Tong-Jie Zhang- Effect of the Thermal Pressure On the Dynamical Evolution of the Universe

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International Journal of Modern Physics D, Vol. 9, No. 5 (2000) 575579c World Scientific Publishing Company

EFFECT OF THE THERMAL PRESSURE ON THE

DYNAMICAL EVOLUTION OF THE UNIVERSE

TONG-JIE ZHANG

Department of Astronomy, Beijing Normal University,

Beijing, 100875, China

Received 25 November 1999Communicated by A. Dolgov

While the pressureless dust assumption has been widely adopted in the standard modelof cosmology, a quantitative estimate of the actual contribution of thermal pressure ofthe baryonic matter and the hot/cold dark matter to the dynamical evolution of theUniverse has not yet been made in literature. In this paper we provide a simple scenarioof how the solution, e.g. the age of the Universe, is affected by the inclusion of thethermal pressure of the baryon and dark matter particles.

1. Introduction

In the standard model of cosmology, the thermal pressure of cosmic matter is pre-

sumably taken to be zero. Consequently, the Universe is usually envisioned as a

reservoir of the pressureless dust particles.1 Yet, the rationality of this conven-

tional scenario has not been quantitatively justified. In fact, it remains to be unclear

as to how significant the thermal pressure of the (non)relativistic matter would be

at different cosmic epoch. In particular, it deserves to be understood the contribu-

tion of the hot/cold dark matter particles to the thermal pressure in addition to

that of the baryonic matter because the former may play a dominant role in the

dynamical evolution of the Universe. The possible effect of thermal pressure of the

cosmic matter was recalled once in a while in history (e.g. Peebles2), but whether

the effect can predict any observational feature or motivate any theoretical interest

are still poorly known. In this paper, we present a quantitative yet simple analysis

of the thermal pressure contributions to the dynamical solution of the Universe in

a very conventional way. As an example, we demonstrate the effect of the thermal

pressure by baryons, hot and cold dark matter particles on the evaluation of the

age of the Universe.

E-mail: [email protected]

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576 T.-J. Zhang

2. The Age of the Universe

2.1. The Conventional Model: pM = 0

According to Einsteins general relativity, the equations which govern the homoge-

neous and isotropic Universe are written in the following form

a2

a2=

8G

3+

3

k

a2, (1)

a

a=

4G

3( + 3p) +

3, (2)

where a is the Universe expansion factor, k determines the spatial curvature of the

Universe and is the cosmological constant. The pressure p and density satisfy

the equation of state, p = p(). In the absence of thermal pressure of matter, the

state of equations can be expressed as: p = where is independent of time or

redshift. Combining Eqs. (1) and (2) with p = yields

d

= 3(1 + )

da

a. (3)

This gives a3(1+) where M = 0 and pM = 0 for pressureless dust matter,

R = 1/3 for radiation, and = 1 for the cosmological constant. Therefore,

the matter density, the radiation density and the vacuum energy density (or the

cosmological constant) evolve, respectively, as

M a3 ; R a

4 ; = C . (4)

Introducing the cosmological density parameters at the present epoch such that

=8G

3H200 ; =

3H20; k =

k

a2H20, (5)

and defining the present Hubble constant H0 (a/a)0, we have

+ + k = 1 , (6)

where = M + R. Alternatively, from Eq. (1) we get

H2 = H20E2(z) , (7)

where

E(z) =

M(1 + z)3 + R(1 + z)4 + + k(1 + z)2 . (8)

Meanwhile, Eq. (7) can be written to be

H0dt =da

aE(z). (9)

The lookback time from the present epoch is thus

t0 tz = H10

z0

(1 + z)1E(z)1 dz . (10)


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Effect of the Thermal Pressure on the Dynamical Evolution of the Universe 577

Finally, the age of the Universe reads

t0 = H

100

(1 + z)

1E(z)

1 dz . (11)

2.2. The case of pressure pM = 0

Unlike for the case of pressureless particles pM = 0, in the equation of state

p = will no longer be a constant but is dependent on the redshift z. If we still

assume the matter of the Universe to be ideal-gas, the equation of state becomes

pM = nMkTM

=BM

BmH kBTBM +

DM

MD kBTDM

= M

fB

BmHkBTBM +

fDMD

kBTDM

, (12)

where fB = BM/M, fD = DM/M, mH is the mass of hydrogen atom, B and MDare the mean molecular weight of the gas and the mass of dark matter particles,

respectively. As described above, the general form of equation of state still reads as

p = . So, we can write

M =pM

Mc2 =kB

c2 f

BBmHT

BM +fD

MDTDM

. (13)

Note that the above expression contains a term c2 because the natural unit c = 1

has been adopted in the original equation of state p = . After matter decouples,

the evolution of temperature follows TM = TM0(1 + z)2 where TM0 is the tempera-

ture at present epoch, and z is the redshift. Due to the different decoupling epoch

between baryonic matter and dark matter, there are different values of TM0 for

the baryon and dark matter particles. In particular, the different species of dark

matter particles correspond to the different decoupling temperatures Tdec and the

corresponding present temperatures TM0. So, Eq. (13) givesM = M0(1 + z)

2 , (14)

where

M0 =kBc2

fB

BmHTBM0 +

fDMD

TDM0

. (15)

Substituting Eq. (14) into Eq. (3), we get

M(z) = M0(1 + z)3EM(z) , (16)

where EM(z) = exp[3M0(2z + z2)/2]. Apparently, EM(0) = 1 and M(0) = M0.

With the help of Eq. (16), we can rewrite Eq. (8) as

E(z) =

M(1 + z)3EM(z) + R(1 + z)4 + + k(1 + z)2 . (17)


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578 T.-J. Zhang

The present value of E(z) can be estimated from EM(0) = 1:

E

2

(0) = MEM(0) + R + + k

= M + R + + k

= 1 , (18)which is the same as the result for pM = 0.

2.3. Estimate ofM0

The decoupling time for the baryonic matter occurred at (1 + zdec) 1100. We

can estimate the temperature of the matter component at present epoch through

TBM0 = TR0/(1 + zdec) 2.7 K/1100 0.00245 K, where TR0 is the present

temperature of the cosmic background radiation. For the hot dark matter particles

(HDM), the rest mass is of order of magnitude M

10

2

eV, corresponding to adecoupling temperature of Tdec = (1 3) Mev. We adopt MHDM = 10 eV and

Tdec(HDM) = 1 Mev for our illustration below. The decoupling redshift is thus

1 + zdec(HDM) = Tdec(HDM)/TR0 and the present temperature of HDM is TDM0 =

TR0/(1 + zdec(HDM)) = (2.7K)2/1 Mev 6.28 1010 K. Using mP = 1.67

1024 g, B = 1.21, fB 0.17 and fD 0.8, we finally obtain M0 4.56 1015,

which is indeed extremely small!

3. Discussion and Conclusions

Due to the fact that Rh2105 at present-day3, where h=H0/100 kms1 Mpc1,the role of radiation in the evolution of the Universe can be neglected. So, we roughly

have M + + k = 1. It is not difficult to show that an analytic resolution can

be found for Eq. (11) in some special cases45 of = 0 and k = 0 without

the pressure of matter. In general, Eq. (11) can only be solved numerically. As

motivated by inflation, space curvature is negligibly small. For this reason, we take

k = 0, thus M + = 1 which is a flat cosmological model. The free parameter

is M or . Given a value of M, we can integrate Eq. (11) numerically for the

cases with and without matter pressure pM respectively. The results reveal that

the age of the Universe t0 varies with the matter density parameter M : t0 dropsremarkably with the increase of M from 0 to 2 for flat models. It is unlikely that

we can observe the difference between the pM = 0 model and the one for pM = 0.

As for the cold dark matter model (CDM), it satisfies Tdec(CDM)

MCDM/19 52 Mev and MCDM 1 Gev. Taking MCDM = 1 Gev as an example,

we find that the fraction of CDM in M0 is negligibly small as compared with that

of HDM. In addition, for a flat model described above, k = 0 and R = 0, Eq. (17)

reduces to

E(z) = M(1 + z)3EM(z) + 1M . (19)

Provided that = 0 but k = 0, namely, a open or close model, Eq. (17) will

take another form

E(z) =

M(1 + z)3EM(z) + (1M)(1 + z)2 . (20)


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Effect of the Thermal Pressure on the Dynamical Evolution of the Universe 579

A comparison of Eqs. (19) and (20) reveals that the effect of the term EM(z) in

the open or close model, which characterizes the thermal pressure of matter, is less

than the one in a flat model, because the term (1 M) in Eq. (20) is magnifiedby a factor of (1 + z)2.

References

1. S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972).2. P. J. E. Peebles, Principles of Physical Cosmology (Princeton University Press, Prince-

ton, 1993).3. T. Padmanabhan, Structure Formation in the Universe (Cambridge University Press,

Cambridge, 1993).

4. S. M. Carroll, W. H. Press and E. L. Turner, Annu. Rev. Astron. Astrophys.30

, 499(1992).5. A. Sandage, Astrophys. J. 133, 355 (1961).

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Tong-Jie Zhang- Effect of the Thermal Pressure On the Dynamical Evolution of the Universe