Onset of inflation amid backreaction from inhomogeneities

We analyze the onset of inflation for a simple single-field model when the system begins with significant inhomogeneities on length scales shorter than the initial Hubble radius. We incorporate certain nonlinear interactions among the coupled degrees of freedom by using the nonperturbative Hartree approximation. Consistent with recent, more computationally intensive numerical-relativity studies, we find inflation to be robust for large-field models, even when the system begins with significant structure on sub-Hubble scales. We consider the space of initial conditions $(\phi (t_0),\dot{\phi }(t_0))$, where $\phi$ is the vacuum expectation value of the quantized field that drives inflation. Although some regions of $(\phi (t_0),\dot{\phi }(t_0))$ that would have yielded sufficient inflation in the absence of inhomogeneities fail to do so when backreaction from inhomogeneities is incorporated, an equal volume of such regions succeeds in producing sufficient inflation which did not do so in the absence of inhomogeneities. For large-field models, in other words, the total volume of the space of initial conditions $(\phi (t_0),\dot{\phi }(t_0))$ that yields sufficient inflation is conserved when we incorporate nonlinear backreaction from inhomogeneities, compared to the case in which inhomogeneities are neglected.

Figure 1

Typical initial surface of $\mathrm{\Psi }(t_0,\mathbf{x})$ in the $x\text{−}y$ plane [dependence on the polar angle $\theta$ in Eq. (2) is suppressed] for ${\phi }_0=25M_{\mathrm{pl}}$ and ${\dot{\phi }}_0=-0.25M_{\mathrm{pl}}^2$. The field fluctuations $\delta \widehat{\varphi }(t_0,\mathbf{x})$ were initialized as in Eq. (35), with the random coefficients ${\gamma }_{n\ell m}$ and ${\delta }_{n\ell m}$ for each mode drawn from the ranges in Eq. (44), which in turn determined the modes ${\mathrm{\Psi }}_{n\ell m}(t_0)$ from Eq. (29). Further details about the construction of the initial surface for $\mathrm{\Psi }(t_0,\mathbf{x})$ are given in Appendix pp3. The blue cylinder has radius equal to $r_H(t_0)=\pi /H(t_0)$, in units of $M_{\mathrm{pl}}^{-1}$.

Figure 2

$|\mathrm{Re}(\delta {\varphi }_{n\ell m})|$ versus $N=\ln a$ for the $\ell =0$, 1 modes in our simulation with $k_{n\ell }\ge a_0{H}_0$.

Figure 3

$|\mathrm{Re}({\mathrm{\Psi }}_{n\ell m})|$ versus $N$ for the $\ell =0$, 1 modes in our simulation with $k_{n\ell }\ge a_0{H}_0$.

Figure 4

The dimensionless power spectrum of the curvature perturbation ${\mathcal{P}}_{\mathcal{R}}(k_{n\ell })$ versus $N$ for the $\ell =0$, 1 modes in our simulation with $k_{n\ell }\ge a_0{H}_0$.

Figure 5

The energy density in fluctuations $\delta {\rho }_{(2)}(t)$ [normalized by the initial value $\overline{\rho }(t_0)$] versus $N$, with $\lambda ={10}^{-10}$. At early times, while most modes are still inside the Hubble radius, $\delta {\rho }_{(2)}(t)$ decays like radiation.

Figure 6

The evolution of the Hubble parameter $H(t)$ versus $N$ for early times, with $\lambda ={10}^{-10}$. In the absence of fluctuations (blue), $H(t)\propto a^{-3}(t)$. For large initial fluctuations (pink), $H(t)\propto a^{-2}(t)$ at early times. For fluctuations that begin in the Bunch-Davies initial state (yellow), the evolution of $H(t)$ falls between the other two cases.

Figure 7

The slow-roll parameter $\epsilon$ versus $N$ for early times with $\lambda ={10}^{-10}$, for the system with large initial fluctuations. Inflation begins at $N\sim 2$ when $\epsilon <1$, and slow-roll begins by $N\sim 3$ with $\epsilon <0.1$. Note that modes with comoving wave number up to the UV regulator scale $\kappa$ remain within the Hubble radius until $N\sim 6$, and the shortest-wavelength mode in the spectrum, with $k_{\max }=4\kappa$, crosses outside the Hubble radius at $N\sim 7$.

Figure 8

The evolution of the slow-roll parameter $\epsilon$ versus $N$ with $\lambda ={10}^{-10}$ when we neglect fluctuations (blue), when we initialize fluctuations in the Bunch-Davies state (yellow), and for a particular simulation that began with large initial fluctuations, with $\mathcal{C}=20$ (pink).

Figure 9

The evolution of the system $(\phi ,\dot{\phi })$ when we fix ${\phi }_0=25M_{\mathrm{pl}}$ and select ${\dot{\phi }}_0=\pm 0.25M_{\mathrm{pl}}^2$, with $\lambda ={10}^{-10}$. Dashed lines indicate evolution of the system for $\epsilon >0.1$. The line along $\dot{\phi }=0$ corresponds to the slow-roll inflationary attractor.

Figure 10

The average value of $\delta {\rho }_{(2)}(t_0)$ at each grid point in $({\phi }_0,{\dot{\phi }}_0)$, when quantum fluctuations are initialized with random coefficients drawn from the ranges in Eq. (44), compared to the value of $\delta {\rho }_{(2)}(t_0)$ when fluctuations are initialized in the Bunch-Davies state, with $\lambda ={10}^{-10}$.

Figure 11

The average value of ${\mathrm{\Psi }}_{\mathrm{rms}}(t_0)$ at each grid point in $({\phi }_0,{\dot{\phi }}_0)$ when quantum fluctuations are initialized with random coefficients drawn from the ranges in Eq. (44), with $\lambda ={10}^{-10}$. Note that ${\mathrm{\Psi }}_{\mathrm{rms}}(t_0)$ is roughly proportional to ${\dot{\phi }}_0^2$.

Figure 12

Average of the ratio of the time $N_{\mathrm{sr}}$ when the system first reaches the slow-roll inflationary attractor with $\epsilon \le 0.1$, to the time $N_{\kappa }$ when the mode with comoving wave number equal to the UV regulator scale crosses outside the Hubble radius, $\kappa =aH$, for the case of large initial fluctuations ($\mathcal{C}\simeq 20$), with $\lambda ={10}^{-10}$. For all simulations across $({\phi }_0,{\dot{\phi }}_0)$, we find $N_{sr}/N_{\kappa }\le 0.55\pm 0.06$.

Figure 13

Contours of constant $N_{\mathrm{infl}}$ in $({\phi }_0,{\dot{\phi }}_0)$ for $\lambda ={10}^{-10}$ when the fluctuations are neglected (left), when the fluctuations are initialized in the Bunch-Davies state, ${\mathcal{C}}_{\mathrm{BD}}=2$ (middle), and when the fluctuations are initialized with random coefficients ${\gamma }_{n\ell m}$ and ${\delta }_{n\ell m}$ for each mode drawn from the ranges in Eq. (44), which yields $\mathcal{C}\simeq 20$ (right). For the case of large initial fluctuations (right), the contours of constant $N_{\mathrm{infl}}$ were evaluated by averaging 32 simulations per grid point. In each plot, regions of dark blue indicate $N_{\mathrm{infl}}<45$ and regions of light yellow indicate $N_{\mathrm{infl}}\ge 70$. In white regions in the lower left, the system never entered slow roll. The critical lines that yield $N_{\mathrm{infl}}=65$ $e$-folds of inflation are indicated in red.

Figure 14

The critical line in $({\phi }_0,{\dot{\phi }}_0)$ that yields sufficient inflation, with $N_{\mathrm{infl}}\ge 65$, for $\lambda ={10}^{-10}$, for the cases of homogeneous evolution with no fluctuations (blue), fluctuations in the Bunch-Davies initial state (yellow), and large initial fluctuations (pink). (Points to the right of the critical lines achieve sufficient inflation.) For the latter, we show the critical line based on averaging across 32 simulations per grid point, as well as $\pm 2\sigma$ contours. The jitter in the pink curves arises from stochastic fluctuations, and it is expected that the curves would become smooth as we increase both the resolution of our sampling and the number of simulations at each point in phase space.

Figure 15

Average of the ratio of the time $N_{\mathrm{sr}}$ when the system first reaches the slow-roll inflationary attractor to the time $N_{\kappa }$ when $\kappa =aH$ for the case of large initial fluctuations ($\mathcal{C}\simeq 20$), with $\lambda ={10}^{-12}$. Across all simulations we find $N_{\mathrm{sr}}/N_{\kappa }\le 0.53\pm 0.04$.

Figure 16

Contours of constant $N_{\mathrm{infl}}$ in $({\phi }_0,{\dot{\phi }}_0)$ for $\lambda ={10}^{-12}$ when the fluctuations are neglected (left), when the fluctuations are initialized in the Bunch-Davies state with ${\mathcal{C}}_{\mathrm{BD}}=2$ (middle), and when the fluctuations are initialized with random coefficients ${\gamma }_{n\ell m}$ and ${\delta }_{n\ell m}$ for each mode drawn from the ranges in Eq. (44), which yields $\mathcal{C}\simeq 20$ (right). For the case of large initial fluctuations (right), the contours of constant $N_{\mathrm{infl}}$ were evaluated by averaging 32 simulations per grid point. In each plot, regions of dark blue indicate $N_{\mathrm{infl}}<45$ and regions of light yellow indicate $N_{\mathrm{infl}}\ge 70$. In white regions in the lower left, the system never entered slow roll. The critical lines that yield $N_{\mathrm{infl}}=65$ $e$-folds of inflation are indicated in red.

Figure 17

The critical line in $({\phi }_0,{\dot{\phi }}_0)$ that yields $N_{\mathrm{infl}}\ge 65$ $e$-folds of inflation for $\lambda ={10}^{-12}$, for the cases of homogeneous evolution (blue); fluctuations in the Bunch-Davies initial state (yellow); and large initial fluctuations (pink). (Points to the right of the critical lines achieve sufficient inflation.) For the latter, we show the critical line based on averaging across 32 simulations per grid point, as well as $\pm 2\sigma$ contours.

Figure 18

The quantity ${\rho }_{\text{total}}=\overline{\rho }+\delta {\rho }_{(2)}$ versus $N$ as we vary $R=c/{\overline{H}}_0$. We fix $({\phi }_0,{\dot{\phi }}_0)=(25M_{\mathrm{pl}},-0.25M_{\mathrm{pl}}^2)$ and initialize the field fluctuations in the Bunch-Davies state. Throughout the simulations described in our paper, we fixed $c=1.5\pi \approx 4.71$. We found similar results as we increased $c$. In particular, for these initial conditions we found $N_{\text{total}}=63.01\pm 0.78$ as we varied $4.71\le c\le 25.85$, indicating minimal dependence of the evolution of the dynamical system on our choice of $R$.