Equilibrium thermodynamics in the expanding universe
17
This exponential expansion inhibits the fonnation of the gravitationally bound
clumps of matter that are presumably a necessary precondition for life to evolve;
once the clumps are formed, the cosmological constant has no further effect.
Thus, the weak anthropic principle requires Acff to be small enough to allow
the fonnation of sufficiently large clumps of matter. Gravitational condensation
began in our universe at a red shift Ze where Ze ~ 4. The energy density of
matter at that time was greater than the present matter density Pm by a factor
of R 3 (to}/R 3 (te) = (l +Ze)3 ~ 125. The cosmological constant has no effect so
long as it is dominated by the matter density. Thus. provided Pvac ;S 125pm. the
vacuum energy density would not inhibit gravitational condensation. (A more
careful treatment [4] gives a further factor of l1r 2 .) We conclude that if the
anthropic principle accounts for the value of the (positive) cosmological constant.
then we should expect Pvac ,.., (10 - 100)Pm because there is no anthropic
reason for it to be smaller. This gives the prediction OA '" (10 - lOO)Om. at
variance with the values (1.92) derived from the supernovae and WMAP data.
Nevertheless, it implies a much smaller value Pvac/ M~ than that given in (1.98)
which was derived from supersymmetry considerations.
In contrast. a negative cosmological constant does not affect gravitational
clumping. We see from the Friedmann equation (1.34) that if A is negative. the
expansion of the universe ceases (for a flat universe (k = 0» when the matter
density tenn is cancelled by the cosmological constant. We have already noted
that the deceleration parameter qO given in (1.90) is positive for A < O. It follows
that after expansion has ceased. the universe begins to contract and, in fact, it
collapses to a singularity in a finite time T. It is easy to show (exercise 4) that
21r
(1.99)
T = J3IAI'
Anthropic considerations would then !:'luire that this leaves sufficient time for
life to evolve, say T ~ !H o - 1 where Ho = J3/81rGNPm is the Hubble time in
our universe. This would give
OA < (41r)2
(1.I00)
Om'" 3
Again, this would entail a much smaller value of Pvac/ M~ than was obtained
from supersymmetry considerations. However, the supernovae data indicate a
universal acceleration rather than a deceleration. Thus, A is positive and the
previous bound is only of academic interest.
1.8 Equilibrium thermodynamics in the expanding universe
It makes sense to discuss eqUilibrium thermodynamics during most of the history
of the universe because reaction rates were much faster than the time scale for
17
This exponential expansion inhibits the fonnation of the gravitationally bound
clumps of matter that are presumably a necessary precondition for life to evolve;
once the clumps are formed, the cosmological constant has no further effect.
Thus, the weak anthropic principle requires Acff to be small enough to allow
the fonnation of sufficiently large clumps of matter. Gravitational condensation
began in our universe at a red shift Ze where Ze ~ 4. The energy density of
matter at that time was greater than the present matter density Pm by a factor
of R 3 (to}/R 3 (te) = (l +Ze)3 ~ 125. The cosmological constant has no effect so
long as it is dominated by the matter density. Thus. provided Pvac ;S 125pm. the
vacuum energy density would not inhibit gravitational condensation. (A more
careful treatment [4] gives a further factor of l1r 2 .) We conclude that if the
anthropic principle accounts for the value of the (positive) cosmological constant.
then we should expect Pvac ,.., (10 - 100)Pm because there is no anthropic
reason for it to be smaller. This gives the prediction OA '" (10 - lOO)Om. at
variance with the values (1.92) derived from the supernovae and WMAP data.
Nevertheless, it implies a much smaller value Pvac/ M~ than that given in (1.98)
which was derived from supersymmetry considerations.
In contrast. a negative cosmological constant does not affect gravitational
clumping. We see from the Friedmann equation (1.34) that if A is negative. the
expansion of the universe ceases (for a flat universe (k = 0» when the matter
density tenn is cancelled by the cosmological constant. We have already noted
that the deceleration parameter qO given in (1.90) is positive for A < O. It follows
that after expansion has ceased. the universe begins to contract and, in fact, it
collapses to a singularity in a finite time T. It is easy to show (exercise 4) that
21r
(1.99)
T = J3IAI'
Anthropic considerations would then !:'luire that this leaves sufficient time for
life to evolve, say T ~ !H o - 1 where Ho = J3/81rGNPm is the Hubble time in
our universe. This would give
OA < (41r)2
(1.I00)
Om'" 3
Again, this would entail a much smaller value of Pvac/ M~ than was obtained
from supersymmetry considerations. However, the supernovae data indicate a
universal acceleration rather than a deceleration. Thus, A is positive and the
previous bound is only of academic interest.
1.8 Equilibrium thermodynamics in the expanding universe
It makes sense to discuss eqUilibrium thermodynamics during most of the history
of the universe because reaction rates were much faster than the time scale for