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Consider a logarithmic band of wavenumbers around $k\sim R^{-1}$, where $R$ is a comoving scale. The dimensionless cosmological power spectrum of the density contrast is ${\Delta }_{\delta }^2(k)=k^3{P}_{\delta }(k)/(2{\pi }^2)$. For $P_{\delta }=Ak$ this scales as $k^4$, so the rms density amplitude in such a band scales as $R^{-2}$.

The cosmological Poisson equation gives $-k^2{\Phi }_{\mathbf{k}}=4\pi G{a}^2{\bar{\rho }}_m{\delta }_{\mathbf{k}}$, with $\Phi$ a dimensionful peculiar gravitational potential. Its spectrum and its potential fluctuations per logarithmic wavenumber are thereforeindependent of $k$. Equivalently, ${\Phi }_R\sim G{\bar{\rho }}_m(aR{)}^2{\delta }_R$ is independent of $R$ because ${\delta }_R\propto R^{-2}$. This is the scale-local meaning of the Harrison-Zeldovich spectrum. An uncut spectrum has a divergent total potential variance; equal amplitude per logarithmic band does not imply a finite unsmoothed variance over all scales. For wavelengths beyond the Hubble radius, the statement is formulated using the corresponding conserved primordial curvature mode and its adiabatic matter-era potential, rather than treating the subhorizon Poisson approximation as universally valid.

On angular scales larger than roughly a degree, the Sachs-Wolfe effect relates the last-scattering potential to the observed Cosmic microwave background anisotropy: $\Delta T/T\simeq \Phi /(3c^2)$ for adiabatic modes in matter domination. Thus the observed large-angle amplitude constrains $A$. Scale-invariant potential bands produce a Sachs-Wolfe plateau, $\ell (\ell +1)C_{\ell }$ approximately constant, rather than constant $C_{\ell }$. Its amplitude and any tilt test the normalization and spectral shape. Cosmic variance, Integrated Sachs-Wolfe effect and assumptions about adiabatic initial conditions limit this inference; the density spectrum's late-time transfer must also be included when comparing it with galaxy clustering.

In the incoming Bunch-Davies vacuum, only $⟨a_{\mathbf{k}}{a}_{\mathbf{q}}^{†}⟩={\delta }_{\mathbf{kq}}$ contributes to the coincident vacuum expectation value. ThusReplacing the box sum by $\mathcal{V}\int d^{3}k/(2\pi {)}^3$ givesThe subhorizon vacuum contribution is the flat-space zero-point term; at fixed comoving $k$ it redshifts as $a^{-2}$. The other term gives frozen scalar power on superhorizon scales. Specifically, the dimensionless cosmological power spectrum per logarithmic interval isIts late-time value is independent of $k$, producing a scale-invariant inflationary power spectrum. Each fixed Fourier mode freezes separately as $\tau \to 0$.

The coincident integral is formal: the vacuum part diverges at large $k$, and an unlimited scale-invariant part diverges at small $k$. The conclusion concerns the power in each finite logarithmic band. A finite inflationary history supplies a largest generated wavelength, as described by the infrared qualification of a scale-invariant covariance; taking the late-time limit inside an unregulated integral would not be justified.

Put $\theta =i\mathbf{k}\cdot \mathbf{v}$. The synchronous perfect-fluid density equation and Cosmological Euler equation in synchronous gauge giveDifferentiate the first equation and use the second and the scalar trace metric equation. Before making a nonrelativistic approximation, the result isFor $w=w_m=c_s^2\ll 1$, discard relative pressure corrections to the background gravitational and damping terms, while retaining $c_s^2{k}^2$ because a large wavenumber can compensate a small sound speed. Matter domination and the flat Friedmann equation give $\frac{3}{2}{\mathcal{H}}^2=4\pi G{\bar{\rho }}_m{a}^2$. Thus the leading nonrelativistic density equation in synchronous gauge isThe approximation sign records the small finite-$w$ terms just displayed; the printed Newtonian-form equation is not an exact relativistic equation at nonzero $w$.

The comoving Jeans wavenumber balances pressure and gravity: $c_s^2{k}_J^2=4\pi G{\bar{\rho }}_m{a}^2$. Defining physical wavelength as $2\pi a/k$, the physical Jeans length isFor wavelengths longer than this scale, gravity can overcome pressure and density modes grow; shorter modes are pressure-supported acoustic oscillations, with expansion modifying both growth and damping. Before cosmological recombination, the tightly coupled photon-baryon fluid has substantial photon pressure, $c_{s,\gamma b}^2=1/[3(1+R_b)]$, where $R_b=3{\bar{\rho }}_b/(4{\bar{\rho }}_{\gamma })$. The baryonic Jeans scale is correspondingly large, so baryons oscillate instead of freely collapsing on small scales. Cold dark matter has a much smaller pressure scale and can supply gravitational potential wells. After photon decoupling, baryons lose photon pressure support and their sound speed falls to its thermal-gas value; the baryon Jeans length across recombination drops, allowing baryons to fall into those wells. The earlier photon-coupled fluid is not itself a constant-small-$w$ fluid; its role is a physical comparison of pressure support, not an extension of the preceding approximation through recombination.

Define a normalized smoothing window on physical radius $R$, with comoving radius $r=R/a$. If $⟨{\delta }_{\mathbf{k}}{\delta }_{{\mathbf{k}}^{\prime }}^{\ast }⟩=(2\pi {)}^3{\delta }_D(\mathbf{k}-{\mathbf{k}}^{\prime })P(k,\tau )$, the smoothed matter density variance isThus ${\sigma }_R$ is the root-mean-square amplitude. For a Gaussian window, $W(u)=e^{-u^2/2}$; this choice makes the scale-free integral ultraviolet convergent.

On scales where pressure is negligible, a flat matter-dominated universe has $a\propto {\tau }^2$, $\mathcal{H}=2/\tau$, and $4\pi G{\bar{\rho }}_m{a}^2=6/{\tau }^2$. The density equation has powers satisfying $s(s-1)+2s-6=0$, soSelect the growing mode and exclude residual synchronous gauge modes. Starting with $P(k,{\tau }_{\mathrm{eq}})=Ak$ gives $P(k,\tau )=Ak(\tau /{\tau }_{\mathrm{eq}}{)}^4$. The dimensionless cosmological power spectrum is thenAt cosmological horizon crossing, $k\sim aH=\mathcal{H}$, hence $k\tau \sim 2$, andFor the smoothed variance, change variables to $u=kr$:The Gaussian integral equals $1/2$. At a physical Hubble-radius window, $R=H^{-1}$ and $r=\tau /2$, giving the Gaussian horizon-crossing variance for a Harrison-Zeldovich spectrumThe window-dependent numerical constant is secondary; cancellation of the scale dependence is the Harrison-Zeldovich spectrum property. This demonstration uses the growing pressureless matter-era idealization and therefore applies to modes crossing in that era. Modes which crossed before equality require radiation-era transfer and cannot be evolved through that period with the matter equation. A pure $P\propto k$ spectrum with a real-space top-hat window also has a logarithmically divergent ultraviolet variance unless a physical high-$k$ cutoff or transfer function is supplied; the Gaussian choice makes the stated idealized variance well defined.

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.

Let $\delta =(\rho -\bar{\rho })/\bar{\rho }$ and define the cosmological density power spectrum by $⟨{\delta }_{\mathbf{k}}{\delta }_{{\mathbf{k}}^{\prime }}^{\ast }⟩=(2\pi {)}^3{\delta }_{\mathrm{D}}(\mathbf{k}-{\mathbf{k}}^{\prime })P(k)$. The contribution per logarithmic wave-number interval is the dimensionless cosmological power spectrum ${\Delta }^2(k)=k^{3}P(k)/(2{\pi }^2)$. The linear cosmological mass variance on mass scale $M$ iswith $R$ a comoving smoothing radius and $W$ the top-hat filter in Fourier space. On halo scales the observed spectrum is consistent with greater variance at smaller masses. Growth therefore brings smaller objects to the collapse threshold earlier, statistically; larger structures assemble by accretion and dark-matter halo mergers. This statistical growth from smaller bound systems to larger ones is hierarchical galaxy formation. It is an ordering of dark-matter assembly, not a rule that every small visible galaxy must precede every large visible galaxy.

For linear modes the linear growth factor gives $P(k,z)=D^2(z)P(k,0)$. During matter domination $D\propto a$, so $P\propto a^2$ and the characteristic nonlinear mass increases. Once modes become nonlinear, coupling between scales and halo formation change the shape, so the same multiplication cannot be used for the entire late-time spectrum. At low redshift accelerated expansion suppresses linear growth. Galaxy clustering traces the matter spectrum with galaxy bias; it should not be equated directly with an unbiased matter measurement.

The broad linear shape was set by the cosmological transfer function before and around matter-radiation equality, with baryonic acoustic structure also imprinted before recombination. A nearly scale-invariant primordial curvature spectrum has ${\mathcal{P}}_{\mathcal{R}}(k)\propto k^{n_s-1}$, with $n_s$ close to one. After converting curvature perturbations to matter-density perturbations,Thus nearly scale-invariant primordial curvature does not mean a constant density $P(k)$. Modes with $k\ll k_{\mathrm{eq}}$ enter the horizon after equality and have $T\simeq 1$. Modes with $k\gg k_{\mathrm{eq}}$ enter during radiation domination, when radiation controls the expansion and cold-matter perturbations grow only slowly. The cold-dark-matter transfer function behaves approximately as $T\propto \ln (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$ at large $k$. Hence the density spectrum turns over near $k_{\mathrm{eq}}=a_{\mathrm{eq}}{H}_{\mathrm{eq}}/c$:The turnover records the equality horizon, while the late nonlinear excess records gravitational clustering. On galactic scales the effective slope is greater than $-3$, giving the growing small-scale variance needed for hierarchical galaxy formation.

Cold dark matter has negligible primordial thermal velocities and a very short collisionless free streaming length on galactic scales. Warm dark matter has appreciable residual velocities while structure is being seeded; particles stream across small fluctuations and reduce their contrast. Its cosmological transfer function is consequently cut off below a characteristic length, suppressing low-mass halos and delaying their formation. Above that cutoff its assembly can still be hierarchical. The important distinction is the free-streaming scale, rather than the present temperature or an arbitrary particle-mass label.

Linear evolution assumes $|\delta |\ll 1$. When $\delta$ becomes of order unity, overdense regions depart strongly from the Hubble flow and can turn around and collapse. Collisionless dark matter develops multistream motion after trajectories cross; gravitational mixing redistributes energy and produces a bound dark matter halo. The formal infinite-density collapse of an ideal spherical pressureless solution is not the physical endpoint. A roughly virialized halo has $2K+W\simeq 0$ and a characteristic virial velocity $V_{\mathrm{vir}}^2=G{M}_h/r_{\mathrm{vir}}$. Its gas virial temperature is conventionallyIt measures the thermal energy associated with the gravitational potential; the numerical factor depends on the velocity-dispersion convention. It does not imply that the collisionless dark matter has a thermodynamic gas temperature.

The unheaded request about baryon conversion efficiency of a halo is also answered here. Define $f_{\ast }=M_{\mathrm{stars}}/(f_b{M}_h)$ using the cosmic baryon fraction $f_b$. In small halos, shallow potentials let stellar feedback drive outflows or repeatedly heat star-forming gas; supernova energy per stellar mass is roughly fixed while binding energy per gas mass scales as $V_{\mathrm{vir}}^2$. Photoheating during reionization also prevents very small halos from retaining or accreting cool gas. Molecular/atomic cooling thresholds further reduce star formation in the smallest systems. These effects make $f_{\ast }$ fall toward low mass.

Near $M_h\sim 10^{12}{M}_{\odot }$, gas can cool efficiently and the potential is deep enough to retain more of it, while a long-lived hot atmosphere and maintenance heating are less effective than in larger systems. At high mass, higher virial temperature and lower cooling efficiency let a substantial hot atmosphere persist. Active-galactic-nucleus feedback can prevent that atmosphere from supplying cold gas and can expel some gas; the cooling-time bottleneck alone is not an adequate explanation for the low stellar fractions of massive groups and clusters. The peak reflects a competition between gas supply/cooling and feedback, rather than complete conversion of all baryons at a sharply universal mass. Its exact location and height depend on epoch, metallicity, gas history and the stellar population included.

The quantum harmonic oscillator has $\hat{H}=({\dot{\hat{q}}}^2+{\omega }^2{\hat{q}}^2)/2$, with $\omega >0$. Using $\hat{a}|0⟩=0$ and the canonical commutation relation givesFor $q=r{e}^{is}$, the Wronskian normalization isConsequentlyEquality requires $\dot{r}=0$ and $r^2=1/(2\omega )$. The minimum-energy normalized oscillator mode is thereforeThe constant phase $\alpha$ is irrelevant. This mode also solves the oscillator equation of motion; an arbitrary squeezed mode would have larger vacuum energy.

For inflation, introduce the Mukhanov-Sasaki variable $v=z\mathcal{R}$. Since $z$ depends only on conformal time,Integrating the cross term by parts gives the canonical bulk actionup to the boundary term $-\frac{1}{2}\int d^{3}x[(z^{\prime }/z)v^2{]}_{\mathrm{boundary}}$. The resulting Euler-Lagrange field equation and its Fourier transform areAt leading order in the slow-roll approximation, retain a small positive, nearly constant $ϵ$ in $z=a\sqrt{2ϵ}$, while approximating $a=-1/(H\tau )$ and $H$ as constant. Then $z^{\prime \prime }/z\simeq 2/{\tau }^2$, soStrict exact de Sitter spacetime would have $ϵ=0$ and would not supply the nonzero curvature kinetic coefficient assumed here; the calculation is the leading quasi-de Sitter limit, not a substitution of zero into $z$.

For $-k\tau \gg 1$, the expansion correction is negligible and each canonical mode is a quantum harmonic oscillator of conformal frequency $k$. The Bunch-Davies vacuum selects its positive-frequency, minimum-energy mode $e^{-ik\tau }/\sqrt{2k}$ in that early subhorizon regime. It does not minimize an instantaneous Hamiltonian after the effective squared frequency has become negative outside the horizon.

A basis of exact solutions to the leading mode equation is $e^{-ik\tau }(1-i/(k\tau ))$ and its complex conjugate. Write the normalized Bogoliubov transformation combination asThe Bunch-Davies vacuum boundary condition sets ${\beta }_k=0$ and ${\alpha }_k=1$ up to phase. HenceSubstitution verifies the equation, and $v_k{v}_k^{\ast \prime }-v_k^{\prime }{v}_k^{\ast }=i$ verifies the Wronskian normalization.

Dividing by $z$ gives the comoving curvature perturbation varianceThis is the dimensional slow-roll curvature power spectrum in the printed normalization, with the reduced Planck mass set to one. The corresponding dimensionless cosmological power spectrum is $k^3{P}_{\mathcal{R}}/(2{\pi }^2)=H^2/(8{\pi }^2ϵ)$, which is independent of $k$ at this order. Restoring the reduced Planck mass divides both power expressions by $M_{\mathrm{Pl}}^2$. Slowly varying background quantities are evaluated near each mode's horizon exit; their variation generates the small departure from exact scale invariance.

A scale-invariant primordial potential has $P_{\Phi }(k)\propto k^{-3}$. On subhorizon linear scales, the cosmological Poisson equation and the cold-dark-matter transfer function give ${\delta }_c(k,a)\propto D(a)k^{2}T(k){\Phi }_{\mathrm{prim}}(k)$, where $D(a)\propto a$ during matter domination. ConsequentlyLarge scales enter only after equality and have $T(k)\simeq 1$. Small scales enter during radiation domination; their approximately logarithmic growth up to equality gives $T(k)\propto (k_{\mathrm{eq}}/k{)}^2\ln (k/k_{\mathrm{eq}})$. ThereforeIf the slow logarithm is suppressed in a rough sketch, the slopes are $+1$ and $-3$, with a turnover near $k_{\mathrm{eq}}$. The logarithmic correction is real and should not be mistaken for a different primordial spectral index. In the matter-only late-time approximation with $D(0)=1$, the cosmological redshift $z=1$ means $a=1/2$ and $D(1{)}^2=1/4$; it changes the amplitude, not these asymptotic shapes. Neither the initial normalization nor cosmological parameters needed for an absolute power are supplied.

This is a schematic smooth interpolation with the derived asymptotes, not a precision transfer-function fit. The linear ideal-fluid model excludes baryonic acoustic structure and small-scale nonlinear evolution.

There is also a gauge and horizon qualification. For a mode still outside the Hubble radius at $z=1$, the printed Newtonian-gauge density has ${\delta }_c\simeq -2\Phi$, and its formal dimensional spectrum is instead proportional to $k^{-3}$. The conventional large-scale $k$ branch describes modes that are large relative to the equality scale but already subhorizon at the observation time. Alternatively, the comoving matter density, ${\Delta }_c={\delta }_c+3\mathcal{H}{\theta }_c/k^2$ with ${\theta }_c=\nabla \cdot {\mathbf{v}}_c$, removes that constant gauge term and obeys ${\Delta }_c=-2k^2\Phi /(3{\mathcal{H}}^2)$ in the growing matter solution. The usual matter-spectrum sketch can be continued to small $k$ in this comoving-density convention. The dimensional spectrum requested here is $P$, rather than the dimensionless cosmological power spectrum $k^{3}P/(2{\pi }^2)$, whose slopes would differ by three.