Effect of the trace anomaly on the cosmological constant

It has been argued that the quantum (conformal) trace anomaly could potentially provide us with a dynamical explanation of the cosmological constant problem. In this paper, however, we show by means of a semiclassical analysis that the trace anomaly does not affect the cosmological constant. We construct the effective action of the conformal anomaly for flat Friedmann-Lemaître-Robertson-Walker spacetimes consisting of local quadratic geometric curvature invariants. Counterterms are thus expected to influence the numerical value of the coefficients in the trace anomaly and we must therefore allow these parameters to vary. We calculate the evolution of the Hubble parameter in quasi-de Sitter spacetime, where we restrict our Hubble parameter to vary slowly in time, and in Friedmann-Lemaître-Robertson-Walker spacetimes. We show dynamically that a universe consisting of matter with a constant equation of state, a cosmological constant, and the quantum trace anomaly evolves either to the classical de Sitter attractor or to a quantum trace anomaly driven one. When considering the trace anomaly truncated to quasi-de Sitter spacetime, we find a region in parameter space where the quantum attractor destabilizes. When considering the exact expression of the trace anomaly, a stability analysis shows that whenever the trace anomaly driven attractor is stable, the classical de Sitter attractor is unstable, and vice versa. Semiclassically, the trace anomaly does not affect the classical late time de Sitter attractor, and hence it does not solve the cosmological constant problem.

Figure 1

Dynamics of the Hubble parameter in quasi-de Sitter spacetime in the presence of a nonzero cosmological constant, the trace anomaly, and matter ($\omega =0$). Depending on the initial conditions, the Hubble parameter evolves to either the classical de Sitter or the quantum anomaly driven attractor. We have used $\lambda =1/50$, $b^{\prime }=-0.015$ (standard model value) and $b^{\prime \prime }=0$.

Figure 2

Validity of the assumption $ϵ\ll 1$. For the various initial conditions in Fig. 1, one can clearly see that this approximation is well justified.

Figure 3

Validity of the assumption $\dot{ϵ}=0$. For the various initial conditions in Fig. 1, we have calculated $\dot{ϵ}/(Hϵ)$. Clearly, this condition is violated at all times.

Figure 4

Instability of the quantum anomaly driven attractor in quasi-de Sitter spacetimes. We have used $\lambda =1/50$, $\omega =1/3$, $b^{\prime \prime }=7b^{\prime }/6<5b^{\prime }/6$.

Figure 5

Phase space flow in quasi-de Sitter spacetime. The phase space consists of two lines. The flow, indicated by the arrows, is towards the classical or quantum attractor represented by two dots. In this regime, both attractors are stable. The vertical dashed line indicates the branching point and the dashed-dotted line the classical evolution. We have used $\omega =0$, $\lambda =1/50$, $b^{\prime }=-0.015$ (standard model value) and $b^{\prime \prime }=0$.

Figure 6

Phase space flow in quasi de Sitter spacetime for $b^{\prime \prime }=2b^{\prime }/3$ such that the $\square R$ does not contribute in the trace anomaly. Qualitatively, the dynamics does not change when compared to Fig. 5. Again, both attractors are stable. Apart from $b^{\prime \prime }$, the value of the parameters are identical to Fig. 5.

Figure 7

Dynamics of the Hubble parameter taking the full trace anomaly into account. We took $b^{\prime \prime }=0$ such that $b^{\prime \prime }-2b^{\prime }/3>0$ yielding an unstable classical attractor. We have used $\omega =0$, $\lambda =1/50$, $b^{\prime }=-0.015$ (standard model value).

Figure 8

Parametric phase space plot for Fig. 7 in which the classical attractor is unstable. We have indicated the quantum anomaly driven attractor as a small black sphere.

Figure 9

Dynamics of the Hubble parameter taking the full trace anomaly into account. We took $b^{\prime \prime }=b^{\prime }$ such that $b^{\prime \prime }-2b^{\prime }/3<0$ yielding a stable classical attractor. We have used $\omega =0$, $\lambda =1/50$, $b^{\prime }=-0.015$ (standard model value). Clearly, the classical attractor is under-damped, resulting in various oscillations around $H_0^{\mathrm{C}}$.

Figure 10

Parametric phase space plot for Fig. 9 in which the classical attractor is stable. Because of under-damping, one spirals towards the classical attractor. For clarity, we have only included the flow for two initial conditions $H(0)=\sqrt{3/\Lambda }/2$ and $H(0)=6\sqrt{3/\Lambda }$. The twister like structure is clearly visible. Qualitatively, the phase space flow resulting from other initial conditions is identical.

Figure 11

Parameter space for oscillations around the quantum anomaly driven attractor. The gray region (light shaded) is excluded either because $b^{\prime }>0$ or because the quantum attractor has become unstable. The red region (dark shaded) in parameter space shows oscillatory behavior, whereas the white region is either critically damped or over-damped. We used $\lambda =1/50$.