Some Conceptual Some Conceptual Problems in CosmologyProblems in Cosmology

Prof. J. V. NarlikarProf. J. V. NarlikarIUCAA, PuneIUCAA, Pune

2. 2. The very early UniverseThe very early Universe

The time - temperature relationship

In a radiation-dominated Friedmann model, Einstein’s Field equations lead to the equation:

3H(t)2 + 3k/S2 = [8G/c4] u / c2

Here u is the energy density of relativistic particles.

The time - temperature relationshipIf there are gb bosonic degrees of freedom and gf fermionic degrees of freedom, then we have the following relationship in thermodynamic equilibrium:

u = ½gaT4 ,where, the effective number of degrees of

freedom are given byg = gb + (7/8) gf

Here T is the radiation temperature: that is the common temperature of all relativistic particles in thermodynamic equilibrium.

The field equations give u S 4,

and S t 1/2 , so that we get,

t = Kg1/2T 2, where,

K = [ 3c2 / 16Ga]1/2.This is the famous time-temperature relation.

The time - temperature relationship

We substitute values for the constants G, a, c in these equations to get

tsecond = 2.4 g1/2TMeV2 = 2.4 g1/2 106 TGeV

2

The time - temperature relationship

In this equation we have used the ‘energy’ units for temperature. The rationale is that in an equilibrium situation, the typical energy per particle will be kT where k is the Boltzmann constant.

The time - temperature relationship

For a direct comparison,

k = 1.38 1016 erg per deg Kelvin.

Note that we are using the statistical mechanics of high temperature mixture of particles in flat spacetime.

Is this justified in curved spacetime?

A reminder

What does the principle of equivalence say?

A reminder

The ‘flat earth’ approximation: the earth may be taken as flat over distances short compared to its radius RE. Thus we may build townships over several kilometres, assuming the Earth as flat, because the typical size RT of a town is small compared to the radius of the earth:

RT = RE , 103 102 << 1.

A reminder In general relativity, we assume that over

regions of size short compared to the radius of curvature of space-time, the spacetime may be approximated by the ‘flat’ special relativistic version of physics.

Thus, to find out how physics works in the curved spacetime of general relativity, we first see what the physics is like in the ‘nearly flat’ spacetime approximation of a small region and then extend it globally by using the ‘covariance’ of the basic equations.

This is the well-known principle of equivalence

Size of a flat region in curved spacetime The early universe has the scale factor S

proportional to t1/2 and a calculation of the components of the Riemann Christoffel tensor gives the typical component as

R 1/c2 t2,

Implying that the radius of curvature of the spacetime at epoch t is of the order ct.

Size of a flat region in curved spacetime Now if we wish to apply physics at this

epoch, our equations should be based on extrapolations of the well known ‘flat space’ equations which are supposed to hold in a ‘nearly flat region’.

From our analogy of the flat earth, we assume that the flat region in the universe has a characteristic size of

L ct, where 1.

How small should be? From our solar system tests of GR, we

know that non-Newtonian and non-Euclidean effects start getting detected at of the order of 106.

Thus we need to keep our region at least as small as given by this factor.

So we now need to know how many particles of any species exist in a volume of the size L ct at the epoch t ?

A basic requirement of statistical mechanics is that the number of such particles be large compared to 1. [Otherwise the rules of statistics will not apply.]

Calculation of Since the flat spacetime physics is valid in

the small region we have chosen, we use its well known formulae.

At temperature T, the relativistic species in thermodynamic equilibrium has total number density given by

N = 2.4 g / 2 [2kT/ch]3.Here g is the total number of spin-degrees

of freedom as defined earlier.

Calculation of Multiply by the volume of the ‘locally flat’

sphere: V = 4L3/3 = 4(ct)3/3

to get

g[2kT/ch]3(ct)3.

Calculation of We now use the time-temperature

relationship derived earlier:t = [ 3c 2/16Ga]1/2g1/2T 2

We also use the concept of Planck temperature TP and Planck time tP , which are given by

TP = h / 2ktP , tP = { Gh/2c 5}1/2

Calculation of Substituting all these into the formula for

, we get finally, (3/37g1/2) [TP /T]3.

Notice that 3 is small and so we will not get a large value for , unless TP is very large compared to T. In other words, the whole analysis fails as we get closer to the Planck temperature.

Calculation of

Let us apply the result to specific epochs in the early universe.

The era of primordial nucleosynthesisThe Planck temperature is around 1019 GeV, whereas the temperature of electron-positron annihilation is of the order of 1 MeV. Thus the ratio TP/T is of the order of 1022. In this case we have, for = 106, g = 10, and

1018 102 1066 = 1046,

which is a large number! Thus the use of flat spacetime statistical mechanics is justified in this case.

The GUT epoch

This epoch is characterized by temperature around 1016 GeV, and for this caseTP/T 103. Taking = 106, g = 100 we get

1018 3 103 109 = 3 1012

Even if we relax the ‘flat spacetime’ requirement to, say, = 103, we can raise still, but we get no more than the very small value of 3 103.

Even for = 103, we have of the order unity. For statistical mechanics to be valid, we therefore need a larger volume of space and that forces us to use the subject abinitio in curved spacetime.

The GUT epoch

In short the concepts of thermal equilibrium, temperature and particle interactions cannot be simply generalized using the principle of equivalence.

Padmanabhan, T. and Vasanthi, M.M. 1982, Phys. Letters A, 89, 327

Additional conceptual problems

1. The time scale for GUT and inflation is of the order of 1036 second. What is the operational definition of such a time scale? We have atomic clocks stable at around nanoseconds. Pulsars can do somewhat better. What is the physical process that generates a time scale of the order of 1036 second ?

2. The Compton wavelength of a particle of energy 1016 GeV is 1.3 1029 cm. The size of a horizon at the GUT epoch is

2310102.4106g1/2 TGeV 2

1.41028 cmThus we are on the threshold of quantum

theory.

Additional conceptual problems

3. The density of matter today is of the order of 1030 g/cm3. The present temperature of the universe is 2.7 K, which corresponds to 2.3104eV. This falls as inverse of scale factor as the universe expands. So the universe has expanded since the GUT epoch by a factor

= 1016 GeV/ 2.3104eV 41028.

Additional conceptual problems

Consequently, the density of matter at the GUT epoch would have been at least 3 6.41085

times the present value. In short the density at the GUT epoch was at least 6.41055 g/cm3.

What was the equation of state for such matter?

[Recall that when dealing with neutron stars, astrophysicists spent a lot of time discussing the equation of state of matter with density 1015 g/cm3.]

Concluding remarks The extrapolation of basic physics to GUT

energies and the astronomical picture of the presently expanding universe to the very early epochs far exceeds anything attempted so far in physics and astronomy.

Conceptual issues like these warn us about the dangers of such sweeping extrapolations.