Effects of quantized scalar fields in cosmological spacetimes with big rip singularities

Effects of quantized free scalar fields in cosmological spacetimes with big rip singularities are investigated. The energy densities for these fields are computed at late times when the expansion is very rapid. For the massless minimally coupled field it is shown that an attractor state exists in the sense that, for a large class of states, the energy density of the field asymptotically approaches the energy density it would have if it was in the attractor state. Results of numerical computations of the energy density for the massless minimally coupled field and for massive fields with minimal and conformal couplings to the scalar curvature are presented. For the massive fields the energy density is seen to always asymptotically approach that of the corresponding massless field. The question of whether the energy densities of quantized fields can be large enough for backreaction effects to remove the big rip singularity is addressed.

Figure 1

The dashed line represents the phantom energy density ${\rho }_{\mathrm{ph}}$ in the case that $w=-1.25$. The two solid lines represent the energy density $⟨\rho {⟩}_r$ for two states of the massless minimally coupled scalar field. The central line is the Bunch-Davies attractor state. The oscillating line is the fourth-order adiabatic state with $k_0=0.2$ and ${\eta }_0=-100$. For both states, the mass scale $\mu =1$ has been used.

Figure 2

The dashed line represents the phantom energy density ${\rho }_{\mathrm{ph}}$ in the case that $w=-1.25$. The upper solid line is $⟨\rho {⟩}_r$ for a massive ($m=0.005$) conformally coupled scalar field in the fourth-order adiabatic state with $k_0=0.1$ and ${\eta }_0=-100$; the lower one is $⟨\rho {⟩}_r$ for the massless conformally coupled scalar field.

Figure 3

The dashed line represents the phantom energy density ${\rho }_{\mathrm{ph}}$ in the case that $w=-1.25$. The central solid line is the energy density $⟨\rho {⟩}_r$ for a massless minimally coupled scalar field in the Bunch-Davies state with mass scale $\mu =0.001$. The oscillating line is the energy density for a massive ($m=0.001$) minimally coupled scalar field in the fourth-order adiabatic state with $k_0=1$ and ${\eta }_0=-50$. Note that the quantity $|\rho |$ rather than $\rho$ has been plotted since the energy density of the massive field is negative at $\eta ={\eta }_0$, oscillates between positive and negative values for a period of time, and then becomes positive definite at late times. Note that, except at the initial time, the times where the curve is nearly vertical when it reaches a lower limit are times at which the energy density goes through zero, which is $-\infty$ on the scale of this plot. Thus the curve should really extend down to $-\infty$ at these points.