Adopted two-derivative multi-scalar cosmology EFT

Adopt the following model premise for a datum of Four-dimensional Einstein-frame multi-scalar cosmology EFT data, in the classical low-energy regime and in units $\mathrm{\hslash }=c=1$.

(i) Action and variation class. On a smooth spacetime region $\mathcal{U}\subseteq \mathcal{X}$, use action differences from $S_{\mathrm{E}\mathrm{F}\mathrm{T}}={\int }_{\mathcal{U}}\sqrt{-g}[\frac{M_P^2}{2}R-\frac{1}{2}{G}_{ij}(\varphi )g^{\mu \nu }{\mathrm{\partial }}_{\mu }{\varphi }^i{\mathrm{\partial }}_{\nu }{\varphi }^j-V(\varphi )+{\mathcal{L}}_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}]d^{4}x$ only through differences for variations supported in a fixed compact subset of the open region $\mathcal{U}$. An admissible scalar variation at $\varphi$ is a jointly smooth map $\mathrm{\Phi }:(-\epsilon ,\epsilon )\times \mathcal{U}\to \mathcal{T}$, $\epsilon >0$, with $\mathrm{\Phi }(0,x)=\varphi (x)$ and $\mathrm{\Phi }(s,x)=\varphi (x)$ for every $x\mathrm{\notin }K$ and all $s$, for some compact $K\subset \mathcal{U}$; hold the metric and other fields fixed. Write ${\varphi }_s(x):=\mathrm{\Phi }(s,x)$ and define the action difference by integrating the difference of the displayed Lagrangian volume forms at ${\varphi }_s$ and $\varphi$ over $\mathcal{U}$. Stationarity means that the derivative of this difference at $s=0$ is zero for every such variation. In particular, on any open patch whose field image lies in a target chart, every smooth vector of test functions ${\eta }^i$ compactly supported in that patch gives the variation $q^i({\varphi }_s)=q^i(\varphi )+s{\eta }^i$, extended by $\varphi$ elsewhere. For sufficiently small $\mathrm{\mid }s\mathrm{\mid }$ it remains in the chart: the image of the compact support is compact inside the open chart image and $\eta$ is bounded. Thus these local variations are admitted. The common compact support and joint smoothness justify differentiating their action difference under the integral. In the matter-free branch ${\mathcal{L}}_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}=0$, the scalar configuration is dynamically admissible only if it is stationary under every compactly supported smooth target-coordinate variation. Stationarity is adopted as a model premise; the field equation obtained from it is derived in Conditional homogeneous multi-field FLRW equations.

(ii) Gravitational field law. Separately adopt the direct Einstein equation of Adopted Einstein field equation in the direct-equation presentation with cosmological parameter $\mathrm{\Lambda }=0$ and scalar Hilbert stress-energy, with the convention of The Hilbert stress-energy tensor of a metric-dependent action, as source: $G_{\mu \nu }=M_P^{-2}{T}_{\mu \nu }\mathrm{.}$ This is the natural-unit form of the cited law with $M_P^2=(8\pi G_N{)}^{-1}$. For a matter-free model, $T_{\mu \nu }$ is only the stress tensor of the scalar sector. If ${\mathcal{L}}_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}$ is nonzero, its stress tensor and any direct scalar source terms must be stated and the matter-free deductions on the paired page do not apply without a separate argument. The Einstein equation is adopted directly here; the action display fixes the scalar dynamics and Einstein-frame normalization, and this item does not infer the metric equation by varying the displayed curvature term.

(iii) Regime and epistemic status. The premise is restricted to the declared EFT field and energy range. Whenever retained-mode frequencies or spatial momenta are compared with ${\mathrm{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}$ or an omitted-mode threshold, their values are defined by the application-specific observer and mode prescription required by clause (d) of Four-dimensional Einstein-frame multi-scalar cosmology EFT data, and all scales are expressed in the same named metric frame. The prescribed quantities must lie below the declared cutoff and every omitted-mode threshold. Additional derivative, curvature, string-loop, ${\alpha }^{\mathrm{\prime }}$, source, or backreaction bounds are application-specific assumptions, not consequences of this postulate. The adopted action and field law are not derived predictions or reported observations. No existence or uniqueness of solutions, observational fit, or global string compactification is asserted.

Four-dimensional Einstein-frame multi-scalar cosmology EFT data, SI unit and coordinate conventions of the classical gravitational framework, Adopted Einstein field equation in the direct-equation presentation, Adopted spacetime model and classical domain of the gravitational framework, The Hilbert stress-energy tensor of a metric-dependent action