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Normalize $a(t_0)=1$. In the matter era, horizon crossing $c{t}_H=2\pi a(t_H)/k$ with $a=(t/t_0{)}^{2/3}$ gives $t_H/t_0=(k_0/k{)}^3$. In the radiation era, match $a=(t_{\mathrm{eq}}/t_0{)}^{2/3}(t/t_{\mathrm{eq}}{)}^{1/2}$ to obtainThe boundary mode therefore has $k_{\mathrm{eq}}=k_0(1+z_{\mathrm{eq}}{)}^{1/2}$. For $k<k_{\mathrm{eq}}$, the growing mode multiplies the primordial amplitude by $(t_0/t_H{)}^{2/3}=(k/k_0{)}^2$. For $k>k_{\mathrm{eq}}$, growth is approximately frozen until equality, then multiplied by $(t_0/t_{\mathrm{eq}}{)}^{2/3}=1+z_{\mathrm{eq}}=(k_{\mathrm{eq}}/k_0{)}^2$. Squaring these factors gives the matter power spectrumThe large-$k$ estimate omits the logarithmic radiation-era transfer correction, consistently with the stipulated approximation.

The comoving distance to the Cosmic microwave background last-scattering surface is of order the present horizon, ${\chi }_{\ast }\sim 1.3\times 10^4\mathrm{Mpc}$. At recombination the particle horizon is much smaller, so replacing ${\chi }_{\ast }$ by today's horizon is adequate for the requested estimate. A transverse Fourier mode projects approximately to angular multipole $\ell \sim k{\chi }_{\ast }$. If an angular feature's width is estimated by $\theta \sim 1/\ell$, the resolution ${\theta }_{\min }\sim 10^{-3}$ givesUsing the common half-wavelength convention $\theta \sim \pi /\ell$ instead gives ${\ell }_{\max }\sim 3000$ and $k_{\max }\sim 0.2$--$0.3\mathrm{Mp}{\mathrm{c}}^{-1}$. These express the same order-of-magnitude resolution estimate with different angular-scale conventions. On the graph, a useful indicative band is therefore from about $10^{-4}\mathrm{Mp}{\mathrm{c}}^{-1}$ to order $10^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, potentially a few times $10^{-1}$; it is not a sharp instrumental boundary. The lowest useful multipoles are of order a few, with substantial cosmic variance.

The cosmic microwave background constrains the matter power spectrum indirectly through the primordial spectrum and radiation/matter cosmological transfer functions. Beam resolution, Cosmic microwave background diffusion damping, foreground emission, and projection of three-dimensional fluctuations into angular anisotropy limit the recoverable small-scale information. Resolving a given angle alone does not guarantee a precise matter-spectrum measurement there.

Inflation produces the primordial comoving curvature perturbation while modes exit the Hubble radius. Its adiabatic super-Hubble conservation allows that initial amplitude to be carried through the end of inflation and subsequent changes of the background equation of state without solving each mode's full evolution at every intervening time. After Hubble re-entry, cosmological transfer functions relate it to the cosmic microwave background temperature and polarization fluctuations and to the matter power spectrum. The conserved curvature perturbation connects inflationary initial conditions to later cosmological observables. This bridge uses the adiabatic conditions stated in part (v); entropy perturbations can supply an additional evolving source.

The broad turnover near $k\sim 10^{-2}{\mathrm{Mpc}}^{-1}$ marks the matter-radiation equality scale. Modes entering the Hubble radius after matter-radiation equality grow efficiently during matter domination. Modes entering earlier have their cold-matter growth delayed by radiation domination: their subhorizon growth is logarithmic rather than proportional to $a$. This is the Mészáros effect. The smaller undulations on the declining side are baryon acoustic oscillations inherited from the photon-baryon fluid, with an oscillatory wavenumber spacing set approximately by $\pi$ divided by the sound horizon. They are damped at sufficiently short scales by photon diffusion and the transfer of the baryon perturbation into the total matter field. The plotted spectrum is a linear prediction; nonlinear evolution changes the actual sufficiently small-scale spectrum at redshift zero.

For an adiabatic growing mode, the comoving curvature perturbation transfers to the matter overdensity asup to an irrelevant overall sign convention for $\mathcal{R}$. Normalize the cosmological transfer function by $T\to 1$ at small $k$ and the linear growth factor by $D\to a$ in matter domination. The relation distinguishes a dimensional matter power spectrum from a dimensionless cosmological power spectrum: the dimensional curvature spectrum is proportional to $k^{-3}{\mathcal{P}}_{\mathcal{R}}(k)$. ThereforeFor locally constant scalar spectral index $n_s$, the asymptotic cold-dark-matter power spectrum isbecause the cold-dark-matter transfer function tends to one at small $k$ and scales as $\ln (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$ at large $k$. The asymptotes refer to the smooth envelope, not the individual baryonic wiggles. The large-$k$ local logarithmic slope is $n_s-4+2/\ln (k/k_{\mathrm{eq}})$. For the quadratic potential at $N\simeq 60$, derived below, $n_s\simeq 0.967$, so the limiting powers are approximately $k^{0.967}$ and $k^{-3.033}{\ln }^{2}k$. Multiplying by $k^3$ to form dimensionless matter power would change both exponents by three; that is not the plotted quantity.

Redshift-space distortions arise because observed galaxy redshifts include peculiar velocity as well as cosmological redshift. Distances inferred solely from expansion therefore distort clustering along the line of sight. Coherent large-scale motion and small-scale velocity dispersion produce different effects on the inferred matter power spectrum.

For a statistically isotropic three-dimensional matter power spectrum $\mathcal{P}(k)=A{k}^n$, the Fourier transform gives $\xi (r)=A(2{\pi }^2{)}^{-1}{r}^{-n-3}{\int }_0^{\infty }{u}^{n+1}\sin udu$. The improper integral converges at zero precisely when $n>-3$ and at infinity when $n<-1$; it is absolutely convergent only when $n<-2$. Within this range its positive coefficient is $\Gamma (n+2)\sin [\pi (n+2)/2]$, interpreted continuously as $\pi /2$ at $n=-2$. Outside this range, cutoffs, smoothing or an explicitly stated distributional interpretation are necessary; dimensional scaling alone does not establish an ordinary positive power-law correlation function.

For a matter power spectrum $P(k)=A{k}^n$ at a fixed epoch, the Poisson equation gives potential Fourier amplitudes proportional to ${\delta }_k/k^2$, and the growing-mode peculiar velocity is proportional to ${\delta }_k\mathbf{k}/k^2$. With a spherical top-hat window function, their variances are proportional to $\int k^{n-2}W(kR{)}^{2}dk$ and $\int k^{n}W(kR{)}^{2}dk$, respectively. Rescaling $kR$ gives the displayed powers when the integrals converge: $1<n<5$ for the averaged potential and $-1<n<3$ for the bulk velocity. Absolute averaged potential has infrared sensitivity; band-limited fluctuations or potential differences can have different convergence requirements. The density variance converges only for $-3<n<1$, so these three absolute averages cannot all converge for one unbroken power law at every scale.