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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.

For an isotropic matter power spectrum $P(k)=Ak$ on $0\le k\le k_{\max }$ and zero elsewhere, put $K=k_{\max }r$. Angular integration of the Fourier transform gives $\xi (r)=A(2{\pi }^{2}r{)}^{-1}{\int }_0^{k_{\max }}{k}^2\sin (kr)dk$, and two integrations by parts yield the displayed answer. Its apparent origin singularity is removable, with $\xi (0)=A{k}_{\max }^4/(8{\pi }^2)$. The sharp cutoff creates oscillatory, including negative, correlations without violating positivity of the spectrum.

The cosmological two-point correlation function measures excess pairing or correlated density fluctuations at a separation. For a statistically homogeneous and isotropic mean-zero density contrast, $\xi (r)=⟨\delta (\mathbf{x})\delta (\mathbf{x}+\mathbf{r})⟩$. For distinct points in a discrete galaxy population it instead specifies the factorial pair density, ${\bar{n}}^2[1+\xi (r)]$. The latter excludes self-pairs; an unsmoothed discrete-density covariance also contains a Dirac delta at zero separation. These definitions connect large-scale galaxy clustering with the matter power spectrum while keeping galaxy power spectrum shot noise separate.

The parameter ${\sigma }_8$ is the present linear rms density contrast averaged in a spherical top-hat of comoving radius $8h^{-1}\mathrm{Mpc}$, with $h=H_0/(100\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1})$. Thus ${\sigma }_8^2=(2{\pi }^2{)}^{-1}{\int }_0^{\infty }{k}^2{P}_m(k,0)|W(k{R}_8){|}^{2}dk$. It specifies an amplitude on a chosen scale rather than the complete matter power spectrum; converting cluster abundance into this parameter also requires a spectral shape and background cosmology.

The nonuniform spatial distribution of galaxies, quantified for example by a two-point correlation function or a matter power spectrum related through galaxy bias. Survey geometry and selection must be separated from physical clustering.

A galaxy survey measures positions and other properties of a selected galaxy sample. A spectroscopic survey uses cosmological redshift to estimate radial distance, with corrections for peculiar velocity. Its volume, selection, galaxy bias, and sampling determine which matter power spectrum modes can be inferred.

The Harrison-Peebles-Zeldovich spectrum is the scale-invariant primordial growing-mode spectrum: in a fixed conformal time slice before cosmological horizon crossing, the matter power spectrum is proportional to ${\tau }^{4}k$ in the usual growing-mode density convention. At horizon entry ${\tau }_H\sim k^{-1}$, its dimensionless mass variance $k^{3}P(k,{\tau }_H)$ is independent of $k$. This statement fixes a spectral shape, not its amplitude. Radiation domination subsequently suppresses small-scale density growth through the Mészáros effect, producing the asymptotic matter power spectrum $P\propto k^{-3}{\ln }^2(k/k_{\mathrm{eq}})$ after matter-radiation equality.

At a fixed horizon-exit e-fold count and with the later cosmology fixed, rescaling a canonical single-field inflationary potential by a positive constant leaves its potential slow-roll parameters and field solution unchanged. The slow-roll curvature power spectrum is proportional to $V/{ϵ}_V$, so the matter power spectrum scales by the same factor while its leading tilt and transfer-function shape remain unchanged. A changed specified reheating history can also shift the horizon-exit count assigned to a fixed physical wavenumber.

The ordinary Sachs-Wolfe effect combines intrinsic last-scattering temperature with the gravitational redshift of photons escaping the last-scattering gravitational potential. For adiabatic large-scale perturbations in matter domination, with the potential convention in the question, the combination is $\Theta =\varphi /(3c^2)$, up to the observer monopole. It is distinct from the Integrated Sachs-Wolfe effect caused by a time-varying potential along the later line of sight.

Use the Fourier transform convention in which $P_{\delta }(k)=Ak$ is the matter power spectrum and $⟨{\delta }_{\mathbf{k}}{\delta }_{{\mathbf{k}}^{\prime }}^{\ast }⟩=(2\pi {)}^3{\delta }_D(\mathbf{k}-{\mathbf{k}}^{\prime })P_{\delta }(k)$. The Poisson equation becomes $-k^2{\varphi }_{\mathbf{k}}=4\pi G{\bar{\rho }}_0{\delta }_{\mathbf{k}}$, givingWith the matter-dominated critical-density approximation ${\bar{\rho }}_0=3H_0^2/(8\pi G)$,independent of $k$. Thus the Harrison-Zeldovich spectrum gives scale-invariant potential fluctuations and the requested $H_0^4$ dependence. Keeping a general present matter fraction would insert ${\Omega }_m^2$; the answer here follows the stipulated critical matter-dominated normalization. In this Fourier transform convention, $A$ has dimensions of length to the fourth power.

Let $p(z)$ be the normalized galaxy probability density per unit cosmological redshift, $\int p(z)dz=1$, and let $b_g$ be its linear galaxy bias. In a spatially flat background with $c=1$, $d\chi =dz/H(z)$. Thus the projected galaxy density contrast isso its radial kernel is $q_g=Hbp$. The cosmological Poisson equation in Fourier transform variables givesHere ${\Omega }_m$ denotes today's matter density parameter, not its time-dependent value. For a growing linear cosmological density perturbation, write $\delta (\mathbf{k},a)=D(a){\delta }_{\mathrm{ini}}(\mathbf{k})$, where $D$ is the linear growth factor, and define the logarithmic linear growth rate $f=d\ln D/d\ln a$. At fixed comoving coordinate,ConsequentlyIn particular, $f=1$ during matter domination makes the linear potential constant and eliminates this Integrated Sachs-Wolfe effect.

To use the supplied Limber approximation without hiding its scale dependence in a radial kernel, take $U=\delta$, $V={\psi }^{\prime }$, $q_g=Hbp$, and $q_T=2$. The corresponding three-dimensional cross-power spectrum iswhere $P$ is the matter power spectrum. Substitution givesThe Limber approximation relation $k\chi =\ell +1/2$ cancels ${\chi }^{-2}{k}^{-2}$, yieldingBoth powers of $H$ are necessary: one converts the redshift selection to a distance kernel, and the other differentiates the potential in conformal time. The factor two in the Integrated Sachs-Wolfe effect cancels the factor $1/2$ from the cosmological Poisson equation. For positive galaxy bias and $f<1$, the predicted correlation is positive: an overdense region whose negative potential becomes shallower gives a positive integrated temperature shift. The result assumes linear, scale-independent growth and the Newtonian cosmological Poisson equation; the Limber approximation becomes less accurate at the very lowest angular multipoles.

The $z\simeq 0.3$ sample is expected to show the stronger late Integrated Sachs-Wolfe effect correlation, for comparable galaxy bias and selection quality. In the stated flat Lambda-CDM model, neglecting radiation in cosmology,At $z\simeq 4$ the universe is nearly matter-dominated universe, $D\propto a$ and $f\simeq 1$, so the cosmological Newtonian potential $\psi \propto D/a$ is nearly constant. A photon then receives little net Integrated Sachs-Wolfe effect shift. At $z\simeq 0.3$, dark energy has substantially suppressed the linear growth factor relative to $a$, so $f<1$ and the potentials decay appreciably.

For a quantitative indication, the approximation $f\simeq {\Omega }_m(z{)}^{0.55}$ gives $1-f\simeq 0.328$ at $z=0.3$ and $1-f\simeq 0.0101$ at $z=4$. Thus fractional potential evolution, $d\ln |\psi |/d\ln a=f-1$, is much larger for the nearby sample. The high-cosmological redshift matter power spectrum also has a smaller growth amplitude. The entire correlation still depends on $H$, galaxy bias, $p(z)$, and the matter power spectrum at the sampled wavenumber; redshift alone does not fix a universal numerical ratio or guarantee an ordering at every multipole. The robust expectation is

cold dark matter is nonrelativistic and effectively collisionless on the scales of interest. Its small velocity dispersion limits free streaming, allowing small-scale gravitational wells to survive. baryons coupled to photons undergo acoustic motion and damping before recombination; dark matter does not share that photon pressure, and provides potential wells into which baryons can later fall.

For a subhorizon pressureless perturbation, once a Newtonian treatment is applicable,During radiation domination the matter contribution to the background gravity is small; neglecting it gives $\delta \simeq C_1+C_2\log t$. This slow subhorizon growth is the Mészáros effect. During matter domination,in an Einstein-de Sitter universe. Once $\delta$ approaches unity, linear theory fails and collapse, virialization and merging become important.

Modes entering the horizon during radiation domination therefore grow less before equality than longer modes entering later. This scale dependence is encoded in a transfer function, with a schematic linear matter power spectrum $P(k,a)=A{k}^{n_s}{T}^2(k)D^2(a)$. cold dark matter supports hierarchical assembly; a sufficiently large warm- or hot-dark-matter free-streaming length suppresses small-scale structure. Superhorizon density contrasts change under a gauge transformation in cosmological perturbation theory, so the subhorizon growth laws should not be applied indiscriminately outside the horizon. The original cold-dark-matter formation model explicitly connects weak free streaming to the spectrum that seeds galaxies and clusters.^[ref]

Small primordial fluctuations seed later structure. For adiabatic cosmological perturbations on scales outside the horizon, the uniform-density curvature perturbation is conserved when the non-adiabatic pressure perturbation and the decaying mode can be neglected. A density contrast outside the horizon is gauge-dependent, so its apparent power-law growth should not be confused with directly measured local collapse.

For subhorizon, pressureless matter, the linearized continuity equation, Euler equation and Poisson equation combine intoHubble expansion damps peculiar motion; self-gravity amplifies overdensities. During radiation domination, radiation pressure drives acoustic oscillations and cold dark matter growth after horizon entry is only logarithmic. During matter domination the growing and decaying solutions are $\delta \propto a$ and $\delta \propto a^{-3/2}$. Dark-energy domination later suppresses growth relative to the matter-era law.

For a pressure-supported fluid the Fourier-mode equation also contains $+c_s^2{k}^2\delta /a^2$, giving a Jeans length separating acoustic support from gravitational growth. Before recombination, baryons are tightly coupled to photons and oscillate rather than freely following dark matter. Silk damping from photon diffusion damps small-scale photon-baryon fluctuations; after decoupling, baryons can fall into dark-matter potential wells. free streaming smooths species-dependent scales and distinguishes cold from warm or hot dark matter.

Schematically, the linear matter power spectrum can be written $P_{\delta }(k,a)=A{k}^{n_s}{T}^2(k)D^2(a)$ under the usual primordial-curvature normalization. The cosmological transfer function records horizon entry, matter-radiation equality, acoustic physics and damping; the linear growth factor records later amplification. density contrasts approaching unity require nonlinear collapse and halo formation beyond this approximation. cosmic microwave background anisotropies and galaxy clustering test the same initial perturbations through different transfer histories; galaxy bias, redshift-space distortions and observational selection enter the galaxy comparison.

With the specified Fourier transform convention, the inverse is $\delta (\mathbf{x})=\int d^{3}k{\delta }_{\mathbf{k}}{e}^{-i\mathbf{k}\cdot \mathbf{x}}/(2\pi {)}^3$. Inserting both inverse transforms into the cosmological two-point correlation function and applying the given Dirac delta covariance yieldsFor an isotropic matter power spectrum, align the polar axis with $\mathbf{r}$. The angular integral is $2\pi {\int }_{-1}^1{e}^{ikr\mu }d\mu =4\pi \sin (kr)/(kr)$. Hence the isotropic cosmological correlation-power-spectrum relation isThe angular kernel is the order-zero Spherical Bessel function $j_0(kr)$. At $r=0$, replace it by its limit one, giving the unsmoothed variance when the integral converges. The argument assumes an ordinary integrable spectrum, or an explicitly specified distributional Fourier transform interpretation otherwise.

Apply the isotropic cosmological correlation-power-spectrum relation and put $K=k_{\max }$, $q=Kr$. The band-limited linear density correlation isTwo integrations by parts give the primitive $-k^2\cos (kr)/r+2k\sin (kr)/r^2+2\cos (kr)/r^3$. Evaluating both endpoints yieldsThe apparent singularity at the origin is removable. Either the Taylor series or the original integral givesIn fact $\xi (r)=\xi (0)[1-q^2/9+q^4/240+O(q^6)]$. The sharp spectral cutoff produces oscillations and negative correlations at some nonzero separations, even though the matter power spectrum is everywhere nonnegative. A covariance function need not be pointwise nonnegative; its required positivity is that of the covariance quadratic form.

Let $\delta (\mathbf{x})$ be a mean-zero density contrast. For a statistically homogeneous and isotropic field, its two-point correlation function is $\xi (r)=⟨\delta (\mathbf{x})\delta (\mathbf{x}+\mathbf{r})⟩$, depending only on $r=|\mathbf{r}|$. For a galaxy point process the corresponding distinct-pair definition is $d{P}_{12}={\bar{n}}^2[1+\xi (r)]d{V}_{1}d{V}_2$, the excess pair probability relative to independent uniform positions. Self-pair shot noise is separated when using the density-field version.

Specify the Fourier convention byThe diagonal covariance follows from statistical homogeneity, and isotropy makes the matter power spectrum a function of $k$ alone. Substitute the inverse transforms into the correlation and integrate the delta function:Choose the polar axis parallel to $\mathbf{r}$. Angular integration gives $\int d{\Omega }_k{e}^{ikr\cos \theta }=4\pi \sin (kr)/(kr)$. HenceConversely, when the transform is defined, $P(k)=4\pi {\int }_0^{\infty }{r}^2\xi (r)\sin (kr)/(kr)dr$. The stated convention fixes every $2\pi$ factor.

The Press-Schechter formalism starts from a statistically homogeneous Gaussian initial density contrast with variance specified by its linear matter power spectrum. Average it on mass scale $M=(4\pi /3){\bar{\rho }}_{m0}{R}^3$, usually with a spherical top-hat. Let $\sigma (M,t)=b(t){\sigma }_0(M)$, normalizing $b$ consistently. Assume that a region becomes a collapsed halo when its linearly extrapolated smoothed density exceeds a mass-independent linear spherical-collapse threshold ${\delta }_{msc}\simeq 1.686$ in the Einstein-de Sitter approximation. This extrapolates a linear Gaussian statistic to a nonlinear collapse criterion, and ignores the detailed nonspherical dynamics.

The one-scale Gaussian upper tail only assigns half the mass to overdense regions when the variance grows without limit. The conventional factor of two is inserted to account for material in larger collapsed regions even if it falls below the threshold on a smaller scale, the cloud-in-cloud issue. A first-crossing excursion-set argument can justify that factor for an ideal sharp-k filter; it is not a theorem that an arbitrary top-hat one-scale probability alone supplies it. With this prescription, the mass fraction in objects above $M$ isConvert mass fraction to comoving number density by assigning mass $M$ to each halo, using $dn/dM=({\bar{\rho }}_{m0}/M)|dF/dM|$. This yields the Press-Schechter halo mass functionEquivalently one can use a present-extrapolated threshold ${\delta }_c(t)={\delta }_{msc}/b(t)$ and present variance; using both that threshold and an already grown variance would double-count growth. The model assumes approximate mass conservation, a smoothing-to-halo-mass assignment and a spherical universal barrier. It is an approximation to gravitational halo formation, not an exact partition of the real cosmic web.

In hierarchical formation, ${\sigma }_0(M)$ decreases with mass: smaller regions become nonlinear first and are subsequently incorporated into larger haloes. Define a characteristic mass by $b(t){\sigma }_0(M_{\ast })={\delta }_{msc}$. It increases as growth proceeds, moving the exponential cutoff towards larger masses. For a scale-free spectrum with well-defined top-hat variance, $-3<n<1$, ${\sigma }_0\propto M^{-(n+3)/6}$ and hence $M_{\ast }\propto b^{6/(n+3)}$. At a fixed mass, the derivative of $\ln (dn/dM)$ with respect to $\ln b$ is ${\nu }^2-1$: rare high-mass abundances grow, while the small-mass abundance can decline as objects merge into larger systems. Thus one must not claim that every fixed-mass bin increases forever. The growth of $M_{\ast }$ is the scale-free characteristic mass in the Press-Schechter formalism.

For the temperature test, virial equilibrium and gas hydrostatic equilibrium give $k_{B}T\sim \mu m_{p}GM/R$. Defining the halo radius by $M=(4\pi /3){\Delta }_v{\rho }_c(z)R^3$ yieldsA calibrated monotone mass-temperature relation converts the mass function to $dn/dT=(dn/dM)|dM/dT|$, with temperature scatter treated by convolution when necessary. The cluster temperature function normalization is strongly sensitive to the linear fluctuation amplitude: at fixed mass and spectral shape, $\partial \ln n/\partial \ln {\sigma }_8={\nu }^2-1$. Rare hot clusters occupy the exponential tail, so even a modest amplitude change can produce a large abundance change. Here sigma eight is the rms linear density contrast today in an $8h^{-1}\mathrm{Mpc}$ top-hat, not the full spectrum.

Compare observed cluster-temperature counts and their redshift evolution with these predictions to constrain that normalization jointly with matter density and growth. The mass-temperature calibration, selection volume, temperature scatter, nonthermal support and deviations from the ideal halo mass function limit a temperature-only inference. The abundance of rare hot clusters is a sensitive amplitude probe once their mass calibration and cosmology are specified.

Define the matter power spectrum by the Fourier transform conventionStatistical isotropy then givesFor the growing mode found above, write $P_{\delta }(k,\tau )={\tau }^4{P}_A(k)$. At cosmological horizon entry, ${\tau }_H\propto k^{-1}$, so the variance contribution per logarithmic interval is proportional to $k^3{k}^{-4}{P}_A(k)=P_A(k)/k$. Making it independent of $k$ requires $P_A(k)\propto k$. HenceThis is the Harrison-Zeldovich spectrum. Scale invariance applies at horizon entry, not to the unprocessed density variance on a common late-time slice. In the notation ${\delta }_{\mathbf{k}}\propto k^{n/2}{\tau }^2$, it means $n=1$, with random Fourier coefficients supplying the ensemble statistics.

Neglecting the rapidly oscillating radiation source on subhorizon scales, stress-energy conservation and the trace of the Einstein field equations giveEquivalently in cosmic time, ${\delta }_{m,tt}+2H{\delta }_{m,t}-4\pi G{\rho }_m{\delta }_m=0$. Radiation still contributes to the background expansion, even when its perturbation source is averaged away.

Put $\eta =a/a_{\mathrm{eq}}$ and let ${\rho }_{\mathrm{eq}}$ be the density of either component at matter-radiation equality. Then ${\rho }_m={\rho }_{\mathrm{eq}}{\eta }^{-3}$ and ${\rho }_r={\rho }_{\mathrm{eq}}{\eta }^{-4}$. With $C=(8\pi G/3){\rho }_{\mathrm{eq}}{a}_{\mathrm{eq}}^2$, the Friedmann equation givesApply the chain rule to the density equation:Thus the Mészáros equation isSubstituting $D_{+}=1+3\eta /2$ gives $(2+3\eta )(3/2)-3(1+3\eta /2)=0$, sois indeed a solution. It is approximately constant during radiation domination and grows as $a$ during matter domination. The second independent solution iswhich behaves as $\log (4/\eta )-3$ at small $\eta$. Thus smoothing radiation does not eliminate every logarithmic solution; matching through actual horizon entry fixes the combination excited.

Modes with $k\ll k_{\mathrm{eq}}$ enter after equality and retain the $P(k)\propto k$ shape. Modes with $k\gg k_{\mathrm{eq}}$ enter during radiation domination: their horizon-entry spectrum is $P_H\propto k^{-3}$ and they gain little growth until equality. The late matter power spectrum therefore turns over near $k_{\mathrm{eq}}$ and has a large-scale $k$ branch and a small-scale $k^{-3}$ branch, with the more accurate radiation-era matching givingEquivalently the cold-dark-matter transfer function falls approximately as $(k_{\mathrm{eq}}/k{)}^2$, up to the logarithm. The turnover is the consequence of the Mészáros effect; it is not a change of the primordial scale-invariance condition.

Choose the Fourier transform conventionStatistical homogeneity makes the Fourier covariance diagonal, and isotropy makes the matter power spectrum depend only on $k=|\mathbf{k}|$. The two-point correlation function $\xi (r)=⟨\delta (\mathbf{x})\delta (\mathbf{x}+\mathbf{r})⟩$ follows by inserting the transform and using this covariance:Integrating the polar angle relative to $\mathbf{r}$ gives $\int d{\Omega }_k{e}^{ikr\cos \theta }=4\pi \sin (kr)/(kr)$. HenceThe inverse isotropic transform is $P(k)=4\pi {\int }_0^{\infty }{r}^2\xi (r)\sin (kr)/(kr)dr$ whenever these transforms exist, with distributions or regularization otherwise.

For a spherical average of comoving radius $R$, the spherical top-hat window function isAt a fixed epoch let $P(k)=A{k}^n$. The scale-free smoothed density variance isThus the RMS fractional mass fluctuation scales as $R^{-(n+3)/2}$; the RMS mass fluctuation itself scales as $\bar{\rho }{R}^3{\sigma }_{\delta }\propto R^{(3-n)/2}$ at fixed epoch.

The comoving Poisson equation gives ${\Phi }_{\mathbf{k}}=-4\pi G{a}^2\bar{\rho }{\delta }_{\mathbf{k}}/k^2$. For the growing irrotational mode the linearized cosmological continuity equation gives ${\mathbf{v}}_{\mathbf{k}}=-iaHf\mathbf{k}{\delta }_{\mathbf{k}}/k^2$, where $f=d\log D_{+}/d\log a$. The averaged-potential and bulk-velocity variances are thereforeFor a single Cartesian component of the isotropic bulk velocity, divide its variance by three. Rescaling $kR$ givesThese are dimensional scaling laws, with an important convergence qualification. Since $W(y)\to 1$ at small $y$ and has large-$y$ envelope $O(y^{-2})$, the density average converges for $-3<n<1$, the absolute potential average for $1<n<5$, and the bulk velocity for $-1<n<3$. No one unbroken power law makes all three absolute averages finite on the entire range $0<k<\infty$. In particular, long-wavelength modes can dominate the potential. Physical infrared/ultraviolet cutoffs, potential differences, or fluctuation amplitudes in a band around $k\sim R^{-1}$ resolve the relevant divergences; with cutoffs the simple powers need not describe the full integrated variance. The scale-free potential and bulk-flow variance laws should be understood with these conditions, not as a claim that divergent RMS integrals are finite.