Astro 321: Halo Model Final

Non-Linear Matter Power Spectrum

Put together the various pieces of the last few problem sets to form the two and one halo terms of the non-linear power spectrum P_{nl}(k, z) from the linear power spectrum P (k, z):

P_{nl}(k, z) = I_{2}(k, z)P (k, z) + I_{1}(k, z)  (1)

where

I_{2}(k, z) = ∫ d ln M (\frac{M}{ρ_{m}(z = 0)}) \frac{dn}{d ln M} b(M)y(k, M)  (2)

I_{1}(k, z) = ∫ d ln M (\frac{M}{ρ_{m}(z = 0)})^{2} \frac{dn}{d ln M} y^{2}(k, M)  (3)

Show that P_{nl}(k, z) → P (k, z) for k → 0. Plot the logarithmic power spectra Δ_{m}^{2}(k) at z = 0 and z = 1 for the cosmology of the previous problem sets. Separate the total contribution into the two and one halo terms.

Extra credit: add in galaxy occupation to form the galaxy power spectrum. See the Physics Reports review by Sheth & Cooray for details. For definiteness, take N(M) = C + S(M) where C = 1, S(M) = M/30M_{th}, if M > M_{th} = 10^{12}h^{−1}M_{⊙} (else C = S = 0).