Semiclassical Friedmann Equations

A semiclassical Friedmann equation is closed only after specifying more than a scale factor. One needs renormalized gravitational couplings, a quantum field theory and state compatible with the spatial symmetry, a prescription for its stress tensor, and enough initial data for every retained derivative. The result couples a mean FLRW geometry to a conserved expectation value; it is not a mode sum inserted into an otherwise classical equation.

Helpful background. Quantum Fields in Curved Spacetime supplies Hadamard states and stress renormalization; Einstein Equations supplies the constraint structure; Black-Hole Geometry and Mechanics supplies horizon normalization analogies; Semiclassical Gravity supplies the mean-field approximation; and Causality and Global Structure supplies domains of dependence.

FLRW data and the renormalized source

Take cosmic proper time $t$ and a time-independent spatial metric $d\Sigma_k^2$ of normalized constant curvature $k=0,\pm1$:

$$ds^2=-dt^2+a^2(t)d\Sigma_k^2, \qquad H=\frac{\dot a}{a}, \qquad X=H^2+\frac{k}{a^2}.$$

Spatial homogeneity and isotropy require

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

That tensor depends on the field content, masses and couplings, the state and its initial-time prescription, and finite local counterterms. A useful effective action through curvature-squared order is

$$S_{\rm grav}^{\rm ren}=\int d^4x\sqrt{-g} \left[\frac{R-2\Lambda_{\rm ren}}{16\pi G_{\rm ren}} +\alpha_{\rm ren}C_{abcd}C^{abcd} +\beta_{\rm ren}R^2\right].$$

Its variation gives conserved local tensors. In conformally flat FLRW the Weyl tensor itself vanishes, although the full renormalized stress can still contain anomaly and state-dependent pieces. Wald’s axiomatic analysis explains why differences between admissible prescriptions are conserved local curvature tensors rather than arbitrary functions Wald 1977, pp. 1–19.

The constraint and evolution equations

Move the curvature-squared variations to an effective local density and pressure, $\rho_{\rm loc}$ and $p_{\rm loc}$. At one-loop mean-field order the $00$ equation is

$$3X=\Lambda_{\rm ren} +8\pi G_{\rm ren}(\rho_{\rm state}+\rho_{\rm loc}),$$

while the spatial equation may be written

$$-2\dot H+\frac{2k}{a^2} =8\pi G_{\rm ren} \bigl(\rho_{\rm state}+p_{\rm state} +\rho_{\rm loc}+p_{\rm loc}\bigr).$$

The first is a constraint on initial data. Where $H\ne0$, the second follows by differentiating it and using conservation, provided the retained local tensors and state-dependent stress are separately or jointly conserved. At a bounce or turnaround $H=0$, this argument divides by zero; use the spatial equation directly, or a smooth limit from a neighboring solution known to obey it. Inserting a time-dependent density without its corresponding pressure generally violates the Bianchi identity.

Curvature-squared terms add higher time derivatives. When those terms are small perturbative corrections, order reduction evaluates them on the lower-order solution and removes the nonperturbative high-frequency branches. Parker and Simon derive this procedure Parker and Simon 1993, §§ II–IV, PDF. A deliberately large $R^2$ coefficient can instead introduce an additional scalar degree of freedom below the cutoff; retaining it requires an explicit theory and its initial data. It should not be discarded merely because the equations are fourth order. Nor should every formal root of a perturbatively truncated action be treated as physical.

State, scheme, and initial-data checklist

To close the system, record:

• $a(t_0)$ and $H(t_0)$ subject to the $00$ constraint, plus only the derivatives required by the chosen order-reduced formulation;
• the field species, masses, curvature couplings, and quantum state, including the initial hypersurface and adiabatic or Hadamard order;
• $G_{\rm ren}$, $\Lambda_{\rm ren}$, $\alpha_{\rm ren}$, and $\beta_{\rm ren}$ at a stated subtraction scale;
• the decomposition of the stress into state-dependent and local terms, without treating that decomposition as uniquely observable; and
• a control hierarchy such as $|R|/M_*^2\ll1$, physical frequencies below $M_*$, stress fluctuations small enough for a mean equation, and secular corrections below the leading solution.

Only the total left-plus-right equation is invariant under a finite counterterm shift. If $\beta_{\rm ren}$ changes by $\Delta\beta$, the stress definition changes by the negative variation of $\Delta\beta\int\sqrt{-g}R^2$ so that predictions at fixed measured couplings agree.

State data cannot be replaced by an equation of state guessed from symmetry. Two homogeneous Hadamard states may have the same local geometry but different particle content and therefore different $\rho_{\rm state}(t)$. In a free-field calculation, one may evolve mode functions from normalized adiabatic initial data and subtract the same adiabatic orders used to define the gravitational couplings. In an interacting theory, an in-in or closed-time-path effective equation is needed so that the source at time $t$ depends causally on earlier data. Either construction must verify the Wronskian or canonical normalization and the renormalized conservation law. This is what turns the displayed Friedmann equation from a formal relation into a predictive initial-value problem.

For example, in the spatially flat case a free scalar mode $u_q(t)e^{i\mathbf q\cdot\mathbf x}$ obeys

$$\ddot u_q+3H\dot u_q+ \left(\frac{q^2}{a^2}+m^2+\xi R\right)u_q=0, \qquad a^3(u_q\dot u_q^*-\dot u_q u_q^*)=i.$$

Here $q$ is comoving wave number, $q/a$ is physical wave number, and $R=6(\dot H+2H^2)$. With the standard $d^3q/(2\pi)^3$ mode measure, the Wronskian fixes the canonical commutator. Initial mode values and occupation/correlation data specify the state; evolving only $a$ without these data does not close the quantum problem. Fourth-order adiabatic subtraction renormalizes the stress in this four-dimensional setting, while full Hadamard regularity is a stronger ultraviolet condition on the state than merely naming a finite adiabatic order.

Independent limits

With $\rho_{\rm state}=\rho_{\rm loc}=0$, $k=0$, and $\Lambda_{\rm ren}=0$, the constraint gives $H=0$: Minkowski space is recovered. With local quantum terms omitted and a conserved perfect fluid retained, the standard classical Friedmann pair returns. Dimensional consistency also works: every term in the constraint has dimension length$^{-2}$ because $G\rho$ does.

The approximation fails if curvature or state frequencies approach the EFT scale, if the chosen state ceases to be regular, if stress fluctuations rival the mean, or if an order-reduced correction becomes comparable to the leading solution. A formal solution beyond any of those points is not a self-consistent cosmology.

Exercise

For $k=0$ and constant $G$, suppose the total renormalized source has $p=w\rho$ and $\Lambda=0$. Derive $\dot H$ from the constraint and conservation. What happens for $w=-1$?

Solution

Differentiate $3H^2=8\pi G\rho$ to obtain $6H\dot H=8\pi G\dot\rho$. Conservation gives $\dot\rho=-3H(1+w)\rho$. For $H\ne0$, substitution and the constraint yield

$$\dot H=-4\pi G(1+w)\rho=-\frac32(1+w)H^2.$$

For $w=-1$, the density is constant and $\dot H=0$. Positive $\rho$ gives an expanding or contracting de Sitter flat patch; $\rho=0$ gives Minkowski space, and negative $\rho$ has no real flat solution with $\Lambda=0$. At an isolated point with $H=0$, use the spatial equation before division. The derivation verifies the relative sign of that equation within its stated domain.

References

• 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.
• Wald, R. M. (1977). “The Back Reaction Effect in Particle Creation in Curved Spacetime.” Communications in Mathematical Physics 54, 1–19. DOI.