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In an Einstein-de Sitter universe, $H(z)=H_0(1+z{)}^{3/2}$ and the radial comoving distance is $2H_0^{-1}[1-(1+z{)}^{-1/2}]$. A physical Hubble radius $H(z{)}^{-1}$ has comoving size $(1+z)/H(z)$. Dividing by the radial distance gives the displayed small angle, in radians. A Hubble diameter has twice this angle; the matter-era particle-horizon radius also equals twice the Hubble radius. These distinctions matter when comparing approximate CMB angular scales.

During a decelerating expansion the comoving Hubble radius grows. A fixed comoving wavenumber then passes from a superhorizon scale to a shorter scale when $k\simeq aH$. In radiation domination, $aH=1/\tau$, so this occurs at $k\tau \simeq 1$. Here horizon entry refers to Hubble radius crossing; a true causal horizon instead depends on a conformal-time integral.

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.

Start with an inflationary patch of physical radius $H_i^{-1}$ in units $c=1$. After $N$ e-folds and subsequent expansion, its present radius is $H_i^{-1}{e}^N{a}_0/a_{\mathrm{reh}}$. Requiring it to cover the present Hubble radius gives the displayed inequality. With instantaneous conversion to radiation, fixed relativistic degrees of freedom and a fixed ratio $H_i/H_{\mathrm{end}}$, energy conservation gives $H_i\propto T_{\mathrm{reh}}^2$, whereas adiabatic subsequent expansion gives $a_0/a_{\mathrm{reh}}\propto T_{\mathrm{reh}}$. Thus the required $N$ increases approximately as $\log T_{\mathrm{reh}}$. Holding $H_i$ fixed while varying the end energy independently is a different comparison and reverses that simple temperature-only trend. The inflationary energy history and post-inflationary entropy history must be specified.^[ref]

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.

Within the equal-expansion construction of a, subtract the flat-background Friedmann equation to obtainThe background density scales as ${\rho }_b\propto a^{-4}$ during radiation domination and ${\rho }_b\propto a^{-3}$ during matter domination. ThereforeThese are the regular adiabatic growing-mode scalings on scales outside the Hubble radius, with a specified time slicing. In a consistent comoving description the numerical relation between curvature and density contrast differs from the naive subtraction, but the powers agree. The result does not imply that every gauge's density contrast, every decaying mode, or subhorizon dark matter during radiation domination has these powers.

For $r_s\gg d_H$, a regular adiabatic perturbation follows the large-scale density contrast growth in b; this does not require causal communication across the perturbation. Since $d_H=c/H$, the Friedmann equation gives $d_H\propto a^2$ during radiation domination and $d_H\propto a^{3/2}$ during matter domination. A fixed comoving perturbation has physical size $r_s=aR$, so its ratio to $d_H$ decreases as $a^{-1}$ or $a^{-1/2}$ until cosmological horizon entry.

The numerical “exactly at equality” assertion is inconsistent with the supplied parameters and $d_H=c/H$. Normalize $a_0=1$ and use the supplied ${\Omega }_{\gamma }$ as the whole radiation component. ThenBoth terms must be retained at matter-radiation equality. The present comoving size corresponding to its Hubble radius isIts physical size then is $0.0355\mathrm{Mpc}$. A mode with $R=79\mathrm{Mpc}$ has physical size only $0.0263\mathrm{Mpc}$ at equality, already inside the Hubble radius. More explicitly, putting $q=R{H}_0/c$ in $aR=c/H(a)$ givesThis is a direct numerical counterexample using the stated data. Neglecting one component exactly at equality instead gives $150.5\mathrm{Mpc}$, which also does not produce $79\mathrm{Mpc}$. Late-time dark energy is negligible at these early times.

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.

A horizon separates regions according to a specified causal, dynamical or geometrical criterion. The criterion must be stated: an event horizon, an apparent horizon, a Killing horizon and a Cauchy horizon answer different questions. Some horizons describe a black hole, others an observer's acceleration or the universe's expansion. A vanishing coordinate metric component alone does not establish any of these definitions.

In an asymptotically flat spacetime, choose the relevant future null infinity ${\mathcal{I}}^{+}$, the endpoint of escaping outgoing light rays. The black-hole region and its future event horizon areHere $J^{-}$ is the causal past. Events in $\mathcal{B}$ cannot send a causal signal to the chosen infinity. This is a global definition: identifying the horizon requires the full future development, so a local measurement need not reveal whether an event lies on it. Smooth portions of an event horizon are null hypersurfaces generated by null geodesics. Generators may join at past caustics or crease sets, so the horizon need not be smooth everywhere. A past event horizon is defined by the time-reversed escape criterion; in the maximally extended Schwarzschild solution it bounds a white-hole region.

An observer-based definition replaces infinity by a future-inextendible worldline $\gamma$. Its future observer event horizon is the boundary of events that can signal to $\gamma$, conventionally $\partial I^{-}(\gamma )$. A past observer horizon similarly bounds events reachable from its past worldline. Such horizons depend on the observer and need not imply a black hole.

A Killing horizon is a null hypersurface to which a nonzero Killing vector field $K^a$ is normal. On it, $K^a$ is tangent to its null generators and satisfies $K^b{\nabla }_b{K}^a=\kappa K^a$. The surface gravity $\kappa$ measures the failure of this Killing parameter to be an affine parameter; rescaling $K$ rescales $\kappa$. A degenerate Killing horizon has $\kappa =0$. Stationary black-hole horizons often are Killing horizons, whereas a general evolving event horizon need not have any Killing symmetry. The condition $K^a{K}_a=0$ alone is insufficient: the null vector must also be normal to a null hypersurface.

The Schwarzschild spacetime illustrates several of these notions. In units $G=c=1$, let $f=1-2M/r$. Its metric is $d{s}^2=fd{t}^2-f^{-1}d{r}^2-r^{2}d{\Omega }^2$, and $r=2M$ is the future event horizon of the black-hole region in the appropriate extension. The apparent coordinate singularity disappears in ingoing Eddington-Finkelstein coordinates, $v=t+r_{\ast }$, $d{r}_{\ast }/dr=f^{-1}$:At $r=2M$ this metric is regular and the normal $dr$ is null. Future outgoing radial rays satisfy $dr/dv=f/2$; inside the horizon their areal radius decreases even on the outgoing family. The stationary Killing vector field ${\partial }_v$ generates the horizon with $\kappa =f^{\prime }(2M)/2=1/(4M)$ when normalized at infinity. The Kretschmann scalar is $48M^2/r^6$, finite at the horizon and divergent at $r=0$. Thus the event horizon is not the curvature singularity. The photon sphere at $r=3M$, which supports circular null orbits, is also not the horizon.

Quasi-local horizon notions use null expansions. For a closed spacelike two-surface, choose future outgoing and ingoing null normals $\ell ,n$, with $\ell \cdot n=1$ in signature $(+---)$. Each expansion is the logarithmic rate of its positive area element along the corresponding null normal. A future trapped surface has ${\theta }_{\ell }<0$ and ${\theta }_n<0$: both future light fronts initially shrink. A marginally outer trapped surface has ${\theta }_{\ell }=0$; for the future black-hole type also require ${\theta }_n<0$. On a chosen spacelike slice, the outer boundary of the future trapped region is an apparent horizon, ordinarily an outermost marginally outer trapped surface when regularity permits this characterization. It depends on the slice, and a slicing need not display an apparent horizon even when the global spacetime contains an event horizon.

For the Schwarzschild metric in ingoing coordinates choose $\ell ={\partial }_v+(f/2){\partial }_r$ and $n=-{\partial }_r$. The spherical null expansion in ingoing coordinates givesThe round spheres inside $2M$ are future trapped, while $r=2M$ is marginal. On standard horizon-penetrating slices its sections give the apparent horizon, coinciding with the stationary event and Killing horizon. This coincidence in a symmetric stationary example does not identify the notions in a general spacetime.

A trapping horizon is a hypersurface foliated by marginal surfaces. Its future outer type obeys ${\theta }_{\ell }=0$, ${\theta }_n<0$ and ${\mathcal{L}}_n{\theta }_{\ell }<0$. Unlike an event horizon, this can be diagnosed from local expansion data and their variation. The null energy condition explains its causal character. At a marginal surface, the Null Raychaudhuri equation, with the present curvature sign and vanishing null twist, givesin units $c=1$. A possible nonaffinity term vanishes here because it is proportional to the zero expansion. Write a horizon evolution vector as $V=\ell -Cn$. Preserving ${\theta }_{\ell }=0$ gives $C=({\mathcal{L}}_{\ell }{\theta }_{\ell })/({\mathcal{L}}_n{\theta }_{\ell })\ge 0$. Hence $V^2=-2C\le 0$: the future outer horizon is spacelike or null. Its area expansion is ${\theta }_V=-C{\theta }_n\ge 0$. These conclusions use the outer condition and the energy condition, not merely marginality.

For the global event horizon, the future-complete horizon focusing proof of the area theorem uses the same Null Raychaudhuri equation. Under the null energy condition, a negative generator expansion forces a future caustic in finite affine parameter. Future completeness and the usual regularity or predictability assumptions exclude such a caustic on the future event-horizon boundary. Its generator expansion is therefore nonnegative and its classical area cannot decrease. These assumptions matter: this is not an unconditional result for every null surface or every matter model.

Rotating and charged solutions provide further distinctions. A nonextremal Kerr black hole has $r_{\pm }=M\pm \sqrt{M^2-a^2}$, with $a=J/M$. Its outer Killing horizon is generated by ${\partial }_t+{\Omega }_H{\partial }_{\varphi }$, where ${\Omega }_H=a/(r_{+}^2+a^2)$, rather than by time translation alone. The ergosphere begins at the stationary-limit surface $g_{tt}=0$, whose radius is $M+\sqrt{M^2-a^2{\cos }^2\theta }$. Away from the axis this lies outside the event horizon: it forbids static observers but does not forbid escape to infinity. In Reissner-Nordstrom spacetime, the nonextremal radii are $r_{\pm }=M\pm \sqrt{M^2-Q^2}$. The outer surface is an event horizon, whereas the inner surface is a Cauchy horizon in the exact analytic extension.

A Cauchy horizon concerns predictability from initial data. The future domain of dependence $D^{+}(\Sigma )$ of a partial Cauchy surface consists of events for which every past-inextendible causal curve intersects $\Sigma$. Its future Cauchy horizon isBeyond it, causal influences can arrive without meeting the specified initial surface; data on that surface no longer determine the solution there. This is distinct from the inability to send signals to infinity. The inner horizon of the exact charged black hole supplies an example, and the definition necessarily specifies which initial surface is being used.

Flat spacetime itself has acceleration horizons. In the right Rindler wedge, set $t=\xi \sinh (a\tau )$ and $x=\xi \cosh (a\tau )$, with $\xi >0$. The Minkowski spacetime metric becomes $d{s}^2=a^2{\xi }^{2}d{\tau }^2-d{\xi }^2-d{y}^2-d{z}^2$. The null boundaries $t=x$ and $t=-x$ are Rindler horizons of the boost Killing flow. A uniformly accelerated observer cannot receive signals from one side of its future horizon, even though the curvature vanishes and inertial observers encounter no such boundary. These are observer and Killing horizons, not a black-hole event horizon defined using the full Minkowski null infinity.

In a spatially flat Friedmann-Lemaître-Robertson-Walker metric with noncompact spatial sections, radial light rays have $d\chi /dt=\pm 1/a(t)$. The particle horizon measures how far signals could have travelled since the initial cosmic time; the cosmological event horizon measures how far signals emitted now could ever travel toward a comoving observer. Their comoving and proper radii areThe respective horizon is present only if its integral is finite. The event-horizon integral uses the actual future endpoint, often $t_f=\infty$, so it is global and observer-dependent. For $a(t)=t^q$ on $0<t<\infty$, a finite particle horizon exists for $0<q<1$, while a finite event horizon exists for $q>1$. Neither integral is finite for $q=1$. In the expanding flat patch of de Sitter spacetime, $a=e^{Ht}$ and the proper event-horizon radius is $H^{-1}$. The static form $d{s}^2=(1-H^2{r}^2)d{t}^2-(1-H^2{r}^2{)}^{-1}d{r}^2-r^{2}d{\Omega }^2$ identifies it as a cosmological Killing horizon; its surface-gravity magnitude is $H$ with the central static time normalization.

Finally, the Hubble radius must not automatically be called an event horizon. For a flat expanding cosmology, the two future radial null expansions of a sphere of proper radius $R$ are ${\theta }_{\pm }=2(H\pm 1/R)$. At $R=H^{-1}$ the ingoing expansion vanishes, while outside it both expansions are positive: this is the cosmological marginal, past-trapped type rather than a future black-hole trapped surface. A dust universe with $a\propto t^{2/3}$ has such a Hubble sphere but no future event horizon. Global causal boundaries, slice-dependent trapped-surface boundaries, symmetry horizons and initial-data boundaries must therefore be distinguished, even when several coincide in a special example.

For a homogeneous isotropic metric, radial light propagation covers comoving distance $\int cdt/a(t)$. The physical particle horizon is $d_p(t)=a(t){\int }_{t_i}^{t}cd{t}^{\prime }/a(t^{\prime })$, when this is finite. A future event horizon exists when $d_e(t)=a(t){\int }_t^{t_f}cd{t}^{\prime }/a(t^{\prime })$ is finite. The Hubble radius is the instantaneous scale $c/H$, with $H=\dot{a}/a$; it is not generally the same as either causal horizon.

For an Einstein-de Sitter universe, $H=2/(3t)$, so $c/H=3ct/2$, while $d_p=3ct$ from a beginning at zero. The future integral diverges, so there is no event horizon. For an eternal-past de Sitter spacetime, $H$ is constant and the particle-horizon integral from $-\infty$ diverges. Its event horizon and Hubble radius both equal $c/H$. Thus a Hubble radius is not automatically a particle horizon, even in these simple isotropic models.

The Jeans length is the physical wavelength at which the restoring pressure term and self-gravity balance. In the elementary static-fluid approximation,Wavelengths longer than ${\lambda }_J$ are unstable to self-gravitating collapse, whereas shorter ones have restoring sound oscillations. Expansion and relativistic effects modify this elementary criterion. Ideal cold dark matter has $c_s=0$, hence no positive pressure-supported Jeans length in these equations. For the specified negative-pressure barotropic string fluid, $c_s^2=-1/3$; there is no real restoring-pressure Jeans threshold.

For $\eta \gg 1$, the matter growth equation has leading termsThe gravitational term is one inverse power smaller than the derivative coefficients. Dropping it at leading order gives $({\eta }^2{D}_{\eta }{)}_{\eta }=0$, hence a constant mode and a decaying ${\eta }^{-1}$ mode. A more controlled check is possible: direct substitution gives $D_{-}=\sqrt{1+\eta }/{\eta }^{3/2}$ as an exact solution. Reduction of order then gives the growing mode, normalized by $D_{+}/\eta \to 1$ at early times,The integral grows as $\eta +O(\log \eta )$, while the prefactor is ${\eta }^{-1}[1+O({\eta }^{-1})]$. Thuswith corrections of order $\log \eta /\eta$. The perturbation freezes because the matter fraction tends to zero while the coasting fluid controls the expansion.

To describe the requested cold-dark-matter transfer function, its normalization must be specified. The early matter-era Hamiltonian constraint, in the negative-expansion convention for $K$, gives $-4k^2\Psi /a^2-4H\kappa =16\pi G\delta \rho$. For the growing comoving dust mode, the continuity equation gives $\kappa ={\dot{\delta }}_C=H{\delta }_C$ in proper time. Since $16\pi G\bar{\rho }=6H^2$, this yields ${\delta }_C=-2k^2\Psi /[5(aH{)}^2]$. On early superhorizon scales $\Psi$ is the conserved curvature amplitude, up to the chosen overall sign, and $(aH{)}^2\propto a^{-1}$. This establishes the $k^{2}a$ initial growing behavior rather than assuming identical horizon-entry amplitudes at all epochs. For regular adiabatic modes outside the horizon during early matter domination, $A_k=\mathcal{A}{k}^2{\mathcal{R}}_{\ast }(k)$, with a $k$-independent constant $\mathcal{A}$ and primordial curvature amplitude ${\mathcal{R}}_{\ast }$. The matter equation above has no $k$, so at fixed late timeConsequently the density-per-curvature transfer is proportional to $k^2{D}_{+}$ on scales entering after equality. The conventional shape transfer, which factors out $k^2$ and the common growth factor, is a plateau, $T(k)\simeq 1$ on these scales. Comparing with a matter-only universe that continues growing as $\eta$ instead gives a common amplitude suppression $D_{+}/\eta \simeq 5/(2\eta )$, still independent of $k$. A decline caused solely by later horizon entry would contradict the supplied scale-independent matter equation: the regular synchronous growing solution already evolves outside the Hubble radius. This is the scale-independent matter suppression by a coasting fluid.

There is also a limiting entry scale. Since $a^2{\rho }_S$ is constant, define $k_{\ast }^2=(8\pi G/3)a^2{\rho }_S$. ThenA mode enters the Hubble radius only if $k>k_{\ast }$; $k\gg k_{\ast }$ enters during matter domination, whereas $k\downarrow k_{\ast }$ enters arbitrarily late. Modes $k\le k_{\ast }$ never enter in this ideal coasting future. The requested post-equality entrants occupy $k_{\ast }<k<k_{\mathrm{eq}}$. The following original diagram distinguishes the two transfer conventions and marks this restricted interval; it does not attempt to model modes entering during radiation domination.

The adiabatic initial conditions for separately conserved components have equal ${\delta }_N/(1+w_N)$. Therefore the radiation relation is replaced here byTo check dynamically, put $S={\delta }_S-2{\delta }_C/3$. At $k=0$, subtracting two-thirds of the matter equation from the string equation gives $S^{\prime \prime }+2\mathcal{H}{S}^{\prime }=0$. Selecting $S=0$ and no independent relative-velocity mode preserves the relation. Thus the growing adiabatic string density contrast follows the matter mode and is also approximately frozen at late times to leading order in gradients. This is the adiabatic matter-string density relation.

At late times $\mathcal{H}\to k_{\ast }$, while ${\Omega }_C$ and ${\delta }_C^{\prime }$ vanish. The homogeneous string equation becomesFor $k\gg k_{\ast }$, one solution grows rapidly with exponent approximately $k/\sqrt{3}-k_{\ast }$, instead of undergoing radiation-like acoustic oscillations. This is a gradient instability of a negative-pressure perfect fluid. For $k\ll k_{\ast }$, the growing exponent is only $k^2/(6k_{\ast })$, while the other exponent is approximately $-2k_{\ast }$. The leading zero-gradient approximation is frozen, but at fixed nonzero superhorizon $k$ a slow instability can accumulate over sufficiently long conformal time. The strict gradient limit and infinite-future limit therefore need not commute.

These are the apparent behaviors of the equations given. They do not establish rapid collapse of a realistic cosmic string network: tension, anisotropic stress and the constitutive sound response were excluded in replacing the network by this barotropic perfect fluid. Once the modeled string perturbation becomes nonlinear, the linear calculation itself ceases to apply.

Use the flat matter-only Einstein-de Sitter approximation for the distance-redshift relation, neglecting radiation corrections at cosmological recombination. Let $z_{\ast }$ be the cosmological recombination redshift. Adiabatic photon cooling gives $1+z_{\ast }=T_{\ast }/T_0$. The Hubble rate is $H(z)=H_0(1+z{)}^{3/2}$, so the comoving radial distance to last scattering isThe angular diameter distance is $D_A={\chi }_{\ast }/(1+z_{\ast })$. A proper Hubble radius $H_{\ast }^{-1}=H_0^{-1}(1+z_{\ast }{)}^{-3/2}$ therefore subtends the small angleThis is the angular Hubble-radius scale in an Einstein-de Sitter universe; the present Hubble constant cancels.

Taking $T_{\ast }=0.3\mathrm{eV}$ and the literal printed $T_0=2.35\mathrm{meV}$ gives $1+z_{\ast }=127.7$ and ${\theta }_H\simeq 0.0486$ radians, or $2.8$ degrees. As noted above, that present CMB energy is too large by ten. With the corrected $T_0=0.235\mathrm{meV}$, $1+z_{\ast }=1277$ andwhich gives the familiar order-one-degree scale within the assumed cosmology. Using $0.28\mathrm{eV}$ to mark a few-per-cent ionization fraction changes this only modestly. These numbers refer to the Hubble-radius scale, not its diameter. In a matter-only universe the particle-horizon radius is $2H^{-1}$ and would give twice the angle; the acoustic sound horizon is a different length again.

The continuity and Cosmological Euler equation in synchronous gauge combine particularly simply. Differentiate ${\delta }_C^{\prime }=-\nabla \cdot v_C-h^{\prime }/2$ and add $\mathcal{H}{\delta }_C^{\prime }$:The velocity terms cancel by the Cosmological Euler equation in synchronous gauge, so this step does not require setting the initial velocity exactly to zero.

For subhorizon modes deep in radiation domination, take the rapidly oscillating radiation perturbation to be negligible in the averaged gravitational forcing. Also neglect the subdominant matter self-gravity to leading order in ${\rho }_C/{\rho }_r$. With $a\propto \tau$ and $\mathcal{H}=1/\tau$, the resulting equation isIntegrating twice giveswhere the arbitrary reference time ${\tau }_{\ast }$ makes the logarithm dimensionless and can be absorbed into $B$. This is logarithmic growth of matter perturbations during radiation domination.

The neglect of matter self-gravity is needed for the displayed form to be exact within the leading radiation-background approximation. Uniform radiation alone leaves a nonzero term $4\pi G{a}^2{\rho }_C{\delta }_C$ if the cold-matter density is retained. Relative to ${\mathcal{H}}^2$, its coefficient is $3{\Omega }_C(a)/2$, small deep in the radiation era but not near equality. The Mészáros equation retains that effect in a matter-plus-radiation background; its solution basis tends to a constant and a logarithm at $a/a_{\mathrm{eq}}\ll 1$, recovering the present leading result. It connects smoothly to matter-era growth rather than allowing the logarithmic approximation to be extrapolated indefinitely.

Galaxy-scale modes enter the Hubble radius before matter-radiation equality. The earlier radiation forcing and horizon-entry matching ordinarily generate a nonzero logarithmic coefficient, even though subsequent oscillatory radiation forcing is small. A Fourier mode with entry time ${\tau }_h\sim k^{-1}$ accumulates $\log ({\tau }_{\mathrm{eq}}/{\tau }_h)\sim \log (k{\tau }_{\mathrm{eq}})$ before equality. Across a large scale-factor range this factor need not be small, so treating the density contrast as exactly frozen would lose an important part of the galaxy-scale seed amplitude and its scale dependence. Growth is nevertheless much slower than ${\delta }_C\propto a$ in matter domination. This is the Mészáros effect, and the logarithmic matching contributes to the small-scale matter cold-dark-matter transfer function.

Let $w=c_s^2$ be constant and small, and write $\theta =i\mathbf{k}\cdot \mathbf{v}$. The supplied synchronous gauge equations implyDifferentiating the first and eliminating $\theta$ givesFor the dominant single component, the Einstein equation gives $h^{\prime \prime }+\mathcal{H}{h}^{\prime }=-3{\mathcal{H}}^2(1+3w)\delta$. Therefore the exact algebraic intermediate relation isThe requested nonrelativistic density equation in synchronous gauge follows to leading nonrelativistic order: discard pressure corrections to the background, Hubble damping, and metric tensor terms while retaining $c_s^2{k}^2\delta$, which can be comparable to gravity at the Jeans wavenumber. Since $w\ll 1$, that scale lies well inside the Hubble radius, so keeping its enhanced gradient term is consistent. The resulting equation iswhere the Friedmann equation gave $3{\mathcal{H}}^2/2=4\pi G{a}^2{\bar{\rho }}_m$. Now ${\delta }^{\prime }=a\dot{\delta }$ and ${\delta }^{\prime \prime }=a^2(\ddot{\delta }+H\dot{\delta })$, soThis is the specified approximation, not an exact finite-$w$ relativistic identity. The pressure-gradient term stabilizes short wavelengths, while gravity drives the long-wavelength Jeans instability.

Now apply the stipulated equation to $P=K{\rho }^{4/3}$ with $K>0$. In the pressure-negligible matter background,Thus $c_s^2{k}^2/a^2$ is proportional to $t^{-2}$, exactly like the gravitational term. DefineThe comoving Jeans wavenumber and $q$ are constant in this era. The equation becomes the Euler-Cauchy equationSetting $\delta \propto t^p$ gives $p^2+p/3+2(q-1)/3=0$. For distinct real roots the full pair of modes isThese are the Jeans modes of a four-thirds polytropic cosmological fluid. The physical Jeans length isHere ${\lambda }_J\propto a$, so its comoving value is constant. For $\lambda >{\lambda }_J$, $q<1$ and $p_{+}>0$, giving genuine growth; the other solution decays. In the long-wavelength pressure-free limit, $p_{+}=2/3$ and $p_{-}=-1$, the familiar matter-era modes. At $\lambda ={\lambda }_J$, the two powers are $0$ and $-1/3$, so the leading mode is constant rather than growing.

For $1<q<25/24$ both powers are negative: pressure prevents growth, although the modes still decay without oscillation. At $q=25/24$ they coincide and the complete solution isFor $q>25/24$ they become a complex-conjugate pair; a real basis isThese short-wavelength acoustic oscillations have a decaying envelope. The onset of instability is $q=1$, while the threshold for oscillatory time dependence is slightly larger, $q=25/24$; expansion damping explains why these two thresholds are not identical.

Here $H=\dot{a}/a=2/(3t)$, so the perturbation equation becomes $\ddot{\delta }+(4/3t)\dot{\delta }-(2/3t^2)\delta =0$. A trial $t^s$ gives $s(s-1)+(4/3)s-2/3=0$, with roots $2/3,-1$. ThusThe pure growing mode obeys ${\delta }_k(t)/{\delta }_k(t_i)=a(t)/a(t_i)$; for generic data the decaying component eventually becomes negligible, though the growing-mode normalization must be determined from both initial data.

With $a(t_0)=1$, the physical wavelength is $2\pi a(t)/k$. Equating it to the Hubble radius $c/H=3ct/2$, and using $k_0=2\pi H_0/c$, givesGrowing-mode evolution from crossing to today multiplies the perturbation amplitude by $(t_0/t_k{)}^{2/3}=(k/k_0{)}^2$. Hence the supplied horizon-crossing variance givesThis conclusion uses the stated matter-dominated growing-mode approximation over the relevant evolution, rather than an additional radiation-era transfer function.

The inflaton has quantum fluctuations while their wavelengths are shorter than the Hubble radius. Choosing the Bunch-Davies vacuum fixes the short-wavelength positive-frequency modes. During near-de Sitter slow-roll inflation, expansion carries each mode through $k=aH$, leaving a field fluctuation of amplitude approximately $H/(2\pi )$ on spatially flat slices.

The inflaton clock-shift origin of curvature perturbations converts this field displacement into a local time shift of magnitude $|\delta \Phi /\dot{\Phi }|$, and hence into a comoving curvature perturbation of magnitude $H|\delta \Phi /\dot{\Phi }|$. For the single-field adiabatic growing mode on an attractor, the superhorizon conservation of single-field comoving curvature makes this perturbation approximately constant at $k\ll aH$. These vacuum-generated perturbations seed the later density fluctuations.

The physical particle horizon is the greatest physical distance at time $t$ from which a light signal could have reached the observer since the beginning of the cosmological model. If that beginning is $t=0$, radial null propagation has comoving distance ${\int }_0^{t}cd{t}^{\prime }/a(t^{\prime })$, soFor $a(t)=t^{\alpha }$ and $0<\alpha <1$, the integral converges at zero and equals $t^{1-\alpha }/(1-\alpha )$. ConsequentlyThe horizon problem is that widely separated regions of the observed cosmic microwave background have nearly equal temperatures although their past causal regions did not overlap before recombination in the usual decelerating hot-big-bang history. That history offers no causal equilibration mechanism for their observed uniformity. The particle horizon used here is distinct from the Hubble radius $c/H$.

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.

Use $\hslash =c=1$. For a canonical light scalar field with effective mass $m^2\ll H^2$, define the canonical Fourier mode $u_k=a\delta {\Phi }_k$. In the nearly de Sitter approximation it obeysDeep inside the Hubble radius, $k\gg aH$, the mode behaves like a Minkowski oscillator. Choosing the Bunch-Davies vacuum sets $u_k\to e^{-ik\eta }/\sqrt{2k}$. The normalized de Sitter solution isInflation stretches the physical wavelength until it exits the Hubble radius. For $|k\eta |\ll 1$, the growing scalar solution freezes with $|\delta {\Phi }_k{|}^2=|u_k/a{|}^2\to H^2/(2k^3)$. The dimensionless power spectrum is thereforeHere $H_k$ is evaluated when $k=aH$, since slow evolution near horizon exit fixes the frozen amplitude. Nearly constant $H$ and a light field make this spectrum almost scale-invariant; slow background evolution and small mass corrections produce a small tilt. The stretched vacuum modes supply the primordial stochastic fluctuations.