Effect of the trace anomaly on the cosmological constant

TL;DR

Using a semiclassical analysis, the authors show that the quantum trace anomaly does not affect the late-time effective cosmological constant; the classical de Sitter attractor is not dynamically altered. They derive the anomaly effective action in flat FLRW and find either classical or anomaly-driven attractors depending on parameters, but conclude the trace anomaly does not solve the cosmological constant problem.

Key findings

• The trace anomaly can be generated by an infinite effective action in flat FLRW consisting of local quadratic curvature invariants \(R^2\) and the Gauss-Bonnet term \(E\); counterterms imply the coefficients \(b,b',b''\) are not uniquely fixed, so the authors vary them.
• Late-time asymptotes are a classical de Sitter attractor \(H^0_C\) and a quantum anomaly-driven attractor \(H^0_A\), independent of \(b''\); in the small-\(\lambda\) expansion \(H^0_C \simeq \sqrt{\Lambda/3}[1-8\pi b'\lambda]\) and \(H^0_A \simeq \sqrt{-1/(16\pi G b')-\Lambda/3}\).
• In quasi de Sitter spacetime, both attractors can be stable, but the quantum attractor destabilises when inequality (50) is violated; they illustrate this with \(b''=7b'/6<5b'/6\) for radiation \(w=1/3\), \(\lambda=1/50\), \(b'=-0.015\).
• In exact FLRW spacetimes, stability depends on the sign of \(b''-2b'/3\): if \(>0\), the quantum attractor is stable and the classical de Sitter attractor is unstable; if \(<0\), the classical attractor is stable and the quantum attractor is unstable; at \(b''=2b'/3\), both are stable. The classical attractor shows under-damped oscillations, with frequency independent of the equation of state.

Methods

• Semiclassical effective-field-theory approach: trace of the Einstein field equations plus stress-energy conservation, including matter with constant equation of state \(w>-1\), a cosmological constant, and the trace anomaly.
• Derives the anomaly effective action in flat FLRW from local quadratic curvature invariants \(R^2\) and Gauss-Bonnet \(E\), then studies two regimes: quasi de Sitter spacetime with slowly varying \(H\) and truncated higher derivatives, and exact FLRW spacetime including all higher-derivative contributions via numerical and linear stability analysis.
• Varies the anomaly coefficients \(b,b',b''\) because counterterms can shift their physical values; uses Standard Model values \(N_S=4\), \(N_F=45/2\), \(N_V=12\), with \(b'=-0.015\) and \(\lambda=1/50\) in figures.

Limitations

• Stated: the authors do not consider a new conformal degree of freedom; they do not know whether the truncated quasi de Sitter or exact FLRW treatment is correct, and note that higher-derivative terms may destabilise the system (Ostrogradsky). The quantum attractor is of order the Planck mass, where perturbative general relativity may break down and quantum fluctuations may affect or remove it.
• Stated: the physical coefficients in the trace anomaly have not been measured, so they must be allowed to vary.
• Evident from the text: the analysis assumes flat, homogeneous, isotropic FLRW spacetimes and constant \(w>-1\); stability conclusions depend on the sign of \(b''-2b'/3\) and on \(\lambda\ll 1\).