Measuring structure — the power spectrum and the correlation function

Write the density contrast as a sum of plane waves, exactly as for the single modes on the earlier pages:

By construction the average of $\delta$ over space is zero — it measures departures from the mean density. The simplest non-trivial number is therefore the variance, the mean of $\delta^2$:

[The notes write the averages as plain integrals $\int d^3x$; an average needs a factor $1/V$, the volume, so strictly $P(k) = |\hat\delta(k)|^2/V$; the same applies to $\xi$ in eq. (408) below. The step to $k$-space is Parseval's theorem.]

If the universe is statistically homogeneous and isotropic, $|\hat\delta|^2$ can depend only on the length $k = |\vec k|$, not on direction, and $d^3k = 4\pi k^2\,dk$. Then

using $4\pi/(2\pi)^3 = 1/(2\pi^2)$. The power spectrum $P(k)$ says how much of the total variance sits in waves of wavenumber $k$ — that is, on length scales around $2\pi/k$. Two remarks from the notes:

• $P(k)$ and $\sigma$ change with time, because every $\hat\delta(k)$ grows. In the linear regime all large-scale modes grow by the same growth factor $D$ from the earlier page, so $\sigma = D\,\sigma_0$.
• The starting point is the primordial power spectrum left at the end of inflation. Its general form is a power law,

which has no preferred scale ("scale-free"). Inflation predicts $n \approx 1$; section 2.D below shows why that value is natural.

We cannot measure $\delta$ for the dark matter directly, but we can count galaxies. Suppose galaxies trace the mass, with mean number density $n_0$. The probability of finding one galaxy in a small volume $dV_1$ and another in $dV_2$ a distance $r$ away is

On averaging, the terms linear in $\delta$ vanish, and the product $\langle\delta(\vec x)\,\delta(\vec x + \vec r)\rangle$ is left. It depends only on the distance $r$ (isotropy), and is called the two-point correlation function $\xi(r)$. For a random (Poisson) scattering of galaxies $\xi = 0$; $\xi > 0$ means pairs at that separation are more common than random — the galaxies cluster.

Substituting the Fourier expansion twice and using $\int d^3x\,e^{i(\vec k - \vec k')\cdot\vec x} = (2\pi)^3\delta_D(\vec k - \vec k')$ collapses the double integral to a single one:

So ξ and P are a Fourier pair: the same information, one in real space and one in wavenumber space. (Here $\delta_D$ is the Dirac delta, not the density contrast.) At $r = 0$ the correlation function is just the variance, $\xi(0) = \sigma^2$.

Galaxy surveys find, over a wide range of separations, a power law:

The correlation length $r_0$ is where $\xi = 1$: pairs are twice as common as random. ($h \approx 0.7$ is the Hubble constant in units of 100 km/s/Mpc.) Different kinds of object have different $r_0$, and more massive objects cluster more strongly: the notes quote $\xi_{cc} \approx 20\,\xi_{gg}$ for the cluster–cluster compared with the galaxy–galaxy correlation function.

Section 2.D asks what $n$ and $A$ are. First, how big are the fluctuations on a given scale? Smoothing the field over a region of size $R$ removes waves shorter than $R$, so roughly only modes up to $k_\text{max} \sim 1/R$ contribute (small $k$ = large scales):

(the integral converges at $k = 0$ only for $n > -3$). A region of size $R$ holds a mass $M \propto R^3 \propto k^{-3}$, so $k \propto M^{-1/3}$ and

For $n = 1$, $\sigma \propto M^{-2/3}$: a region with 1000 times more mass has fluctuations 100 times smaller. Small masses fluctuate most — the root of hierarchical ("bottom-up") structure formation, in which small objects form first.

Outside the horizon every mode grows. The notes' argument is that no mass scale should be special: every mode should have the same fluctuation strength $\sigma$ at the moment it enters the horizon. What does that demand of $n$?

The mass inside the horizon is $M_h \propto \rho_m r_h^3$, with $\rho_m \propto (1+z)^3$. The physical horizon grows as $r_h \propto t$, i.e. as $a^2$ in the radiation era and $a^{3/2}$ in the matter era, so

Meanwhile, outside the horizon, $\sigma$ grows like $\delta$ — as $a^2$ in the radiation era and $a$ in the matter era (the super-horizon growth of the previous page):

[The notes print $a^{3/2}$ for the matter case; $(1+z)^{-1}$, which they also print, is $a$, consistent with $\delta \propto a$ in the matter era.]

Take a mode of mass $M$ that enters during the radiation era, at redshift $z_h$. From the end of inflation, at $z_p$, until then it grows as $a^2$:

The primordial amplitude is $\sigma(z_p) \propto M^{-(n+3)/6}$ from eq. (412), and the mode enters when $M_h = M \propto (1+z_h)^{-3}$, so $(1+z_h)^{-2} \propto M^{2/3}$:

For a mode entering in the matter era the notes say only that the calculation is the same, giving the same result (eq. 419). Filling in the step (this page's own): $M_h \propto (1+z_h)^{-3/2}$ gives $(1+z_h)^{-1} \propto M^{2/3}$ again, and the earlier growth through the radiation era is the same factor for every such mode, so it does not change the $M$-dependence. The last form, $M^{-(n-1)/6}$, is this page's simplification of the notes' exponent. For the horizon-entry amplitude to be the same for every mass, the exponent must vanish: no special scale means $n = 1$.

This is the Harrison–Zel'dovich (scale-invariant) spectrum, proposed around 1970 by Edward Harrison and Yakov Zel'dovich (and independently Peebles and Yu), before inflation. [The notes say σ(z_h) "should not depend on n"; the requirement is that it should not depend on $M$, which then fixes $n$.] Inflation, on the 8.286 pages, produces almost exactly this spectrum.

Inflation predicts the shape, but in practice the amplitude $A$ is measured. The standard yardstick is $\sigma_8$, the rms density fluctuation in spheres of radius $8\,h^{-1}$ Mpc. Galaxies need not trace the mass exactly; their fluctuations differ by the bias $b$:

Observed galaxies have $\sigma_{8,\text{gal}} \approx 1$. The notes quote the combined WMAP and SDSS values

So $n$ is close to 1 but slightly smaller — a small "tilt" that simple inflation models predict. Since the mass $\sigma_8$ is below 1 while $\sigma_{8,\text{gal}} \approx 1$, these galaxies are biased, $b > 1$. (A standard explanation, not given in the notes: galaxies form preferentially in the higher peaks of the density field, and clusters, in the highest peaks, are even more biased.)

The primordial spectrum is not what we see today. On the previous page, dark matter modes that entered the horizon during the radiation era stagnated until matter–radiation equality, while those entering later grew without interruption. Small scales entered earliest and lost the most growth. The notes describe this with a transfer function $T(k)$:

where $L_0$ is the comoving horizon at matter–radiation equality, and $n = 1$ has been used. [The notes write $T(K)$; the second case is meant up to normalisation, $T \propto (kL_0)^{-2}$.] Large scales ($k \ll 1/L_0$) entered after equality and keep the primordial $P \propto k$. Small scales are suppressed by $T^2 \propto k^{-4}$, giving $P_0 \propto k^{-3}$. The spectrum therefore rises, peaks near $k_0 \sim 1/L_0$, and falls — the turnover is a fossil of the moment matter took over from radiation.

• The density field is described statistically: $\sigma^2 = \frac{1}{2\pi^2}\int P(k)\,k^2\,dk$ with the power spectrum $P(k) = |\hat\delta(k)|^2$. It grows in time as $\sigma = D\sigma_0$.
• The two-point correlation function $\xi(r)$ is the excess probability of finding pairs at separation $r$; it is the Fourier transform of $P(k)$. Galaxies show $\xi \approx (r/5\,h^{-1}\text{Mpc})^{-1.8}$; massive objects cluster more.
• For $P = Ak^n$, $\sigma \propto M^{-(n+3)/6}$, and the amplitude at horizon entry is $\propto M^{-(n-1)/6}$. Requiring no special scale gives the Harrison–Zel'dovich spectrum $n = 1$; measurements give $n$ slightly below 1.
• The amplitude is measured as $\sigma_8$; galaxies are biased tracers, $\sigma_{8,\text{gal}} = b\,\sigma_{8,\text{mass}}$.
• Stagnation of modes that entered the horizon before matter–radiation equality bends the spectrum: $P_0 = Ak\,T^2(k)$ rises as $k$ on large scales and falls as $k^{-3}$ on small ones, peaking near the horizon scale at equality.