Cosmic vacuum energy with thermodynamic and gravodynamic action power

Abstract

In this paper we investigate the suspected effect of cosmic vacuum energy on the dynamics of cosmic space, while nevertheless still now the phenomenon of vacuum energy is not yet physically settled in a rigorous form. In view of what one needs for general relativistic approaches, we start here with considerations of the specific energy-momentum tensor of cosmic vacuum energy in the standard hydrodynamical form, and derive relations between vacuum energy density and vacuum pressure. With the help of fundamental thermodynamic relations we then find relations of the two quantities, vacuum pressure and energy density, to the scale R of the universe. These, however, allow for a multitude of power exponents n , including the case of a constant vacuum energy density with n=0 and Rn=const. Then we argue that for spaces of cosmic dimensions not only thermodynamical relations have to be fulfilled, but also, as we call them "gravodynamical relations", meaning that vacuum pressure has to work against the inner gravitational binding of space, mostly due to the gravitating masses distributed in this cosmic space. When we include this effect in addition to the thermodynamics we find that the vacuum energy density ρΛ then can not anymore be considered as constant, but unavoidably as falling off with the scale of the universe according R−2 . At the end of this article we then suspect, since vacuum energy even nowadays is not yet a physically well founded and understood quantity, that the Hubble expansion of the present universe is not driven by vacuum pressure, but by the change of gravitational binding energy at the ongoing structure formation of cosmic matter during the Hubble expansion.

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Introduction
A. Formulation of the thermodynamics and gravodynamics of
empty space?
The question what means “empty space” - or synonymous for that
- “vacuum” - has not yet been satisfactorily answered. In fact this
question appears to be a very fundamental one which has already been
put by mankind since the epochs of the greek natural philosophers
till the present times of modern quantum eld theoreticians. The
changing opinions given as answers to this fundamental question over
the changing epochs have been reviewed for example by Weinberg,1
Overduin and Fahr2 or Peebles and Ratra,3 but here we do not want
to repeat all of these dierent answers that have been given in the
past, we only at the begin of this article want to emphasize a few
fundamental aspects of the present-day thinking with respect to the
physical constitution of empty space.
Especially challenging in this respect is the possibility that empty
space could despite of its conceptual “emptiness” - nevertheless
unavoidably be “energy-loaded”, perhaps simply as property of
physical space itself. This strange and controversial aspect we shall
investigate further below in this article. In a brief and rst denition
we want to denote empty space as a spacetime without any topied or
localized energy representations, i.e. without energy singularities in
form of point masses like baryons, leptons, darkions (i.e. dark matter
particles) or photons, even without point-like quantum mechanical
vacuum uctuations. The latter condition, however, as stated by
modern quantum theoreticians, anyway cannot be fullled, since
vacuum uctuations cannot be forbidden or be suppressed as learned
from the basics of quantum mechanical principles.
If then nevertheless there should be a need to discuss that such
empty spaces could be still energy-loaded, then this energy of empty
space has to be seen as a pure volume-energy, somehow connected
with the magnitude of the volume or perhaps with a scalar quantity
of spacetime metrics, like for instance the global or local curvature of
this space. In a completely empty space of this virtue, of course, no
specic space points can be distinguished from any others, and thus
volume-energy or curvature, if existent, are numerically identical at
all space coordinates.
As it was shown by Fahr4 vacuum energy conservation can be
formulated as constancy of the proper energy of a co-moving cosmic
proper volume. Nevertheless an invariance of this vacuum energy per
co-moving proper volume,
,vac
e
can of course only then be expected
with some physical sense, if this quantity does not do any work on the
dynamics of the cosmic geometry, especially by physically or causally
inuencing the evolution of the scale factor
()Rt
of the universe.
If to the contrary, for example such a work in fact is done,
and vacuum energy inuences the dynamics of the cosmic
spacetime (perhaps either by ination or deation), e.g. as in case
of a non-vanishing energy-momentum tensor, then automatically
thermodynamic requirements need to be respected and fullled, for
example relating vacuum energy density and vacuum pressure by the
standard thermodynamic relation.5
33
()
vac v ac
dd
RpR
dR dR
ε
= −
This above thermodynamic request is shown to be fullled by the
following expression for the vacuum pressure.
3
3
vac vac
n
p−
=−∈
(A)
Here by the vacuum energy density itself is represented by a
scale-dependence of the form
vac
n
R∈
. Then, however, it turns out
that the above thermodynamic condition, besides for the trivial case
3n=
when the vacuum does not at all act as a pressure (since the
latter is vanishing according to Equ.(A);
( 3) 0!)
vac
p
n= =
, is only non-
trivially fullled for exponents , thus allowing also for
0n=
, i.e.
describing a constant vacuum energy density
.
vac
const∈
=
Phys Astron Int J. 2022;6(2):62‒66. 62
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Cosmic vacuum energy with thermodynamic and
gravodynamic action power
Volume 6 Issue 2 - 2022
Hans J Fahr
Argelander Institut für Astronomie, Universität Bonn, Germany
Correspondence: Hans J Fahr, Argelander Institut für
Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn,
Germany, Email
Received: June 22, 2022 | Published: June 29, 2022
Abstract
In this paper we investigate the suspected eect of cosmic vacuum energy on the dynamics
of cosmic space, while nevertheless still now the phenomenon of vacuum energy is not
yet physically settled in a rigorous form. In view of what one needs for general relativistic
approaches, we start here with considerations of the specic energy-momentum tensor of
cosmic vacuum energy in the standard hydrodynamical form, and derive relations between
vacuum energy density and vacuum pressure. With the help of fundamental thermodynamic
relations we then nd relations of the two quantities, vacuum pressure and energy density,
to the scale
R
of the universe. These, however, allow for a multitude of power exponents
n
, including the case of a constant vacuum energy density with
0n=
and
.
n
R const=
Then
we argue that for spaces of cosmic dimensions not only thermodynamical relations have
to be fullled, but also, as we call them “gravodynamical relations”, meaning that vacuum
pressure has to work against the inner gravitational binding of space, mostly due to the
gravitating masses distributed in this cosmic space. When we include this eect in addition
to the thermodynamics we nd that the vacuum energy density
ρ
Λ
then can not anymore
be considered as constant, but unavoidably as falling o with the scale of the universe
according
2
R
−
. At the end of this article we then suspect, since vacuum energy even
nowadays is not yet a physically well founded and understood quantity, that the Hubble
expansion of the present universe is not driven by vacuum pressure, but by the change of
gravitational binding energy at the ongoing structure formation of cosmic matter during the
Hubble expansion.
Physics & Astronomy International Journal
Research Article Open Access

Cosmic vacuum energy with thermodynamic and gravodynamic action power 63
Copyright:
©2022 Fahr
Citation: Fahr HJ. Cosmic vacuum energy with thermodynamic and gravodynamic action power. Phys Astron Int J. 2022;6(2):62‒66.
DOI: 10.15406/paij.2022.06.00253
B. Restricted vacuum conditions under gravitational selfbinding
A more rigorous and highly interesting restriction for exponent
n is, however, obtained after recognition that the above thermodynamic
expression
()A
under large-scale cosmic conditions needs to be
enlarged by a term representing the work that the expanding volume
does against the internal gravitational binding of matter or vacuum
energy in this volume.
For mesoscale gas dynamics (like aerodynamics, meteorology
etc.) this term does of course not play a role and can tacitly be
neglected. On cosmic scales, however, there is a severe need to take
into account this term. Under cosmic perspectives binding energy is an
absolutely necessary quantity to be brought into the gravodynamical
and thermodynamical energy balance of stellar matter, of interstellar
cloud matter, or of cosmic matter. As worked out in quantitative terms
by Fahr and Heyl,6 this then leads to the following more completed
relation
33
2
25
4
8
) [( 3 ) ]
1
(5
vac vac vac vac
ppRR
Gd R
d
dd
R RdR c
d
π
∈∈
=−− +
where the last term on the right-hand side accounts for the internal,
gravitational self-binding energy of the vacuum.
This completed equation describing the variation of the vacuum
energy with the scale R of the universe, as one can easily show, is
again solved by the expression of the afore mentioned relation
()A
:
3
3
vac vac
n
p−
=−∈
, but now - dierent from before - leading to the
following new requirement
33
2
2 25
4
32
3
8
) [ ( )]
15
(
vac vac vac
RnnR
Gd R
dR
c
dd
dR dR
π
−
∈ ∈ ∈−
=
Now, as one can see, in its above form, the upper, extended relation,
however, is only fullled by the power exponent:
2!n=
, - meaning
that the corresponding cosmic vacuum energy density in order to meet
the above requirements must vary - and needs to vary - like.
2
vac R−
∈ (B)
This consequently furthermore means that, if it has to be
consistently taken into account that vacuum energy acts upon
spacetime both in a thermodynamical and gravodynamical sense,
then the only reasonable assumption for the vacuum energy density
is that
vac
∈
drops o at the cosmic expansion inversely proportional to
the square of the cosmic scale, i.e.
,
2
0
.( / )
vac vac o
RR
∈∈
=
- rather than
being a constant.6,7 The question then, however, arises, how under
these latter, new circumstances structure formation does inuence the
cosmic expansion, a problem recently discussed for the rst time by
Fahr8 and here, under the new auspices given now by relation (B)
above, is taken up once again.
C. The evolution of the Hubble parameter
The above result unavoidably leads to the important question of
what Hubble parameter
()H Ht=
and what temporal change of it,
i.e. dH/dt, one has to expect as prevailing at the dierent cosmologic
evolution periods or dierent world times t. For Friedman-Lemaı
tre-
Robertson-Walker cosmologies (FLRW) the Hubble parameter
() ()/ ()Ht Rt Rt=
generally is not a constant, but is given in form of
the following dierential equation (derived from the 1. Friedman
equation; e.g.:3-5
2
2
2
8[]
3
BDv
H
RG
R
πρ ρ ρρ
Λ
= = + ++

where
G
is Newton‘s gravitational constant, and
, ,,
BDv
ρ ρ ρρ
Λ
denote the relevant equivalent cosmic mass densities of
baryons, of dark matter, of photons, and of the vacuum energy.
In case that all of these quantities count at the same cosmologic
period, then this complicates to nd a closed solution for
()Ht
and
()Rt
over these cosmic times, because
B
ρ
may vary proportional to
3
R
−
,
D
ρ
most probably also according to
3
R−
, but
v
ρ
is generally
thought to vary according to R-4 (see Goenner, 1996, but also Fahr
and Heyl, 2017, 2018).5 A solution for the Hubble parameter in this
general case is shown in Figure 1 below.4
Figure 1 The Hubble Parameter H(x) (yellow curve) and the expansion
velocity R (x) (blue curve) are shown as functions of the normalized Hubble
scale x = R/R0 on the basis of best-tting values for ρm, ρd, ρv,.9
Amongst these quantities the cosmic vacuum energy density
ρ
Λ
certainly is the physically least certain quantity, but on the other
hand - if described with Einstein’s cosmological constant
,Λ
then it
represents a positive, constant energy density, i.e its mass equivalent
ρ
Λin connection with a constant and positive vacuum energy density
,Λ
would consequently as well be a positive, constant quantity not
dependend on the scale R or cosmic time t. This in fact would oer
for the later phases of cosmic expansion, i.e. when at late times
0
tt≥
evidently
, ,,
BDv
ρ ρρρ
Λ
an easy and evident solution of the above
equation for the late Hubble parameter
00
() ()H Ht t H= ≥ = Λ
:
00
8
() 3
H
G
H const
πρ
Λ
=
Λ= =
As support for this to be true already now it has been concluded
from recent supernova SN1a redshift observations (Perlmutter, 2003,
Riess et al., 1998, Schmidt et al., 1998)10 that in fact at the present
cosmic era, most probably already sometimes ago, we were and are
in a coasting, perhaps even an accelerated expansion phase of the
universe.
Now, however, when taking it further on serious that this is due
to the term
Λ
connected with cosmologic vacuum energy density
ρ
Λ
, - however this time, not being a constant, but falling o like
2
R
−
, as
discussed above in case the vacuum is thermodynamically and gravo
dynamically active -, this then expresses the complicating fact that
ρ
Λis not a constant anymore, but nevertheless sooner or later along
the evolution of the universe at ongoing expansion must unavoidably
become the dominant quantity in the universe amongst the other upper
ingredients, i.e.
, ,,
BDv
ρ ρρρ
Λ

since falling o inversely proportional
with
R
, however, with the smallest power of (1 / R)2.
Then in fact one will certainly also enter a cosmologic time with
, ,,
BDv
ρ ρρρ
Λ

when the above dierential equation for the Hubble
parameter
(t)
H
H
=
can be written not in the earlier form given above,

Cosmic vacuum energy with thermodynamic and gravodynamic action power 64
Copyright:
©2022 Fahr
Citation: Fahr HJ. Cosmic vacuum energy with thermodynamic and gravodynamic action power. Phys Astron Int J. 2022;6(2):62‒66.
DOI: 10.15406/paij.2022.06.00253
but nevertheless in an essentially simplied form, namely dierent
from above, this time by:
0
,0
8 88
[ ] [ ( )]
3 33
BDv
H
R
RG G G
R
RR
π ππ
ρ ρ ρρ ρ ρ
ΛΛ Λ
=
= + ++ =


Under these new auspices of a thermodynamically reacting cosmic
vacuum the expansion of the universe in this phase is then described
by the above expression.
0
,0
8
3
R
RG
RR
πρ
Λ
=

with
0
R
and
0
t
denoting the present-day scale of the universe and the
present cosmic time, and denoting the equivalent vacuum mass
density at this time
0
t
. This, however, expresses the astonishing fact
that from that time onwards into the future of the universe for
0
tt≥
the
cosmic expansion will be characterized - neither by an acceleration
nor by a deceleration -, but by a constant expansion velocity with
/ 0!R dR dt= =
 
, since:
0 0 ,0
8.
3
RR G
R const
πρ
Λ
== =

This means the cosmic expansion would naturally and necessarily
sooner or later enter into a so-called “coastal” phase of the universal
expansion. For such a coastal phase cosmologists since long ago
were hunting (see e.g. Kolb,1989, Dev et al., 2001, Gehlaut et al.,
2003),11 and on the other hand were hoping for12,13 to also t distant
supernovae-SN1a redshift measurements equivalently well as with
“the accelerated universe” (Perlmutter et al., 1999, Schmidt et al.,
1999, Riess et al., 1999).10
For that reason we shall now describe the Hubble parameter in the
period
0
tt≥
at times with
00
1
()
H
tt
≥
−
nding:
00 0
0
000 00
8
3
() ()1 ()
H
G
RH
tt R Rtt Htt
πρ
Λ
≥= =
+− + −

where the Hubble parameter at time t=tois denoted by
0
0
8 /3
H
G
πρ
Λ
=
. The important question then remains whether or
not, even under these new perspectives, i.e of a “coastal cosmic
expansion”, the vacuum energy could still be understood as response
to the change of negative gravitational binding energy of the universe
connected with the ongoing expansion of matter in cosmic space, as
demonstrated recently by Fahr?8
D. How operates a thermo-reactive vacuum under ongoing
cosmologic structure formation?
Cosmic structure formation denotes the phenomenon of growing
clumpiness of the cosmic matter distribution in cosmic space during
the ongoing evolution of the expanding universe, i.e. the origin of
larger and larger mass structures like galaxies, clusters or super-
clusters of galaxies. Usually one does start cosmology with the
assumption that at the beginning of cosmic time and the evolution of
the universe cosmic space has a uniform deposition with matter and
energy, justifying the use of the famous Robertson-Walker geometry.
The question for the evolved universe then may arise whether or not the
later cosmic expansion dynamics and the scale evolution
/R dR dt=

may perhaps be inuenced by the ongoing structure formation, as
it has to happen in order to create out of its earlier uniformity that
hierarchically structured present-day universe manifest to us today?
The question now is whether this process of a structuration perhaps
inuences the ongoing Hubble expansion of the universe, perhaps
either accelerating or decelerating, or stagnating its expansion with
respect to the solutions of the standard Friedmann universe?5 This,
however, could simply be due to the fact that under the new conditions
of a self structuring cosmic matter the eective mass density
()
eff eff t
ρρ
=
of the universe does not behave like it normally does in a Friedmann
universe like 3
00
.( / )
RR
ρρ
=, but rather like
3
,0 0
( )( / )
eff eff tR R
ρρ
=
.
The manifest universe, as it is, is not a homogeneous material
structure, but stellar matter is distributed in space in form of
galaxies, clusters of galaxies, and superclusters, i.e. it is structured
in hierarchies. This can be described up to supercluster-scales by a
point-related correlation function with an observationally supported
correlation index of
1.8
α
=
( see Bahcall and Chokski, 1992). This
two-point correlation structure seen in cosmic galaxy distributions
can be expressed through the underlying cosmic mass distribution
given by an equivalent mass density of
0,
()
.( / )
l
ll
α
α
ρρ
−
=
.14
It is then most interesting to see from recent results by Fahr8 that the
gravitational binding energy in this hierarchically structured universe,
and its change with time, is described by a function
(,)
pot pot
R
α
∈∈
=
not
only dependend on the outer scale
()R Rt=
, but also on the correlation
coecient
()t
αα
=
of the structured cosmic matter in this cosmic
system, namely given in the form:
2
25
(4 ) (3 )
(,) 9(5 2 )
pot
R GR
πα
αρ
α
∈−
=−
where obviously the permitted range of the structure coecient is
given by values
2.5
α
≤
. Here
G
is Newton‘s gravitational constant,
and
()R
ρρ
=
denotes the average mass density in the associated, re-
homogenized universe. It is interesting to recognize that for
0
α
=
(i.e. homogeneous matter distribution) in fact the potential energy of
a homogeneously matter-lled sphere with radius
R
is found, which
does not vanish, but has a nite value, namely6,7
2
25
(4 )
( 0) 15
pot
GR
π
αρ
∈= =
This latter binding energy, however, is fully incorporated by the
Friedmann-Lemaı
tre cosmology as the one normally reponsible for
the deceleration of the “normal” Hubble expansion of the universe
without the action the vacuum via
ρ
Λ.
If in contrast the cosmic deceleration turns out to be smaller than
the “normal” Hubble deceleration or it even indicates an acceleration
which normally is ascribed to the action of vacuum energy, then in
our view this must be ascribed to the increased production of binding
energy due to the upcome of structure formation with cosmic time. That
means what really counts is the dierence
( ) ( 0)
pot pot pot
αα
∆∈ =∈ −∈ =
between a structured and an unstructured universe. The value
( 0)
pot
α
∈=
hereby serves as reference value for that potential energy in
the associated, re-homogenized universe. What really counts in terms
of binding energy of a structured universe causing a deviation from the
Friedmann-Lemaitre expansion of the universe is the dierence
pot
∆∈
between the structured and the unstructured universe, since evidently
the unstructured universe has its own, but nonvanishing amount of
binding energy. For general cases one therefore obtains:
2 22
25 25 25
(4 ) (3 ) (4 ) (4 ) (3 ) 1
(,) [ ]
9(5 2 ) 15 3 3(5 2 ) 5
pot
R GR GR GR
πα π π α
α ρρ ρ
αα
−−
∆∈ = − = −
−−

The question now poses itself: Is the change of binding energy
(,)
pot R
α
∆∈
per cosmic time t or scale increment dR balanced by a
corresponding unphysical change in thermal energy
(,)
therm
R
α
∆∈
of
normal cosmic matter? This we shall investigate in the next section
down here.
E. The thermal energy of cosmic matter in the expanding
universe
Starting from the assumption that the cosmic dynamics can

Cosmic vacuum energy with thermodynamic and gravodynamic action power 65
Copyright:
©2022 Fahr
Citation: Fahr HJ. Cosmic vacuum energy with thermodynamic and gravodynamic action power. Phys Astron Int J. 2022;6(2):62‒66.
DOI: 10.15406/paij.2022.06.00253
be represented by a Hubble expansion with a Hubble parameter
/H RR=
it can be shown4,8,15,16 that cosmic gases subject to such
an expansion undergo a so-called Hubble drift in velocity space
while moving with their own velocities v from place to place. This
unavoidable Hubble drift
H
v vH=−⋅
will enforce the change per time
of the velocity distribution function
(,)f vt
of the cosmic gas atoms
which is described by the following kinetic transport equation:
( )H
ff
vH f
tv
∂∂
= −
∂∂

(40)
This above partial dierential equation allows to derive the
resulting distribution function
(,)f vt
as function of the velocity
v
and
of the cosmic time
t
, and as well its velocity moments, like e.g. the
density
()nt
and the temperature
()Tt
of the cosmic gas.
As it was shown already by Fahr,15,16 the above kinetic transport
equation does not allow for a solution in the form of a separation
of variables, i.e. putting
( ,) () ( )
tv
f vt f t f v= ⋅
, but one rather needs a
dierent, non-straight forward ad-hoc method of nding a kinetic
solution of this above transport equation Equ.(40). It turns out that
under the assumptions a): that at time
0
tt=
a Maxwellian distribution
00
(, ) (,T)f v t Max v=
is valid, and b): that since that time a Hubble
parameter
00
( ) (1 ( ))Ht H t t= ⋅−−
prevails like it was derived in the
section before given by:
0
0
00
()
1 ()
H
Ht t Ht t
≥=
+−
with
00
8 / 3,HG
πρ
Λ
=
0
ρ
Λ
denoting the equivalent mass energy
density of the cosmic vacuum at the time
0
tt=
, one can then write the
actual distribution function at times
0
tt≥
, derived on the basis of the
above partial dierential kinetic equation, in the following form (see
Fahr, 2021):
3
22
00
0 00 00
3/2 3
0
(1 ( ) )
( , ) exp[ 3 ( ) exp[ (1 ( )) ]
Ht t
fvt n Htt x Ht t
v
π
−−
= − − ⋅ − ⋅− −
where
0
v
denotes the thermal velocity by
2
00
/v kT m=
at the time
0
t
,
when a temperature
00
()Tt T=
prevails. Hereby the normalized velocity
coordinate x was introduced by
0
/x vv=
. Furthermore it turns out that
one can interprete the actually prevailing distribution function
(,)f vt
as an actual Maxwellian with the time-dependent temperature
0
()Tt t≥
given by:
0
2
00
() (1 ( ))
T
Tt Ht t
=−−
and a time-dependent density
3
0 00 0
0
()
( ) exp[ 3 ( )] ( )
Rt
nt n H t t n R
−
= − −=⋅
One therefore nds that under the given cosmologic prerequisites
of a Hubble expansion with the Hubble parameter
0
H( )tt≥
the thermal
energy
therm
∈
of matter in this universe thus increases with time t like:
3
30 00
2
00
(3 / 2)
4 34
( ).( ( ))
3 23
(1 ( ))
therm
n kT R
R n t kT t Htt
ππ
∈= ⋅ = −−
Meaning that the total thermal energy of the matter in this whole
Hubble universe apparently increases with the expansion - obviously
violating standard thermodynamical principles according to which the
temperature of matter decreases with the increase of cosmic space
volume.
At this point of the argumentation Fahr4,8 had recently developed a
new idea to explain this mysterious, unphysical increase of the thermal
energy on a physical basis: Namely he suspected that this increase in
thermal energy of cosmic matter in this expanding universe is just
compensated by the increase in negative-valued, cosmic binding
energy
(, )
pot m
l
α
∆∈
in case of a specic level of structure formation,
measurable as a specic level of the correlation coecient
0
()t
αα
=
.
The hope was that this negative binding energy is the genuine physical
reason for the action of a so-called “vacuum pressure”, corresponding
to an equivalent mass density
2
0/8
cG
ρπ
Λ= Λ . Since we now have a new
request derived in section 4 of how vacuum energy density should
behave with the scale
R
, this idea needs to be re-checked here putting
the question whether or not this argumentation can still stand.
To pursue a little more this idea, we again start from the two
competing quantities, i.e. the potential binding energy dierence
between the structured and the unstructured universe on one hand:
2
25
(4 ) (3 / ) 1
(,) [ ]
3 3(5 2 ) 5
pot
R GR
πα
αρ
α
∆∈ = −
−
and the thermal energy dierence between the non-
thermodynamical and the thermodynamical universe of cosmic matter
on the other hand:
33
0 00 2
00
4 34 1
() ( ()) (3/2) [ 1]
3 23 (1 ( ) )
therm
R n t kT t n kT R Ht t
ππ
∈= ⋅ ⋅ = −
−−
Now, in order to guarantee energy conservation, we shall require
that the change with cosmic time t of the rst quantity
(, )
pot m
l
α
∆∈
is
equal to the negative change of the second quantity
therm
∆∈
- a question
that advices to specically study the following quantity:
2
3 25
0 00 2
00
4 1 (4 ) (3/ ) 1
( ) (3 / 2 ) [ 1] [ ]
3 3 3( 5 2 ) 5
(1 ( ))
therm pot n kT R G R
Ht t
π πα
αρ
α
∆∈=∈ −∆∈ = − − −
−
−−
In the following part we consider the times with
0
1 ()Htt
Λ
−
and describe the temporal evolution of the structure index
α
by:
0 00
( ) exp[ ( )]t Ht t
α
αα ε
= −
with
1
α
ε
≤
. Then one can simplify the above
expression into the form:
2
3 25
0 00
0 00 0 0
0 00
(3 / [1 ( )])
4 (4 ) 1
( ) (3 / 2) [ 2 ( )] [ ]
3 3 3(5 2 [1 ( )]) 5
Ht t
t n kT R H t t G R
Ht t
α
α
αε
ππ ρ
αε
+−
∆∈ − − −
−+ −

or furthermore - when identifying:
3
0 0 0 ,0
4(3 / 2)
3
therm
n kT R
π
=∈
as the total
thermal energy at time
0
tt=
, and:
2 25 2
0 0 ,0
(4 / 3) /
tot
G R GM R
πρ
= =∈
as the total binding energy at
0
tt=
, one
obtains:
0 00
,0 0 0 ,0
0 00
(3 / [1 ( )]) 3
( ) [2 ( )] [ ]
(5 2 [1 ( )]) 5
therm tot
Ht t
t Ht t Ht t
α
α
αε
αε
+−
∆∈ ∈ − −∈ −
−+ −

Conclusion
First of all, the condition that the non-thermodynamical increase
of the thermal energy
therm
∈
in a universe with Hubble expansion
is compensated just by the negative binding energy
(,)
pot R
α
∆∈
of the structured mass in the universe can only be fullled, if this
gravitational binding energy dierence between the structured and the
unstructured universe becomes negative. The required condition can
in fact be fullled, if at the time
0
tt=
the gravitational binding energy
of the cosmic masses, i.e.
2 25 2
,0 0 0
(4 / 3) /
tot G R GM R
πρ
∈= =
equals the actual
thermal energy
3
,0 0 0 0
4(3 / 2)
3
therm n kT R
π
∈=
of the particles. This then leads to
the following request:
0 00
0 00
0 00
(3 / [1 ( )]) 3
( )/ [2 ( )] [ ]
(5 2 [1 ( )]) 5
Ht t
t Ht t Ht t
α
α
αε
αε
+−
∆∈ ∈ − − −
−+ −

What concerns the needed and necessary correlation coecient
α
, one nds, however, as further restriction that only when this
coecient has attained a value of
1.5
c
αα
≥=
, then the resulting
mathematical sign of
,0ther m
∈
allows a physical solution in the expected
form (see our Figure 2 below). This means that only when the
structure formation process in the universe has progressed far enough,
then the above required equality can in fact be achieved. But then, at
times after that, when an accelerated expansion of the universe with a
Hubble parameter
0
HH=
prevails, then in fact the increase in negative
potential energy of cosmic matter
(,)
pot R
α
∆∈
is exactly balanced by the
increase of thermal cosmic energy
()
therm
R∆∈
. During this phase of the
expansion of the universe one is obviously justied to assume that the
creation of negative binding energy is the reason for the accelerated

Cosmic vacuum energy with thermodynamic and gravodynamic action power 66
Copyright:
©2022 Fahr
Citation: Fahr HJ. Cosmic vacuum energy with thermodynamic and gravodynamic action power. Phys Astron Int J. 2022;6(2):62‒66.
DOI: 10.15406/paij.2022.06.00253
expansion of the universe, a phenomenon which in present-day
cosmology is condently always ascribed to the action of vacuum
energy.
Figure 2 The quantity
∆
ϵpot(α) as a function of the correlation coefcient α.
In Figures 3 and 4 we show how the normalized total energy
0
( )/t∆∈ ∈
varies with cosmic time t before and after the time
0
tt=
demonstrating clearly how the actual correlation index
0
()t
αα
=
inuences the situation. Though this clearly shows the importance
of the actual correlation index, we at this point of the paper can not
hide the fact that we actually do not have a clear description of the
evolution of
α
and of the cosmic matter structure with world time
t
. This is clearly appears as a point that remains to be theoretically
derived in the upcoming time of cosmologic theory.
Figure 3 The quantity
∆
ϵ(t)/ϵ0 is shown as function of x = H0(t – t0) from
x=-0.5 to x=+0.5 for different correlation coefcients (from top to bottom):
α1 = 2. 1; α2 = 1. 8; α3 = 1. 3
Figure 4 The loarithm Log (10,
∆
ϵ(t)/ϵ0) is shown as function of x = H0(t
– t0) from x=0.1 to x=1.0 for different correlation coefcients (from top to
bottom):
α1 = 2. 1; α2 = 1. 8; α3 = 1. 3
Acknowledgments
None.
Conicts of Interest
None.
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