Cold-dark-matter transfer function (CDM transfer function)

At fixed time after equality, the comoving cold-dark-matter density contrast per unit primordial curvature grows as $k^2$ for $k\ll k_{\mathrm{eq}}$ and only logarithmically for $k\gg k_{\mathrm{eq}}$. Small-scale modes entered during radiation domination and could grow only logarithmically until equality.

Assume all relevant modes begin outside the horizon at common conformal time ${\tau }_i$, grow as ${\tau }^2$ until radiation-era horizon entry, then freeze until matter-radiation equality, and again grow as ${\tau }^2$ in matter domination. With horizon entry ${\tau }_h=2\pi /k$, $k_{\mathrm{eq}}=2\pi /{\tau }_{\mathrm{eq}}$ and final time ${\tau }_0$, direct multiplication of the growth factors gives the displayed transfer with $G=({\tau }_0/{\tau }_i{)}^2$. A primordial density power spectrum $P_i=Ak$ becomes $P_0=A{G}^{2}k$ below equality wavenumber and $A{G}^2{k}_{\mathrm{eq}}^4{k}^{-3}$ above it. This approximation neglects the true logarithmic radiation-era matter growth, baryonic acoustic structure and late acceleration; a conventionally normalized transfer divides out $G$.

The matter equation in the barotropic matter-string model contains no wavenumber. Growing initial data therefore evolve as ${\delta }_c(k,\eta )=A_{k}D(\eta )$ with the same $D$ for every mode. For regular adiabatic modes initially outside the horizon during matter domination, $A_k$ is proportional to $k^2$ times the primordial curvature amplitude. The density-per-curvature transfer is consequently proportional to $k^{2}D(\eta )$ for modes entering after equality. Factoring out both $k^2$ and $D$ gives a constant shape transfer on those scales. Comparing with continued matter-era growth instead introduces the common suppression $D(\eta )/\eta$, not a new wavenumber dependence.

In the idealization that subhorizon cold matter freezes during radiation domination, early-entering modes have an entry density of order the primordial curvature, retain it until equality, and then share the same matter-era growth factor. Their density-per-curvature transfer becomes approximately independent of $k$. Matching this plateau to the matter-era $k^2$ transfer gives the displayed sharp-transition model, with its high-$k$ coefficient dependent on the matching convention. A scale-invariant curvature spectrum then gives density power proportional to $k$ below equality and $k^{-3}$ above it. Actual cold matter grows logarithmically in the radiation era, adding the familiar high-$k$ logarithms of the Mészáros effect.

For the regular growing mode in a purely matter-dominated flat background, $h=-2{\delta }_c$ and ${\delta }_c=C{\tau }^2$. The synchronous Einstein constraint $\mathcal{H}{h}^{\prime }+k^2{h}^{-}/3=3{\mathcal{H}}^2{\delta }_c$ with $\mathcal{H}=2/\tau$ gives $h^{-}=60C/k^2$. Hence $\zeta =h^{-}/6+{\delta }_c/3=10C/k^2+C{\tau }^2/3$, conserved to leading order on superhorizon scales. Identifying its early constant value gives the displayed density transfer; it includes a $k^2$ factor that is factored out of the conventionally normalized matter transfer function.

The dimensional matter power spectrum contains a factor $k^{n_s}$, rather than $k^{n_s+3}$. The cosmological transfer function tends to one below the matter-radiation equality scale and to $\ln (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$ above it. The smooth spectrum therefore scales as $k^{n_s}$ at small wavenumber and $k^{n_s-4}{\ln }^2(k/k_{\mathrm{eq}})$ at large wavenumber. The logarithmic factor records slow cold-matter growth during radiation domination. Multiplying by $k^3$ instead produces dimensionless matter power.

Modes with $k\gg k_{\mathrm{eq}}$ enter the horizon during radiation domination. Their matter perturbations grow only logarithmically until matter-radiation equality, rather than as $a$. Matching the radiation-era logarithmic growth onto the matter-era growing mode gives the large-$k$ cosmological transfer function scaling shown, up to constants and the matching scale inside the logarithm. For a primordial scalar spectral index $n_s$, the late-time linear cosmological density power spectrum consequently changes from $P(k)\propto k^{n_s}$ at small $k$ to $P(k)\propto k^{n_s-4}{\log }^2(k/k_{\mathrm{eq}})$ at large $k$. These scalings omit baryonic acoustic structure and nonlinear evolution.

Here the cold-dark-matter transfer function is normalized so that ${\delta }_m(k,a)=A(a)k^{2}T(k)\mathcal{R}(k)$, where $\mathcal{R}$ is the primordial comoving curvature perturbation and $A(a)$ is independent of $k$. During radiation domination, horizon entry occurs at $a_{\mathrm{ent}}\propto k^{-1}$. An adiabatic entry amplitude of order $\mathcal{R}$ then acquires a factor $\log (a_{\mathrm{eq}}/a_{\mathrm{ent}})\sim \log (k/k_{\mathrm{eq}})$ by matter-radiation equality. Subsequent matter domination supplies a common growth factor; division by the conventional $k^2$ prefactor therefore gives $T(k)\propto k^{-2}\log (k/k_{\mathrm{eq}})$.

A scale-invariant curvature seed gives wavenumber-independent dimensionless matter power at radiation-era horizon entry. Horizon entry at ${\tau }_h\sim 1/k$, followed by logarithmic growth to equality, gives a density transfer proportional to $\ln (k{\tau }_{\mathrm{eq}})$. Subsequent matter-era growth is proportional to ${\tau }^2$, so the dimensionless density power is proportional to $(\tau /{\tau }_{\mathrm{eq}}{)}^4{\ln }^2(k{\tau }_{\mathrm{eq}})$. This is a high-wavenumber asymptote, not an expression at the transition point.