Slow-roll approximation

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,