Nonsingular cosmology with a scale-invariant spectrum of cosmological perturbations from Lee-Wick theory

We study the cosmology of a Lee-Wick type scalar field theory. First, we consider homogeneous and isotropic background solutions and find that they are nonsingular, leading to cosmological bounces. Next, we analyze the spectrum of cosmological perturbations which result from this model. Unless either the potential of the Lee-Wick theory or the initial conditions are finely tuned, it is impossible to obtain background solutions which have a sufficiently long period of inflation after the bounce. More interestingly, however, we find that in the generic noninflationary bouncing cosmology, perturbations created from quantum vacuum fluctuations in the contracting phase have the correct form to lead to a scale-invariant spectrum of metric inhomogeneities in the expanding phase. Since the background is nonsingular, the evolution of the fluctuations is defined unambiguously through the bounce. We also analyze the evolution of fluctuations which emerge from thermal initial conditions in the contracting phase. The spectrum of gravitational waves stemming from quantum vacuum fluctuations in the contracting phase is also scale-invariant, and the tensor to scalar ratio is not suppressed.

Figure 1

Evolution of the background fields $\varphi$, $\tilde{\varphi }$ and of the background equation of state parameter $w$ in a noninteracting model as a function of cosmic time (horizontal axis). The background fields are plotted in dimensionless units by normalizing by the mass $M_{\mathrm{rec}}={10}^{-6}{m}_{\mathrm{pl}}$. Similarly, the time axis is displayed in units of $M_{\mathrm{rec}}^{-1}$. The mass parameters $m$ and $M$ were chosen to be $m=5M_{\mathrm{rec}}$ and $M=10M_{\mathrm{rec}}$. The initial conditions were ${\varphi }_i=1.74\times {10}^3{M}_{\mathrm{rec}}$, ${\dot{\varphi }}_i=1.44\times {10}^4{M}_{\mathrm{rec}}^2$, ${\tilde{\varphi }}_i=8.98M_{\mathrm{rec}}$, ${\dot{\tilde{\varphi }}}_i=-14.08M_{\mathrm{rec}}^2$.

Figure 2

Evolution of the background fields $\varphi$, $\tilde{\varphi }$ and of the background equation of state parameter $w$ in a noninteracting model as a function of cosmic time (horizontal axis). The background fields are plotted in dimensionless units by normalizing by the mass $M_{\mathrm{rec}}={10}^{-6}{m}_{\mathrm{pl}}$. Similarly, the time axis is displayed in units of $M_{\mathrm{rec}}^{-1}$. The mass parameters $m$ and $M$ were chosen to be $m=1.4\times {10}^{-1}{M}_{\mathrm{rec}}$ and $M=10M_{\mathrm{rec}}$. The initial conditions were ${\varphi }_i=-3.57\times {10}^3{M}_{\mathrm{rec}}$, ${\dot{\varphi }}_i=5.56\times {10}^2{M}_{\mathrm{rec}}^2$, ${\tilde{\varphi }}_i=2.98\times {10}_{-6}{M}_{\mathrm{rec}}$, ${\dot{\tilde{\varphi }}}_i=-1.39\times {10}^{-6}{M}_{\mathrm{rec}}^2$.

Figure 3

Plot of the duration of the deflationary phase as a function of the ratio of energy densities of $\varphi$ and $\tilde{\varphi }$ (horizontal axis). The duration (vertical axis) is shown in terms of the e-folding number of deflation.

Figure 4

A sketch of the evolution of scales in a bouncing universe. The horizontal axis is a comoving spatial coordinate, the vertical axis is conformal time. Plotted are the Hubble radius $|\mathcal{H}{|}^{-1}$ and the wavelength $\lambda$ of a fluctuations with comoving wave number $k$.

Figure 5

Result of the numerical evolution of the curvature perturbations with different comoving wave numbers $k$ in the Lee-Wick bounce. The horizontal axis in the left panel is cosmic time, and in the right panel it is comoving time. The initial values of the background parameters are the same as in Fig. 1. The units of the time axis are $M_{\mathrm{rec}}^{-1}$, the comoving wave number $k$ is unity for $k=M_{\mathrm{rec}}$, as in Fig. 2.

Figure 6

Plot of the power spectrum of the curvature perturbation $\Phi$ (lower panel) and of the spectral index (upper panel) as functions of comoving wave numbers $k$ in the Lee-Wick bounce. The initial values of the background parameters are the same as in Fig. 1.

Figure 7

Power spectrum of the Lee-Wick modes $\delta \tilde{\varphi }$ as a function of cosmic time in the Lee-Wick bounce model (the blue dashed line). The comoving wave number is taken to be $k=10$. The initial values of the background parameters are the same as in Fig. 1.

Figure 8

Plot of the evolution of the absolute value of the ratio of the right-hand side to the $k^2\Phi$ term in Eq. (19). The comoving wave number is taken to be $k=10$. The initial values of the background parameters are the same as in Fig. 1.