Vacuum fluctuations in an ancestor vacuum: A possible dark energy candidate

We consider an open universe created by bubble nucleation, and study possible effects of our “ancestor vacuum,” a de Sitter space in which bubble nucleation occurred, on the present universe. We compute vacuum expectation values of the energy-momentum tensor for a minimally coupled scalar field, carefully taking into account the effect of the ancestor vacuum by the Euclidean prescription. We pay particular attention to the so-called supercurvature mode, a non-normalizable mode on a spatial slice of the open universe, which has been known to exist for sufficiently light fields. This mode decays in time most slowly, and may leave residual effects of the ancestor vacuum, potentially observable in the present universe. We point out that the vacuum energy of the quantum field can be regarded as dark energy if mass of the field is of order the present Hubble parameter or smaller. We obtain preliminary results for the dark energy equation of state $w(z)$ as a function of the redshift.

Figure 1

Left panel: The potential for the field $\mathrm{\Phi }$. The local minimum at $\mathrm{\Phi }={\mathrm{\Phi }}_A$ is the false vacuum, whose geometry is de Sitter space (the ancestor vacuum). On the true vacuum side, it is assumed that a plateau region exists, on which slow-roll inflation occurs. The true minimum of the potential is taken to be zero, since we expect the present cosmological constant to be realized as the vacuum energy of a quantum field on this background. Right panel: Penrose diagram for the Coleman-De Luccia geometry. The spacetime is divided into five regions by the red lines. Region I is an open FLRW universe. The green line represents the bubble wall. On its left (right) is the true (ancestor) vacuum. The blue curves indicate orbits of the $SO(3,1)$ symmetry. The directions of the coordinates ($\eta$, $R$ for Region I; $\tau$, $X$ for Region III) are indicated by arrows. Region I is drawn with the future null infinity, because the present cosmological constant will relax to zero in the future if it is due to the vacuum energy of a quantum field.

Figure 2

Left panel: The Euclidean geometry for a CDL instanton in the thin-wall limit. A flat space for the true vacuum (whose vacuum energy is assumed to be zero) is patched to a piece of a sphere $S^4$ for the false vacuum with a positive vacuum energy. The $S^3$ part is represented by $S^1$ in the figure. Right panel: The potential $a^{\prime \prime }/a$ in the massless scalar equation of motion (3.3) on a CDL background in the thin-wall limit. Asymptotically, $a^{\prime \prime }/a\to 1$ for $X\to \pm \infty$. On the true-vacuum side ($X<X_0$), the potential is flat. At the bubble wall ($X=X_0$), there is a negative delta function. If mass is nonzero, the potential will be lifted by the additional term $+m^2{a}^2$.

Figure 3

Left panel: The definition of the contour ${\mathcal{C}}_1$ for the $k$-integration. Middle panel: A contour equivalent to ${\mathcal{C}}_1$. The residue at the $k=(1-\epsilon )i$ pole is equal to the bound state contribution (the second line) in (3.17). Right panel: The contour ${\mathcal{C}}_2$ used at the end of Section 4 to show the finiteness of $⟨{\varphi }^2⟩$ in the early-time ($\eta ,{\eta }^{\prime }\to -\infty$) limit. By deforming ${\mathcal{C}}_1$ to ${\mathcal{C}}_2$, the $k$-integral is expressed as a sum over the residues at $k=in$ ($n=1,2,\dots$).

Figure 4

The equation of state $w(z)$ as a function of the redshift $z$. The parameters for the left panel are $\epsilon =0.1$, $m_0/H_0=0.1$, $r_{c0}=20$, for which the present equation of state is $w_0=-0.968$. Those for the right panel are $\epsilon =0.1$, $m_0/H_0=0.1$, $r_{c0}=40$, for which $w_0=-0.992$. Note that these are preliminary results, obtained by ignoring the time derivative terms in $\rho$ and $p$ (though this approximation can be justified for these choices of parameters).