Cosmological Entropy and Temperature Dependent G

Cosmological entropy and temperature dependent G

Corrado Massa

Via Fratelli Manfredi 55

42124 Reggio Emilia

ITALY

ABSTRACT

The Einstein–like field equations E a b = 8 π G ( T ) S a b with

gravitational “constant” G function of the temperature T

of the environment are solved in a Robertson–Walker radiation

dominated universe. The divergenceless Einstein tensor E a b

and the varying G ( T ) forbid a divergenceless energy –

momentum tensor S a b and a particle creation process occurs.

If the process involves mainly massless particles the entropy

per baryon at present time results ( G C / G D ) 3 / 4 where G C

= G ( T C ) and G D = G ( T D ) ; T C , T D are (respectively)

the temperature of the universe at the beginning of the classical

(i.e. non quantum) era and at the end of the decoupling era.

The temperature dependence law G( T ) = G ( 0 )( 1 – ζ 2 ) – 1

suggested by a number of gauge theory models employed in the

unification program ( ζ = T / τ where τ is a new universal

constant with dimension of a temperature ) gives the observed

value 10 8 to 10 9 of the entropy per baryon provided τ lies in

the range 10 24 to 10 28 K, consistent with the energy scale of a

typical grand unified theory. The possibility ζ > 1 ( G < 0, anti –

gravity ) shortly discussed in Sec 6 , implies hyperbolic space.

1. INTRODUCTION

The possibility of a temperature – dependent gravitational

constant G has been suggested by a number of authors.

In this paper I show that a temperature – dependent G =

G ( T ) may explain the observed huge value 10 9 of the

ratio Σ between the number of photons in the cosmo-

logical thermalized radiation field at 2.7 K and the number

of baryons in the universe. Other explanations have been

proposed in grand unified theories (GUTs) (Narlikar,1983 )

and in cosmological models with time–dependent cosmo-

logical “constant” ( Massa 2003 ). Natural units such as

c = h = k = G ( 0 ) = 1 are employed throughout unless

otherwise indicated; c, h, k and G( 0 ) are the speed of

light, the reduced Planck constant, the Boltzmann constant,

and the zero–temperature value of G, very close to the

usually observed value 6.7 x 10 – 8 cm 3 g – 1 s – 2 .

2. BASIC ASSUMPTIONS

In investigating the early, radiation dominated universe let

us make eight assumptions:

A 1) the cosmological spacetime in the radiation – dominated

era is described by the spatially isotropic Robertson – Walker

( R W ) metric with space curvature index Z and cosmic

scale factor R = R ( t ) function of the cosmic time t ;

A 2) in the cosmic fluid the Planck distribution law holds, with

the Stefan law U = αT 4 for the energy density U, and the law

N = β( R T ) 3 for the number N of photons in a comoving

volume of radius R; α = α (B) + α (F) is the number of species

of radiation, a number which depends on T and running from

about unity to about 100; α (B) and α (F) are respectively the

number of helicity states of bosons and fermions; β ~ 0.24

is a constant numerical factor springing from the Planck law.

A 3 ) the cosmic fluid is described by the energy–momentum

tensor S a b of a perfect fluid with energy density U and

isotropic pressure U / 3 ,

S a b = ( 4 / 3 ) α T 4 V a V b – ( α T 4 / 3 ) g a b ( 2 . 1 )

V a is the four–velocity and g a b is the fundamental tensor.

In the RW spacetime we obtain

S 0 0 = U = α T 4 , S 1 1 = S 2 2 = S 3 3 = – U / 3 ( 2 . 2 )

A 4 ) the gravitational field equations have an Einstein–like form;

in other words, the Einstein tensor E a b is 8 π G( T ) times the

energy momentum tensor, namely

E a b = 8 π G ( T ) S a b ( 2 . 3 )

Assumption A4 is the simplest way to treat a varying G in

a general relativistic context and has been adopted by a number

of authors ( Linde 1980, Davies 1981, Pollock 1982, Raychaudhury

and Bagchi 1983, Rauch 1984, Massa 1994, 1989 a, b ).

A 5 ) quantum effects are negligible provided

T 2 G ( T ) < 1 ( 2 . 4 )

This assumption is reasonable because in the standard model,

where g is constant, quantum gravity effects become important

when the parameter G T 2 exceeds unity (Weinberg, 1972 ).

According to inequality ( 2 . 4 ) a critical temperature T C exists,

and obeys the equation

T C 2 G C = 1 ( 2. 5 )

where G C stands for G ( T C ). When T > T C quantum gravity

effects dominate, the spacetime loses its usual structure and is

expected to be replaced with a quantum foam with no definite

geometry and topology; in such a situation the very notion of

particle loses its meaning. Accordingly I make the sixth assumption,

A 6 ) in the early universe, particles begun to exist when T = T C .

The seventh assumption is

A 7 ) the number N B of baryons in a comoving volume is

independent of time (i.e., independent of temperature) namely

NB = B R 3 / m = γ ( 2 . 6 )

where B is the density ( = mass per unit 3–volume) of baryonic

matter in the universe, m is a typical baryon mass, and γ is a

constant.

The eighth and final assumption is

A 8 ) The photon–to–baryon ratio at the critical time (i. e. when

the temperature of the cosmic fluid was T = T C ) was not very

far from unity. I stress that the observed huge value of Σ at

present time Σ* ~ 10 9 is commonly considered peculiar by

cosmologists; why a value so far from unity? Assumption ( A 8 )

with Σ C standing for Σ ( T C ) can be written as

1 / 1000 < Σ C < 1000 ( 2 . 7 )

This number is not peculiar, because there is nothing strange in

a value of Σ C not far from unity.

3. COSMOLOGICAL CONSEQUENCES OF THE

FIELD EQUATIONS ( 2 . 3 )

For the well known identities E a b / b = 0 ( slash stands for

covariant differentiation ) eq ( 2 . 3 ) demands

[ G ( T ) S a b ] / b = 0 ( 3 . 1 )

which for the RW metric and for the energy – momentum

tensor ( 2 . 2 ) implies (d G / G ) + (d U / U ) + 4 (d R / R ) = 0

which integrates to ( remember, U = α T 4 )

( R T ) 4 G ( T ) = δ ( = an integration constant ) ( 3 . 2 )

The dynamic of cosmic expansion is just the same as in the

standard model; to see this fact consider the field equations

( 2 . 3 ), the energy–momentum tensor( 2 . 2 ) and the RW

metric; straightforward computation yields

3 ( d R / d t ) 2 + 3 Z = 8 π αG ( T ) R 2 T 4 ( 3 . 3 )

that for eq ( 3 . 2 ) implies ( just as in the standard model )

3 ( d R / d t ) 2 R 2 + 3 Z R 2 = 8 π α δ ( 3 . 4 )

According to assumption ( A 2 ) the number of photons in

a comoving volume is N = β( R T ) 3 and for eq ( 3 . 2 )

we obtain:

G ( T ) 3 / 4 N ( T ) = ε ( 3 . 5 )

ε = βδ 3 / 4 is ( of course ) a constant. Note, eq ( 3 . 5 ) implies

N = N ( T ) namely a temperature dependent N, at variance with

the standard model which requires N = constant. The compatibility

of assumption (A 2) with a continuous photon creation is discussed

in Wichoski and Lima (1999); they find that in a RW universe

with photon creation the standard Planckian spectral distribution

law has to be replaced with a more general law; however if RT

greatly exceeds ( N / α ) 1 / 3 or if the radiation is decoupled, the

spectrum is not destroyed by the Hubble expansion and the

relations U = α T 4 , N = β( R T )3 still hold. This condition,

tantamount to inequality α > β , is satisfied with sufficient

approximation in this model.

4. THE PHOTON – TO – BARYON RATIO

Let N B denote the number of baryons in a comoving volume.

The entropy per baryon, defined by

Σ = N / N B ( 4 . 1 )

depends on T because eq ( 3 . 5 ) and eq ( 2 . 6 ) require

Σ = Σ ( T ) = ( ε / γ ) [ 1 / G ( T ) ] 3 / 4 ( 4 . 2 )

At the critical time the entropy per baryon was

Σ C = ( ε / γ ) ( 1 / G C ) 3 / 4 ( 4 . 3 )

Remember, Σ C stands for Σ ( T C ). Let T D denote the

temperature of the cosmic fluid at the decoupling, when

the thermal interaction between matter and radiation ends,

the universe becomes transparent and photons cannot

thermalize with matter. The entropy per baryon is then

Σ D = ( ε / γ ) ( 1 / G D ) 3 / 4 ( 4 . 4 )

where Σ D stands for Σ ( T D ) . For ( 2 . 5 ) this implies

Σ D / Σ C = ( G C / G D ) 3 / 4 ( 4 . 5 )

Note,

Σ D = Σ * ( = the value of Σ at present epoch) ( 4 . 6 )

because all “ tardy ” photons ( i.e. the photons generated

after the decoupling ) cannot thermalize with matter and give

no contribute to the thermalized cosmic background radiation

we observe today. For eq ( 4 . 6 ) we find

Σ * = Σ C ( G C / G D ) 3 / 4 ( 4 . 7 )

5. CALCULATION WITH A SPECIAL G ( T )

This Section considers a special temperature–dependence

law of the form

G = G ( T ) = ( 1 – ζ 2 ) – 1 ( 5 . 1 )

required by a variety of gauge theories used in the unification

program; ζ = T / τ where τ is a universal constant with the

dimensions of a temperature. Law( 5 . 1 ) has been investigated

both at the local and cosmological level by Linde (1980) Davies

(1981) Pollock (1982) Raychaudhury and Bagchi ( 1983 ) and

Massa (1989 a, b); possible consequences in particle physics and

in cosmic ray physics are pointed out in Massa ( 1989 b, 2000 ).

The numerical value of τ is unknown and depends on the details

of the gauge theory employed. It is reasonable to set τ > 10 13 K

because the standard model works well when T < 10 13 K .

Insert eq ( 5 . 1 ) into eq ( 4 . 7 ) and into eq ( 2 . 5 ) and obtain

respectively ( with ζ D = T D / τ , and ζ C = T C / τ )

( Σ * / Σ C ) 4 / 3 ~ ( 1 – ζ D 2 ) / ( 1 – ζ C 2 ) ( 5 . 2 )

T C 2 [ 1 + ( 1 / τ ) 2 ] = 1 ( 5 . 3 )

The observed decoupling temperature, T D ~ 5000 K , is much less

than 1013 K and thus ζ D 2 < < 1 ; eq ( 5 . 2 ) reduces with excellent

approximation to

( Σ * )( 1 – ζ C 2 ) 3 / 4 = Σ C ( 5 . 4 )

which for eq ( 5 . 3 ) and ζ = T / τ yields

τ = [ ( Σ * / Σ C ) 4 / 3 – 1 ] – 1 / 2 ~ ( Σ C / Σ * ) 2 / 3 ( 5 . 5 )

namely, if we display c, h, k, and G ( 0 ),

τ = T 0 [ ( Σ * / Σ C ) 4 / 3 – 1 ] – 1 / 2 ~ T 0 ( Σ C / Σ * ) 2 / 3 ( 5 . 5 a)

where T 0 = √ [ h c 5 / k 2 G ( 0 ) ] ~ 10 3 2 K

Insert into eq ( 5 . 5 a ) the numerical values 1/1000 < Σ C < 1000

required by assumption A 8, and Σ * ~ 10 9 required by observations,

and find

10 24 K < τ < 10 2 8 K ( 5 . 6 )

The value 10 15 K of the electroweak model ( Davies 1981 ) is

totally ruled out. Interestingly, the energy scale of a typical

GUT, from 10 14 GeV ~ 10 27 K to 10 15 GeV ~ 10 2 8 K ,

lies just in the range ( 5 . 6 ).

6. FINAL REMARK

At the first glance equation (5 . 1) suggests that at T = τ

the gravitational coupling G passes through an infinite

discontinuity and becomes negative at T > τ , i.e. antigravity

results, such a possibility is maintained by Linde (1980) and

negated by Pollock (1982). If we set T > τ , namely ζ > 1 , and

repeat the procedure of Sections 1 to 4, we find again eq( 5 . 5 )

while eq ( 3 . 3 ) and eq ( 5 . 1 ) give

Z = – [ ( d R / d t ) 2 + ( 8 π α R 2 T 4 / 3 )( ζ 2 – 1 ) – 1 ] ( 6 . 1 )

that for ζ > 1 implies a negative Z, namely hyperbolic space.

REFERENCES

Davies, P.C.W. , 1981, Phys. Lett. 101 B, 399.

Linde, A.D., 1980, Phys. Lett, 93 B, 394.

Massa, C., 1989 a, Helvetica Phys. Acta 62, 424 .

Massa, C., 1989 b , Nuovo Cimento B 104, 477 .

Massa, C., 1994, Nuovo Cimento B 109, 95 .

Massa, C., 2000 , Astroph. Space Sci. 271, 365.

Massa, C., 2003 , Astroph. Space Sci. 286, 313

Narlikar, J.V., 1983, Introduction to Cosmology, Jones and Bartlett,

Boston.

Pollock, M., 1982 , Phys. Lett. 108 B, 386 .

Rauch, R. T., 1984, Phys. Rev. Lett. 52, 1843 .

Raychaudhury, A.K., & Bagchi, B., 1983, Phys. Lett. B, 124, 168.

Wichoski, U.F. , & Lima, J.A.S., 1999, Phys. Lett. A 262, 103 .