Page 1

arXiv:gr-qc/0106078v1 25 Jun 2001

Thermodynamics of Expansive

Nondecelerative Universe

Miroslav S´ uken´ ık and JozefˇSima

Slovak Technical University, Faculty of Chemical Technology

Radlinsk´ eho 9, SK-812 37 Bratislava, Slovakia

e-mail: sukenik@minv.sk, sima@chtf.stuba.sk

Abstract

The present contribution deals with thermodynamic aspects of the

model of Expansive Nondecelerative Universe. In this model, in the

matter era a dependence TCBR≈ ECBR≈ a−3/4holds for the energy

of cosmic background radiation, ECBR and its temperature, TCBR,

while the proportionality of the energy density εCBRto the gauge fac-

tor a can be expressed as εCBR≈ a−3. The given relationships comply

with experimental observations of the cosmic background radiation as

well as with a surprising finding that the Universe expansion is not

decelerated by gravitational forces. It is rationalized that the specific

entropy is proportional to a−1/4, i.e. it is gradually decreasing in time.

1Introduction

In the model of Expansive Nondecelerative Universe (hereinafter ENU) it

holds [1, 2]

a = c.tc=2G.mU

c2

(1)

where a is the gauge factor, tcis the cosmological time, and mUis the mass

of the Universe.The present-time values provided by calculations based

on ENU do not differ substantially from the generally accepted values and

are as follows: apt∼= 1.3 × 1026m, tc(pt)∼= 1.4 × 1010yr, mU(pt)∼= 8.6 ×

1052kg. It stems from (1) that the Universe mass is time-increasing. Since

the mass-energy of the Universe must be time-independent (and equal to

zero), simultaneously with the matter creation also an equivalent amount of

gravitational field energy is formed which is, however, negative. This is why

1

Page 2

the Universe can expand with a constant velocity equal to the velocity of

light c in ENU. In ENU, Schwarzschild metric is replaced by Vaidya metric

[3] originally elaborated to solve the problems of radiating stars, latter shown

by Virbhadra [4, 5] and by us [6] that its application is more general.

It is postulated in ENU that the energy density of the Universe is just

critical and it is expressed as

εcrit=

3c4

8π.G.a2

(2)

Till the end of the radiation era, there was a thermodynamic equilibrium

of matter and radiation, and energy, temperature and gauge factor were

related as follows

ECBR≈ TCBR≈ a−1/2

The fact that the energy density in (2) is proportional to a−2and not

to a−3can be rationalized by matter creation. Thus, ENU describes the

Universe in which eternal inflation occurs. In classical inflationary models of

universe, after completing its inflation stage the Universe should decelerate

due to effects of gravitational forces. As a consequence, in the models of

inflationary universe a new matter incessantly emerges from behind the event

horizon and in this way the proportionality ε ≈ a−2stated by (2) is explained.

In the ENU model relation (2) holds permanently, i.e. also in the matter

era and the energy density matches well with the accepted gauge factor value.

Detailed and precise observations performed in the last few years have led

to a conclusion that the predicted decrease in the Universe expansion rate

has not occurred. Contrary, a nonzero value of the cosmological constant

Λ or a newly elaborated quintessential model [7] gave rise to a presumption

stated that the Universe expansion accelerates and that such an acceleration

started at the beginning of the matter era. The postulate on the Universe ac-

celeration leads, however, directly to a very important conclusion concerning

impossibility of relation (2) validity in the matter era and also to a conclusion

on impossibility of critical energy density preserving.

(3)

2Thermodynamics of ENU

It is generally accepted that the radiation era ended approximately in the

time

tr∼= 7 × 105yr(4)

when the temperature of radiation approached to

Tr∼= 5 × 103K (5)

2

Page 3

(In the contribution, the subscripts pt, r and m refer to the present-time,

the end of radiation era, and matter era, respectively). The present-time

temperature is

Tpt∼= 2.735 K(6)

Taking into account that the Universe expansion did not decelerate in the

matter era, a presumption emerges stating that not only event horizon but

also original part of the Universe extended in about four orders

tpt

tr

=apt

ar

∼= 104

(7)

Based on the fact that from the end of the radiation era the of cosmic

background radiation temperature has decreased by three orders but the

gauge factor increased by four orders, a relation between the energy of cosmic

background radiation ECBRand its temperature TCBRfollows

TCBR≈ ECBR≈ a−3/4

(8)

Introducing the value of aptinto calculation of the present-time critical

energy density of the Universe it follows that

εcrit(pt)∼= 8.577 × 10−10J/m3

(9)

The energy density of radiation can be extracted from Stefan-Boltzmann

law and for cosmic background radiation is generally given as

εCBR=4σ.T4

c

(10)

Based on (6) and (10) the present-time energy density value of cosmic back-

ground radiation reaches

εCBR(pt)∼= 4.229 × 10−14J/m3

(11)

It follows from (8) and (10) that during the matter era

εCBR(m)≈ a−3

(12)

and at the same time, stemming from (2), (9), (10), (11) and (12) it follows

apt

ar

=

εcrit(pt)

εCBR(pt)

(13)

Based on (13) the gauge factor value at the end of the radiation era can be

calculated

ar∼= 6.41 × 1021m(14)

3

Page 4

together with the cosmological time value at the same time

tr∼= 6.6 × 105yr (15)

Treatment of relations (6), (8) and (14) leads to

Tpt

Tr

=

?ar

apt

?3/4

(16)

Stemming from (16) the temperature at the end of the radiation era is directly

calculable and it reaches the value of

Te∼= 4650 K

(17)

being in excellent agreement with the generally accepted value obtained using

other independent modes of calculations.

3Specific Entropy

Total average number of relict photons n(hν)min a cubic meter during the

matter era relates to the gauge factor according to (8) and (12) as follows

n(hν)m≈ a−9/4

(18)

and that of protons n(p)m(representing the matter particles) based on (2)

as

n(p)m≈ a−2

(19)

Dependence of the specific entropy S, defined as a number of relict pho-

tons per one proton, on the gauge factor is in the matter era expressed as

S ≈ a−1/4

(20)

At the time being, the temperature of cosmic background radiation (2.735

K) leads to the following number of relict photons in a volume unit

n(hν)pt=εCBR(pt)

ECBR(pt)

∼= 4 × 108

(21)

where ECBR(pt)is the mean energy of actual relict photons.

Given the present time energy density (2) and gauge factor values, a

number of protons in a volume unit reaches

n(p)pt∼= 5

(22)

4

Page 5

The present-time specific entropy calculated as a ratio of values provided

in (21) and (22) is of the order

Spt∼= 108

(23)

At the end of the radiation era, the specific entropy value approached

Sr∼= 109

(24)

Comparison of (23) and (24) verifies the correctness of (20), i.e. a slow de-

crease in specific entropy with time.

Within a discussion on a time-dependence of specific entropy some contra-

dictions emerge. If the specific entropy is constant, i.e. if relation (24) is valid

at the present-time number of relict photons and gauge factor, the Universe

density would have to have subcritical value. The assumption of permanent

critical (nearly critical) density, however, excludes a constant value of specific

entropy. The majority of current cosmological models take, however, critical

mass-energy density a priori into account and tries to solve this discrepancy

introducing some “exotic” nonbaryonic forms of matter.

4Thermodynamics and Gravitation

In the period of time starting with the Universe expansion and finishing

with the end of the radiation era, energy densities given by (2) and (10)

are identical which corresponds to (3). It must therefore hold for the mean

energy value of the photons of cosmic background radiation

ECBR= EPc

?lPc

a

?1/2

=

?¯ h3.c7

G.a2

?1/4

(25)

where lPcand EPcare the Planck length and Planck energy [8], respectively

lPc =

?G.¯ h

c3

?1/2

?1/2

= 1.606151 × 10−35m(26)

EPc =

?¯ h.c5

G

= 1.2211 × 1019

(27)

Since proportionality (8) holds in the matter era, the mean energy value of

the photons of relict radiation is expressed as

ECBR=

?¯ h3.c7.ar

G.a3

?1/4

(28)

5

Page 6

The ENU model allows to localize gravitational field energy. The wave

function of gravitational field is described [9] as

Ψg= exp

i.t

?m.c5

¯ h.a.r2

?1/4

(29)

where Ψgis the wave function of gravitational field quanta created by a body

with the mass m at the distance r. The mentioned thermodynamic equilib-

rium in the radiation era means that the total mass of the relict radiation

is equal to the total mass of matter particles. This is why this mass can be

expressed as

m∼=a.c2

G

(30)

When taking the Universe as one system into account, the following general

equation must always hold

r = a (31)

Introducing (30) and (31) into (29), we obtain the Universe wave function in

the form [9]

Ψg= expi.t(tPc.tc)−1/2?

Here we can formulate a hypothesis stating that the mean energy of a

photon of relict radiation is modulated by the energy of gravitational quanta

Egand we will intent to justify this hypothesis and prove its correctness. In

such a case the absolute values of the corresponding mean energies must be

identical, i.e.

|Eg| = ECBR

?

(32)

(33)

Writing a Schr¨ odinger-like equation for the energy of gravitational quanta

Eg

Eg.Ψg= i.¯ hdΨg

dt

(34)

it comes from (32) and (34) that

|Eg| =

¯ h

(tPc.tc)1/2=

?¯ h3.c7

G.a2

?1/4

(35)

which is a relation identical to that of (25).

Situation is quite different in the matter era. Stemming from the validity

of (12) for the total mass of the photons of relict radiation, mCBR(total) it

must hold

mCBR(total)≈ V.a−3= const.(36)

6

Page 7

At the same time

mCBR(total)∼=ar.c2

G

∼= 1049kg(37)

This value is really constant. At the present-time volume of the Universe

being

VU∼= 1079m3

(38)

and the energy density of relict radiation (11) one can easily come, in an

independent way, to the mass given by (37). This mass generates the gravi-

tational field that modulates the mean value of the energy of relict photons

in the matter era. This allows to introduce (31) and (37) into (29) and obtain

Ψg= exp

i.t

?

ar.c7

¯ h.G.a3

?1/4

(39)

The mean energy of photons in the matter era, stemming from (34) and (39)

is

|Eg| = ECBR=

?¯ h3.c7.ar

G.a3

?1/4

(40)

Relation (40) is identical to postulated relation (28). In this way, using

the ENU model we are able to harmonize theory and observation. It holds

that the mean energy of the photons of relict radiation is equal to the absolute

value of the energy of gravitational quanta that are generated by the total

mass of relict radiation mCBR(total).

5Conclusions

All values presented in this contribution comply with experimentally ob-

served or generally accepted values.

Stemming from a supposed acceleration of the Universe expansion hy-

pothesed in the models including nonzero value of cosmological constant Λ

in those introducing quintessence (or other “exotic” forms of matter and en-

ergy), values of the Universe-related quantities might differ in a substantial

extent in the future. Some of the open questions can be answered by exact

measurements of the parameter ω (defined as the pressure-to-energy density

ratio)

ω =p

ε

(41)

that in the models accepting the nonzero Λ should reach the value

ω = −1(42)

7

Page 8

in case of quintessential models it should be of the range

ω = (−1;−1/3) (43)

and in our ENU model [10]

ω = −1/3 (44)

It is worth mentioning, however, that to obtain exact value of ω, the exact

value of Hubble constant must be known.

Summarizing the conclusions offered in the present contribution it should

be pointed out that in the majority of conventional models it is postulated

that ECBR≈ TCBR≈ a−1, εCBR≈ a−4,S = const.

In the ENU, ECBR ≈ TCBR ≈ a−3/4, εCBR ≈ a−3,S ≈ a−1/4. Obser-

vations are in accord with the values derived by ENU model. Other mod-

els explain the above mentioned observed relations as a consequence of the

matter emerging from behind event horizon due to the Universe expansion

deceleration. The latest measurements, however, suggest that the Universe

expansion might accelerate and exclude its deceleration. In the ENU the

expansion neither decelerates nor accelerates, it is constant and equal to the

velocity of light.

References

[1] V. Skalsk´ y, M. S´ uken´ ık, Astrophys. Space Sci., 178 (1991) 169

[2] V. Skalsk´ y, M. S´ uken´ ık, Astrophys. Space Sci., 209 (1993) 123

[3] P.C. Vaidya, Proc. Indian Acad. Sci., A33 (1951) 264

[4] K.S. Virbhadra, Phys. Rev., D60 (1999) 104041

[5] K.S. Virbhadra, Pramana – J. Phys., 38 (1992) 31

[6] M. S´ uken´ ık, J.ˇSima, Preprint gr-qc/0101026

[7] G. Huey, J.E. Lidsey, Preprint astro-ph/0104006

[8] D.E. Groom et al., Eur. Phys. J., C15 (2000) 1

[9] M. S´ uken´ ık, J.ˇSima, J. Vanko, Preprint gr-qc/0010061

[10] J.ˇSima, M. S´ uken´ ık, Preprint gr-qc/0105090

8