Scrutinizing the cosmological constant problem and a possible resolution

We suggest a new perspective on the cosmological constant problem by scrutinizing its standard formulation. In classical and quantum mechanics without gravity, there is no definition of the zero point of energy. Furthermore, the Casimir effect only measures how the vacuum energy changes as one varies a geometric modulus. This leads us to propose that the physical vacuum energy in a Friedmann-Lemaître-Robertson-Walker expanding universe only depends on the time variation of the scale factor $a(t)$. Equivalently, requiring that empty Minkowski space is gravitationally stable is a principle that fixes the ambiguity in the zero-point energy. On the other hand, if there is a meaningful bare cosmological constant, this prescription should be viewed as a fine-tuning. We describe two different choices of vacuum, one of which is consistent with the current universe consisting only of matter and vacuum energy. The resulting vacuum energy density ${\rho }_{\mathrm{vac}}$ is constant in time and approximately $k_c^2{H}_0^2$, where $k_c$ is a momentum cutoff and $H_0$ is the current Hubble constant; for a cutoff close to the Planck scale, values of ${\rho }_{\mathrm{vac}}$ in agreement with astrophysical measurements are obtained. Another choice of vacuum is more relevant to the early universe consisting of only radiation and vacuum energy, and we suggest it as a possible model of inflation.

Figure 1

The Hubble constant as a function of redshift $z$ for the solution to our model Eq. (29) including radiation, versus the Friedmann equation Eq. (6) with radiation, matter, and a standard cosmological constant with ${\Omega }_m=0.266$, ${\Omega }_{\mathrm{rad}}=8.24\times {10}^{-5}$, ${\Omega }_{\Lambda }=1-{\Omega }_m-{\Omega }_{\mathrm{rad}}$.

Figure 2

The vacuum energy ${\rho }_{\widehat{\mathrm{vac}}}/{\rho }_i$ as a function of $Ht$ for ${\Omega }_{\mathrm{rad}}=0.8$ and ${\Omega }_m=0.15$.