Semiclassical Cosmology and Cosmological Horizons

This chapter applies the semiclassical Einstein equation to homogeneous cosmology and to the causal horizons of cosmological observers. Its organizing principle is closure: a scale factor becomes predictive only after the quantum state, renormalized couplings, constraint data, approximation order, and horizon algebra have been specified. The same discipline separates de Sitter equilibrium from evolving FLRW thermodynamics and controlled backreaction from an infrared extrapolation.

Helpful background. Quantum Fields in Curved Spacetime supplies states, anomalies, and renormalized stress; Einstein Equations supplies the Friedmann constraints; Black-Hole Geometry and Mechanics supplies Killing and causal horizons; and Semiclassical Gravity supplies mean-field validity and generalized entropy.

The state–geometry system

For an FLRW line element, symmetry reduces the renormalized source to an energy density and pressure,

$$\langle T^a{}_b\rangle_{\rm ren} =\operatorname{diag}(-\rho,p,p,p), \qquad \dot\rho+3H(\rho+p)=0.$$

The constraint has the schematic form

$$3\left(H^2+\frac{k}{a^2}\right) =\Lambda_{\rm ren} +8\pi G_{\rm ren}\bigl(\rho_{\rm state}+\rho_{\rm local}\bigr).$$

The split on the right is useful but scheme dependent; a finite curvature counterterm shifts local stress and gravitational couplings together. Only the complete equation at fixed measured couplings is invariant. Higher-derivative terms are retained perturbatively or order-reduced so that cutoff-scale roots are not mistaken for new low-energy cosmologies Parker and Simon 1993, §§ II–IV, PDF.

The state supplies more than an equation of state. Mode normalization, initial hypersurface, Hadamard or adiabatic order, and any anomaly auxiliary data determine the state-dependent stress. The geometry then feeds back into that state evolution. Stress conservation and the Friedmann constraint must be monitored together; satisfying only a trace equation or only a mass-shell relation does not close the system.

Six steps through cosmology and horizons

The pages follow the order in which a controlled calculation needs its inputs:

1. Semiclassical Friedmann Equations derives the constraint and evolution equations and inventories state, scheme, coupling, and initial-data requirements.
2. Vacuum Energy and Renormalized Cosmological Constant explains why a regulated zero-point sum is not the observed $\Lambda$ while retaining the genuine radiative-sensitivity problem.
3. Trace-Anomaly-Driven Cosmology derives a state-selected de Sitter fixed-point relation, distinguishes its roots, and tests a declared linear stability model with a four-resolution reproducible benchmark.
4. de Sitter Temperature and Entropy fixes the observer and generator, derives $T=H/(2\pi)$ by Euclidean regularity, and tracks the cosmological first-law sign.
5. Backreaction of Cosmological Quantum Fields compares particle production, infrared accumulation, anomaly stress, causal response, and fluctuation limits.
6. Generalized Laws for Cosmological Horizons specifies the causal horizon and accessible algebra needed for a generalized entropy statement and separates it from apparent-horizon proposals.

The sequence moves from local equations to global observer structure. The anomaly page’s numerical ODE is deliberately a branch and integrator diagnostic, not a simulation of the nonlinear universe. The horizon pages likewise distinguish exact stationary de Sitter thermality from slowly evolving or multi-horizon systems.

Reader routes and cross-checks

For a renormalization route, read pages 1–3 and track how $\Lambda$, $R^2$, the anomaly, and state-dependent stress change under a finite scheme translation. For a backreaction route, read pages 1, 3, and 5 and compare the mean shift with stress fluctuations, response poles, secular parameters, and the first omitted term. For a horizon route, read pages 4 and 6, keeping the observer, normalized generator, horizon type, entropy side, and state fixed.

Every route uses four independent checks:

• conservation must reproduce the evolution equation from the Friedmann constraint where $H\ne0$, with the spatial equation checked separately at a turnaround;
• dimensions and the $H\to0$ limit must recover the appropriate flat-space behavior;
• a finite counterterm shift must leave predictions invariant after couplings are translated; and
• a claimed quantum effect must exceed errors before perturbation, mean-field, adiabatic, or EFT control fails.

Pure de Sitter provides an equilibrium benchmark. The central static observer normalizes $\xi=\partial_t$, obtains $\kappa=H$, and hence $T=H/(2\pi)$; the leading entropy is $A/(4G)$ Gibbons and Hawking 1977, pp. 2739–2743. A generic FLRW apparent horizon lacks this exact Killing normalization and Euclidean construction. Its thermodynamic interpretation is an additional proposal, not a change of notation.

Review the chapter

Consider a flat background with $3H_0^2=8\pi G\rho_0$. A renormalized quantum source has $\delta\rho/\rho_0=0.06$, numerical and subtraction uncertainty $0.004$ in the same ratio, loop parameter $0.12$, and stress-fluctuation ratio $0.10$. Find $\delta H/H_0$ at linear order. Which checks pass, and what would still be needed for a long-time claim?

Solution

Linearizing the constraint gives

$$\frac{\delta H}{H_0} =\frac12\frac{\delta\rho}{\rho_0}=0.03.$$

The mean shift is $0.03$ with uncertainty $0.002$ from the stated density error, so it is numerically distinguishable in this approximation. The loop parameter $0.12$ is below unity but is not by itself an error bound for the observable. Likewise the stress ratio $0.10$ does not determine metric fluctuations: propagate the noise through the retarded response and compare the resulting expansion variance with both the background and the claimed precision. A long-time claim still needs conserved pressure, constraint evolution, state regularity, response stability, refinement, and control of accumulated secular and EFT errors. These data alone neither certify deterministic backreaction nor define a horizon entropy law.

What completion enables

A reader can now formulate a closed semiclassical initial-value problem, distinguish a renormalization convention from the cosmological-constant naturalness problem, interpret anomaly stability coefficients, and normalize de Sitter thermodynamics for a stated observer. The chapter also provides a stop rule: when the state, fluctuations, response, derivative expansion, or horizon definition leaves its declared regime, report a controlled boundary rather than a cosmological conclusion.

References

• Gibbons, G. W., and Hawking, S. W. (1977). “Cosmological Event Horizons, Thermodynamics, and Particle Creation.” Physical Review D 15, 2738–2751. DOI.
• Parker, L., and Simon, J. Z. (1993). “Einstein Equation with Quantum Corrections Reduced to Second Order.” Physical Review D 47, 1339–1355. DOI. Open PDF.