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A horizon-length patch on the matter-radiation equality surface has physical size $a_{\mathrm{eq}}{\tau }_{\mathrm{eq}}$ in a flat universe with conformal time measured from the Big Bang. Dividing by its angular diameter distance $a_{\mathrm{eq}}({\tau }_0-{\tau }_{\mathrm{eq}})$ gives its small angular size. For the radiation-matter scale factor in conformal time, this is $(\sqrt{2}-1)\sqrt{{\Omega }_{r,0}}/[1-\sqrt{2{\Omega }_{r,0}}]$. A patch diameter is twice the angle associated with one horizon radius. The particle horizon is an integral over past evolution and must not be silently replaced by the instantaneous Hubble radius.

When physical wavelengths greatly exceed the Hubble radius, spatial derivatives are small compared with temporal expansion scales. A cosmological gradient expansion then approximates neighbouring regions as locally homogeneous universes, while retaining finite perturbation amplitudes. This does not alone discard frozen tensor cosmological perturbations.

In matter domination with the constant growing potential, the full Newtonian-gauge energy constraint gives the stated density contrast. The second term dominates inside the Hubble radius and grows as $a$, because ${\mathcal{H}}^2\propto a^{-1}$. Outside the Hubble radius, the constant $-2\Phi$ term dominates and is gauge dependent. The comoving density combination ${\Delta }_c={\delta }_c+3\mathcal{H}{\theta }_c/k^2$ cancels it for ${\theta }_c=2k^2\Phi /(3\mathcal{H})$, leaving the cosmological Poisson equation relation. Matter-spectrum power laws must specify whether they refer to this comoving density or to subhorizon Newtonian-gauge density.

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.

In matter domination, the Einstein field equations with constant $\Phi$ give the Newtonian-gauge matter density from a constant gravitational potentialWell inside the Hubble radius, the first term is negligible. Since ${\mathcal{H}}^2\propto a^{-1}$ and $\Phi$ is constant,Equivalently, the matter-era growing and decaying density modes follow from ${\delta }_c^{\prime \prime }+\mathcal{H}{\delta }_c^{\prime }-(3/2){\mathcal{H}}^2{\delta }_c=0$: they are ${\tau }^2\propto a$ and ${\tau }^{-3}\propto a^{-3/2}$.

The lower panel of the preceding figure gives the three requested density contrast sketches in Newtonian gauge in cosmology. Their early superhorizon density is nearly constant, rather than proportional to $a^2$ in this gauge. A large-$k$ mode enters first and grows only approximately logarithmically during radiation domination. Once the rapid radiation forcing has subsided, ${\delta }_c^{\prime \prime }+\mathcal{H}{\delta }_c^{\prime }\simeq 0$ with $\mathcal{H}\simeq 1/\tau$, so ${\delta }_c\simeq C+D\ln \tau \simeq C+D\ln a$. This is the Mészáros effect; forcing around entry determines the coefficients. After equality its growing part becomes proportional to $a$.

A mode near $k_{\mathrm{eq}}$ begins substantial growth around equality. A small-$k$ mode keeps its superhorizon constant term until entering during matter domination, then follows the same $a$ growth law. Their entry scale factors satisfy $a_{\mathrm{ent}}\propto k^{-1}$ during radiation domination and $a_{\mathrm{ent}}\propto k^{-2}$ during matter domination. At a common late time, the scaled amplitudes are therefore of orderup to common dimensional constants and order-one matching terms. Thus the three late growth curves have the same logarithmic slope one as functions of $a$, but different amplitudes. These amplitude statements require the modes to have entered the Hubble radius; the full Newtonian-gauge formula above remains available for modes that have not.

Take regular adiabatic initial conditions and a common scale-invariant primordial amplitude, $k^{3/2}{\Phi }_{\mathrm{prim}}(k)=\text{constant}$. Let $k_{\mathrm{eq}}={\mathcal{H}}_{\mathrm{eq}}$. The three histories are distinguished by their time of cosmological horizon crossing:

The $9/10$ factor follows from conserving the large-scale adiabatic curvature: the constant potential is proportional to $3(1+w)/(5+3w)$, whose matter-to-radiation ratio is $(3/5)/(2/3)=9/10$. Define the cold-dark-matter transfer function by ${\Phi }_{\mathrm{late}}=(9/10)T(k){\Phi }_{\mathrm{prim}}$. Then $T\to 1$ on large scales, while $T\sim (k_{\mathrm{eq}}/k{)}^2\ln (k/k_{\mathrm{eq}})$ far below the equality length, as explained by the density growth below.

The upper panel sketches $k^{3/2}\Phi$ for the three modes. The lower panel provides the corresponding density contrast histories used in the next part. A negative common primordial potential was chosen so the growing density is positive; this arbitrary phase has no effect on a cosmological density power spectrum.

The curves integrate the ideal coupled radiation-fluid and pressureless-matter equations, rather than patching a pure-radiation solution onto a matter solution. They neglect baryons, free-streaming anisotropic stress, dark energy and nonlinear evolution, consistently with the stated mixture. Dots mark $k=\mathcal{H}$; equality is the dashed vertical line.

For a single barotropic perfect fluid in general relativity, the perturbations obey $\delta P=w\delta \rho =w\bar{\rho }\delta$. This adiabatic closure is needed: constant background $P/\rho$ alone would not eliminate an independent entropy perturbation. The absence of scalar anisotropic stress allows the common potential used in Newtonian gauge in cosmology.

Substituting the density constraint into the pressure equation gives the gravitational potential evolution of a barotropic fluidSince $a\propto {\tau }^{2/(1+3w)}$, the conformal Hubble parameter is $\mathcal{H}=2/[(1+3w)\tau ]$. The bracket cancels identically, leavingFor a Fourier transform mode, ${\nabla }^2$ becomes $-k^2$.

During radiation domination, $w=1/3$ and ${\Phi }_k^{\prime \prime }+4{\Phi }_k^{\prime }/\tau +(k^2/3){\Phi }_k=0$. Set $x=k\tau /\sqrt{3}$ and write $\Phi =u/x$. The resulting equation is $u_{xx}+2u_x/x+(1-2/x^2)u=0$, so the two Spherical Bessel functions in the hint giveAt $x\ll 1$, the two solutions approach a constant and a mode proportional to ${\tau }^{-3}$. The regular adiabatic mode, normalized to its primordial potential, isAfter entry into the sound horizon, $x\gg 1$, the potential oscillates at cosmological sound speed $1/\sqrt{3}$ with envelope $x^{-2}\propto a^{-2}$. The Hubble radius and sound horizon differ by the sound-speed factor; outside the Hubble radius the regular potential is constant, while well inside it radiation supports acoustic oscillations.

During matter domination, $w=0$ gives ${\Phi }_k^{\prime \prime }+6{\Phi }_k^{\prime }/\tau =0$ at every wavenumber. HenceThe growing density mode has a constant potential both outside and inside the Hubble radius; the other potential mode decays. Pressureless matter has zero cosmological sound speed, so horizon entry does not produce the radiation acoustic decay. These formulas cover both independent solutions, while the subsequent sketches select the regular adiabatic growing mode.

The slow-roll approximation neglects $\ddot{\varphi }$ relative to $3H\dot{\varphi }$ and kinetic energy relative to the potential. It is self-consistent on the attractor when the potential slow-roll parameter $ϵ\ll 1$ and the second potential slow-roll parameter $|\eta |\ll 1$. ThusHere, as in the inflaton equations, $M_{\mathrm{pl}}$ denotes the reduced Planck mass, not the unreduced mass used in the thermal calculation. The slow-roll curvature power spectrum becomesso

To differentiate with respect to horizon-exit scale, write $\mathcal{N}=\ln a$, increasing with physical time. Along the slow-roll trajectory,The difference between differentiating with respect to $\ln k$ and $\mathcal{N}$ contributes only at second slow-roll order to the tilt. Since $\ln {\Delta }_{\mathcal{R}}^2=3\ln V-2\ln |V^{\prime }|+\mathrm{constant}$,Therefore the scalar spectral index in potential slow-roll parameters isAll background quantities in these expressions are evaluated when the particular mode exits the Hubble radius.

A light inflaton behaves approximately as a scalar field in de Sitter space while a wavelength is well inside the Hubble radius. Its Bunch-Davies vacuum has quantum fluctuations. Cosmic inflation stretches each Fourier mode until $k=aH$, after which its physical wavelength exceeds the Hubble radius. The nearly constant growing field mode has a typical fluctuation per logarithmic wavenumber interval $\delta \varphi \sim H/(2\pi )$.

A field fluctuation changes the local position on the rolling background trajectory. Neighboring regions therefore reach the same field value, and the end of inflation, at slightly different times. A clock displacement of magnitude $|\delta \varphi /\dot{\varphi }|$ becomes a difference in local expansion of order $H|\delta \varphi /\dot{\varphi }|$. Equivalently, in a conventional sign choice the comoving curvature perturbation is related to the field fluctuation on spatially flat slices byChanging the sign convention for spatial curvature changes the sign of $\mathcal{R}$, but not its spectrum. This is the inflaton clock-shift origin of curvature perturbations. For a single-field slow-roll attractor there is no independent entropy mode. On a super-Hubble scale, gradient terms are negligible and the superhorizon conservation of single-field comoving curvature preserves the growing adiabatic mode, so fluctuations generated near exit persist as primordial curvature perturbations. This conservation requires the attractor and adiabatic assumptions; a freely chosen non-attractor background would not have the same conclusion.

For the growing adiabatic mode in an epoch of constant $w$, the superhorizon potential is constant. The Friedmann equation gives $4\pi G{a}^2(\bar{\rho }+\bar{P})=\frac{3}{2}(1+w){\mathcal{H}}^2$, so the given relation impliesThus ${\varphi }_{\mathrm{rad}}=-2\mathcal{R}/3$ and ${\varphi }_{\mathrm{mat}}=-3\mathcal{R}/5$. Photon continuity, neglecting its gradient term on superhorizon scales, gives $({\Theta }_0-\varphi {)}^{\prime }=0$. The radiation-era initial condition fixes ${\Theta }_0-\varphi =\mathcal{R}$. Through the matter-radiation transition it follows that ${\Theta }_{0\ast }=\mathcal{R}+{\varphi }_{\mathrm{mat}}=2\mathcal{R}/5$. Since $\psi =\varphi$, the matched large-scale emission perturbation isKeeping the radiation value of ${\Theta }_0$ unchanged across the transition would give the wrong coefficient. This is the Sachs-Wolfe radiation-to-matter matching for the paper's sign convention for comoving curvature perturbation.

Neglect the Integrated Sachs-Wolfe effect, the superhorizon-suppressed Doppler term, observer monopole, and any observer kinematic dipole. Put ${\mathbf{x}}_0=0$ by statistical homogeneity, and defineThe plane-wave expansion on ${\mathbf{x}}_{\ast }=-{\chi }_{\ast }\mathbf{e}$ gives angular multipolesStatistical isotropy and orthonormal spherical harmonics then imply $⟨a_{\ell m}{a}_{{\ell }^{\prime }{m}^{\prime }}^{\ast }⟩={\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}{C}_{\ell }$. Radial integration supplies $(4\pi {)}^2(2{\pi }^2)/(2\pi {)}^3=4\pi$, giving the large-angle angular power spectrumThis assumes adiabatic growing modes, matter domination at emission, negligible anisotropic stress, and wavenumbers that are outside the Hubble radius then. It treats recombination as instantaneous, omits late potential evolution and rescattering, and omits lensing at this linear order. The formal primordial contribution is defined for $\ell >0$; the observed dipole has a large additional kinematic contribution, so the clean large-angle primordial comparison is usually $\ell \ge 2$. A scale-invariant spectrum also yields the Sachs-Wolfe plateau $\ell (\ell +1)C_{\ell }=2\pi {\mathcal{P}}_{\mathcal{R}}/25$ for $\ell \ge 1$.

The printed $\mathcal{H}$ is the conformal Hubble parameter, $\mathcal{H}=aH=\dot{a}$; ${\mathcal{H}}^{-1}$ is the comoving Hubble radius, rather than the physical radius $H^{-1}$. Since the derivative requested is with respect to cosmic time,Thus the cosmic-time derivative of the comoving Hubble radius hasFor an expanding universe, the same factor $1+3\omega$ determines whether a small departure from flatness grows and whether the comoving Hubble radius grows. Ordinary matter with $\omega >-1/3$ therefore has both signs associated with the conventional Flatness problem and Horizon problem. Scales whose physical wavelengths now exceed the Hubble radius were even farther outside it, in relative terms, earlier in such an era, rather than being brought inside for causal equilibration. Accelerated expansion with $\omega <-1/3$ reverses both signs.

There is a qualification to the word “always”: an actual Horizon problem depends on the complete past history, not just this local sign. In a flat, constant-$\omega$ hot Big Bang model with $\omega >-1/3$, $a\propto (t-t_B{)}^p$ with $p=2/[3(1+\omega )]<1$, so the comoving particle horizon ${\int }_{t_B}^{t}d{t}^{\prime }/a(t^{\prime })$ is finite. The conventional causal problem then accompanies the flatness instability. An earlier accelerated era or a different past boundary can change the causal conclusion even when the present-era comoving Hubble radius is growing. Thus the requested correspondence is valid in that usual expanding hot Big Bang setting, not a universal logical equivalence between global horizons and local stability. At $\omega =-1/3$ both local effects are marginal.

A quantum fluctuation is a nonzero variance of an observable in a quantum state, even when its mean vanishes. A vacuum field mode has such zero-point variance. In an inflationary background, the expanding geometry transfers short-wavelength vacuum field fluctuations into primordial perturbations as their wavelengths cross the Hubble radius.

For an adiabatic single-field attractor, gradient terms are negligible well outside the Hubble radius and the growing comoving-curvature mode is constant. The separate-universe interpretation identifies it with the difference of local expansion histories. The conservation statement can fail for a non-attractor phase or an independent entropy mode.