Four-dimensional Einstein-frame multi-scalar cosmology EFT data

Fix the Lorentzian and FLRW conventions of FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter and the equation names of The Friedmann, acceleration and continuity equations of an FLRW model datum. A four-dimensional Einstein-frame multi-scalar cosmology EFT datum consists of the following explicitly supplied data.

(a) Spacetime and cosmological branch. In the general branch supply an oriented, time-oriented smooth Lorentzian four-manifold as the data $(\mathcal{X},g,\mathcal{N},\mathfrak{o})$: $\mathcal{X}$ is a smooth four-manifold without boundary, $g$ is a smooth symmetric nondegenerate covariant $2$-tensor of signature $(-,+,+,+)$, $\mathcal{N}$ is a smooth vector field with $g(\mathcal{N},\mathcal{N})<0$ selecting the future cones, and $\mathfrak{o}$ is an orientation of $\mathcal{X}$. The time orientation and orientation are independent supplied choices, as in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model and Oriented smooth manifolds and oriented charts. When the cosmological branch is FLRW, give a nonempty open interval $I\subseteq \mathbb{R}$, an oriented smooth three-manifold $\mathrm{\Sigma }$, a Riemannian metric $\gamma$ of constant normalized curvature $k\in \{-1,0,1\}$, and a smooth positive scale factor $a:I\to (0,\mathrm{\infty })$; then $\mathcal{X}=I\times \mathrm{\Sigma }$ and $g=-c^{2}d{t}^2+a(t{)}^2\gamma$ in product charts, as in FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter. Use the product orientation determined by the standard increasing orientation on $I$ and the supplied orientation on $\mathrm{\Sigma }$, and take $\mathcal{N}=n=c^{-1}{\mathrm{\partial }}_t$ from FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter, which is smooth, future-directed and timelike. A spatially flat branch means $k=0$. The choice of spacetime metric and its regularity are part of the input, not a consequence of the scalar data.

(b) Scalar target and fields. A finite-dimensional smooth manifold $\mathcal{T}$ of dimension $N\ge 0$ (Smooth manifolds and their smooth charts), a smooth positive-definite Riemannian metric $G$ on $\mathcal{T}$ (Riemannian metric and riemannian manifold), a smooth potential $V:\mathcal{T}\to \mathbb{R}$, and a smooth field map $\varphi :\mathcal{X}\to \mathcal{T}$ ($C^r$ and smooth maps between smooth manifolds). In a target chart $(q^1,\dots ,q^N)$ write $G_{ij}(q)$ for the metric components and ${\varphi }^i=q^i\circ \varphi$. The target metric, its inverse $G^{ij}$, and the potential are chart data for tensorial objects on $\mathcal{T}$; the field path or map need not lie in one chart globally. This is the finite-dimensional sigma-model datum; the single-real-scalar definition in The scalar-field matter model of the classical gravitational framework supplies only the one-field convention and is not a definition of this multi-field target.

(c) Normalization and frame. Adopt $\mathrm{\hslash }=c=1$ for this EFT notation. Supply a constant reduced Planck mass $M_P>0$ and use the Einstein-frame two-derivative density ${\mathcal{L}}_{\mathrm{E}\mathrm{F}\mathrm{T}}=\frac{M_P^2}{2}R-\frac{1}{2}{G}_{ij}(\varphi )g^{\mu \nu }{\mathrm{\partial }}_{\mu }{\varphi }^i{\mathrm{\partial }}_{\nu }{\varphi }^j-V(\varphi )+{\mathcal{L}}_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}\mathrm{.}$ Here $R$ is the scalar curvature of the metric-selected connection of $g$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors), $g^{\mu \nu }$ is the inverse spacetime metric (The inverse metric, musical isomorphisms and index raising), $\sqrt{-g}{d}^{4}x$ is the local expression for the metric volume density (The metric volume form, current flux through 3-faces and finite-chain charges), the target indices are contracted with $G$, while ${\mathcal{L}}_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}$ is a real scalar function on $\mathcal{X}$ of mass dimension four, specified by a local rule as follows. Supply the complete additional field list, each field's domain, codomain or component transformation law, regularity, couplings and units. Supply an integer $m\ge 0$ and a coordinate formula for ${\mathcal{L}}_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}[g,\varphi ,\mathrm{\Psi }](x)$ depending smoothly on the values at $x$ of those fields and their derivatives through order $m$ on a stated open set of component data. Here $\mathrm{\Psi }$ denotes that supplied additional field list. Require the formulas in overlapping spacetime and field charts to agree as a scalar, and require field regularity at least $C^m$, so the resulting scalar is continuous. If an application differentiates this term or its action, it must supply the stronger regularity and admissible variations needed for that operation. The zero-sector choice is the identically zero rule with an empty additional field list. A stress tensor or equation-of-state list alone is not an action term: a sector supplied only by equations must be declared separately as an equation-level source and is not inserted into ${\mathcal{L}}_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}$. In target coordinates of mass dimension one, the declared convention is $[G_{ij}]=1$, $[{\varphi }^i]=[M_P]=[{\mathrm{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}]=\text{mass}$, $[V]=[{\mathcal{L}}_{\mathrm{E}\mathrm{F}\mathrm{T}}]={\text{mass}}^4$, and the action is dimensionless. If dimensions are assigned differently to target coordinates, the dimensions of $G_{ij}$ are transformed accordingly. The Einstein-frame condition is that the coefficient of $R$ is the constant $M_P^2\mathrm{/}2$; any frame transformation from another description is separately specified as part of the input.

(d) EFT domain and auxiliary data. Give a positive cutoff ${\mathrm{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}$, the spacetime or field range on which the truncation is used, and the omitted-mode condition. Express cutoff and threshold scales in the same spacetime metric used to define the observer and modes; by default this is the Einstein-frame metric $g$ of clause (c), unless a separately supplied frame transformation is applied. Whenever an application compares local mode frequencies or spatial momenta with these scales, it must supply the observer and mode prescription. One permitted local prescription on a Lorentzian region is a smooth future-directed unit timelike vector field $n$ with $g(n,n)=-1$. Its maximal integral curves, parametrized by proper time with tangent $n$, are the chosen observer worldlines; together they form the chosen observer congruence. Specify a local oscillatory mode by its real phase $\mathrm{\Theta }$ and wave covector ${\xi }_{\mu }:={\mathrm{\partial }}_{\mu }\mathrm{\Theta }$ in the approximation being used. Define the measured frequency and spatial momentum magnitude by ${\omega }_n:=-n^{\mu }{\xi }_{\mu },\mathrm{\mid }p{\mathrm{\mid }}_n:=\sqrt{(g^{\mu \nu }+n^{\mu }{n}^{\nu }){\xi }_{\mu }{\xi }_{\nu }}\mathrm{.}$ If the modes are not assigned local wave covectors, instead give the precise spectral operator, its domain and boundary conditions, and the spectral subset retained by the EFT. In either case the application states which frequency/momentum or spectral quantities are required to lie below ${\mathrm{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}$ and every omitted-mode threshold. For a symmetric cutoff on modes of both phase signs, compare $\mathrm{\mid }{\omega }_n\mathrm{\mid }$; any one-sided frequency convention is stated explicitly. For example, in the spatially flat FLRW branch with $g=-d{t}^2+a(t{)}^2{\delta }_{ij}d{x}^{i}d{x}^j$ and dimensionless $x^i$, the comoving observers have $n={\mathrm{\partial }}_t$ and a Fourier mode of dimensionless wave number $k$ has physical spatial momentum $k\mathrm{/}a(t)$. A general Lorentzian metric alone selects no such observer or mode prescription. Any stronger derivative, curvature, string-loop, ${\alpha }^{\mathrm{\prime }}$ or backreaction bound needed by a particular application is additional model data. List all initial and boundary data and every retained fluid, gauge field and source explicitly; absence means that the sector is set to zero, not that it is unspecified.

(e) Well-definedness and status. Under a target-chart change, $G_{ij}(\varphi ){\mathrm{\partial }}_{\mu }{\varphi }^i{\mathrm{\partial }}_{\nu }{\varphi }^j$ is the pullback-metric contraction and $V(\varphi )$ is a scalar, so the displayed scalar density is independent of target coordinates. The oriented metric volume form $\sqrt{-g}{d}^{4}x$ is coordinate invariant. On a noncompact spacetime, use the action only through differences under compactly supported smooth variations; the local rules above make the difference density continuous and compactly supported, hence integrable with finite integral. This definition fixes data and conventions only. It asserts neither stationarity nor field equations, existence or uniqueness of solutions, a likelihood, an observation, or a string compactification realizing the data.

FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter, The Friedmann, acceleration and continuity equations of an FLRW model datum, The scalar-field matter model of the classical gravitational framework, Smooth manifolds and their smooth charts, Riemannian metric and riemannian manifold, $C^r$ and smooth maps between smooth manifolds, The inverse metric, musical isomorphisms and index raising, The metric volume form, current flux through 3-faces and finite-chain charges, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Oriented smooth manifolds and oriented charts