Primordial scalar and tensor observables for inflation models

Fix the noncompact, spatially flat FLRW branch $\mathrm{\Sigma }={\mathbb{R}}^3$ of the cosmological EFT data in Four-dimensional Einstein-frame multi-scalar cosmology EFT data, with ${\gamma }_{ij}={\delta }_{ij}$ and metric $d{s}^2=-d{t}^2+a(t{)}^2{\delta }_{ij}d{x}^{i}d{x}^j\mathrm{.}$ Here $I\subseteq \mathbb{R}$ is the cosmic-time interval, $a\in C^{\mathrm{\infty }}(I,(0,\mathrm{\infty }))$ carries length, $\mathbf{x}\in {\mathbb{R}}^3$ is dimensionless, and $n={\mathrm{\partial }}_t$ is the background comoving unit timelike observer field, $g(n,n)=-1$, from FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter. A Fourier covector $\mathbf{k}$ is dimensionless in this chart; its spatial momentum magnitude measured by $n$ is $k\mathrm{/}a$, where $k=\mathrm{\mid }\mathbf{k}\mathrm{\mid }$. All linear perturbations below are evaluated on one declared primordial slice $t=t_{\ast }$, before any separately specified late-time transfer to CMB angular observables. The formulas use the continuous Fourier convention on ${\mathbb{R}}^3$; a compact flat spatial quotient instead requires its discrete Laplacian modes and corresponding covariance normalization.

(a) Scalar curvature perturbation. Let $J\subseteq I$ be an open time interval containing the chosen slice $t=t_{\ast }$. On $J\times {\mathbb{R}}^3$, take $\tilde{g}$ to be a smooth Lorentzian metric of signature $(-,+,+,+)$ and write ${\tilde{g}}_{\mu \nu }=g_{\mu \nu }+\delta g_{\mu \nu }$, where $\delta g$ is a smooth symmetric covariant $2$-tensor retained only to first order. Choose the time orientation of $\tilde{g}$ continuously from the background observer $n={\mathrm{\partial }}_t$. Assume $t$ is a time function for $\tilde{g}$: its level sets are spacelike and ${\tilde{g}}^{\mu \nu }{\mathrm{\partial }}_{\mu }t{\mathrm{\partial }}_{\nu }t<0$, with $-{\tilde{g}}^{\mu \nu }{\mathrm{\partial }}_{\nu }t$ future-directed. Let ${\tilde{T}}_{\mu \nu }$ be the specified smooth symmetric total perturbed stress tensor. Write the spatial components' scalar/vector/tensor decomposition, for nonzero Fourier modes, as ${\tilde{g}}_{ij}=a^2[(1-2\psi ){\delta }_{ij}+2{\mathrm{\partial }}_i{\mathrm{\partial }}_{j}E+{\mathrm{\partial }}_i{F}_j+{\mathrm{\partial }}_j{F}_i+h_{ij}],{\mathrm{\partial }}_i{F}_i=0,{\mathrm{\partial }}_i{h}_{ij}=0,{\delta }_{ij}{h}_{ij}=0.$ The scalar functions $\psi ,E$ are the scalar components in this decomposition; $F_i$ is transverse and $h_{ij}$ is transverse-traceless. The future-directed unit normal to the perturbed time slices is ${\tilde{n}}^{\mu }=-\frac{{\tilde{g}}^{\mu \nu }{\mathrm{\partial }}_{\nu }t}{\sqrt{-{\tilde{g}}^{\alpha \beta }{\mathrm{\partial }}_{\alpha }t{\mathrm{\partial }}_{\beta }t}}\mathrm{.}$ Define the slice energy density by $\rho :={\tilde{T}}_{\mu \nu }{\tilde{n}}^{\mu }{\tilde{n}}^{\nu }$. Write $\bar{\rho }(t)$ for its homogeneous background value and $\delta \rho (t,\mathbf{x})$ for its first-order perturbation in the same gauge. Let $H:=\dot{a}\mathrm{/}a$. On a background interval where $\dot{\bar{\rho }}\mathrm{\ne }0$, define $\zeta :=-\psi -H\frac{\delta \rho }{\dot{\bar{\rho }}}\mathrm{.}$ Adopt the linear time-slicing convention $\psi \mapsto \psi +H\mathrm{\Delta }t$ and $\delta \rho \mapsto \delta \rho -\dot{\bar{\rho }}\mathrm{\Delta }t$ for a smooth slicing shift $\mathrm{\Delta }t(t,\mathbf{x})$. The two shift terms cancel, so $\zeta$ is gauge invariant at linear order in this convention; on a uniform-density slice, where $\delta \rho =0$, $\zeta =-\psi$. This definition does not identify $\zeta$ with a comoving-slice curvature variable $\mathcal{R}$. A model using $\mathcal{R}$ must define it from its momentum perturbation and state and establish any claimed equality under the model's exact gauge, matter, and scale assumptions.

(b) Tensor perturbation. For each $\widehat{\mathbf{k}}=\mathbf{k}\mathrm{/}k$, $k>0$, choose a local polarization basis $e_{ij}^s(\widehat{\mathbf{k}})$, $s\in \{+,\times \}$, spanning the two-dimensional transverse-traceless subspace and satisfying ${\widehat{k}}_i{e}_{ij}^s=0,{\delta }_{ij}{e}_{ij}^s=0,e_{ij}^s{e}_{ij}^{s^{\mathrm{\prime }}\ast }=2{\delta }_{s{s}^{\mathrm{\prime }}}\mathrm{.}$ With $h_{ij}(\mathbf{k})$ the Fourier transform of the tensor part in (a), reality requires $h_{ij}(-\mathbf{k})=\overline{h_{ij}(\mathbf{k})}$; choose paired polarization bases at opposite directions so their amplitudes obey this relation. Define the polarization amplitudes by $h_{ij}(\mathbf{k})={\sum }_{s=+,\times }{e}_{ij}^s(\widehat{\mathbf{k}})h_{\mathbf{k}}^s\mathrm{.}$

(c) Fourier and spectrum convention. Let $\mathcal{S}({\mathbb{R}}^3)$ denote the complex smooth functions $f$ for which every seminorm ${\sup }_x\mathrm{\mid }{x}^{\alpha }{\mathrm{\partial }}^{\beta }f(x)\mathrm{\mid }$ is finite, for multi-indices $\alpha ,\beta \in {\mathbb{N}}_0^3$, with the topology generated by those seminorms. A tempered distribution is a continuous complex linear functional on this test space; a real distribution obeys $Z(\bar{f})=\overline{Z(f)}$. On the fixed slice, treat $\zeta$ and $h_{ij}$ as real tempered distributions on ${\mathbb{R}}^3$ (or random tempered distributions); Fourier transforms and the equalities below are understood distributionally. For any perturbation $f$, define $f_{\mathbf{k}}:={\int }_{{\mathbb{R}}^3}{e}^{-i\mathbf{k}\cdot \mathbf{x}}f(\mathbf{x})d^{3}x$ in the tempered-distribution sense and use the inverse convention $f(\mathbf{x})=\int \frac{d^{3}k}{(2\pi {)}^3}{e}^{i\mathbf{k}\cdot \mathbf{x}}{f}_{\mathbf{k}}\mathrm{.}$ The omitted zero mode is part of the background. Let the specified classical ensemble be a probability law on these fields, with $⟨\cdot ⟩$ its expectation. Require measurable smeared fields, finite second moments, and continuous one-point and jointly continuous bilinear two-point functionals on $\mathcal{S}$; require a supplied tempered two-point distribution on ${\mathbb{R}}^3\times {\mathbb{R}}^3$ for each component pair whose pairing with $f\otimes g$ equals the corresponding second moment. For a quantum model, let $\mathfrak{A}$ be its specified unital $\ast$-algebra of perturbation observables and $\omega :\mathfrak{A}\to \mathbb{C}$ its specified state, meaning a positive linear functional with $\omega (1)=1$; supply complex linear smearing maps $Z,H_{ij}:\mathcal{S}({\mathbb{R}}^3)\to \mathfrak{A}$ with $Z(f{)}^{\ast }=Z(\bar{f})$ and $H_{ij}(f{)}^{\ast }=H_{ij}(\bar{f})$, representing the quantized linear perturbations $\zeta ,h_{ij}$ of (a)-(b), with the tensor symmetry, transverse and trace constraints imposed distributionally. Require $f\mapsto \omega (Z(f))$ and its tensor analogues to be continuous, and $(f,g)\mapsto \omega (A(f)B(g))$ jointly continuous for every pair of components $A,B\in \{Z,H_{ij}\}$. Also supply a tempered distribution $W_{AB}$ on ${\mathbb{R}}^3\times {\mathbb{R}}^3$ for every component pair with $W_{AB}(f\otimes g)=\omega (A(f)B(g))$. These are explicit correlation-data requirements, not an asserted existence theorem for an arbitrary algebra and state; no pointwise mode is an algebra element. Write $⟨A{⟩}_{\omega }:=\omega (A)$ in the order displayed. In the classical branch use $Z(f):=\zeta (f)$ and $H_{ij}(f):=h_{ij}(f)$ as random variables. For either branch define the Fourier-smeared field by $\widehat{Z}(f)=Z(\mathcal{F}f)$, where $(\mathcal{F}f)(x)=\int e^{-ik\cdot x}f(k)d^{3}k$, and likewise for $H$. The notation $⟨{\zeta }_{\mathbf{k}}{\zeta }_{{\mathbf{k}}^{\mathrm{\prime }}}⟩$ below denotes the distribution whose pairing with $f(k)g(k^{\mathrm{\prime }})$ is $\omega (\widehat{Z}(f)\widehat{Z}(g))$ (classical expectation in the ensemble branch). Define connected pairings by subtracting the product of one-point pairings. Polarization amplitudes mean contraction of ${\widehat{H}}_{ij}$ with $e_{ij}^{s\ast }\mathrm{/}2$ on smooth paired polarization patches away from $k=0$, using compactly supported tests there; all displayed polarization identities are in this patchwise distributional sense. In either case the formulas use connected two-point expectations, $⟨AB{⟩}_c:=⟨AB⟩-⟨A⟩⟨B⟩$. For the classical ensemble and quantum-state branches, the slice quantities in (a) are represented at the level of their smearings by the supplied distributional fields and correlation data; this does not construct a pointwise perturbed metric or normal from arbitrary distributional data. Statistical homogeneity and isotropy mean invariance of these two-point distributions under translations and rotations of ${\mathbb{R}}^3$. Assume their nonzero-mode spectral measures have the displayed nonnegative, locally integrable density functions ${\mathcal{P}}_{\zeta }(k)$ and ${\mathcal{P}}_t^s(k)$ for $k>0$, with specified pointwise representatives wherever pivot values are used; any infrared regulator or singular spectral part is separately stated. The Dirac distribution is normalized by ${\int }_{{\mathbb{R}}^3}{\delta }^3(\mathbf{k})\phi (\mathbf{k})d^{3}k=\phi (0)$ for test functions $\phi$. The covariance conventions are $⟨{\zeta }_{\mathbf{k}}{\zeta }_{{\mathbf{k}}^{\mathrm{\prime }}}{⟩}_c=(2\pi {)}^3{\delta }^3(\mathbf{k}+{\mathbf{k}}^{\mathrm{\prime }})\frac{2{\pi }^2}{k^3}{\mathcal{P}}_{\zeta }(k),$ $⟨h_{\mathbf{k}}^s{h}_{{\mathbf{k}}^{\mathrm{\prime }}}^{s^{\mathrm{\prime }}}{⟩}_c=(2\pi {)}^3{\delta }^3(\mathbf{k}+{\mathbf{k}}^{\mathrm{\prime }}){\delta }_{s{s}^{\mathrm{\prime }}}\frac{2{\pi }^2}{k^3}{\mathcal{P}}_t^s(k)\mathrm{.}$ The displayed tensor covariance is the adopted polarization-diagonal convention. In the real $+,\times$ basis, rotational invariance together with this diagonal form entails ${\mathcal{P}}_t^{+}(k)={\mathcal{P}}_t^{\times }(k)$. If tensor polarizations have a cross-covariance, specify the full polarization matrix and define the total spectrum as its trace under this normalization; an isotropic parity-odd component must be represented in that full matrix. If tensor polarizations are cross-correlated or have a non-diagonal covariance, specify the full polarization matrix and define the total spectrum as its trace under this normalization. This convention sets scalar-tensor cross-correlations to zero; any model retaining them must state the cross-spectrum separately. Define ${\mathcal{P}}_t(k):={\sum }_{s=+,\times }{\mathcal{P}}_t^s(k)$, so both tensor polarizations are included exactly once. All these spectra are dimensionless in the stated Fourier convention. Under a coordinate rescaling $\mathbf{y}=L_0\mathbf{x}$, $\tilde{a}=a\mathrm{/}{L}_0$, $\tilde{k}=k\mathrm{/}{L}_0$ for $L_0>0$, the spectral arguments and pivot rescale together while ${\mathcal{P}}_{\zeta }$ and ${\mathcal{P}}_t$ remain unchanged.

(d) Pivot observables. Choose and report a pivot $k_{\ast }>0$ in the same Fourier convention at which the selected representative of ${\mathcal{P}}_{\zeta }$ is positive and differentiable with respect to $\ln k$, and the selected representative of ${\mathcal{P}}_t$ has a finite value. A locally integrable spectral density specified only up to almost-everywhere equality does not determine $r$ at a single pivot; the pointwise representative there is part of the supplied spectral data. Define $A_s:={\mathcal{P}}_{\zeta }(k_{\ast }),n_s-1:={\frac{d\ln {\mathcal{P}}_{\zeta }}{d\ln k}\mid }_{k=k_{\ast }},r:=\frac{{\mathcal{P}}_t(k_{\ast })}{{\mathcal{P}}_{\zeta }(k_{\ast })}\mathrm{.}$ For example, if a source quotes ${\tilde{k}}_{\ast }$ in inverse-length coordinates $\mathbf{y}=L_0\mathbf{x}$, then the corresponding dimensionless pivot here is $k_{\ast }=L_0{\tilde{k}}_{\ast }$. Any slow-roll expansion, approximation order, number-of-e-folds convention, reheating map, late-time transfer function, likelihood, prior, data set and uncertainty convention is separate explicitly declared input. These are primordial model definitions, not an inflationary prediction, CMB measurement or fit.

Four-dimensional Einstein-frame multi-scalar cosmology EFT data, FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter