination
period till now, i.e.
today r
HH H
Λ
= =
. That would mean concerning
Phys Astron Int J. 2022;6(4):135‒140. 135
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The cosmic pendulum: Kepler‘s laws representing a
universal cosmic clock
Volume 6 Issue 4 - 2022
Hans J Fahr
Argelander Institut für Astronomie, Universität Bonn, Germany
Correspondence: Hans J Fahr, Argelander Institut für
Astronomie, Universität Bonn, Auf dem Huegel 71, 53121 Bonn,
Germany, Email
Received: Sepetmebr 30, 2022 | Published: October 17, 2022
Abstract
Recent observations of the James Webb Space Telescope (JWST) seem to show that
structure formation and the build-up of planetary systems in the universe already must have
started astonishingly enough at a time of 0.1 Gigayears after the Big-Bang. The question
thus arises whether these earliest planetary systems did originate under similar conditions
as did our solar system about 4.1 Gigayears later? In this article we are looking onto this
fundamental problem and show that for the context of the origin of solar systems it very
much counts how the Hubble expansion of the universe has developed over cosmic eons.
If the cosmic expansion dynamics is too large, no solar systems at all would have been
produced, if it would be too small, solar systems would have originated just shortly after
the cosmic matter recombination, but not anymore since then.
In other words, the Keplerian laws, derivable with the help of Newton‘s gravitational law,
would they perhaps reect the changes in an expanding universe over the cosmic eons? And
if yes, - how would they do it? In this article we conclude that in fact Newton‘s pendulum
or Kepler‘s planetary revolution periods represent a perfect cosmic clock indicating the
actual status of the expanding universe. Only in case, however, that Newton‘s gravitational
constant
G
would vary with the scale
R
of the universe like
GR
, then this clock
astonishingly enough would be synchronized for the whole cosmic evolution not serving
anymore as a cosmic tracer.
Physics & Astronomy International Journal
Review Article Open Access
The cosmic pendulum: Kepler‘s laws representing a universal cosmic clock 136
Copyright:
©2022 Fahr
Citation: Fahr HJ. The cosmic pendulum: Kepler‘s laws representing a universal cosmic clock. Phys Astron Int J. 2022;6(4):135‒140.
DOI: 10.15406/paij.2022.06.00266
the above relation that r today
HH=
! If, however, the Hubble
parameter at present times, as well as at the recombination time
r
t
, is
purely determined by baryonic matter, i.e. by the rest-mass density of
baryonic matter
()
BR
ρρ
=
, then one could use the following relation
taken from the rst of the two Friedman equations7 and obtain:
28
() ()
3B
G
HR R
πρ
=
Concerning the corresponding Hubble parameter at the
recombination epoch at
r
tt=
with
3
10 odayr t
RR −
=
8one would then
obtain a value:
3
3/ 2 3/ 2 4.5
10
( ) ( ) 10
today
r today tod
r
r
ay today
r
R
HH R
RR
HH
=⋅=⋅ =
meaning that the Hubble parameter at the recombination era could
under this prerequisites certainly have been much larger than the
present day Hubble parameter
today
H
.
For a more general study of the historic evolution of the Hubble
parameter
()H Ht=
one should, however, start from a broader, more
general analytic basis by again looking back to the rst of the Friedman
equations,7 when expressing the fact that the Hubble parameter in a
more general outline is given by:.
2
2
2
8
[]
3
BDv
R
HG
R
π
ρ ρ ρρ
Λ
+ ++= =
where all quantities denote equivalent mass densities
[]
BDv
ρ ρ ρρ
Λ
+ ++
of baryonic matter, of dark matter, of photons,
and of the vacuum energy. These quantities are thought to be known
as functions of time
t
, or equivalently, of the scale of the universe
()R Rt=
, though at least the quantities D
ρ
and
ρ
Λare physically
not at all well conceived, neither by its concrete meaning nor by its
dependence on the scale
R
of the universe.
By introduction of
2
00
3 /8HG
π
Ω=
with 0
H
denoting the present-
day Hubble parameter one can write the upper equation in the
following form:
0
[ ][ ]
1
1BDv BD v
ρ ρ ρρ
ΛΛ
+ + + = Ω +Ω +Ω +Ω
Ω
=
For the present cosmic epoch one has obtained observational
best-t values for the above quantities
BDvΛ
Ω +Ω +Ω +Ω
given by2,6
with the following numerical values:
0.04;.... 0.23;.... 0.01;.... 0.72
BDvΛ
Ω = Ω = Ω= Ω =
Inserting now in addition the expected dependences of
; ;;
BDv
ρ ρ ρρ
Λ
on the scale
R
of the universe leads us then to the
following expression:
2
2
2
23 3 4
0() () () ]
o oo
BDv
R RR
HRR
R
HRRΛ
Ω +Ω +Ω⋅+Ω= =
Hereby the equivalent mass energy density
v
ρ
of the cosmic
photons has been taken into account by its value corresponding to
a cosmologically redshifted Planck radiation.9 When introducing the
present-day
Ω
-values into the upper equation, one then obtains the
R
-dependence of the Hubble parameter in the following form:
34
00 0
( ) 0.27( / ) 0.01( / ) 0.72]HR H RR RR=⋅ ++
Figure 1 Hubble parameter
()Hx
(yellow curve) and the expansion velocity
()Rx
(blue curve) as functions of the normalized Hubble scale
0
/x RR=
.
Going back to the expected recombination point at
0/ 1000
r
RR=
one thus learns that the Hubble parameter
()
rr
H HR=
for this time is
given by:
34
00
0.27(1000) 0.01(1000) 0.72] 0.84
r
HH H
−−
=⋅ ++
or expressing the surprising fact that at the expected
recombination time
r
tt=
the photon eld does contribute the utmost
to the Hubble parameter and amounts at that time
r
tt=
to a value:
0
0.84
r
HH
.
Taking as our basis such a “coasting universe” which prevails for
the case of
2
R
ρ
−
Λ
(
Λ
denoting the mass density equivalent of the
vacuum energy,
R
denoting the scale of the universe,4 and taking the
period when the vacuum energy in the later phases of cosmic expansion
unavoidably becomes the dominant ingredient of the cosmic mass
density
,,
bdv
ρ ρρρ
Λ
(indices
,,bdv
standing for baryons, dark
matter, and photons, respectively), then one unavoidably nds:
dR
R const
dt
= =
(1)
which in fact because of
0R=
necessarily implies: a “coasting
expansion” of the universe! Then consequently a Hubble parameter
must be expected that falls o with the cosmic scale
R
like:
0
0
() ( )
R
R
HR H
RR
= = ⋅
(2)
meaning that the Hubble parameter
()HR
in case of a coasting
cosmic expansion permanently decreases like
1
HR
−
, and
consequently the inverse of it,
1
H
−
, i.e. the expansion time period
1/ ( )
ex
HR
τ
=
of cosmic matter, permanently grows proportional to
R
!
Structure formation in the cosmic gas
As discussed in Fahr and Zönnchen9 in a homogeneous expanding
cosmic gas cosmic matter structures can form due to selfgravitational
interactions in density perturbations of this cosmic gas. These self-
generating structures are persistent phenomena of cosmic sound
waves, however, when selfgravity of the oscillatory matter is included.
The cosmic pendulum: Kepler‘s laws representing a universal cosmic clock 137
Copyright:
©2022 Fahr
Citation: Fahr HJ. The cosmic pendulum: Kepler‘s laws representing a universal cosmic clock. Phys Astron Int J. 2022;6(4):135‒140.
DOI: 10.15406/paij.2022.06.00266
The typical dispersion relation for such self-gravitating, accoustic
waves is given in the following form:10
2 22
() 4
sr
k vk G
ω πρ
= −
with
ω
as the wave frequency,
2/k
πλ
=
as the wave vector
and wave length
λ
, and
s
v
as the eective, local sound velocity at
recombination era.
G
is Newton‘s gravitational constant, and
r
ρ
is the
actual local matter density at the recombination time
r
tt=
.
As evident from the above dispersion relation, there exists a
critrical wave number
c
k
with
2
4r
c
s
G
kv
πρ
=
and the property that all waves with wavenumbers
c
kk≤
lead
to unstable, standing waves with imaginary values for associated
frequencies
ω
, i.e. with growing wave amplitudes and hence ongoing
of structure formation.
From that fact one can conclude that the characteristic wavelengths
of standing wave structures at the recombination epoch are given by:
2
2
22
4
s
c
cr
r
s
v
kG
G
v
π
ππ
λρ
πρ
= = =
Calculating the value of
c
λ
one obtains with
/
s rr
vP
γρ
=
and
5/3
γ
=
,
r rr
P n KT=
and
r
T
denoting pressure and temperature of the
cosmic H-gas:
22
( ) ()
2.3
r rr r r
c
rr
rr
P n KT KT KT
mG mG
GG
πγ πγ πγ
λρρ
ρρ
= = = =
The temperature at the recombination era is expected to be about
3000K
, and due to the redshift cooling of the present
CMB
(
3K
-radiation) one obtains the redshift relation:
0
(1 ) ( / ) 1000
r
z RR+=
. This means that the present cosmic density of the universe
31 3
0
10 /g cm
ρ
−
=
should have been larger at the recombination era by
a factor 3
(1000) yielding an actual value at
r
tt=
of
22 3
10 /
r
g cm
ρ
−
=
.
This argumentation is based on the assumption that cosmic photons
are subject to redshifts which are due to the expansion of the universe.
If this cosmic mainstream basis is questioned, then, as we shall show
at the end, this would change all of our above conclusions.
The baryon gas temperature
r
T
, solely due to the inuence of the
Hubble drift at the recombination era, should develop according to a
linear approach for
0.1 ( )
rr
Ht t≥−
by:11
2
(t) (1 ( ) )
Hr
H
rr
T
THt t
=
−−
and the density is given by :
3
()
(t) ( )
()
r
Hr
Rt
Rt
ρρ
= ⋅
Covering a time period
t∆
after the recombination point
r
tt−
, over
which the Hubble parameter
r
HH=
can be considered as constant,
permits then to write
( ) ( ) exp[ ( )]
r rr
Rt Rt H t t= −
and consequently yielding the following density as function of
time:
3
()
( ) ( ) exp[ 3 ( )]
( )exp[ ( )
r
H r r rr
r rr
Rt
t Ht t
Rt H t t
ρρ ρ
=⋅ = −−
−
The critical mass
c
M
of a collapse-critical gas package is then
given by:
3 3 3/2 3/2 1/2
44
2.3 ( ) 541.3 ( )
33
HH
c cH H H
H
KT KT
MmG mG
ππ
λρ ρ ρ
ρ
−
= = = ⋅
If now one introduces the above expressions for
()
H
Tt
and
()
Ht
ρ
as functions of
t
, one then can see the marginally possible,
selfgravitational collapse mass
()
cc
M Mt=
as function of the cosmic
time
t
after the recombination point as given by:
,
3 3/2 1/2
,0
3
4 exp[(3 / 2) ( )]
() () () [51.3 ( ) ] ()
3(1 ( ) )
Hr rr
c c H Hr c
rr
KT
Ht t
Mt t t M t
mG Ht t
π
λρ ρ µ
−−
==⋅=⋅
−−
The above expression
()t
µ
describing the growth factor of
the mass condensate in time is shown in Figure 2. The three
curves represent solutions for three Hubble parameters namely
0
70 / / ;
today
H H km s Mpc= =
10
2HH=
; and
20
4HH=
. One can
see that the critical mass substantially increases and also reaches
an expected magnitude of
6
10
, meaning that masses of the order of
11
10
c
M>
M
, i.e. solar masses, within a time of several Billions
of years are possible, however, it must be realized that the results of
Figure 2 are based on the assumption that within the considered time
the actual Hubble parameter is not varying, but keeps a xed value of
000
;2 ;4HHHH=
.
Figure 2 The mass growth factor
()t
µ
as function of cosmic time in
Megayears in a linear approach with
1,2,4H=
0
H
.
The above expression shows that possible critical masses
()
c
Mt
are growing with cosmic time
t
, however, one should keep in mind,
to produce elementary cosmic cornerstones like galaxies, one would
need a growth factor of about
6
10
. Furthermore there exists a severe
limitation for this mass growth given through a comparison between
gravitational free-fall times
ff
τ
and expansion times
ex
τ
. The time
ff
τ
is the time it takes to condense the gravitationally unstable mass
()
c
Mt
to a stable structure by its free-fall in the genuine gravitational
eld, without the pressure action taken into account, and is given by:
1
4
ff
r
G
τπρ
=
The expansion time
ex
τ
is the typical time needed to expand the
mass
()
c
Mt
with the ongoing Hubble expansion to innity or say:
back to the whole universe!, and it is simply given by:
The cosmic pendulum: Kepler‘s laws representing a universal cosmic clock 138
Copyright:
©2022 Fahr
Citation: Fahr HJ. The cosmic pendulum: Kepler‘s laws representing a universal cosmic clock. Phys Astron Int J. 2022;6(4):135‒140.
DOI: 10.15406/paij.2022.06.00266
1
ex
r
R
RH
τ
= =
The critical mass can only survive as a cosmic structure, as long as
ff
τ
is smaller than
ex
τ
, meaning that one should numerically have the
following relation fullled:
11
4r
rH
G
πρ
≤
Creation of solar-type collapse centers in a coastingly expanding
universe
We shall ask now under which conditions stars like our Sun with
masses of
MM
°
can have formed over the epochs of cosmic
expansion. This addresses the question whether or not “solar systems”
(i.e. planetary systems with a central mass
1MM
°
= ⋅
like our Sun)
over the cosmic epochs have had dierent orbital parameters and
consequently might have looked dierent over the cosmic eons. We
start from a specic cosmic expansion state characterized by the actual
cosmic scale
00
() RRt =
and the actually prevailing homogeneous
cosmic mass density
0 00
( ) (R )t= =
of this epoch.
Let us assume that in this cosmic phase by a locally induced
gravitational collapse instability a mass center with a central mass
M
, just equal to one solar mass
M
, is formed from all the matter
originally uniformly distributed inside the mass-generating source
vacuole with a linear dimension
()D DR=
, obtained by the following
request:
3 33
0
0
44
() () () ( )( )
33
R
DR R DR R M
R
ππ
= =
This makes evident that the actual linear dimension
()D DR=
forming one solar mass unit
MM=
in the expanding universe is
given by:
1/3
3
00
() ]
4
3
M
DR R
R
π
= ⋅
expressing the fact that the characteristic solar mass-vacuole with
a linear dimension
()DR
is just growing proportional to the cosmic
scale
R
of the universe. Hereby it has tacitly been assumed that the
universe has a Euclidean geometry with a curvature parameter of
0k=
.
As motivated in the beginning of this article, we now assume to
have a universe with a “coasting expansion” , i.e. with the property
that the Hubble constant is given by
00
() / ( / )
cc
H R RR H R R= = ⋅
.
Then producing via collapse a mass unit of one solar mass
M
in the
center of the sphere with radius
()DR
might mean that any massive
object at the periphery of the originating vacuole now is attracted
in Newton‘s sense by the gravitational eld of the central mass
M
, but at the same time with respect to this mass center it is subject
to the dierential Hubble drift
() ()
c
H
v DR H R= ⋅
due to the coasting
expansion dynamics. This dierential Hubble drift with respect to the
mass center supplies the necessary kinetic energy of the peripheral
object for its orbital motion around the central mass
M
.
Looking now both for the specic kinetic energy
kin
E
of this object
with respect to the mass center, and for the specic gravitational
binding energy
bind
E
of this object with respect to the central mass
M
one nds:
2
1[() ()]
2
c
kin
E DR H R= ⋅
and:
()
bind
GM
EDR
=
where
G
denotes Newton‘s gravitational constant. Considering
the ratio
/
kin bind
EE∈=
of kinetic over binding energy of such a
“Keplerian” object would then lead to the following expression:
2
232 00
]
4
1[() ()] () () 3
2
() 22
()
c
o
cc
H
R
R
DR H R DR H R
RGM GM G
DR
π
⋅
⋅
∈= = =
This shows that the ratio
()R∈=∈
linearly grows with the scale
R
of the universe which means that the actually arising Kepler
problem: “motion of a planet around its sun” all the time in the
universe would change its character with the cosmic scale
R
, in the
sense that the appearing Kepler object has higher and higher kinetic
energy, while in contrast a bound system can only exist for
() 1R∈≤
. This is unavoidable, unless
G
is assumed to vary proportional to
R
as discussed in Fahr and Heyl.5 In fact for
00
() ( / )GR G R R= ⋅
permanently during the cosmic evolution the same “Kepler”-problem
then would arise.
Without a variable
G
this , however, means that the ratio
()R∈
of kinetic over binding energy of the Kepler object is permanently
increasing with the increase of the scale of the universe
R
. In
order, however, to have a bound Kepler object , one thus should
have
1,
c
∈≤∈ ≤
which never after achieving a critical scale
c
R
of the
universe will be realizable anymore. Hereby this critical scale
c
R
is
given by
2
0
00
33
3
00
23
3
2
3
00
2
[]
4
3
() ()
2 2 2[]
4
()
3
2[ ]
4
3
c
G
RH
R
R
DR D R R M
g R GM GM R
R
RM
GM R
R
π
τπ π π π
τ
τπ π
=
= = =
=
meaning that bound planetary systems with a central mass of
1MM=
will after that cosmic time
()
cc
t tR=
never anymore newly
appear during the ongoing expansion of the universe.
This would have the interesting consequence that the “Kepler
pendulum” (with the specic acceleration
2
() / ()gR G M D R= ⋅
at
a distance
()DR
from a solar mass object with
1M=
M
would act
as “a cosmic clock” with a “cosmic oscillation/revolution period of:
The cosmic pendulum: Kepler‘s laws representing a universal cosmic clock 139
Copyright:
©2022 Fahr
Citation: Fahr HJ. The cosmic pendulum: Kepler‘s laws representing a universal cosmic clock. Phys Astron Int J. 2022;6(4):135‒140.
DOI: 10.15406/paij.2022.06.00266
33
33
00 00
() 2 (R)/g(R) 2 ()/ () 2 ()/ 2 [ ]/ 2
44
33
M RM
R L D R g R D R GM R GM R GM
RQ RQ
τπ π π π π
ππ
= = = = =
As already mentioned in Fahr and Heyl5 again this period would
change into a “linear” cosmic clock
()RR
τ
when one could assume
a scale variable Newton parameter as:
00
() ( / )G GR G R R= = ⋅
.
The more interesting point in this context, however, is that the
above derived ratio
()R∈
would under this latter assumption in fact
be! a cosmologic constant, i.e.:
2
0
0
00
0
0
4
3
() 2
H
R
R
rG
π
⋅
∈ =∈=
, if the Newton gravitational coupling coecient
G
seen over the
cosmic eons would not be a constant, but instead would scale with
R
according to the formula
00
() ( / )G GR G R R= = ⋅
!
This anyway becomes manifest, also without the assumption of
the scale variable
G
, when writing the Kepler pendulum period in
the form ( taking
()DR
as the length of the pendulum, and
()gR
as the
gravitational acceleration of the central Sun) :
33
3
00
() ()
2 2 2[]
4
()
3
DR D R R M
g R GM GM R
τπ π π π
= = =
and seeing that Kepler‘s third law (i.e.
23
R
τ
,
()DR
taken as
the main axis of the planetary ellipse) would come out quite naturally
from the above:
3
2
3
00
2[ ]
4
3
RM
GM R
τπ π
=
So there are obviously two options immaginable: Either
under variable
G
-conditions, like those discussed above, planetary
systems can be produced at all cosmic times with the same character
as at Newton‘s times, - or without variable
G
-conditions the Kepler
problem is specic for all cosmic evolution periods and it even exists
a critical cosmic scale
c
RR=
after passing the latter no planetary
systems can be built and be expected as arising at all anymore.
In addition assuming that the planetary object at the periphery of
the solar mass vacuole starts orbiting the central mass on a circular
orbit (i.e. before
c
RR=
is reached!), then at each of its orbital positions
with an orbital velocity
v
the centripetal force equals the gravitational
attraction force of the central mass and thus it is required that:
2
2
v GM
RR
=
meaning that the kinetic energy of the object
2
(1 / 2)
kin
mv∈=
equals
just half the binding energy
(1 / 2) (1 / 2 ) /
kin bind mGM R∈= ∈ =
. This
also again leads to Kepler‘s third law concerning the dependence of
orbital periods
τ
and the main ellipse axis
R
of the orbit:
3
22
2
( )4
RR
v GM
π
τπ
= =
Conclusion
In constrast to the case given in a static universe10 processes of
structure formation evidently run very dierent in an expanding
universe. This is because then structure formation denitely will
depend on the specic form of the prevailing cosmic expansion (e.g.
decelerated, accelerated or coasting expansion etc.,).12-14 To explain the
SN-1a luminosities as function of redshifts Perlmutter et al.,2 Schmidt
et al.,1 or Riess et al.3 have prefered as basis an accelerated expansion
of the universe connected with the action of a constant vacuum energy
density. Such a constant vacuum energy is as yet a physically non-
understood quantity and is problematic from its physical nature and
action.15-19 There are, however, more recent attempts by Casado20 and
Casado and Jou21 showing that a “coasting”, non-accelerated universe
can equally well explain these supernovae luminosities. If in fact
vacuum pressure and vacuum energy play a cosmologic role, and if it
must be assumed that the universe expands under the thermodynamic
and gravodynamic action of vacuum pressure, then as shown by Fahr22
the unavoidable consequence is a “coasting expansion” of the universe
with
/ .,R dR dt const= =
R
denoting the scale of the universe.
As we do show in this article under the conditions of a coasting,
instead of an accelerated, expansion of the universe the origin of new
solar-type systems is possible all the time after the cosmic matter
recombination, while under the condition of an accelerated expanion
of the universe new solar systems cannot be built up after some critical
cosmci time period. This could perhaps be a criterion to exclude the
possibility of universes with an accelerated Hubble expansion.
Acknowledgments
None.
Conicts of interest
None
Funding
None.
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