11.4 Correlation functions and power spectra

Correlation Functions and Power Spectra in Large-Scale Structure

Galaxies aren't scattered randomly across the cosmos. They cluster in patterns, forming filaments, walls, and voids that stretch across hundreds of megaparsecs. The correlation function and power spectrum are the two primary statistical tools cosmologists use to quantify this clustering. They encode the same information but in complementary ways, and together they connect the large-scale structure we observe today back to conditions set in the very early universe.

Correlation Function vs. Power Spectrum

The two-point correlation function $\xi(r)$ measures the excess probability of finding a pair of galaxies separated by distance $r$, compared to what you'd expect from a purely random (Poisson) distribution. If $\xi(r) > 0$, galaxies are more clustered than random at that separation. If $\xi(r) = 0$, the distribution is indistinguishable from random at scale $r$. This is a real-space quantity: you're directly measuring how galaxy positions relate to each other as a function of physical distance.

The power spectrum $P(k)$ captures the same clustering information but in Fourier space. Here, $k$ is the wavenumber (with units of inverse length), so small $k$ corresponds to large spatial scales and large $k$ to small scales. $P(k)$ tells you the amplitude of density fluctuations at each scale. A peak at some $k$ means there's a lot of structure at the corresponding wavelength $\lambda \sim 2\pi / k$.

Calculations Between Correlation and Power

These two quantities are a Fourier transform pair. Converting between them is straightforward in principle:

Correlation function → Power spectrum:

$P(k) = \int \xi(r)\, e^{-i\vec{k} \cdot \vec{r}}\, d^3r$

You integrate the correlation function over all space, weighted by the Fourier kernel, to obtain the power spectrum.

Power spectrum → Correlation function:

$\xi(r) = \frac{1}{(2\pi)^3} \int P(k)\, e^{i\vec{k} \cdot \vec{r}}\, d^3k$

This is the inverse Fourier transform. Because the universe is statistically isotropic (no preferred direction), both $\xi$ and $P$ depend only on the magnitudes $r$ and $k$, which simplifies these 3D integrals into 1D integrals involving spherical Bessel functions. In practice, going from observed data to clean estimates of either quantity requires careful handling of survey geometry, selection effects, and shot noise.

Interpreting and Connecting the Matter Power Spectrum

Features of the Matter Power Spectrum

The shape of $P(k)$ encodes a remarkable amount of physics. Three features are especially important:

Large scales (small $k$): The spectrum rises roughly as $P(k) \propto k^{n_s}$ with $n_s \approx 1$. This nearly scale-invariant slope reflects the primordial fluctuations generated during inflation. These modes entered the Hubble horizon after matter-radiation equality, so they grew without suppression and preserve the primordial shape.

Turnover at $k_{\text{eq}}$: There's a broad peak in $P(k)$ at the wavenumber corresponding to the comoving horizon scale at matter-radiation equality. Modes with $k > k_{\text{eq}}$ entered the horizon during the radiation-dominated era, when radiation pressure prevented dark matter perturbations from growing efficiently (a process called Mészáros suppression). Beyond this turnover, the spectrum falls off approximately as $P(k) \propto k^{-3}$. The location of this turnover depends on $\Omega_m h^2$, making it a direct probe of the matter density.

Baryon Acoustic Oscillations (BAO): Superimposed on the smooth shape are small wiggles caused by acoustic oscillations in the baryon-photon plasma before recombination ($z \approx 1100$). Sound waves propagated outward from initial overdensities until photons decoupled from baryons, freezing in a characteristic scale of about 150 Mpc (comoving). These oscillations act as a standard ruler: measuring the BAO scale at different redshifts constrains the expansion history, the Hubble constant $H_0$, and the dark energy equation of state.

Power Spectrum and the Universe's Initial Conditions

Primordial power spectrum: Inflation predicts that quantum fluctuations were stretched to cosmological scales, producing a nearly scale-invariant spectrum of density perturbations. This is parameterized as:

$P_{\text{primordial}}(k) \propto k^{n_s}$

where $n_s$ is the scalar spectral index. A perfectly scale-invariant (Harrison-Zel'dovich) spectrum has $n_s = 1$. Inflationary models generically predict $n_s$ slightly less than 1, and observations from Planck measure $n_s \approx 0.965$, confirming this prediction at high significance.

Transfer function: The primordial spectrum gets modified as the universe evolves. The transfer function $T(k)$ encapsulates all the physics between inflation and the epoch when structure begins forming: radiation pressure suppression, Silk damping, baryon loading, and the transition from radiation to matter domination. The observed matter power spectrum is related to the primordial one by:

$P(k) = T^2(k)\, P_{\text{primordial}}(k)$

The transfer function is scale-dependent. It's roughly unity for modes that entered the horizon well after matter-radiation equality (large scales) and falls off for smaller-scale modes that were suppressed during the radiation era.

Constraining cosmological parameters: The shape, amplitude, and features of $P(k)$ are sensitive to multiple cosmological parameters simultaneously:

• The overall amplitude constrains $\sigma_8$ (the normalization of fluctuations on 8 $h^{-1}$ Mpc scales)
• The turnover location constrains the matter density $\Omega_m h^2$
• The BAO wiggles constrain the baryon density $\Omega_b h^2$ and serve as a geometric probe of expansion
• The large-scale slope constrains $n_s$ and thereby inflationary physics

Fitting the observed power spectrum (from surveys like SDSS, DESI, and Euclid) against theoretical predictions is one of the primary ways cosmologists test the $\Lambda$CDM model and search for deviations that might point to modified gravity or alternative dark energy models.