Trace-Anomaly-Driven Cosmology

The trace anomaly converts a classically traceless matter sector into a local curvature source. In an FLRW spacetime it can support approximately de Sitter branches and introduce higher-derivative perturbation modes. Their stability is not fixed by one universal “anomaly sign”: it depends on field-content coefficients, the scheme-dependent $R^2$ coupling, the quantum state or auxiliary-field data, and which roots remain within the effective theory.

Helpful background. Quantum Fields in Curved Spacetime supplies the anomaly and state; Einstein Equations supplies the constraint; Black-Hole Geometry and Mechanics supplies curvature conventions; Semiclassical Gravity supplies order reduction and response; and Causality and Global Structure supplies the initial-value domain.

The anomaly in a conformally flat universe

For conformal matter in four dimensions, write

$$\langle T^a{}_a\rangle_{\rm ren} =\frac{1}{(4\pi)^2} \left(c\,C_{abcd}C^{abcd}-a_{\rm A}\,E_4+b\,\Box R\right),$$

where $E_4=R_{abcd}R^{abcd}-4R_{ab}R^{ab}+R^2$ and $a_{\rm A}$ denotes the type-A anomaly coefficient, distinct from the scale factor $a(t)$. FLRW is conformally flat, so $C_{abcd}C^{abcd}=0$. The coefficient $b$ can be shifted by a finite $R^2$ counterterm; the matching shift of the gravitational coupling leaves the complete field equation unchanged. Physical stability can depend on the measured $R^2$ coupling, but not on a mere change of subtraction convention. The type-A coefficient is fixed by field content, while the complete stress also requires conservation and state data. Shapiro reviews these distinctions Shapiro 2008, §§ 3–6, PDF.

For a homogeneous stress tensor, conservation and the trace $\mathcal A$ give

$$\dot\rho_{\rm anom}+4H\rho_{\rm anom}=-H\mathcal A.$$

The integration constant is a radiationlike $a^{-4}$ contribution. Thus the trace alone does not select a unique state. An auxiliary-field representation makes this missing homogeneous data explicit, as discussed by Mottola and Vaulin Mottola and Vaulin 2006, §§ II–IV, PDF.

In cosmic time on a flat FLRW background, the signs and derivative content can be checked directly from

$$R=6(2H^2+\dot H),\qquad E_4=24H^2(H^2+\dot H),\qquad \Box R=-\ddot R-3H\dot R.$$

The last minus sign follows from the metric signature. For a de Sitter-invariant state with constant $H$ and no radiationlike integration term, conservation gives $p=-\rho$ and hence

$$\mathcal A=-\frac{3a_{\rm A}}{2\pi^2}H^4, \qquad \rho_{\rm anom}=-\frac{\mathcal A}{4} =\frac{3a_{\rm A}}{8\pi^2}H^4.$$

In Einstein gravity at this order, $3H^2=\Lambda+8\pi G\rho_{\rm anom}$ therefore reduces to

$$x=\frac{\Lambda}{3}+\epsilon x^2, \qquad x=H^2,\qquad \epsilon=\frac{Ga_{\rm A}}{\pi}.$$

For $a_{\rm A}\gt0$, $\Lambda\ge0$, and $\epsilon\Lambda\ll1$, the small root is $x=\Lambda/3+\epsilon\Lambda^2/9+\cdots$. The exact quadratic requires $4\epsilon\Lambda/3\le1$ for real roots. Positive $\Lambda$ gives a low-curvature de Sitter branch; at $\Lambda=0$ that branch is Minkowski. The other formal root has $\epsilon x\sim1$; its existence does not establish control of the perturbative curvature expansion. This is a state-selected fixed-point check, not a derivation of a full cosmological history.

Fixed points and linearized branches

Substitution into the semiclassical Friedmann equation yields terms involving $H^4$, $H\ddot H$, $\dot H^2$, and $H^2\dot H$. A constant-$H$ solution sets derivative terms to zero and solves an algebraic equation whose low-curvature root is continuously connected to the classical cosmological-constant branch. Additional roots can lie at curvature of order the inverse higher-derivative coefficient and should not automatically be retained in a truncated EFT.

To illustrate the subsequent stability calculation, consider a declared reduced homogeneous model for a perturbation of a selected expanding branch $H(t)=H_0+\delta H(t)$:

$$\delta\ddot H+3H_0\delta\dot H+\mu_{\rm eff}^2\delta H=0.$$

In an actual anomaly calculation, the coefficients of such an equation must be derived from the full stress, constraint, and selected state; this second-order model is not a universal consequence of the trace alone. The benchmark treats $\mu_{\rm eff}^2$ as an input. Its characteristic roots are

$$s_\pm=\frac{-3H_0\pm\sqrt{9H_0^2-4\mu_{\rm eff}^2}}{2}.$$

If $\mu_{\rm eff}^2\lt 0$, then $s_+\gt 0$ and the branch has a growing homogeneous mode. If $\mu_{\rm eff}^2\gt 0$, both roots have negative real part for $H_0\gt 0$. This statement concerns the declared reduced linear equation. Gauge modes, inhomogeneous perturbations, nonlocal response kernels, and cutoff-scale roots require separate analysis.

Before normalization, an equation can have the form $A\,\delta\ddot H+B\,\delta\dot H+C\,\delta H=0$. Dividing by $A$ requires $A\ne0$; the effective mass is $C/A$, so the sign of $C$ alone is insufficient. The background must satisfy both the Friedmann constraint and the trace equation, with compatible radiationlike state data. In the pure Einstein–$\Lambda$ limit, the flat vacuum constraint fixes $H$: it does not leave an arbitrary additional homogeneous damping mode. An extra mode can disappear or leave the cutoff as the higher-derivative coupling vanishes. Check this limit before identifying any benchmark root with a physical cosmological mode.

Retained numerical benchmark

The benchmark was rerun with the locked reference environment: CPython 3.9.6, NumPy 2.0.2, SciPy 1.13.1, IEEE-754 binary64 arithmetic. The retained script uses fixed-step classical fourth-order Runge–Kutta on the first-order form of the equation. Inputs are $H_0=1$, $\mu_{\rm eff}^2=-0.4$, $t_f=5$, and pure growing-mode initial data $(\delta H,\delta\dot H)=(1,s_+)$. Download the MIT-licensed benchmark, then use the standalone download-and-run recipe:

curl -fL https://blackhole.org/code/semiclassical-gravity/semiclassical-gravity-benchmarks.py -o semiclassical-gravity-benchmarks.py
python3 semiclassical-gravity-benchmarks.py --case trace-anomaly-driven-cosmology

For this numerical test the tabulated $\delta H$ is a unit-normalized mode amplitude. A physical perturbation is $\delta H_{\rm phys}=A_0\delta H$ with $A_0\ll H_0$, and linearization requires $A_0e^{s_+t}\ll H_0$ throughout the interval. Setting the test amplitude to one does not assert that an order-one change of the Hubble parameter is perturbative.

Steps $N$	$\Delta t$	Final $\delta H$	Measured exponent	State error	Characteristic residual
10	5.0000000e-01	1.8953627e+00	1.2788204e-01	1.6136463e-07	5.4989081e-08
20	2.5000000e-01	1.8953628e+00	1.2788206e-01	1.0357277e-08	3.5295045e-09
40	1.2500000e-01	1.8953628e+00	1.2788206e-01	6.5600488e-10	2.2355051e-10
80	6.2500000e-02	1.8953629e+00	1.2788206e-01	4.1273749e-11	1.4065304e-11

Successive halving of $\Delta t$ reduces the state error by factors $15.6$, $15.8$, and $15.9$, consistent with fourth-order convergence. The independently evaluated analytic roots are

$$s_+=0.127882059610,\qquad s_-=-3.127882059610.$$

The finest measured exponent agrees with $s_+$ at the printed precision, and substituting it into $s^2+3H_0s+\mu_{\rm eff}^2$ gives the reported decreasing residual. Changing only $\mu_{\rm eff}^2$ to a positive value makes both analytic root real parts negative, providing a branch-sign check independent of time integration.

What the computation does not establish

This benchmark verifies the integrator, characteristic equation, refinement behavior, and sign interpretation for one linear ODE. It is not a numerical solution of the full nonlocal anomaly effective action and does not determine physical anomaly coefficients. It initializes a pure analytic eigenmode, omits inhomogeneous perturbations, and has no expanding-grid or renormalized constraint error beyond the displayed characteristic residual.

In a physical calculation, failure modes include selecting different auxiliary-field data while calling the state unchanged, shifting the finite $R^2$ coefficient without recomputing $\mu_{\rm eff}^2$, retaining a cutoff-scale runaway, under-resolving a stiff decaying mode, or evolving until the perturbation is no longer linear. A credible nonlinear result must monitor the Friedmann constraint, trace equation, stress conservation, and curvature hierarchy while refining both solver tolerances and initial-state resolution.

Exercise

Using $H_0=1$ and $\mu_{\rm eff}^2=-0.4$, compute the characteristic residual of the coarse measured exponent $s_{\rm num}=0.12788204$. Why is agreement of $\delta H(5)$ alone a weaker test?

Solution

Substitution gives

$$\left|s_{\rm num}^2+3s_{\rm num}-0.4\right| \approx6.38\times10^{-8},$$

This differs from the table’s $5.50\times10^{-8}$ because the exercise rounds the exponent to fewer digits before substitution. Near an exact root, the polynomial residual is sensitive to this last-digit rounding. A final amplitude can agree accidentally through compensating phase, initial-data, or normalization errors. The characteristic residual tests whether the measured exponent satisfies the differential operator, while refinement of the full state checks the integration order. Both are needed.

References

• Mottola, E., and Vaulin, R. (2006). “Macroscopic Effects of the Quantum Trace Anomaly.” Physical Review D 74, 064004. DOI. Open PDF.
• Shapiro, I. L. (2008). “Effective Action of Vacuum: Semiclassical Approach.” Classical and Quantum Gravity 25, 103001. DOI. Open PDF.