PHY 7097
November 14, 2012
Assignment #12
Reading:
Finish chapter 8 of Mukhanov.
Problems:
Due in class on Monday, 11/26/12.
(1) Recall again the metric perturbation of problem 1
Fourier transform at a certain instant is,

1 2 3

2 5 6
(δgµν )k = a2 
3 6 8
4 7 9
of Assignment #11 whose spatial

4
7 
.
9 
10
Suppose the numerical value of the wave vector is k = (3,2,1). Recall that the rotation
matrix for a rotation of angle θ about axis θb is,
Rij = (δ ij − θbi θbj ) cos(θ) + ǫijk θbk sin(θ) + θbi θbj .
a) Work out the rotation matrix — that is, give the nine components — for a rotation
of angle θ about axis b
k.
b) Work out the rotated metric perturbation.
c) What are the spatial Fourier transforms of the four rotated scalars, φ, ψ, B and
E?
d) What are the spatial Fourier transforms of the two transverse rotated vectors, Si
and Fi ?
e) What is the spatial Fourier transform of the rotated transverse-traceless tensor hij .
(2) This problem concerns the results you obtained in parts d) and e) of the previous
problem.
a) Break the two transverse vectors Si and Fi into their helicity eigenstates.
b) Break up the transverse-traceless tensor into its helicity eigenstates.
(3) Answer the following questions in the context of the model discussed in problems 2
and 3 of Assignment #11.
a) What is the scalar power spectrum ∆2R (k)?
b) What is the tensor power spectrum ∆2h (k)?
c) What is the tensor-to-scalar ratio r?
d) What is the scalar spectral index ns ?
e) What is the tensor spectral index nt ?