Comoving curvature perturbation ($\mathcal{R}$)

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 adiabatic scalar perturbations without anisotropic stress, define $A=4\pi G{a}^2(\bar{\rho }+\bar{P})={\mathcal{H}}^2-{\mathcal{H}}^{\prime }$ and $c_a^2={\bar{P}}^{\prime }/{\bar{\rho }}^{\prime }$. In conformal Newtonian gauge, $\mathcal{R}=-\varphi -\mathcal{H}({\varphi }^{\prime }+\mathcal{H}\varphi )/A$. Background conservation gives $A^{\prime }/A=-\mathcal{H}(1+3c_a^2)$. Differentiating the expression for $\mathcal{R}$ and using the scalar-potential equation gives the displayed identity. For regular adiabatic modes with nonzero background enthalpy, evolution is suppressed on scales $k\ll \mathcal{H}$.

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 describes relative composition changes between cosmic components, rather than only a common displacement along the background evolution. For cold dark matter and photons, define fractional density perturbations ${\delta }_c=\delta {\rho }_c/{\bar{\rho }}_c$ and ${\delta }_{\gamma }=\delta {\rho }_{\gamma }/{\bar{\rho }}_{\gamma }$. Their relative entropy variable isThe factor $3/4$ follows from the different background dilution laws ${\bar{\rho }}_c\propto a^{-3}$ and ${\bar{\rho }}_{\gamma }\propto a^{-4}$, with scale factor $a$. Under a first-order time shift $\tilde{\eta }=\eta +T$, the density perturbation gauge transformation gives ${\tilde{\delta }}_c={\delta }_c+3\mathcal{H}T$ and ${\tilde{\delta }}_{\gamma }={\delta }_{\gamma }+4\mathcal{H}T$, where $\mathcal{H}=a^{\prime }/a$. These shifts cancel in $S_{c\gamma }$.

Thus this relative entropy is gauge invariant at first order. The condition $S_{c\gamma }=0$ gives the common adiabatic relation ${\delta }_c=3{\delta }_{\gamma }/4$. A pure isocurvature perturbation initial condition is chosen with vanishing initial comoving curvature perturbation; a mixed initial state can instead have both nonzero curvature and nonzero $S_{c\gamma }$. Vanishing initial curvature is therefore a condition on a pure initial mode, not a requirement for defining the relative entropy variable.

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.

The tensor spectrum is proportional to $H^2$, but the single-field slow-roll curvature spectrum is proportional to $H^2/{ϵ}_H$. Thus $n_s-1\simeq -2{ϵ}_H-d\ln {ϵ}_H/dN$. A decreasing $H$ gives a red leading tensor tilt, while the evolving scalar conversion factor can change the sign of the scalar tilt.