Ack! Theory!
• Recall the definitions of density contrast:

δ(x) = Δρ/ρ0

where ρ0 is the average density of the universe and Δρ is the deviation from that average
• Recall the 2-point correlation function:

ξ(r) = <δ(x)δ(x+r)>

Another way to think about ξ(r) is as the excess probability from a Poisson case of finding a neighbour galaxy at a distance r. Thus probability of finding a pair of objects separated by distance r, each occupying volume elements dV is:

dP = (n0)2[1+ξ(r)]dV1dV2

where n0 is the average number density
• ξ(r) is dependent on galaxy type: "red"/early-type galaxies tend to be more clustered (thus greater ξ(r)) than "blue"/late-type galaxies (Lahav et al. 2002). This results makes perfect sense if we accept the notion that elliptical galaxies are formed from spiral galaxy collisions. A clustering of galaxies would produce a greater number of ellipticals, and therefore early-type galaxies should be more heavily clustered
• For 0.1 < r < 10 h-1 Mpc, we usually observe:

ξ(r) = (r/r0)

where r0 ∼ 5 h-1 Mpc and γ = 1.8, though these values vary slightly depending on the galaxy type
• The Power Spectrum P(k) is simply the Fourier transform of ξ(r) (recall derivation in class)
• It turns out P(k) and δ(k) (FT of δ(x)) are related (recall derivation in class):

P(k) = |δ(k)|2/V
P(k) ∝ kn

where V is volume. P(k) is therefore a featureless power law.
• It has been theorized that the fluctuations δ(k) arose from quantum fluctuations amplified during inflation. Since these quantum fluctuations have random "phase", the probability disribution of δ(k) is Gaussian
• The primordial density fluctuations were scale-invariant (i.e. n=1) based on inflation theory (recall class notes) and verified with early CMB measurements
• We can go ahead and measure ξ(r) from galaxy surveys. Obviously we need good statistics, therefore a large sample is necessary. The 2dF survey has roughly 250,000 galaxies in its data set. From measured ξ(r), one can perform a FFT to obtain a measured P(k) that can be compared with theoretical models.
• One important question to address in galaxy surveys is: Do galaxies trace mass? We need to understand bias

δg = b (δm)

• where δg is the density contrast as measured in galaxies, and δm is the true mass density contrast. There are several statistical tests (such as the bispectrum) that can be used to quantify the bias b
• We must also understand survey selection effects very well, such as completeness, observational limitations, etc. and correct data appropriately

Power Spectrum Today

• In radiation dominated era, once perturbations enter horizon, growth ceases until matter-radiation equality time. We saw in class that fluctuations during the radiation dominated era cannot grow. As the horizon gets larger with time, fluctuations on bigger and bigger scales enter the horizon and cease to grow. Fluctuations on scales larger than the horizon, however, can continue to grow
• This process continues until radiation-matter equality, after which all fluctuations can continue to grow. For large enough scales, fluctuations never entered the horizon and thus never halted their growth
• Thus initial (i.e. primordial) P(k) is processed to yield P(k) today. Large scale fluctuations (small k) should retain their primordial shape, whereas small scale fluctuations (large k) were processed as they entered the horizon. The turnover point in the processed power spectrum is an indication of the horizon size at equality time.
• Nonlinear gravitational collapse will affect shape of P(k) at large k (recall class discussion)
• Shape of P(k) today is very dependent on cosmology. It is sensitive to dark matter and baryon content, the nature of dark matter (HDM, WDM or CDM), the Hubble constant, but is less sensitive to the cosmological constant. We can define a transfer function T(k) such that:

P(k)today = T2(k)*P(k)primordal

Here are some examples of transfer functions for different cosmological models:

Peacock (1999)

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