Zero-point quantum fluctuations and dark energy

In the Hamiltonian formulation of general relativity, the energy associated to an asymptotically flat space-time with metric $g_{\mu \nu }$ is related to the Hamiltonian $H_{\mathrm{GR}}$ by $E=H_{\mathrm{GR}}[g_{\mu \nu }]-H_{\mathrm{GR}}[{\eta }_{\mu \nu }]$, where the subtraction of the flat-space contribution is necessary to get rid of an otherwise divergent boundary term. This classic result indicates that the energy associated to flat space does not gravitate. We apply the same principle to study the effect of the zero-point fluctuations of quantum fields in cosmology, proposing that their contribution to cosmic expansion is obtained computing the vacuum energy of quantum fields in a Friedmann-Robertson-Walker space-time with Hubble parameter $H(t)$ and subtracting from it the flat-space contribution. Then the term proportional to ${\Lambda }_c^4$ (where ${\Lambda }_c$ is the UV cutoff) cancels, and the remaining (bare) value of the vacuum energy density is proportional to ${\Lambda }_c^2{H}^2(t)$. After renormalization, this produces a renormalized vacuum energy density $\sim M^2{H}^2(t)$, where $M$ is the scale where quantum gravity sets is, so for $M$ of the order of the Planck mass a vacuum energy density of the order of the critical density can be obtained without any fine-tuning. The counterterms can be chosen so that the renormalized energy density and pressure satisfy $p=w\rho$, with $w$ a parameter that can be fixed by comparison to the observed value, so, in particular, one can choose $w=-1$. An energy density evolving in time as $H^2(t)$ is however observationally excluded as an explanation for the dominant dark energy component that is responsible for the observed acceleration of the Universe. We rather propose that zero-point vacuum fluctuations provide a new subdominant “dark” contribution to the cosmic expansion that, for a UV scale $M$ slightly smaller than the Planck mass, is consistent with existing limits and potentially detectable.

Figure 1

The functions ${\rho }_{\Lambda }(z)/{\rho }_c(z)$ for $w_{\Lambda }=(w_{\Lambda }{)}_{\min }$ and $(w_{\Lambda }{)}_{\max }$ (black solid lines) compared to the function ${\rho }_{\mathrm{DE}}(z)/{\rho }_c(z)$ given by Eq. (88) with ${\Omega }_Z=+0.10$ (upper dashed curve, red) and ${\Omega }_Z=-0.10$ (lower dashed curve, blue).