PHY 7097
November 5, 2012
Assignment #11
Reading:
Begin chapter 8 of Mukhanov.
Problems:
Due in class on Wednesday, 11/14/12.
(1) Suppose the numerical value of the spatial Fourier transform of the metric perturbation
at a certain instant is


1 2 3 4
2 5 6 7 
(δgµν )k = a2 
.
3 6 8 9 
4 7 9 10
Suppose the numerical value of the wave vector is k = (3,2,1). Find the numerical
values of the spatial Fourier transforms of the following metric components:
a) The four scalars, φ, ψ, B and E.
b) The two transverse vectors, Si and Fi .
c) The tensor hij .
Warning: This problem involves fractions and high school algebra. It may
be harmful to the self-esteem of American citizens.
(2) Consider a simplified model of cosmology in which the slow roll parameter ǫ ≡ −Ḣ/H 2
is
n
0.05 ∀t < t1
ǫ(t) =
.
2.00 ∀t > t1
Suppose that the values of the scale factor and the Hubble parameter at time t1 (which
is the end of inflation) are a(t1 ) = a1 and H(t1 ) = H1 .
a) What is the Hubble parameter H(t) for t < t1 and for t > t1 ?
b) What is the scale factor a(t) for t < t1 and for t > t1 ?
c) Suppose that the current values of the scale factor and the Hubble parameter are
a(t0 ) = 1 and H(t0 ) = H0 . What are the values of a1 and t1 in terms of H0 , H1
and t0 ?
(3) Suppose k = 2π/λ is the current physical wave number of some perturbation. Recall
the criterion for horizon crossing: k = H(tk )a(tk ). Answer the following questions in
the context of the model described in the previous problem.
a) What is the time of first horizon crossing?
b) What is the time of second horizon crossing?
c) What is the wave number k0 of the largest observable perturbation?