Preheating after multifield inflation with nonminimal couplings. II. Resonance structure

This is the second in a series of papers on preheating in inflationary models comprised of multiple scalar fields coupled nonminimally to gravity. In this paper, we work in the rigid-spacetime approximation and consider field trajectories within the single-field attractor, which is a generic feature of these models. We construct the Floquet charts to find regions of parameter space in which particle production is efficient for both the adiabatic and isocurvature modes, and analyze the resonance structure using analytic and semianalytic techniques. Particle production in the adiabatic direction is characterized by the existence of an asymptotic scaling solution at large values of the nonminimal couplings, ${\xi }_I\gg 1$, in which the dominant instability band arises in the long-wavelength limit, for comoving wave numbers $k\to 0$. However, the large-${\xi }_I$ regime is not reached until ${\xi }_I\ge \mathcal{O}(100)$. In the intermediate regime, with ${\xi }_I\sim \mathcal{O}(1–10)$, the resonance structure depends strongly on wave number and couplings. The resonance structure for isocurvature perturbations is distinct and more complicated than its adiabatic counterpart. An intermediate regime, for ${\xi }_I\sim \mathcal{O}(1–10)$, is again evident. For large values of ${\xi }_I$, the Floquet chart consists of densely spaced, nearly parallel instability bands, suggesting a very efficient preheating behavior. The increased efficiency arises from features of the nontrivial field-space manifold in the Einstein frame, which itself arises from the fields’ nonminimal couplings in the Jordan frame, and has no analog in models with minimal couplings. Quantitatively, the approach to the large-${\xi }_I$ asymptotic solution for isocurvature modes is slower than in the case of the adiabatic modes.

Figure 1

Left: potential in the Einstein frame, $V({\varphi }^I)$, for a two-field model with ${\lambda }_{\chi }=1.25{\lambda }_{\varphi }$, $g={\lambda }_{\varphi }$, and ${\xi }_{\chi }=0.8{\xi }_{\varphi }$. Right: field trajectories for different couplings and initial conditions (from Ref. [25]). Open circles indicate fields’ initial values (in units of $M_{\mathrm{pl}}$). We set the fields’ initial velocities to zero and adjust the initial angle in field space, ${\theta }_0=\arctan ({\varphi }_0/{\chi }_0)$. We fix ${\lambda }_{\varphi }={10}^{-2}$ and ${\xi }_{\varphi }={10}^3$ and vary the other parameters $\{{\lambda }_{\chi },g,{\xi }_{\chi },{\theta }_0\}$ as follows: $\{0.75{\lambda }_{\varphi },{\lambda }_{\varphi },1.2{\xi }_{\varphi },\pi /4\}$ (red), $\{{\lambda }_{\varphi },{\lambda }_{\varphi },0.8{\xi }_{\varphi },\pi /4\}$ (blue), $\{{\lambda }_{\varphi },0.75{\lambda }_{\varphi },0.8{\xi }_{\varphi },\pi /6\}$ (green), and $\{{\lambda }_{\varphi },0.75{\lambda }_{\varphi },0.8{\xi }_{\varphi },\pi /3\}$ (black). In each case, the initial transient motion damps out within a few efolds, yielding effectively single-field evolution during inflation.

Figure 2

Left: rescaled background solution $\delta (t)=\sqrt{{\xi }_{\varphi }}\varphi (t)/M_{\mathrm{pl}}$ as a function of $\tilde{t}=\sqrt{{\lambda }_{\varphi }}{M}_{\mathrm{pl}}t/{\xi }_{\varphi }$ after the end of inflation, with ${\xi }_{\varphi }=100$. Shown in green is the evolution of $\delta (t)$ when Hubble expansion is included self-consistently. The other curves show the rigid-spacetime solution with initial condition $\delta (0)=(\alpha \times 0.8)\sqrt{{\xi }_{\varphi }}{\varphi }_0/M_{\mathrm{pl}}$, with $\alpha =1$, 0.7, 0.5, 0.3 (blue, red, brown, and black, respectively). Right: rescaled period of background oscillation in a static universe [in units of $(\sqrt{{\lambda }_{\varphi }}{M}_{\mathrm{pl}}{)}^{-1}$] for ${\xi }_{\varphi }=10,{10}^2,{10}^3,{10}^4$ (top to bottom: blue, red, green, and brown, respectively) as a function of the amplitude parameter $\alpha$. The analytic approximation for the period in the large-${\xi }_{\varphi }$ regime is shown in black, and is derived under the assumption that $6{\xi }_{\varphi }{\alpha }^2\gg 1$.

Figure 3

Fast Fourier transforms of $\varphi (\tau )$ for ${\xi }_{\varphi }=0$ (left) and ${\xi }_{\varphi }=10$ (right). For ${\xi }_{\varphi }>0$, the background field’s oscillations show a richer spectral content, including more non-negligible harmonics, which may drive additional resonances compared to the ${\xi }_{\varphi }=0$ case. All higher harmonics are normalized with respect to the amplitude of the fundamental mode and $\omega$ is measured in units of $\sqrt{{\lambda }_{\varphi }}{M}_{\mathrm{pl}}$.

Figure 4

Magnitudes of the first seven nonzero Fourier coefficients $a_n$ of $\varphi (\tau )$ as functions of ${\xi }_{\varphi }$. The $a_n$ have been normalized by the magnitude of the dominant mode $a_2$ for fixed ${\xi }_{\varphi }$, so $a_2$ is not displayed. The Fourier coefficients corresponding to the asymptotic solution in the ${\xi }_{\varphi }\to \infty$ limit are marked by black dots at the right side of the plot.

Figure 5

The behavior of the Fourier coefficients $a_{4n+2}$ for $n\ge 1$ within the range $0\le {\xi }_{\varphi }\le 1$. For a given initial amplitude, ${\varphi }_0$, there exists a single value ${\xi }_{\varphi }$ at which all higher harmonics vanish identically, leaving purely harmonic evolution for $\varphi (\tau )$.

Figure 6

Rescaled effective mass for the adiabatic perturbations, ${\tilde{m}}_{\mathrm{eff},\varphi }^2={\xi }_{\varphi }^2{m}_{\mathrm{eff},\varphi }^2$ (in units of $M_{\mathrm{pl}}$) versus $\tilde{t}\equiv \sqrt{{\lambda }_{\varphi }}{M}_{\mathrm{pl}}t/{\xi }_{\varphi }$, for ${\xi }_{\varphi }=1,10,{10}^2,{10}^3,{10}^4$ (brown, red, green, blue, and orange, respectively), with $m_{\mathrm{eff},\varphi }^2$ given in Eq. (28). The sharp features become more pronounced in the limit ${\xi }_{\varphi }\gg 1$, and asymptote to a single, self-similar behavior.

Figure 7

Left: normalized Floquet exponent $\mathrm{Re}[{\mu }_k]T$ for the adiabatic mode $v_k$ for $0\le {\xi }_{\varphi }\le 1$. Right: normalized Floquet exponent $\mathrm{Re}[{\mu }_k]T$ for the adiabatic mode $v_k$ for ${\xi }_{\varphi }\ge 1$. The common behavior for ${\xi }_{\varphi }\gg 1$ is evident.

Figure 8

Energy density ${\rho }_k^{(\varphi )}$ versus $\tilde{t}=\sqrt{{\lambda }_{\varphi }}{M}_{\mathrm{pl}}t/{\xi }_{\varphi }$ for ${\tilde{k}}^2=[{\xi }_{\varphi }k/(\sqrt{{\lambda }_{\varphi }}{M}_{\mathrm{pl}}){]}^2=0.03$ and ${\xi }_{\varphi }=1,10,{10}^2,{10}^3,{10}^4$ (bottom to top: brown, red, green, blue, and orange, respectively). The energy density rapidly asymptotes to a single behavior in the limit of large ${\xi }_{\varphi }$.

Figure 9

Left: Floquet chart for adiabatic perturbations with ${\xi }_{\varphi }={10}^2,{10}^3,{10}^4$ (red, blue, and black, respectively). The colored dots correspond to the rescaled wave numbers $\tilde{k}={\xi }_{\varphi }k/(\sqrt{{\lambda }_{\varphi }}{M}_{\mathrm{pl}})$ used in the right panel. Right: energy density for the adiabatic mode with ${\xi }_{\varphi }={10}^4$ and ${\tilde{k}}^2=0$, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06 (black, blue, red, green, purple, brown, and cyan, respectively). The black-dotted lines correspond to exact exponential growth, $\exp (2\mathrm{Re}[{\mu }_k]t)$. Note that the Floquet chart (left) is normalized with respect to ${\xi }_{\varphi }$, to make the comparison with the evolution of the energy density as a function of $\tilde{t}=\sqrt{{\lambda }_{\varphi }}{M}_{\mathrm{pl}}t/{\xi }_{\varphi }$ (right) easier.

Figure 10

The Floquet chart for adiabatic perturations with ${\xi }_{\varphi }={10}^3$, and the amplitude of the background oscillation parametrized as $\delta (0)=(\alpha \times 0.8)\sqrt{{\xi }_{\varphi }}{\varphi }_0/M_{\mathrm{pl}}$, with varying $\alpha$. Left: Floquet exponent versus wave number, each scaled by the period of background oscillations $T$. Right: Floquet exponent scaled by the nonminimal coupling ${\xi }_{\varphi }$, versus the rescaled wave number $\tilde{k}={\xi }_{\varphi }k/(\sqrt{{\lambda }_{\varphi }}{M}_{\mathrm{pl}})$.

Figure 11

Normalized Floquet exponents $\mathrm{Re}[{\mu }_k]T$ for the isocurvature perturbations $z_k$ as functions of $g/{\lambda }_{\varphi }$ and $(kT{)}^2$, for ${\xi }_{\varphi }={\xi }_{\chi }$. Top, left to right: ${\xi }_{\varphi }=0$, 0.01; middle, left to right: ${\xi }_{\varphi }=0.1$, 0.4; bottom, left to right: ${\xi }_{\varphi }=0.7$, 1. The bands are visibly tilting, shifting, and squeezing with increasing nonminimal coupling, indicating less efficient preheating for ${\xi }_{\varphi }\sim \mathcal{O}(1)$ compared to the ${\xi }_{\varphi }=0$ case.

Figure 12

Normalized Floquet exponents $\mathrm{Re}[{\mu }_k]T$ for the isocurvature perturbations $z_k$, as functions of $[(g/{\lambda }_{\varphi })-1]{\xi }_{\varphi }$ and $(kT{)}^2$, for ${\xi }_{\varphi }={\xi }_{\chi }$. Top, left to right: ${\xi }_{\varphi }=10,{10}^2$; bottom, left to right: ${\xi }_{\varphi }={10}^3,{10}^4$. The similarity of the Floquet charts for large ${\xi }_{\varphi }$ is visible.

Figure 13

Normalized Floquet exponents $\mathrm{Re}[{\mu }_k]T$ for the isocurvature perturbations $z_k$, as functions of $[(g/{\lambda }_{\varphi })-1]{\xi }_{\varphi }$ and $(kT{)}^2$, for ${\xi }_{\varphi }={\xi }_{\chi }=10,{10}^2,{10}^3,{10}^4,{10}^5$. Each two-dimensional plot corresponds to a slice through the full Floquet chart of Fig. 12 at a specific normalized wave number: top, left to right: $kT=0$, 1; bottom, left to right: $kT=10$, 100.

Figure 14

Left: slope $y$ of the upper boundary of the primary stability band as a function of ${\xi }_{\varphi }$ for $0\le \xi \le 1$. This plot shows that stability bands tilt upward in the $(kT{)}^2\text{−}(g/{\lambda }_{\varphi })$ plane as ${\xi }_{\varphi }$ increases. Right: width of the primary instability band $\mathrm{\Delta }$ as a function of ${\xi }_{\varphi }$, where $\mathrm{\Delta }$ is defined as the difference between the $g/{\lambda }_{\varphi }$ intercepts of the upper and lower stability boundaries.

Figure 15

The rescaled effective masses, ${\tilde{m}}_{\mathrm{eff}}^2={\xi }_{\varphi }^2{m}_{\mathrm{eff}}^2$, for the adiabatic (blue) and isocurvature (red) modes (in units of $M_{\mathrm{pl}}$) versus $\tilde{t}=\sqrt{{\lambda }_{\varphi }}{M}_{\mathrm{pl}}t/{\xi }_{\varphi }$, for a symmetric potential (${\xi }_{\varphi }={\xi }_{\chi }$, $g={\lambda }_{\varphi }={\lambda }_{\chi }$), with ${\xi }_{\varphi }={10}^3$. The green and black dashed curves show the contributions to ${\tilde{m}}_{\mathrm{eff},\chi }^2$ arising from the potential (${\tilde{m}}_{1,\chi }^2$) and from the nontrivial field-space manifold (${\tilde{m}}_{2,\chi }^2$), respectively.

Figure 16

Energy density ${\rho }_k^{(\chi )}$ for the isocurvature modes $z_k$ in a symmetric potential (${\xi }_{\varphi }={\xi }_{\chi }$, $g={\lambda }_{\varphi }={\lambda }_{\chi }$), as a function of $\tilde{t}=\sqrt{{\lambda }_{\varphi }}{M}_{\mathrm{pl}}t/{\xi }_{\varphi }$. The wave numbers are $\tilde{k}=0$ (left) and $\tilde{k}=0.1$ (right), with $\tilde{k}=k{\xi }_{\varphi }/(\sqrt{{\lambda }_{\varphi }}{M}_{\mathrm{pl}})$. In both plots, the nonminimal couplings are given by ${\xi }_{\varphi }=10,{10}^2,{10}^3,{10}^4$ (bottom to top: blue, red, green, black dashed, respectively). Compare with the growth of adiabatic modes shown in Figs. 8 and 9.

Figure 17

The number of times the isocurvature mode $z_k(t)$ with $k=0$ oscillates per background oscillation ($N_z/N_{\varphi }$) as a function of $a_{\varphi }$ and ${\xi }_{\varphi }$, where $g/{\lambda }_{\varphi }=1+(a_{\varphi }/{\xi }_{\varphi })$: $a_{\varphi }=2$ (blue), $a_{\varphi }=20$ (red), and $a_{\varphi }=200$ (black), from bottom to top.

Figure 18

Left: rescaled effective mass for the isocurvature modes ${\tilde{m}}_{\mathrm{eff},\chi }^2={\xi }_{\varphi }^2{m}_{\mathrm{eff},\chi }^2$ (in units of $M_{\mathrm{pl}}$) versus $\tilde{t}=\sqrt{{\lambda }_{\varphi }}{M}_{\mathrm{pl}}t/{\xi }_{\varphi }$ for a softly broken symmetric potential with ${\xi }_{\varphi }={\xi }_{\chi }={10}^4$ and $g/{\lambda }_{\varphi }=1+(2/{\xi }_{\varphi })$ (blue), $g/{\lambda }_{\varphi }=1+(20/{\xi }_{\varphi })$ (red-dashed), and $g/{\lambda }_{\varphi }=1+(200/{\xi }_{\varphi })$ (black-dotted). Right: effective mass ${\tilde{m}}_{\mathrm{eff},\chi }^2$ for $g/{\lambda }_{\varphi }=1+(2/{\xi }_{\varphi })$ and ${\xi }_{\varphi }=3,10,{10}^2,{10}^3,{10}^4$ (brown, blue, red, green, and black, respectively).

Figure 19

Left: contributions ${\tilde{m}}_{\{1,2\},\chi }^2={\xi }_{\varphi }^2{m}_{\{1,2\},\chi }^2$ to ${\tilde{m}}_{\mathrm{eff},\chi }^2$ versus $\tilde{t}=\sqrt{{\lambda }_{\varphi }}{M}_{\mathrm{pl}}t/{\xi }_{\varphi }$ for a softly broken symmetric potential with ${\xi }_{\varphi }={\xi }_{\chi }$, $g/{\lambda }_{\varphi }=1+(a_{\varphi }/{\xi }_{\varphi })$, and $a_{\varphi }=2$. Solid lines correspond to the contributions that arise from the potential (${\tilde{m}}_{1,\chi }^2$) and dotted lines correspond to the contributions that arise from the nontrivial field-space manifold (${\tilde{m}}_{2,\chi }^2$). Right: energy density ${\rho }_k^{(\chi )}$ for $k=0$ and ${\xi }_{\varphi }={\xi }_{\chi }$, with $g/{\lambda }_{\varphi }=1+(a_{\varphi }/{\xi }_{\varphi })$ and $a_{\varphi }=2$ (dotted line) and $a_{\varphi }=20$ (solid line). In both plots, the nonminimal couplings are given by ${\xi }_{\varphi }=10,{10}^2,{10}^3,{10}^4$ (blue, red, green, and black, respectively).

Figure 20

The rescaled effective mass ${\tilde{m}}_{\mathrm{eff},\chi }^2$ versus $\tilde{t}=\sqrt{{\lambda }_{\varphi }}{M}_{\mathrm{pl}}t/{\xi }_{\varphi }$ for $g/{\lambda }_{\varphi }=2$ (or $a_{\varphi }={\xi }_{\varphi }$), and ${\xi }_{\varphi }=10,{10}^2,{10}^3,{10}^4$ (bottom to top: blue, red, green-dashed, and black-dotted, respectively).

Figure 21

Left: amplitude of the isocurvature mode $|z_k(t)|$ for $k=0$, with $g/{\lambda }_{\varphi }=2$ and varying ${\xi }_{\varphi }$. Note that as ${\xi }_{\varphi }$ increases, $z_k(t)$ oscillates more frequently per oscillation of the background field, entering the broad-resonance regime. Right: isocurvature energy density ${\rho }_k^{(\chi )}$ for $k=0$ and $g/{\lambda }_{\varphi }=2$. For both plots, ${\xi }_{\varphi }=10,{10}^2,{10}^3,{10}^4$ (bottom to top: blue, red, green, and black, respectively).

Figure 22

Floquet exponents for the isocurvature perturbations normalized by the period of background oscillations for ${\xi }_{\varphi }={10}^4$ and varying $ϵ=0.5,0,-1$. Each two-dimensional plot corresponds to a slice of the full Floquet chart at a specific normalized wave number: top, left to right: $kT=0$, 1; bottom, left to right: $kT=10$, 100.

Figure 23

Comparison of the stability boundary of the primary instability band given by $\det U=0$ when $U$ is truncated to $2\times 2$, $3\times 3$, and $10\times 10$ matrices, for ${\xi }_{\varphi }=0$ (left) and ${\xi }_{\varphi }=0.5$ (right). All higher-dimension truncations are too close to the $10\times 10$ truncation to resolve. Hence we find that the $3\times 3$ truncation is sufficient to yield an accurate approximation to the location of the stability boundary in the $(kT{)}^2\text{−}(g/{\lambda }_{\varphi })$ plane.