Power spectrum

Question 1

The power spectrum is ….

Answer 1

the fourier transform of the correlation function

Question 2

Power spectrum equation

Answer 2

P_k = < | delta hat |^2 >

Question 3

rms equation

Answer 3

sigma^2 = < delta ^2 >= ksi(0)

Question 4

Top hat filter

Answer 4

W = 3/(4piR_TH^2 ) or 0

W hat = 3/(kR_TH)^3 (sin(kR_TH)-kR_THcos(kR_TH))

Question 5

Gaussian filter

Answer 5

W(x)=1/((2pi)^(3/2)R_G^3) exp(-|x|^2/(2*R_G^2))

~W(k) = exp(-k^2R_G^2/2)

Question 6

Top hat mass equation

Answer 6

M=(4/3) pi rho_bg R_TH^3)

Question 7

Power spectrum shape

Answer 7

delta ~ a^2 if t < teq < tent
~ 1 if tent < t < teq
~ a if t > teq

Question 8

At t>teq, Power spectrum….

Answer 8

changes! Starts at delta~k^n, then becomes k^n-4 because delta changes at teq. it becomes ~ k^-2 extra
At some point after, there is a free-streaming cutoff

Question 9

sigma_M^2 ~k^3 P(k_M)

Answer 9

M^0 if mass between FS mass and eq mass. High-ish k end
M^-4/3 if mass over eq mass, low k part
I hope
If n=1, low mass fluctuations are suppressed

Question 10

All modes start at 0 because…

Answer 10

the k=0 mode is averaged over the entire universe. the overdensity is 0 then. LArger k means smaller region

Question 11

c tent=

Answer 11

aent/k