Preheating after Higgs inflation: Self-resonance and gauge boson production

We perform an extensive analysis of linear fluctuations during preheating in Higgs inflation in the Einstein frame, where the fields are minimally coupled to gravity, but the field-space metric is nontrivial. The self-resonance of the Higgs and the Higgsed gauge bosons are governed by effective masses that scale differently with the nonminimal couplings and evolve differently in time. Coupled metric perturbations enhance Higgs self-resonance and make it possible for Higgs inflation to preheat solely through this channel. For $\xi \gtrsim 100$ the total energy of the Higgs-inflaton condensate can be transferred to Higgs particles within 3 $e$-folds after the end of inflation. For smaller values of the nonminimal coupling preheating takes longer, completely shutting off at around $\xi \simeq 30$. The production of gauge bosons is dominated by the gauge boson mass and the field-space curvature. For large values of the nonminimal coupling $\xi \gtrsim 1000$, it is possible for the Higgs condensate to transfer the entirety of its energy into gauge fields within one oscillation. For smaller values of the nonminimal coupling gauge bosons decay very quickly into fermions, thereby shutting off Bose enhancement. Estimates of non-Abelian interactions indicate that they will not suppress preheating into gauge bosons for $\xi \gtrsim 1000$.

Figure 1

The components of the effective mass of the Higgs fluctuations $m_{1,h}^2$ and $m_{3,h}^2$ rescaled by the Hubble scale. The blue curves show the numerical curves for $\xi =10$ and the red dashed curves show the approximate analytic expressions of Eqs. (92) and (95), respectively.

Figure 2

The ratio of the components of the effective mass of the Higgs fluctuations $m_{1,h}^2$ (left) and $m_{3,h}^2$ (right) rescaled by the Hubble scale at the end of inflation. The blue, red dashed, and green dotted curves correspond to $\xi =10,{10}^2,{10}^3$, respectively.

Figure 3

The size of the comoving Hubble radius during and after inflation for $\xi =10,{10}^2,{10}^3,{10}^4$ (black, blue, red, and green, respectively). The curves are normalized to unity at the end of inflation.

Figure 4

Left: The effective mass squared (black-dotted curve), along with the contributions from the potential (blue curve) and the coupled metric perturbations (red curve). Right: The energy density in the background Higgs condensate (orange curve) and the Higgs fluctuations (blue curve) for $\xi =10,{10}^2,{10}^3$ (top to bottom). The green curve shows 10% of the background energy density, which is used as a proxy for the limit of our linear analysis. The orange-dashed curve is ${\rho }_0{a}^{-4}$, corresponding to the redshifting of the background energy density during radiation-dominated expansion. The energy density is plotted in units of $M_{\text{Pl}}^4$.

Figure 5

The energy density in the background Higgs condensate (orange curve) and the Higgs fluctuations (blue curve) for the marginal case of $\xi =30$ (top to bottom). The green line shows 10% of the background energy density, which is used as a proxy for the limit of our linear analysis. The orange-dashed curve is ${\rho }_0{a}^{-4}$, corresponding to the redshifting of the background energy density during radiation-dominated expansion.

Figure 6

The number of $e$-folds after inflation when the energy density in the Higgs fluctuations equals the background energy density $N_{\mathrm{reh}}$ (blue solid curve) or 10% of the background energy density $N_{\mathrm{br}}$ (black dashed curve).

Figure 7

The evolution of the longitudinal gauge field mode during inflation for $\xi ={10}^3$. The solid lines correspond to the numerical solution for $k/H_{\mathrm{end}}=1,10,{10}^2,{10}^3,{10}^4$ (blue, green, black, orange, and brown, respectively), along with the approximate solutions of Eqs. (112) and (117) (red-dotted lines). The vertical lines correspond to the points $k|\tau |=x_c$, where the matching between the two asymptotic regimes is performed. For the brown curve this does not occur during inflation.

Figure 8

The mode-function amplitude (left) and frequency (right) for $N=0$, 2, 4 $e$-folds before the end of inflation (blue, green, and black, respectively). Solid lines correspond to the WKB expression of Eq. (108) and red-dotted lines correspond to the approximate solutions for $x\gg x_c$ and $x\ll x_c$. The dots show the full numerical results.

Figure 9

Dominant components of the effective frequency squared for $\xi ={10}^3$ (left) and $\xi ={10}^4$ (right). Color coding is as follows: ${\tilde{m}}_B^2/a^2$ (red), $m_{2,\theta }^2$ (blue), and $k^2/a^2$ (black) for the maximum excited wave number $k_{\max }$. The orange-dotted curve shows the scaling $a^{-3}$.

Figure 10

The particle number density for $k/H_{\mathrm{end}}=1,150,550,2600,28000$ (blue, black, green, red, and purple, respectively). From left to right: $\xi ={10}^2,{10}^3,{10}^4$. If a colored curve is missing from a panel, the corresponding wave number is not excited.

Figure 11

Left: The particle number density after the first inflaton zero crossing for $\xi ={10}^2,{10}^3,{10}^4$ (blue, orange, and green, respectively) Right: The ratio of the energy density in gauge fields to the background inflaton energy density as a function of the nonminimal coupling $\xi$ after the first zero crossing. We see that for $\xi \gtrsim {10}^3$ gauge boson production can preheat the Universe after one background inflaton zero crossing, and hence it is much more efficient than Higgs self-resonance.

Figure 12

Left: Fermion to Higgs mass ratio as a function of the Yukawa coupling for $\xi =10,100,{10}^3$ (blue, red, and green, respectively) at $N=0$, 1.5, 3 (solid, dashed, and dotted curves, respectively). Right: Ratio of the decay rate to the Hubble rate. The color coding is the same.

Figure 13

Loop diagrams that contribute to the scattering of Higgs bosons to gluon pairs.

Figure 14

Energy density per mode (in arbitrary units) with (blue) and without (red) considering particle decays for $\xi ={10}^3$ (left) and $\xi ={10}^4$ (right). The time is rescaled by the period of background oscillations.

Figure 15

Reheat temperature in units of $M_{\mathrm{Pl}}$ as a function of the nonminimal coupling $\xi$. The discontinuity at $\xi \simeq {10}^3$ occurs due to the instantaneous preheating to gauge fields. The light red region represents the uncertainty of the exact threshold of instantaneous preheating to gauge fields. The black-dotted line corresponds to the unitarity scale constraint $T_{\mathrm{reh}}\lesssim M_{\mathrm{Pl}}/3\xi$. The blue-dashed line shows the reheat temperature due entirely to Higgs self-resonance, assuming gauge boson production above the unitarity scale is suppressed due to unknown UV physics.