Cosmic inflation

Cosmic inflation is an early period of accelerated expansion, often modeled by a scalar field whose potential energy dominates its kinetic energy.

Approximately constant positive Hubble parameter gives a nearly de Sitter spacetime expansion in flat slicing. Its scale factor can be approximated by $a=-1/(H\tau )$ with negative conformal time increasing toward zero. Slow variation of the expansion rate limits the accuracy of this approximation.

Reheating transfers the inflaton energy into other particles after cosmic inflation, eventually establishing a hot thermal bath. It requires interactions in addition to a background scalar potential. The duration and averaged equation-of-state parameter of reheating affect the number of e-folds between a chosen scale's horizon exit and the end of inflation.

The reheating temperature characterizes the thermal radiation bath after energy from a nonthermal component has been transferred and thermalized. For sudden decay of a dominant nonrelativistic relic, $T_R\simeq [3M_{\mathrm{Pl}}^2{\Gamma }^2/(c_2{g}_{\star }){]}^{1/4}$. It differs from the residual radiation temperature immediately before decay.

The sudden-decay approximation converts a dominant cold relic's rest-mass density into radiation at $H\simeq \Gamma$, neglecting expansion during the conversion. Its pre-decay radiation temperature obeys $T_{-}^3\simeq 3M_{\mathrm{Pl}}^2{\Gamma }^2/(c_1{g}_{\star }Ym)$, while the reheating temperature is set by the post-decay radiation energy density. Constant relativistic counts and prompt thermalization are additional assumptions.

At fixed scale factor and unchanged relativistic counts, instantaneous conversion of a relic into thermal radiation gives the displayed entropy-density enhancement. It exceeds one because the gas receives energy from outside its prior equilibrium sector. This does not contradict cosmological entropy conservation, whose source-free reversible assumptions fail during decay and thermalization.

The number of e-folds during an interval of expansion is the logarithmic growth of the scale factor:

A primordial perturbation is an early-universe fluctuation that seeds later cosmic microwave background anisotropy and large-scale structure. Inflation generates scalar curvature perturbations and tensor perturbations from quantum fluctuations.

The dimensionless spectrum is the contribution to a statistically isotropic field variance per logarithmic wavenumber interval: $⟨X^2⟩=\int d\log k{\Delta }_X(k)$, with any necessary smoothing or regulators. Some conventions denote this same quantity ${\Delta }_X^2$. The definition, not the superscript convention, fixes its meaning.

The dimensionless spectrum is $\mathcal{P}(k)=k^{3}P(k)/(2{\pi }^2)$. If it rises as $k^4$ below equality and as ${\ln }^{2}k$ above equality, then the dimensional spectrum rises as $k$ and falls as $k^{-3}{\ln }^{2}k$. Thus the dimensional spectrum has a turnover while the dimensionless spectrum only bends to a much slower rise. A schematic transition must smoothly join the two asymptotic regimes.

A canonical homogeneous inflaton $\varphi (t)$ hasand obeys $\ddot{\varphi }+3H\dot{\varphi }+V^{\prime }(\varphi )=0$.

For a homogeneous canonical inflaton in an expanding FLRW metric,The expansion contributes the damping term called Hubble friction.

For a homogeneous canonical scalar field, the cosmological perfect-fluid continuity equation reduces to $\dot{\varphi }(\ddot{\varphi }+3H\dot{\varphi }+V^{\prime })=0$. It implies the inflaton equation of motion wherever $\dot{\varphi }\ne 0$, and by continuity at isolated turning points. It cannot by itself exclude a frozen field on a time interval. Indeed, choose $\varphi ={\varphi }_0$ with $V({\varphi }_0)>0$ but $V^{\prime }({\varphi }_0)\ne 0$; a de Sitter scale factor in flat slicing with $H^2=V({\varphi }_0)/(3M_{\mathrm{Pl}}^2)$ satisfies the homogeneous Friedmann equation and Friedmann acceleration equation, but not the scalar equation. Varying the inflaton action in an expanding universe supplies the missing equation also for a frozen field, requiring $V^{\prime }({\varphi }_0)=0$.

In a flat universe containing a homogeneous canonical scalar field, the ansatzhas the exact solutionIt interpolates between decelerating power-law expansion at early times and exponential accelerated expansion at late times when $2/3<{\varphi }_0^2/M_{\mathrm{Pl}}^2<2$.

For a canonical scalar field in a spatially flat expanding universe,The field Euler-Lagrange equation is

For nearly constant inflaton potential and constant $H_{\inf }$, the spatial Fourier transform obeys

For $a=-1/(H_{\inf }\tau )$, the rescaled modesatisfies

Neglecting metric perturbations, the rescaling $f=a\delta \varphi$ removes the expansion friction from the scalar perturbation action after a boundary term is discarded. Its canonical momentum is $f^{\prime }$, and each real Fourier oscillator has time-dependent squared frequency $k^2+a^2{V}^{\prime \prime }-a^{\prime \prime }/a$. The oscillator can become inverted on superhorizon scales; the canonical commutation relations still apply.

The Bunch-Davies vacuum selects the mode that approaches the positive-frequency ground-state mode of angular frequency $ck$ in the far past $\tau \to -\infty$.

A Fourier mode exits the Hubble horizon during inflation when $k=aH$. On superhorizon scales $k\ll aH$, the growing mode of a sufficiently light scalar perturbation approaches a constant.

A cosmological mode is on a superhorizon scale when its physical wavenumber is smaller than the Hubble expansion rate, $k/a\ll H$, equivalently $k\ll aH$.

A Fourier-space variance $P_k$ in three dimensions is scale invariant when the dimensionless power per logarithmic wavenumber interval,is independent of $k$. For a massless inflaton mode at late times,so ${\Delta }^2(k)$ is constant.

An exact $P(k)=A/k^3$ has divergent unsubtracted point variance. Its finite isotropic covariance difference is the displayed logarithm, unchanged by simultaneous dilation of $r,r_0$. Thus the nonzero-mode scaling statement has a natural subtraction interpretation. Exact dilation invariance of a continuous finite-variance isotropic covariance itself would instead force a constant covariance, corresponding only to a zero-mode spectral measure. Cutoffs or a zero-mode subtraction must accompany the nontrivial spectrum.

For a light scalar in the Bunch-Davies vacuum with metric perturbations neglected, the mode $e^{-ik\tau }(1-i/(k\tau ))/\sqrt{2k}$ gives the displayed equal-time spectrum. The dimensionless power tends to $H^2/(4{\pi }^2)$ on superhorizon scales. The exact massless infinite-volume correlation has an infrared divergence; a finite-volume or finite-duration infrared regulator, and smearing for coincident products, makes the correlators well defined.

At leading slow-roll order, the dimensionless primordial curvature-perturbation amplitude at horizon exit is

The scalar spectral index describes the departure of the primordial scalar spectrum from scale invariance. To first order in potential slow-roll parameters,

The leading scalar spectrum is proportional to $V^3/V^{\prime 2}$. Along a slow-roll trajectory, $d\varphi /d\ln a\simeq -M_{\mathrm{pl}}^2{V}^{\prime }/V$ and $d\ln k/d\ln a=1-{ϵ}_H$. Differentiating at horizon exit and keeping first order gives $n_s-1=-6{ϵ}_V+2{\eta }_V$. The derivative-factor correction contributes at second order.

For a canonical scalar field with potential $V(\varphi )$, slow roll reduces the field and Friedmann equations to

Single-field slow-roll inflation uses one canonical scalar clock whose potential evolves slowly enough for the slow-roll parameters to remain small. Its adiabatic long mode produces the single-field inflation squeezed-limit consistency relation.

Starobinsky inflation is an inflationary model whose Einstein-frame scalar potential can be writtenThe exponentially flat large-field plateau supports slow-roll inflation, while the potential is quadratic near its minimum.

A Hubble slow-roll parameter measures the fractional change of the Hubble rate or of a preceding slow-roll parameter per e-fold. Small values make the inflationary background evolve adiabatically.

For a canonical inflaton under the slow-roll approximation, $\dot{\varphi }/H\simeq -M_{\mathrm{Pl}}^2{V}^{\prime }/V$ and $ϵ\simeq {ϵ}_V=(M_{\mathrm{Pl}}^2/2)(V^{\prime }/V{)}^2$. Differentiating $\log {ϵ}_V$ along this trajectory givesHere $\eta =d\log ϵ/d\log a$, not the alternate convention $-\ddot{\varphi }/(H\dot{\varphi })$. The logarithmic definition requires $ϵ>0$; exact de Sitter expansion is treated as a limit.

The first Hubble slow-roll parameter isFor a canonical inflaton, $ϵ={\dot{\varphi }}^2/(2M_{\mathrm{Pl}}^2{H}^2)\ge 0$.

Using the e-fold variable $N=\log a$, define

The comoving curvature perturbation is the scalar curvature perturbation on hypersurfaces orthogonal to the total matter flow. For single-field adiabatic inflation it is conserved on superhorizon scales.

For single-field adiabatic inflation, the Mukhanov-Sasaki variable rescales the comoving curvature perturbation by $z=a\sqrt{2ϵ}{M}_{\mathrm{Pl}}$, where $M_{\mathrm{Pl}}$ is the reduced Planck mass. Its quadratic bulk action is $\frac{1}{2}\int [(v^{\prime }{)}^2-(\nabla v{)}^2+(z^{\prime \prime }/z)v^2]$, after a boundary term is removed. A Fourier transform mode obeys $v_k^{\prime \prime }+(k^2-z^{\prime \prime }/z)v_k=0$ and has canonical Wronskian normalization. The mass scale and sound-speed factors depend on the underlying scalar theory; this definition assumes the canonical single-field action and positive slow-roll kinetic coefficient.

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.

On a single-field attractor, a field fluctuation changes the local point along the background trajectory. The corresponding time shift has magnitude $|\delta \varphi /\dot{\varphi }|$, and local expansion converts it into curvature of magnitude $H|\delta \varphi /\dot{\varphi }|$. On spatially flat slices a standard sign convention gives $\mathcal{R}=H\delta {\varphi }_{\mathrm{flat}}/\dot{\varphi }$. The vacuum mode amplitude $H/(2\pi )$ then determines the scalar curvature spectrum.

For approximately constant $H$ and $ϵ$, a vacuum-normalized curvature mode is $u_k(\tau )=H(1+ik\tau )e^{-ik\tau }/\sqrt{4ϵM_{\mathrm{Pl}}^2{k}^3}$. Its derivative is $u_k^{\prime }=H{k}^2\tau e^{-ik\tau }/\sqrt{4ϵM_{\mathrm{Pl}}^2{k}^3}$. Ordered unequal-time Wick contractions use the complex conjugate at the earlier vertex, consistently with the vacuum prescription for inflationary in-in integrals.

The non-adiabatic pressure perturbation isOn superhorizon scales it sources the evolution of the comoving curvature perturbation through a convention-dependent signed relation proportional to $H\delta P_{\mathrm{nad}}/(\bar{\rho }+\bar{P})$.

A cosmological entropy perturbation changes the relative composition of cosmic fluids while leaving the total curvature initially unchanged. For cold dark matter and photons, a common relative entropy variable is

At leading order in canonical single-field slow roll and evaluated at horizon exit $k=aH$,

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

Differentiating the curvature spectrum at horizon exit givesto first order in Hubble slow-roll parameters.

For the two graviton polarizations in canonical single-field inflation,at horizon exit.

The graviton power spectrum is the two-point spectrum of a tensor metric perturbation. A convention may quote it per polarization or summed over both polarizations, so that choice must accompany its normalization.

An adiabatic tensor mode is a long-wavelength transverse-traceless metric perturbation whose leading local effect can be produced by a large spatial diffeomorphism. Its effect on short modes is therefore constrained by a Ward identity.

For a single adiabatic long tensor mode, the squeezed scalar-scalar-graviton bispectrum is the anisotropic momentum response of the scalar power spectrum:

The tensor spectral index is the logarithmic slope $n_T=d\log {\Delta }_t^2/d\log k$. Canonical slow-roll inflation gives $n_T=-2ϵ$ to first order.

The tensor-to-scalar ratio is $r={\Delta }_t^2/{\Delta }_{\mathcal{R}}^2$. Canonical single-field slow roll gives $r=16ϵ$.

Combining $r=16ϵ$ with $n_T=-2ϵ$ gives the leading-order consistency relation

A primordial power spectrum is blue when its dimensionless power increases with wavenumber. Canonical single-field slow roll cannot produce a blue tensor spectrum because $ϵ\ge 0$ implies $n_T\le 0$, but its scalar spectrum can be blue when $\eta <-2ϵ$.

For $V(\varphi )=m^2{\varphi }^2$ in units where $3H^2\simeq V$ and for positive $\varphi$,and

For a quadratic inflaton potential ending at the leading estimate ${ϵ}_V=1$, $\Phi /M_{\mathrm{Pl}}=\sqrt{4N+2}$. Cosmological horizon exit gives $d\log k/dN=-1+{ϵ}_H$. A narrow logarithmic interval therefore probes a field span $2\Delta \log k/(\sqrt{4N+2}(1-{ϵ}_H))$, to the accuracy of the slow-roll approximation. At sixty e-folds, five logarithmic wavenumber units probe about $0.64M_{\mathrm{Pl}}$ around a background field near $15.6M_{\mathrm{Pl}}$.

The quadratic potential has equal potential slow-roll parameters, ${ϵ}_V={\eta }_V=1/(2N+1)$. Consequently $n_s=1-4/(2N+1)$ and $r=16/(2N+1)$ at leading order. The relation $r=4(1-n_s)$ is independent of the inflaton mass, so a satisfactory scalar tilt does not by itself guarantee a satisfactory tensor prediction.

For a quadratic potential with the end set by ${ϵ}_V=1$, a mode leaving $N$ e-folds earlier has ${\varphi }_{\ast }^2=(4N+2)M_{\mathrm{pl}}^2$. Its curvature amplitude is $(m/M_{\mathrm{pl}}{)}^2(2N+1{)}^2/(24{\pi }^2)$. Thus the scalar amplitude fixes the inflaton mass while the tilt and tensor ratio depend only on the exit e-fold count at this order.

For $V(\varphi )=\lambda {\varphi }^4/4$ with
$H^2\simeq 8\pi GV/(3c^2)$ and positive $\varphi$,and

The potential slow-roll parameter isUnder the slow-roll equations, ${\dot{\varphi }}^2/(2V)\simeq {ϵ}_V/3$.

The second potential slow-roll parameter isSlow roll usually requires both ${ϵ}_V\ll 1$ and $|{\eta }_V|\ll 1$.

Suppose a homogeneous scalar field oscillates much faster than the Hubble parameter in a monomial potential $V(\varphi )=\lambda |\varphi {|}^n$. The virial theorem averaged over one oscillation gives $⟨{\dot{\varphi }}^2⟩=n⟨V⟩$, and henceA quadratic minimum therefore behaves like pressureless matter, whereas a quartic minimum behaves like radiation in cosmology.

Hilltop inflation begins near a local maximum of the inflaton potential. Fora positive field rolls away from the maximum at $\varphi =0$.

For $V(\varphi )=V_0-m^2{\varphi }^2/2$ with $V_0$ dominant,If $\tilde{N}$ e-folds remain and $n_s-1\simeq 2{\eta }_V$, then

A monomial inflation potential is a positive power of the inflaton. Its potential slow-roll parameter is ${ϵ}_V={\alpha }^2{M}_{\mathrm{Pl}}^2/(2{\varphi }^2)$.

For $V=C_p(\Phi /M_{\mathrm{Pl}}{)}^p$ and the leading endpoint estimate ${ϵ}_V=1$, the slow-roll approximation gives ${\Phi }_N^2/M_{\mathrm{Pl}}^2=2pN+p^2/2$ and ${ϵ}_V=p/(4N+p)$. Thus ${\mathcal{P}}_{\mathcal{R}}=C_p(2pN+p^2/2{)}^{p/2}(4N+p)/(24{\pi }^2{M}_{\mathrm{Pl}}^{4}p)$ and the scalar spectral index obeys $n_s-1=-2(p+2)/(4N+p)$. The coefficient sets the amplitude, whereas the monomial power and exit e-fold count set the leading tilt. Changing the power without specifying the coefficient leaves the overall normalization undetermined.

For $V=\lambda {\varphi }^n$ on $\varphi >0$, slow-roll approximation gives $ϵ\simeq n^2{M}_{\mathrm{Pl}}^2/(2{\varphi }^2)$ and $\eta \simeq 2n{M}_{\mathrm{Pl}}^2/{\varphi }^2$. The slow-roll e-fold count integrates to $N\simeq ({\varphi }_N^2-{\varphi }_e^2)/(2n{M}_{\mathrm{Pl}}^2)$. For $n=2,3,4$, the loose order-one conditions $ϵ\le 1$ and $|\eta |\le 1$ imply ${\varphi }^2\ge 2n{M}_{\mathrm{Pl}}^2$ in this approximation. Genuine slow roll requires a much stronger hierarchy. A prescribed endpoint ${\varphi }_e^2=2n{M}_{\mathrm{Pl}}^2$ gives ${\varphi }_N^2=2n(N+1)M_{\mathrm{Pl}}^2$ andThe actual end criterion for accelerated expansion is $ϵ=1$, whose leading potential estimate instead gives ${\varphi }_e^2\simeq n^2{M}_{\mathrm{Pl}}^2/2$. The prescribed endpoint is a convention, not an exact slow-roll or inflation boundary.

For canonical slow-roll inflation with $V(\varphi )=\lambda {\varphi }^4$, the leading scalar spectral index and tensor-to-scalar ratio obeyIt is a parameter-free consistency check of the quartic potential at leading slow-roll order.

The number of e-folds between field values ${\varphi }_i$ and ${\varphi }_f$ iswhen the field rolls monotonically down the potential.

For $V(\varphi )=\lambda {\varphi }^n/n$ under the stated Cambridge Tripos normalization,If slow roll ends by kinetic-potential equality, then

Natural inflation uses a periodic potentialIts potential slow-roll parameter obeyson the branch where the tangent is positive.

For the cosine potential,

Ultra-slow-roll inflation occurs on an exceptionally flat potential. The homogeneous field velocity decays as $\dot{\varphi }\propto a^{-3}$, so $ϵ\propto a^{-6}$ and the second Hubble slow-roll parameter approaches $-6$.

In ultra-slow-roll inflation, $z=a\sqrt{2ϵ}\propto a^{-2}\propto {\eta }^2$. The comoving curvature mode equation isand its Bunch-Davies mode is proportional to $(1+ik\eta )e^{-ik\eta }/{\eta }^3$. Unlike ordinary slow roll, the superhorizon curvature perturbation grows as $a^3$.

For a perturbation $X$, a normalized commutatorcompares its quantum commutator with its root-mean-square phase-space fluctuations. A limit $\mathcal{C}\to 0$ signals effective classicality.

A curvaton is a light spectator scalar field whose energy density is negligible during inflation but whose fluctuations can later generate the primordial curvature perturbation.

In the curvaton mechanism, the curvaton's relative energy density grows after inflation and its initially isocurvature fluctuation becomes an adiabatic curvature fluctuation when the curvaton decays.