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The particle horizon is the greatest distance from which a light signal could have reached the observer since the initial cosmic time $t_i$. Its comoving radius and its radial proper distance at time $t$ areHere proper distance is measured along the spatial slice, rather than by the transverse area of a sphere. The Hubble parameter, deceleration parameter and cosmological density parameter areThe last expression defines the critical density. For blackbody radiation, $p=\rho /3$, so the Friedmann acceleration equation gives $\ddot{a}/a=-\kappa \rho /3$. ConsequentlyThis is the radiation relation between deceleration and density. It does not require spatial flatness.

For the open radiation universe particle horizon, write $x=a/a_0$. The cosmological continuity equation gives $\rho ={\rho }_0{x}^{-4}$, while the present Friedmann equation gives $-k/a_0^2=H_0^2(1-{\Omega }_0)$. ThusTaking the big-bang endpoint $x=0$ and using $dt=dx/\dot{x}$, the present radial proper distance becomesThereforeThe formula applies to $0<{\Omega }_0<1$. Its flat-radiation limit is $H_0{d}_{h0}\to 1$, agreeing with $a\propto t^{1/2}$ and $d_h=2t=H^{-1}$. The divergence as ${\Omega }_0\to 0$ reflects the unbounded past conformal interval of the limiting empty open model; that endpoint is not a radiation-filled universe.

The horizon problem concerns the nearly uniform temperature of widely separated parts of the cosmic microwave background. In a purely decelerating Hot Big Bang model, their past light cones at last scattering do not overlap far enough to explain this agreement by thermal contact. During cosmic inflation, accelerated expansion shrinks the comoving Hubble radius. A patch initially small enough for causal communication can be stretched to encompass the later observable universe. Reheating converts the inflationary energy into a hot plasma with correlated initial conditions across that patch. Enough inflation must occur before the observable scales leave the Hubble radius; inflation cannot establish contact between regions that were never initially causally related.

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.

The annihilation operators kill the vacuum, and their commutator givesTherefore the scale-invariant inflationary power spectrum has the superhorizon limitThe canonical mode grows as $1/|\tau |$ outside the Hubble radius, while the physical field perturbation $f/a$ freezes. A slowly varying $H$ produces an almost scale-invariant spectrum rather than exact scale invariance.

For a slowly rolling single field, fluctuations in the inflaton clock become the comoving curvature perturbation, $\mathcal{R}\simeq -H\delta \varphi /\dot{\bar{\varphi }}$ up to sign convention. Restoring the reduced Planck mass and using $ϵ={\dot{\bar{\varphi }}}^2/(2M_{\mathrm{pl}}^2{H}^2)$ givesThis conversion presumes nonzero background roll; a strictly constant test scalar in exact de Sitter spacetime does not by itself define a finite comoving curvature perturbation through this formula. The curvature perturbations seed the later density fluctuations.

When metric fluctuations and their Einstein-gravity normalization are restored, the two graviton polarizations have the same massless mode equation. The usual total primordial tensor power spectrum is ${\Delta }_t^2=2H^2/({\pi }^2{M}_{\mathrm{pl}}^2)$, so the single-field leading ratio is $r=16ϵ$. Tensor amplitudes therefore probe the inflationary energy scale, whereas scalar amplitudes also depend on the roll rate. These tensor statements explain the relevance of the mode result; they are not derived from a scalar action with metric perturbations omitted.

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.

For a homogeneous canonical scalar field, ${\rho }_{\varphi }={\dot{\varphi }}^2/2+V$ and $P_{\varphi }={\dot{\varphi }}^2/2-V$. Potential-energy domination gives $P_{\varphi }\simeq -{\rho }_{\varphi }$, accelerated expansion and a decreasing comoving Hubble radius. Hubble friction can permit slow roll down a sufficiently flat potential. Enough inflation stretches an initially small causally connected region to encompass our observable universe, suppresses the relative spatial-curvature term and dilutes unwanted massive relics. quantum fluctuations supply possible seeds for cosmological structure; a successful model must also end inflation and create the hot post-inflationary universe. These statements do not imply that arbitrary initial conditions inevitably inflate.

Use the stated reduced-Planck units throughout the field calculation. Differentiating the Friedmann constraint and substituting the field equation givesDivision assumes the nontrivial expanding branch $H>0$; the vacuum endpoint follows continuously.

During rapid oscillations, require $m\gg H$ and $|\dot{A}|\ll mA$. To leading order, $\dot{\varphi }\simeq -mA\sin mt$, while the Friedmann constraint gives $H_m^2\simeq m^2{A}^2/6$. ThusAveraging over many oscillations on a timescale still short compared with $H^{-1}$ givesThe quadratic-inflaton Hubble oscillations obey, to leading order in $H_m/m$,Differentiating this expression reproduces the cosine term, with smaller corrections from the slowly varying amplitude. Instantaneously $H$ remains nonincreasing because its exact derivative is $-{\dot{\varphi }}^2/2$; an oscillatory correction does not mean oscillatory growth of the total Hubble parameter.

The mean equation integrates toThe shift $t_{\ast }$ is set by matching to the earlier evolution, not necessarily the Big Bang origin of the full solution. Averaged kinetic and potential energies are equal, so $⟨P_{\varphi }⟩\simeq 0$, $⟨{\rho }_{\varphi }⟩\propto a^{-3}$ and $A\propto a^{-3/2}$. The coherent quadratic oscillations behave as pressureless matter, not radiation, even though they are a scalar field rather than a gas of cold particles. radiation would instead give $a\propto t^{1/2}$. The averaged approximation is controlled only after oscillations are fast compared with expansion.

reheating requires interactions beyond the isolated field equations above. The oscillating inflaton can decay into other particles, whose scattering and further decays establish a thermal bath. The perturbative inflaton reheating equations areExpansion redshifts the components differently; transfer becomes efficient around $H\sim \Gamma$, and radiation domination follows when the transferred relativistic energy dominates. For suitable couplings, nonperturbative preheating by parametric resonance can precede thermalization, with backreaction and rescattering affecting its completion. Neither the decay rate nor thermalization is specified by $m$ and $V$ alone. Instantaneous reheating idealizes all this as immediate conversion of the end-of-inflation energy to radiation.^[ref]

For instantaneous-reheating horizon matching, specify the comparison: take the initial patch to be of order $H_i^{-1}$, compare the same-shaped inflationary history with approximately fixed $H_i/H_{\mathrm{end}}$, and assume instantaneous reheating followed by adiabatic expansion. Ignoring changes in relativistic degrees of freedom, $a_0/a_{\mathrm{reh}}\propto T_{\mathrm{reh}}$, while energy conservation and the Friedmann equation give $H_{\mathrm{end}}\propto T_{\mathrm{reh}}^2$. The present size of this inflated patch therefore scales asAt fixed present Hubble radius, a higher instantaneous reheating scale requires more e-folds in this standard comparison, approximately $\Delta N\simeq \ln (T_2/T_1)$, up to slow scale and entropy-degree corrections. Hotter reheating gives more subsequent expansion, but the associated initial Hubble patch is smaller by a stronger power of temperature. If instead one artificially holds the initial physical patch or $H_i$ fixed while varying the end energy independently, the post-inflationary geometric effect alone would give the opposite trend. temperature without an inflation-scale comparison convention does not fix that alternative problem; this is why the matching assumptions must be stated.^[ref]

cosmic inflation is an expanding epoch with $\ddot{a}>0$, equivalently a shrinking comoving Hubble radius $(aH{)}^{-1}=1/\dot{a}$. In general relativity it requires $\rho +3P/c^2<0$ for the total effective fluid. A scalar field whose potential energy dominates its kinetic energy has $P\simeq -\rho c^2$ and can produce approximately exponential expansion. slow-roll inflation is one realization; rapid expansion alone without acceleration does not have the required effect.

The horizon problem is that widely separated regions of the cosmic microwave background have nearly the same temperature despite lying outside one another's causal domains in a purely decelerating hot Big Bang extrapolation. Sufficient inflation stretches a previously small causally connected region across our observable universe; relevant perturbation wavelengths can begin inside the Hubble radius, leave it during inflation and re-enter later. This does not require local signals to outrun light. Inflation supplies a possible earlier causal history, not a proof that every possible initial state becomes homogeneous.

For the flatness problem, the Friedmann equation gives ${\Omega }_{\mathrm{tot}}-1=k{c}^2/(aH{)}^2$ when $a$ is length-valued. Since $aH$ grows during inflation,Pre-existing unwanted massive relics such as magnetic monopoles are also diluted: conserved number density falls as $a^{-3}$. Enough inflation, followed by reheating that does not recreate the relic problem, can make their abundance negligible. These were central motivations of the original proposal, whose first phase-transition implementation also faced a graceful-exit problem.^[ref]

quantum fluctuations of the inflating field and metric supply seeds for linear cosmological density perturbations. Simple single-field slow-roll models predict a dominant adiabatic mode with nearly Gaussian, nearly scale-invariant scalar cosmological perturbations, consistent with the cosmic microwave background; tensor cosmological perturbations are another possible test. These predictions are model-dependent, and consistency is not unique proof of inflation. reheating must connect the accelerated phase to the hot expanding universe. Inflation addresses causal reach, small cosmological spatial curvature and unwanted relic abundance, while providing a mechanism for primordial structure.

The metric uses signature $(+---)$ and a time-valued interval $ds$: a comoving clock has $ds=dt$. Here $t$ is cosmic proper time, $R(t)$ a scale factor with dimensions of length, $c$ the speed of light, $\chi$ a dimensionless comoving radial coordinate, and $\theta ,\varphi$ the usual polar and azimuthal angles. With curvature normalized to $k=+1,0,-1$, $S(\chi )=\sin \chi ,\chi ,\sinh \chi$, respectively. Homogeneity and isotropy apply to each constant-$t$ spatial slice. A comoving object's fixed coordinates do not imply a constant proper separation: spatial lengths scale with $R$.

Radial null propagation obeys $d\chi =cdt/R(t)$ in magnitude. Two neighbouring wave crests between the same comoving emitter and observer have the same radial integral, giving $\delta t_0/R_0=\delta t_e/R_e$. The wavelength redshift is consequentlyPeculiar Doppler shifts are excluded by the comoving assumption. Define angular diameter distance by $d_{\theta }={\ell }_{\perp }/\delta \theta$, where ${\ell }_{\perp }$ is proper transverse size at emission. The transverse metric gives ${\ell }_{\perp }=R_{e}S(\chi )\delta \theta$, henceThis distance is neither the photon travel time times $c$ nor the object's present proper distance.

The comoving particle horizon at $t_1$, with the initial boundary at the Big Bang, is ${\chi }_p(t_1)={\int }_0^{t_1}cdt/R(t)$. Its proper radial distance is $D_p=R(t_1){\chi }_p$; for a power-law scale factor $R=A{t}^p$, $0<p<1$,Thus a matter-only flat Einstein-de Sitter universe, $p=2/3$, has $D_p=3c{t}_1$, and the flat radiation-only case, $p=1/2$, has $D_p=2c{t}_1$. Strictly, Einstein-de Sitter denotes the pressureless matter model; the radiation calculation uses the flat zero-$\Lambda$ radiation analogue requested here.

For the source seen at time $t_1=t_{\mathrm{rec}}$, the radial coordinate is the integral from recombination to today, not the earlier horizon integral. In these flat models,Taking the linear horizon scale in the question to be $D_p$, its small angular extent isThereforeUsing the supplied times gives ${\theta }_{p,m}=0.02684\mathrm{rad}=1.54^{\circ }$ and ${\theta }_{p,r}=0.004244\mathrm{rad}=0.243^{\circ }$. There is a genuine radius/diameter convention to state: the supplied $3c{t}_1$ is a radial horizon distance. If angular diameter means the full diameter of a spherical horizon patch of radius $D_p$, the two answers are instead $2{\theta }_{p,m}=3.08^{\circ }$ and $2{\theta }_{p,r}=0.486^{\circ }$. Both conventions follow from the same distance calculation; a factor of two should not be hidden in the word horizon.

Matter-only evolution is the better of these two approximations because the universe is already matter dominated around recombination and evolves mostly through a subsequent matter era, rather than remaining radiation dominated to the present. Neither is exact: pre-recombination radiation changes the particle horizon, and late-time dark energy changes the distance to the last-scattering surface. No supplied ages have been replaced by later best-fit values.

The cosmic microwave background originates near last scattering and samples many horizon-sized regions. In an always-decelerating hot-Big-Bang model, widely separated last-scattering patches lack overlapping early causal histories, motivating the horizon problem. The measured acoustic angular scale uses the smaller sound horizon, not automatically this light-travel particle horizon; baryon-photon pressure changes the propagation speed. Inflation supplies an earlier accelerated expansion during which the comoving Hubble radius shrinks, allowing a patch with a common earlier causal history to expand to the size of the observable universe. Quantum perturbations can then seed later acoustic and density fluctuations. This does not require local signals to propagate faster than light, and the Hubble radius must not be equated indiscriminately with the particle horizon. CMB temperature and lensing spectra test the resulting perturbation and expansion model.^[ref]

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.