PHY 7097
November 26, 2012
Assignment #13
Reading:
Chapter 9 of Mukhanov.
Problems:
Due in class on Wednesday, 12/5/12.
(1) Recall that, for general a(t), the scalar mode function v(t, k) and the tensor mode
function u(t, k) obey the equations,
2
2
˙
,
ü + 3H u̇ + ka2 u = 0 ,
(1)
v̈ + 3H + v̇ + ka2 v = 0
subject to the normalization conditions,
v v̇ ∗ − v̇v ∗ = i4πG
a3
,
uu̇∗ − u̇u∗ = i32πG
.
a3
It turns out that these equations can be solved whenever the first slow
(t) ≡ −Ḣ/H 2 is a constant (t) = 1 ,
r
2
(1)
u(t,k)
k
8π
G
√
v(t, k) =
, u(t, k) = (1− )H(t)a3 (t) Hν (1− )H(t)a(t) , ν =
81
1
1
(2)
roll parameter
1 3−1
2 1−1
. (3)
Of course a graceful exit for inflation precludes (t) being constant for all time, but
it might be constant for some period during inflation. Assume that (t) = 1 for all
times t ≤ t1 , at which time the Hubble parameter is H1 and the scale factor is a1 .
a) Show that the mode functions (3) obey equations (1) and (2). Hint: Use problem
(2) of Assignment #11 to infer the time dependences of H(t) and a(t).
b) Compute the scalar and tensor power spectra for modes which experience first
(1)
horizon crossing before t1 . Hint: Use the asymptotic small x form for Hν (x)
c) Compute the tensor-to-scalar ratio, and scalar and tensor spectral indices.
d) Compare your results with those derived from the WKB formalism.
(2) Consider single-scalar inflation with a power law potential V (ϕ) = λb φb , where b is an
arbitrary positive constant.
a) Find the scalar field strength ϕ1 at which inflation ends.
b) Use the slow roll approximation to solve for ϕ(t).
c) Use the slow roll approximation to derive approximate formulae for a(t), H(t) and
the two slow roll parameters (t) and η(t).
d) Compute the scalar and tensor power spectra, assuming the smallest observable
wave number k0 experienced first horizon crossing 50 e-foldings before the end of
inflation.
e) Compute the tensor-to-scalar ratio and the two spectral indices.
(3) Next week we will derive the Sachs-Wolfe result for the first order temperature fluctuations imprinted upon a photon as it travels from the surface of last scattering through
the metric perturbations left over from inflation. In this problem you will prepare by
solving the geodesic equation for a light ray moving through perturbations about a
flat background, gµν (x) = ηµν + δgµν (x). We seek the function χµ (τ ) which obeys the
geodesic equation,
µ
(4)
χ̈(τ ) + Γ ρσ χ(τ ) χ̇ρ (τ )χ̇σ (τ ) = 0 .
This is a second order differential equation for the four variables χµ (τ ), so it requires
eight conditions. Four of these conditions are obviously that the geodesic should end
up at the spatial origin now at τ = 1,
χ0 (1) = t0
,
χi (1) = 0 .
(5a)
The remaining conditions involve choosing the final velocity χ̇µ (1). One condition they
must satisfy is that the geodesic is light-like,
gµν (x0 )χ̇µ (1)χ̇ν (1) = 0 .
(5b)
Another condition is that the geodesic began at the time of recombination,
χ0 (0) = trec .
(5c)
The remaining two conditions are arbitrary and correspond to the spatial unit vector n
b
which gives the direction the geodesic is moving at τ = 1. No explicit solution can be
found for arbitrary δgµν (x), but it is simple to work out a few orders of the perturbative
solution in powers of δgµν (x). We merely expand the geodesic in powers of the metric
perturbation,
µ
µ
µ
χµ (τ ) = χ0 (τ ) + χ1 (τ ) + χ2 (τ ) + . . . ,
(6)
µ
µ
where χ0 (τ ) has no metric perturbations, χ1 (τ ) has one power, and so on. The relevant
µ
µ
equations for χ0 (τ ) and χ1 (τ ) are,
µ
χ̈0 (τ ) = 0 ,
µ
χ̈1 (τ ) = η µν −δgνρ,σ χ0 (τ ) + 21 δgρσ,ν χ0 (τ )
µ
a) Find the zeroth order solution χ0 (τ ).
µ
b) Find the first order solution χ1 (τ ).
2
(7a)
ρ
χ̇0 (τ )χ̇σ0 (τ ) .
(7b)