Cosmology and Large-Scale Readout

12.1 Role and Scope

Chapter 12 develops the cosmological readout of the theory: the FLRW background, gauge-invariant perturbations, thermal history, and the routes to CMB, large-scale-structure, lensing and gravitational-wave observables, all on the solved gravitational branch of Chapters 7–8.

Cosmology is the large-scale Tier-1 readout of a compatible geometry, matter/QFT state, any record-exclusive contribution, parameter branch, and initial-data class. It is not assumed at the source level. A physical cosmological branch must solve the background and perturbation equations on the same geometry, preserve the gravitational Ward identities, and connect consistently to thermal history and light-cone observables.

12.2 Effective field equation and exact-once accounting

The cosmological branch inherits

$$G_{\mu\nu}[g]+\Lambda_{\rm br}g_{\mu\nu} +\mathcal Q_{\mu\nu}^{\rm grav} =8\pi G_R\left( T_{\mu\nu}^{M,\rm dyn} +T_{\mu\nu}^{\rm QFT,dyn} +T_{\mu\nu}^{R,\rm tot} +\chi_{\rm DM}T_{\mu\nu}^{\rm DM} +\chi_{\rm DE}T_{\mu\nu}^{\rm DE} \right).$$

The matter and quantum terms are the variations of the disjoint $S_M^{\rm dyn}$ and state-dependent $\Gamma_{\rm QFT,dyn}^{\rm ren}$ defined in Chapter 9. If a species is treated by the quantum functional, it is absent from $S_M^{\rm dyn}$. Pure-metric local terms from QFT renormalization occur only through $\Lambda_{\rm br}$, $G_R$, and $\Gamma_{\rm high}^{\rm grav}$. The record functional is the matched remainder after subtracting every term already represented in $S_M^{\rm dyn}$ or $\Gamma_{\rm QFT,dyn}^{\rm ren}$, with the EFT matching datum $\mathfrak m_R$ fixed before $\operatorname{Sol}_G(\cdot;\mathfrak m_R)$ is evaluated. Archive kinematics does not automatically supply an additional stress: if the record variables merely reparameterize already owned degrees of freedom, then $\Gamma_{R,\rm CTP}=0$ and $T_{\mu\nu}^{R}=0$. Here $\chi_{\rm DM},\chi_{\rm DE}\in\{0,1\}$. A switch is one only when its separately owned covariant action or kinetic functional contains degrees of freedom absent from both the matter and QFT terms, and is zero otherwise. Thus $T_{\mu\nu}^{\rm DM}$ and $T_{\mu\nu}^{\rm DE}$ are action-derived branch stresses rather than additional phenomenological copies.

Here

$$T_{\mu\nu}^{R,\rm tot} :=T_{\mu\nu}^{R}+\chi_{MR}T_{\mu\nu}^{MR}, \qquad \chi_{MR}\in\{0,1\},$$

where $T_{\mu\nu}^{R}$ is only the record-exclusive matched remainder. The interaction switch $\chi_{MR}$ is one only for a separately owned matter–record interaction absent from the matter, QFT and record-exclusive functionals; otherwise it is zero. Likewise $\chi_{\rm DE}=1$ only on a separately owned dynamical dark-energy branch and is zero on a pure-$\Lambda$ branch. The tensor $\mathcal Q_{\mu\nu}^{\rm grav}$ is obtained by varying the pure-metric higher-curvature action $\Gamma_{\rm high}^{\rm grav}$; it is not also inserted as an independent fluid or QFT stress.

Covariance gives

$$\nabla^\mu T_{\mu\nu}^{(A)}=J_\nu^{(A)}, \qquad \sum_AJ_\nu^{(A)}=0,$$

together with the interior Noether identity $\nabla^\mu\mathcal Q_{\mu\nu}^{\rm grav}=0$, with boundary fluxes cancelled by the boundary completion of $\Gamma_{\rm high}^{\rm grav}$. Every dynamical contribution appears exactly once in this equation and once in its perturbation.

12.3 Curvature and Friedmann conventions

Write

$$\mathcal K_{\rm FLRW} = \frac{k_{\rm FLRW}}{a_0^2}, \qquad \Omega_{k0} = -\frac{k_{\rm FLRW}}{a_0^2H_0^2}, \tag{12.1}$$

and, for $\bar a=a/a_0$,

$$\Omega_k(\bar a) = \frac{\Omega_{k0}\bar a^{-2}}{E^2(\bar a)}. \tag{12.2}$$

The canonical background equation is

$$E^2(\bar a) = \Omega_{\Lambda,\rm br} +\sum_{i\in\rm ordinary}\widehat\Omega_i(\bar a) +\Omega_{k0}\bar a^{-2} +\widehat\Omega_{\rm corr}(\bar a). \tag{12.3}$$

A hatted term means an additive contribution to $E^2$. There is no simultaneous $\widehat\Omega_{\rm QG}$, $\widehat\Omega_{\rm rec}$, and $\widehat\Omega_{\rm corr}$ representation of the same action terms.

Explicitly,

$$\rho_{\rm crit,0}=\frac{3H_0^2}{8\pi G_R}, \qquad \widehat\Omega_{\rm corr}(\bar a) =\frac{\rho_{\rm corr}(\bar a)}{\rho_{\rm crit,0}}, \tag{12.4}$$

and the $a=a_0$ closure relation is

$$1 = \Omega_{\Lambda,\rm br} +\sum_{i\in\rm ordinary}\Omega_{i0} +\Omega_{k0} +\widehat\Omega_{\rm corr,0}. \tag{12.5}$$

The acceleration equation is

$$\frac{\ddot a}{a} = \frac{\Lambda_{\rm br}}3 -\frac{4\pi G_R}{3} \sum_{i\in\rm ordinary}(\rho_i+3p_i) -\frac{4\pi G_R}{3} (\rho_{\rm corr}+3p_{\rm corr}). \tag{12.6}$$

Perturbation wavenumbers are $k_{\rm com}$. Tensor equations use $k_{\rm FLRW}$ for background curvature; an unsubscripted $K$ is never used.

With tensor harmonics normalized by $-D^2Q_{ij}^{(\lambda)} =k_{\rm com}^2Q_{ij}^{(\lambda)}$, so that the covariant tensor operator is $-(D^2-2k_{\rm FLRW})$, the canonical tensor equation is

$$h_\lambda''+2\mathcal H h_\lambda' +(k_{\rm com}^2+2k_{\rm FLRW})h_\lambda =16\pi G_Ra^2\Pi_\lambda^{\rm tot} +S_\lambda^{\rm EFT}. \tag{12.7}$$

$\Pi_\lambda^{\rm tot}$ and $S_\lambda^{\rm EFT}$ obey the same exactly-once action ownership as the effective field equation at the head of this chapter.

12.4 Critical Density, Density Parameters, and Closure Relation

The present critical density is

$$\boxed{ \rho_{crit,0} = \frac{3H_0^2}{8\pi G}. } \tag{12.8}$$

For a density branch $i$,

$$\boxed{ \Omega_i = \frac{\rho_{i0}}{\rho_{crit,0}}. } \tag{12.9}$$

The curvature density parameter $\Omega_{k0}$ is fixed by (12.1), and the general closure relation is (12.5). In the seed large-scale branch where QG corrections are negligible or absorbed into $\Omega_C$,

$$\boxed{ 1 = \Omega_{\Lambda,\rm br} +\sum_{i\in\rm ordinary}\Omega_{i0} +\Omega_{k0} \qquad(\widehat\Omega_{\rm corr,0}\ \text{negligible or absorbed by declared branch}). } \tag{12.10}$$

Chapter 12 uses (12.5) as the general accounting form and (12.10) as the seed readout branch.

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Density parameters follow the normalized conventions of the background section: hatted quantities $\widehat\Omega_I(\bar a)=\rho_I(\bar a)/\rho_{\rm crit,0}$ are the present-critical additive contributions entering $E^2(\bar a)=H^2/H_0^2$; instantaneous fractions are $\Omega_I=\widehat\Omega_I/E^2$; the curvature contribution to $E^2$ is $\Omega_{k0}\bar a^{-2}$ with $\Omega_{k0}=-k_{\rm FLRW}/(a_0^2H_0^2)$, while the fractional curvature density is $\Omega_k(\bar a)=\Omega_{k0}\bar a^{-2}/E^2(\bar a)$; and the sole additive correction contribution is $\widehat\Omega_{\rm corr}=\rho_{\rm corr}/\rho_{\rm crit,0}$; any dimensionful correction entering $H^2$ is normalized by $H_0^2$ before it appears; unnormalized equalities such as $\Omega_{\rm QG}=\Delta_{\rm QG}$ are never used.

12.5 Continuity Equation and Scaling Laws

For the declared branch-owned components $A$, including the record and correction accounts, the component continuity equations with exchange are

$$\dot\rho_A+3H(\rho_A+p_A)=Q_A, \qquad \sum_AQ_A=0, \tag{12.11}$$

the FLRW form of the covariant exchange identity of the effective field equation; no additional record or correction exchange term is counted beside the component sum. For each independently conserved perfect-fluid component,

$$\boxed{ \dot\rho_i+3H(\rho_i+p_i)=0. } \tag{12.12}$$

More generally, for an independently conserved component with equation of state $w_I(\bar a)=p_I/\rho_I$,

$$\rho_I(\bar a)=\rho_{I0}F_I(\bar a), \qquad F_I(\bar a) = \exp\left[-3\int_1^{\bar a} \left(1+w_I(\bar a')\right)d\ln\bar a'\right]. \tag{12.13}$$

Using $p_i=w_i\rho_i$ with constant $w_i$,

$$\dot\rho_i+3H(1+w_i)\rho_i=0.$$

Since $H=\dot a/a$,

$$\frac{d\rho_i}{\rho_i} = -3(1+w_i)\frac{da}{a}.$$

Therefore

$$\boxed{ \rho_i(a)=\rho_{i0}a^{-3(1+w_i)}. } \tag{12.14}$$

The standard scalings are:

$$\boxed{ \rho_m(a)=\rho_{m0}a^{-3} \qquad (w_m=0), } \tag{12.15}$$ $$\boxed{ \rho_r(a)=\rho_{r0}a^{-4} \qquad (w_r=1/3), } \tag{12.16}$$ $$\boxed{ \rho_\Lambda(a)=\rho_{\Lambda0} \qquad (w_\Lambda=-1). } \tag{12.17}$$

The curvature contribution to $E^2$ is the additive term $\Omega_{k0}\bar a^{-2}$, while the instantaneous fractional curvature density is $\Omega_k(\bar a)$ of (12.2); the two are distinct objects in the dimensionless Friedmann readout.

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12.6 Scalar, vector, and tensor perturbations

Use conformal time $d\eta=dt/\bar a$ and the covariant derivative $D_i$ of the reference constant-curvature spatial metric $\gamma_{ij}$, for which ${}^{(3)}R[\gamma]=6\mathcal K_{\rm FLRW}$. Decompose

$$\begin{aligned} ds^2=\bar a^2(\eta)\{&-(1+2\phi)d\eta^2 +2(D_iB+S_i)d\eta\,dx^i\\ &+[(1-2\psi)\gamma_{ij}+2D_iD_jE +2D_{(i}F_{j)}+h_{ij}]dx^idx^j\}, \end{aligned}$$

with

$$D^iS_i=D^iF_i=0, \qquad D^ih_{ij}=0, \qquad \gamma^{ij}h_{ij}=0.$$

Under $\eta\mapsto\eta+T$ and $x^i\mapsto x^i+D^iL+L^i$, with $D_iL^i=0$, define $\sigma=B-E'$. The gauge-invariant Bardeen potentials [48, 49, 50] and vector shear are

$$\Phi=\phi+\mathcal H\sigma+\sigma', \qquad \Psi=\psi-\mathcal H\sigma, \qquad V_i=S_i-F_i', \qquad \mathcal H=\frac{\bar a'}{\bar a}.$$

For each species $A$,

$$\delta\rho_A^{\rm gi}=\delta\rho_A+\bar\rho_A'\sigma, \qquad \delta p_A^{\rm gi}=\delta p_A+\bar p_A'\sigma,$$

and the momentum and anisotropic stress split into scalar, transverse-vector, and transverse-traceless parts. The master perturbation equation is

$$\delta G_{\mu\nu} +\Lambda_{\rm br}\delta g_{\mu\nu} +\delta\mathcal Q_{\mu\nu}^{\rm grav} =8\pi G_R\delta T_{\mu\nu}^{\rm owned},$$

where $\delta T_{\mu\nu}^{\rm owned}$ contains each classical, renormalized-QFT, record-exclusive, separately owned interaction, dark-matter, and dynamical-dark-energy contribution exactly once. Scalar, vector, and tensor equations are obtained by the orthogonal SVT projectors on the chosen harmonic domain.

In the spatially flat, uncorrected and noninteracting limit, introduce the scalar velocity potential by $\delta T^0{}_i=(\bar\rho+\bar p)D_iv^{\rm gi}$. The scalar Einstein equations are

$$D^2\Psi-3\mathcal H(\Psi'+\mathcal H\Phi) =4\pi G_R\bar a^2\delta\rho^{\rm gi},$$ $$\Psi'+\mathcal H\Phi =-4\pi G_R\bar a^2(\bar\rho+\bar p)v^{\rm gi},$$ $$\Psi''+\mathcal H(\Phi'+2\Psi') +(2\mathcal H'+\mathcal H^2)\Phi +\frac13D^2(\Phi-\Psi) =4\pi G_R\bar a^2\delta p^{\rm gi},$$ $$\left(D_iD_j-\frac13\gamma_{ij}D^2\right)(\Phi-\Psi) =8\pi G_R\bar a^2\Pi^{S}_{ij}.$$

For illustration, a separately conserved barotropic species with constant $w_A=c_{s,A}^2$ obeys

$$\delta_A'+(1+w_A)(\vartheta_A-3\Psi')=0,$$ $$\vartheta_A'+\mathcal H(1-3w_A)\vartheta_A +D^2\!\left(\Phi+\frac{c_{s,A}^2}{1+w_A}\delta_A\right) -D^2\sigma_A=0,$$

where $\delta_A=\delta\rho_A^{\rm gi}/\bar\rho_A$, $\vartheta_A$ is the scalar velocity divergence, and $\sigma_A$ is the scalar anisotropic-stress potential. Time-dependent equations of state, entropy pressure, massive kinetic species, exchange currents, and EFT/record corrections are governed by the kinetic equation below and by the perturbation of the same covariant conservation law; they are not represented by adding independent duplicate sources.

For transverse momentum $q_i^V$ and vector anisotropic-stress potential $\Pi_i^V$, defined so that its stress contribution is $D_{(i}\Pi^V_{j)}$, the vector constraint and propagation equations take the convention-fixed form

$$(D^2+2\mathcal K_{\rm FLRW})V_i =-16\pi G_R\bar a^2q_i^V+S_i^{V,\rm grav},$$ $$(\bar a^2V_i)' =16\pi G_R\bar a^4\Pi_i^V+\bar a^2\widetilde S_i^{V,\rm grav}.$$

Consequently, an unsourced perfect-fluid vector shear decays according to

$$(\bar a^2V_i)'=0, \qquad V_i\propto\bar a^{-2}.$$

Any sustained vector mode must therefore identify a transverse momentum, anisotropic stress, topological source, or modified-gravity source in the field equation.

For tensor harmonics satisfying $-D^2Q_{ij}^{(\lambda)}=k_{\rm com}^2Q_{ij}^{(\lambda)}$, write $h_{ij}=\sum_\lambda h_\lambda Q_{ij}^{(\lambda)}$. Then

$$h_\lambda''+2\mathcal Hh_\lambda' +(k_{\rm com}^2+2\mathcal K_{\rm FLRW})h_\lambda =16\pi G_R\bar a^2\Pi_\lambda^{\rm tot} +S_\lambda^{\rm grav}.$$

The curvature symbol $\mathcal K_{\rm FLRW}$ is used consistently here; $k_{\rm com}$ is the perturbation wavenumber.

12.7 Species and kinetic evolution

A kinetic species obeys

$$p^\mu\nabla_\mu f_A -\Gamma^i_{\alpha\beta}p^\alpha p^\beta \frac{\partial f_A}{\partial p^i} =C_A[f]+S_A, \qquad T_A^{\mu\nu}=\int dP\,p^\mu p^\nu f_A.$$

Fluids are the appropriate moment closures of this system. Dark-energy sound speed, nonadiabatic pressure, anisotropic stress, and exchange current, and dark-matter velocity dispersion or distribution function, must be specified on any branch that uses them. Physical perturbations require positive kinetic coefficients, absence of gradient instability, hyperbolicity, and a declared effective-theory range.

12.8 Initial conditions, transfer objects, and predictions

Let $q_A(\mathbf k_{\rm com})$ denote the independent primordial or induced scalar, vector, and tensor modes on a selected initial hypersurface $\Sigma_*$. Their covariance is

$$\langle q_A(\mathbf k_{\rm com})q_B^*(\mathbf k_{\rm com}')\rangle = (2\pi)^3\delta^{(3)}(\mathbf k_{\rm com}-\mathbf k_{\rm com}') \frac{2\pi^2}{k_{\rm com}^3}\mathcal P_{AB}(k_{\rm com}).$$

The branch tag records whether $\mathcal P_{AB}$ is source-derived, generated by a specified primordial mechanism, induced by later readout dynamics, bounded, or calibrated. The transfer relation is

$$X_I(\mathbf k_{\rm com},\eta) = \sum_A T_{IA}(k_{\rm com},\eta;\vartheta,\mathfrak b)q_A(\mathbf k_{\rm com}),$$

where $\vartheta$ is the frozen parameter record and $\mathfrak b$ the frozen branch record. Transfer functions are obtained by solving the coupled Einstein–Boltzmann–record–QG system, not declared independently.

The output supports:

$$P_{IJ}(k_{\rm com},z) = \frac{2\pi^2}{k_{\rm com}^3} \sum_{A,B} T_{IA}(k_{\rm com},z)T_{JB}^*(k_{\rm com},z)\mathcal P_{AB}(k_{\rm com}),$$

growth functions $D_I(k_{\rm com},a)$, logarithmic growth $f_I=d\ln D_I/d\ln a$, CMB source functions, lensing potentials $\Phi+\Psi$, and tensor transfer functions. Every prediction carries the initial-condition, parameter, scheme, nonlinear, baryonic, bias, and nuisance assumptions used to obtain it.

Line-of-sight transfer integration follows [51]; curved-model computation follows [52].

12.9 Light-cone and propagation object

The observation map is evaluated on the solved geometry, not on a background coordinate distance by fiat. For a radial null ray,

$$\chi(z)=\int_0^z\frac{dz'}{H(z')},$$

and the transverse comoving distance is

$$f_{\mathcal K}(\chi)= \begin{cases} \mathcal K_{\rm FLRW}^{-1/2}\sin(\sqrt{\mathcal K_{\rm FLRW}}\,\chi),&\mathcal K_{\rm FLRW}>0,\\ \chi,&\mathcal K_{\rm FLRW}=0,\\ (-\mathcal K_{\rm FLRW})^{-1/2}\sinh(\sqrt{-\mathcal K_{\rm FLRW}}\,\chi),&\mathcal K_{\rm FLRW}\lt 0. \end{cases}$$

The luminosity and angular-diameter distances satisfy

$$d_L=(1+z)f_{\mathcal K}(\chi), \qquad d_A=\frac{f_{\mathcal K}(\chi)}{1+z},$$

only when photon number conservation, metric null propagation, and Etherington reciprocity pass. Otherwise the violated assumption and modified propagation kernel are included in $L_C$.

12.10 Perturbation and Thermal-History Handoffs

The cosmological readout must carry:

$$\mathsf{CosOut} = (\mathsf{Background}, \mathsf{SVT}, \mathsf{Init}, \mathsf{Thermal}, \mathsf{Transfer}, \mathsf{Growth}, \mathsf{Lens}, \mathsf{CMB}, \mathsf{LSS}, \mathsf{GW}). \tag{12.18}$$

Scalar, vector, and tensor variables are gauge invariant or accompanied by a gauge certificate. The initialization type is the tagged disjoint union

$$\mathsf{Init} = \mathsf{Reheat}(T_{\rm reh},w_{\rm reh},\Gamma_{\rm reh}) \amalg \mathsf{BounceMatch}(\Sigma_-,\Sigma_+,\mathcal M_{\rm b}) \amalg \mathsf{BoundaryInit}(\Sigma_i,\rho_i,\mathcal C_i), \tag{12.19}$$

and exactly one tagged branch with a complete matching certificate is required. The typed thermal chain then contains baryogenesis/leptogenesis, BBN, recombination, and structure formation [53, 54, 55, 56, 57]. Dark-energy sound speed and anisotropic stress and dark-matter stress/distribution data carry stability gates. Primordial and induced branches are distinguished.

12.11 Dark-Sector Branches and Chapter 13 Handoff

The closure/dark-energy density branch is

$$\boxed{ \rho_C(a)=\rho_{C0}F_C(a). } \tag{12.20}$$

The cosmological constant branch is

$$\boxed{ F_C(a)=1, \qquad w_C=-1. } \tag{12.21}$$

For an evolving dark-energy branch,

$$\boxed{ F_C(a) = \exp \left[ -3 \int_1^a (1+w_C(a'))\,d\ln a' \right]. } \tag{12.22}$$

Chapter 13 owns the dark-energy dynamics, $\Lambda$ magnitude, dark matter microphysics, $\rho_{prim}$, $\rho_{ind}$, and $\mathcal P_\Lambda^{grav}$. Chapter 12 supplies their large-scale slots and branch notation.

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12.12 Low-Boundary Cosmology

The low-boundary condition is

$$\boxed{ \mathcal B_{cos}^{low} \Rightarrow S_{grav}^{early}\ll S_{grav}^{max}. } \tag{12.23}$$

Here:

• Low-boundary objects:

  • S grav early:

    • Symbol: $S_{grav}^{early}$

    • Type: early-universe gravitational entropy measure or placeholder functional

  • S grav max:

    • Symbol: $S_{grav}^{max}$

    • Type: maximum/reference gravitational entropy

  • Hierarchy:

    • Symbol: $\ll$

    • Type: required gravitational entropy hierarchy for arrow orientation

Equation (12.23) is a low-boundary condition and theorem target. Its source-level origin remains routed through $R6$. Its arrow-of-time consequences are handed to Chapter 14.

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12.13 Cosmology-to-Arrow Handoff

The arrow handoff is

$$\boxed{ H_{12\to14} = ( \mathcal B_{cos}^{low}, S_{grav}^{early}\ll S_{grav}^{max}, a(t), H(a), \text{record/entropy orientation data}, R6 ). } \tag{12.24}$$

Chapter 14 may use this as cosmological input for entropy-arrow and record-orientation formalization.

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12.14 Cosmological Theorem Target

R6 begins with the effective field equation, the record-exclusive ownership match fixed before gravitational solution, a selected vacuum/dark branch, admissible background and perturbation initial data, the conservation identities, and the thermal handoffs above. It asks for a nonempty stable cosmological solution whose background, complete SVT system, transfer functions, light-cone observables, and dark-sector variables are mutually compatible. Let $\ell_Q$ denote the admitted short-distance or EFT length scale controlling the leading higher-curvature expansion on the branch. In the limit

$$\ell_Q^2|\mathcal R|\to0, \qquad T_{\mu\nu}^{R,\rm tot}\to0, \qquad J_\nu^{(A)}\to0,$$

the branch must reduce to the standard stable FLRW–Einstein–Boltzmann system. Numerical spectra, the source of the primordial covariance, and empirical comparison are companion-paper calculations.