Introduction to Cosmology

A Cyclic Universe 173

scale of 10−^25 cm. In the cyclic Universe, the fluctuations can be generated a fraction
of a second before the bang when their length scale is thousands of kilometres.
The entropy created in one cycle is expanded and exponentially diluted to near zero
density after the dark energy dominated phase, but the entropy does not increase again
in the contracting phase. The reason is that the branes themselves do not contract,
only the extra dimension contracts. The total entropy on the branes and the number
of black holes increase from cycle to cycle, and increase per comoving volume on our
brane as well. In the vicinity of the black holes, there is no cycling due to their strong
gravitational field. Black holes formed during one cycle will therefore survive to the
next cycle, acting as defects in an otherwise nearly uniform universe.
The kinetic energy and the physical entropy density during each cycle are fed not
only by the interbrane force, but also by gravity which supplies extra energy during
the contraction phase. This kinetic energy of the branes is converted partially into
matter and radiation and blue-shifted by the gravity.
All of this is very speculative, but so is consensus inflation, too. Fortunately, the dif-
ferent models make different testable predictions, notably for gravitational radiation.
A certain discovery of gravitational radiation testified by tensor perturbations in the
CMB would support consensus inflation.

Problems

1. Derive Equation (7.37).


2. Derive휙(푡)for a potential푉(휙)=^14 휆휙^4.


3. Suppose that the scalar field averaged over the Hubble radius퐻−^1 fluctuates
   by an amount휓. The field gradient in this fluctuation is∇휓=퐻휓and the gra-
   dient energy density is퐻^2 휓^2. What is the energy of this fluctuation integrated
   over the Hubble volume? Use the timescale퐻−^1 for the fluctuation to change
   across the volume, and the uncertainty principle to derive the minimum value
   of the energy. This is the amount by which the fluctuation has stretched in one
   expansion time [7].


4. Material observed now at redshift푧=1 is at present distance퐻− 01. The recession
   velocity of an object at coordinate distance푥is푅푥̇. Show that the recession
   velocity at the end of inflation is

푅푥̇ =퐻√^0 푅^0 푥푧r

푧eq

, (7.61)

where푧ris the redshift at the end of inflation. Compute this velocity. The density

contrast has grown by the factor푧^2 r∕푧eq. What value did it have at the end of

inflation since it is now훿≈ 10 −^4 at the Hubble radius [7]?

5. Show that in an exponentially expanding universe (푞=−1) the Hubble sphere
   is stationary. Show that it constitutes an event horizon in the sense that events
   beyond it will never be observable. Show that in this universe there is no particle
   horizon [8].