Inflaton action in an expanding universe

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 $\tau \to -\infty$.

For a minimally coupled massless scalar field in flat de Sitter spacetime, $a=-1/(H\tau )$ and the Canonically rescaled de Sitter scalar mode is $u_k=a{f}_k$. The equation $u_k^{\prime \prime }+(k^2-2/{\tau }^2)u_k=0$ and the positive-frequency condition $u_k\sim e^{-ik\tau }/\sqrt{2k\mathcal{V}}$ select the displayed mode. Direct differentiation verifies its Wronskian normalization, which ensures the equal-time canonical commutation relations with box-normalized creation and annihilation operators. A conformally coupled massless scalar has a different mode equation.

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.

The second term in the equal-time two-point function of a de Sitter scalar is independent of time for constant $H$. It dominates on superhorizon scales and has constant dimensionless power spectrum, giving a scale-invariant inflationary power spectrum. The infrared qualification of a scale-invariant covariance still applies to the formally infinite-volume coincident variance.

The equal-time two-point function of a de Sitter scalar has a contribution $P_{\mathrm{vac}}=1/(2k{a}^2)$, giving the displayed dimensionless power spectrum per logarithmic wavenumber. At fixed comoving wavenumber it decreases as $a^{-2}$; on short physical scales it reproduces the vacuum fluctuations of a massless scalar field in Minkowski spacetime. Its coincident integral is ultraviolet divergent, so the fixed-mode decrease cannot be interchanged with an unregulated integral over all modes.

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 running describes the scale dependence of the scalar spectral index. Differentiating $n_s-1\simeq -6{ϵ}_V+2{\eta }_V$ with the potential slow-roll flow equations gives the displayed second-order slow-roll result. It includes the third potential slow-roll parameter and is generally smaller than the leading spectral tilt.

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.