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Some Conceptual Problems in Cosmology. Prof. J. V. Narlikar IUCAA, Pune. 2. The very early Universe. The time - temperature relationship. In a radiation-dominated Friedmann model, Einstein’s Field equations lead to the equation: 3 H ( t ) 2 + 3 k/S 2 = [8  G/c 4 ]  u / c 2.

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some conceptual problems in cosmology

Some Conceptual Problems in Cosmology

Prof. J. V. Narlikar


the time temperature relationship
The time - temperature relationship

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

3H(t)2 + 3k/S2 = [8G/c4] u / c2

Here u is the energy density of relativistic particles.

the time temperature relationship1
The time - temperature relationship

If there are gb bosonic degrees of freedom andgf 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 by

g = gb + (7/8) gf

the time temperature relationship2
The time - temperature relationship

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 = Kg1/2T 2, where,

K = [ 3c2 / 16Ga]1/2.

This is the famous time-temperature relation.

the time temperature relationship3
The time - temperature relationship

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

tsecond = 2.4 g1/2TMeV2 = 2.4 g1/2 106TGeV2

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 wherek is the Boltzmann constant.

the time temperature relationship4
The time - temperature relationship

For a direct comparison,

k = 1.38 1016 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
A reminder

What does the principle of equivalence say?

a reminder1
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 ,  103 102 << 1.

a reminder2
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
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/c2t2,

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

size of a flat region in curved spacetime1
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
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 106.
  • 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 to1.

[Otherwise the rules of statistics will not apply.]

calculation of
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 [2kT/ch]3.

Here g is the total number of spin-degrees of freedom as defined earlier.

calculation of1
Calculation of 
  • Multiply by the volume of the ‘locally flat’ sphere:

V = 4L3/3 = 4(ct)3/3

to get


calculation of2
Calculation of 
  • We now use the time-temperature relationship derived earlier:

t = [ 3c 2/16Ga]1/2g1/2T 2

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

TP = h / 2ktP ,tP = { Gh/2c 5}1/2

calculation of3
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 of4
Calculation of 

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

the era of primordial nucleosynthesis
The era of primordial nucleosynthesis

The 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  = 106, g = 10, and

 1018 102 1066 = 1046,

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

the gut epoch
The GUT epoch

This epoch is characterized by temperature around 1016 GeV, and for this caseTP/T 103. Taking  = 106, g = 100 we get

 1018 3  103 109 = 3  1012

Even if we relax the ‘flat spacetime’ requirement to, say,  = 103, we can raise  still, but we get no more than the very small value of 3  103.

the gut epoch1
The GUT epoch

Even for  = 103, 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.

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
Additional conceptual problems
  • The time scale for GUT and inflation is of the order of 1036 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 1036 second ?
additional conceptual problems1
Additional conceptual problems
  • The Compton wavelength of a particle of energy 1016 GeV is  1.3  1029 cm. The size of a horizon at the GUT epoch is

2310102.4106g1/2 TGeV2

 1.41028 cm

Thus we are on the threshold of quantum theory.

additional conceptual problems2
Additional conceptual problems
  • The density of matter today is of the order of 1030 g/cm3. The present temperature of the universe is 2.7 K, which corresponds to  2.3104eV. 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/ 2.3104eV  41028.


Consequently, the density of matter at the GUT epoch would have been at least 3  6.41085times the present value. In short the density at the GUT epoch was at least 6.41055 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
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.