Transfer Functions and the Matter Power Spectrum

The late matter spectrum is not a primordial power law carried forward by one number. Each Fourier mode enters the comoving Hubble scale at a different epoch, encounters radiation pressure, baryon drag, diffusion, and free streaming, and only then shares the late pressureless growth. A transfer function isolates that scale-dependent processing from the primordial tilt and the subsequent scale-independent growth.

Helpful background. The Einstein Equation supplies Einstein Equations and their constraints; Relativistic Matter and Compact Objects supplies Relativistic Matter and Compact Objects fluid and kinetic closures; Relativistic Cosmology supplies the Relativistic Cosmology background; Computational Relativity supplies convergence and transfer-function validation; and Probability and Statistical Inference supplies random-field and covariance concepts.

Power-spectrum conventions

Use $a_0=1$ and $e^{i\mathbf k\cdot\mathbf x}$. In the transfer relation below, $\delta_m$ denotes the matter contrast on comoving-orthogonal slices. For pressureless matter its gauge-invariant expression is $\delta_m^{\rm com}=\delta_m^{\rm N}+3\mathcal H\theta_m^{\rm N}/k^2$, where the superscript N denotes Newtonian gauge and $\theta_m=-k^2\delta q_m/\rho_m$. On subhorizon scales this agrees with the usual Newtonian density contrast to the stated approximation. Write

$$\delta_m(\mathbf x,a) =\int\frac{d^3k}{(2\pi)^3} e^{i\mathbf k\cdot\mathbf x}\delta_m(\mathbf k,a).$$

Statistical homogeneity and isotropy define

$$\left\langle \delta_m(\mathbf k,a)\delta_m^*(\mathbf k',a) \right\rangle =(2\pi)^3\delta^{(3)}(\mathbf k-\mathbf k')P_m(k,a),$$

so $P_m$ has dimensions of volume. The dimensionless variance per logarithmic wavenumber is

$$\Delta_m^2(k,a)=\frac{k^3}{2\pi^2}P_m(k,a).$$

For the primordial comoving curvature perturbation, use

$$\mathcal P_{\mathcal R}(k) =A_s\left(\frac{k}{k_*}\right)^{n_s-1}, \qquad P_{\mathcal R}(k)=\frac{2\pi^2}{k^3}\mathcal P_{\mathcal R}(k).$$

The pivot $k_*$ belongs to the amplitude-and-tilt convention; changing it changes the quoted $A_s$ but not the underlying spectrum.

Separating primordial shape, transfer, and growth

For the growing adiabatic mode in general relativity, define

$$\delta_m(\mathbf k,a) =\mathcal M(k,a)\mathcal R(\mathbf k),$$

with

$$\mathcal M(k,a) =\frac{2k^2}{5\Omega_{m0}H_0^2} T_m(k)D_+(a).$$

Here $D_+(a)/a\to1$ in the early matter era and $T_m(k)\to1$ as $k/k_{\rm eq}\to0$. These two normalization conditions are essential: normalizing $D_+(1)=1$ without compensating its early-time amplitude would silently change $T_m$. The matter spectrum is then

$$P_m(k,a)=\mathcal M^2(k,a)P_{\mathcal R}(k),$$

The sign and the $2/5$ follow from the relativistic constraint, not from a choice to make the spectrum positive. Once radiation is negligible and the late matter–$\Lambda$ approximation applies,

$$\Phi(k,a)=-\frac35\frac{D_+(a)}{a}T_m(k)\mathcal R(k), \qquad -k^2\Phi=4\pi Ga^2\rho_m\delta_m^{\rm com},$$

with $4\pi Ga^2\rho_m=3\Omega_{m0}H_0^2/(2a)$. Their combination gives $\mathcal M$ above. This also explains why the explicit $k^2$ is appropriate for the comoving density; a Newtonian-gauge density has a nonzero constant piece in the super-Hubble growing mode and cannot be substituted there.

Equivalently,

$$\Delta_m^2(k,a) =A_s\left(\frac{k}{k_*}\right)^{n_s-1} \left[ \frac{2k^2T_m(k)D_+(a)} {5\Omega_{m0}H_0^2} \right]^2.$$

This factorization assumes one regular adiabatic growing mode. Correlated isocurvature modes require a transfer vector and primordial covariance matrix; multiple clustering species generally have distinct $T_i(k,a)$.

Equality sets the turnover

Define horizon entry by $k=aH$ and

$$k_{\rm eq}\equiv a_{\rm eq}H_{\rm eq}.$$

A different $2\pi$ wavelength convention rescales the named crossing wavenumber but not the physical argument. Modes with $k\ll k_{\rm eq}$ remain outside the comoving Hubble scale until matter domination. Their gravitational potential does not undergo the radiation-era suppression, so $T_m\to1$.

Modes with $k\gg k_{\rm eq}$ enter during radiation domination. Radiation dominates the potential while pressure makes its subhorizon perturbations oscillate; cold-matter perturbations grow only logarithmically until equality. For cold dark matter the asymptotic result is

$$T_m(k)\propto \frac{\ln(k/k_{\rm eq})}{(k/k_{\rm eq})^2}, \qquad k\gg k_{\rm eq},$$

up to a convention-dependent constant inside the logarithm. The elementary horizon-entry argument recovers the decisive $k^{-2}$ suppression; solving the radiation–matter system supplies the logarithm and smooth turnover. Tong gives the horizon-entry derivation in Tong 2019, §3.1.6, printed pp. 146–148, eqs. (3.37)–(3.41) (Official PDF). The transfer, power, equality, and damping dictionary is summarized in Olive and Peacock 2026, §§22.4.4–22.4.5, printed pp. 19–22, eqs. (22.68)–(22.79) (Official PDF).

For a primordial power law, the asymptotic matter spectrum therefore obeys

$$P_m(k)\propto \begin{cases} k^{n_s}, & k\ll k_{\rm eq},\\[2mm] k^{n_s-4}\ln^2(k/k_{\rm eq}), & k\gg k_{\rm eq}. \end{cases}$$

The primordial tilt changes both slopes by the same $n_s-1$ relative to a scale-invariant spectrum. Equality creates the four-power change; the two effects must not be conflated.

Baryons, diffusion, and free streaming

The total matter transfer is more structured than a pure-CDM turnover. Before drag release, photon pressure drives baryon acoustic oscillations; after release, baryons fall into the dark-matter wells, leaving a weak oscillatory feature with the sound-horizon scale. Photon diffusion damps the pre-recombination photon–baryon perturbations. Massive neutrinos erase or suppress clustering below their time-dependent free-streaming scale and make late growth itself scale dependent. Warm or interacting dark matter adds further characteristic scales.

Consequently, $T_m(k)$ is a model output, not a universal fitting curve. Every tabulated or numerical transfer function must state its species, densities, thermal history, gauge, normalization, and redshift. Detailed baryonic feedback and galaxy formation are excluded here; theory-specific gravity tests are developed in their dedicated volumes. Nonlinear evolution and the tracer relation are developed on later pages of this chapter.

From a spectrum to a scale-dependent variance

A power spectrum becomes a real-space fluctuation amplitude only after a window is chosen. For an isotropic filter with Fourier transform $W(kR)$,

$$\sigma_R^2(a) =\int_0^\infty \Delta_m^2(k,a) \lvert W(kR)\rvert^2\,d\ln k.$$

The filter shape and the relation between $R$ and an enclosed mass are part of the definition. A real-space spherical top hat, Gaussian smoothing, and a sharp Fourier cutoff weight the turnover, acoustic feature, and nonlinear tail differently. Thus a quoted $\sigma_R$ cannot be reconstructed from an amplitude alone without the spectrum, units, redshift, and window convention.

This integral also locates two later boundaries. If the selected wavenumbers have become nonlinear, the linear $P_m$ used here is no longer sufficient. If the field is inferred from galaxies rather than total matter, bias, redshift-space mapping, and the survey window intervene before any physical smoothing. The halo page will use a declared matter window and mass mapping; the observational pages will instead forward-model the tracer and survey selection.

Exercises

Starting from $P_m=\mathcal M^2P_{\mathcal R}$, recover the two asymptotic slopes above. Then state how a change $n_s\to n_s+\Delta n$ differs from moving $k_{\rm eq}$.

Solution

Because $P_{\mathcal R}\propto k^{n_s-4}$ and $\mathcal M\propto k^2T_m$ at fixed time,

$$P_m\propto k^{n_s}T_m^2.$$

For $k\ll k_{\rm eq}$, $T_m\to1$, hence $P_m\propto k^{n_s}$. For $k\gg k_{\rm eq}$,

$$T_m^2\propto \frac{\ln^2(k/k_{\rm eq})}{(k/k_{\rm eq})^4},$$

so $P_m\propto k^{n_s-4}\ln^2(k/k_{\rm eq})$ after absorbing the fixed powers of $k_{\rm eq}$ into the amplitude. Changing $n_s$ tilts the spectrum smoothly about the chosen pivot. Changing $k_{\rm eq}$ moves the turnover and the logarithmic argument; it does not add one constant slope at every scale.

References

• Olive, K. A. and J. A. Peacock, “Big-Bang Cosmology,” in F. Takahashi et al. (Particle Data Group), Review of Particle Physics, International Journal of Modern Physics A 41, 2630011 (2026), revised August 2025, Official PDF.
• Tong, D., Cosmology, University of Cambridge Part II Mathematical Tripos lecture notes, Michaelmas 2019, §§3.1.6–3.2.4, printed pp. 146–159, Official PDF.