Irruption of massive particle species during inflation

All species of (nonconformally coupled) particles are produced during inflation so long as their mass $M$ is not too much larger than $H$, the expansion rate during inflation. It has been shown that if a particle species that is normally massive ($M\gg H$) couples to the inflaton field in such a way that its mass vanishes, or at least becomes small ($M<H$), for a particular value of the inflaton field, then not only are such particles produced, but an irruption of that particle species can occur during inflation. In this paper we analyze creation of a massive particle species during inflation in a variety of settings, paying particular attention to models which realize such an irruptive production mechanism.

Figure 1

A graph of $H^{-2}(M_{\mathrm{eff}}^2-a^{-3}{a}^{\prime \prime })$ as a function of time in two models for four choices of parameters. The models are for inflaton-irrupton coupling through either the potential term (discussed in Sec. 2b) or through the kinetic term (discussed in Sec. 2c). Where the lines are dashed, $H^{-2}(M_{\mathrm{eff}}^2-a^{-3}{a}^{\prime \prime })<0$. The dashed vertical lines denoted “$t_{\mathrm{EI}}$” mark the end of the inflationary phase. Time is in units of $m^{-1}$, where $m$ is the inflaton mass, and $t=0$ corresponds to the time when $\varphi =3M_{\mathrm{Pl}}$. The inflaton field has the value $\varphi ={\varphi }_{*}=0.8M_{\mathrm{Pl}}$ at $t=13.6$, indicated by the dashed vertical lines marked “$t_{*}$.” Model parameters are chosen such that in each individual plot the irrupton masses are the same at $\varphi \equiv 0$ and $a^{\prime \prime }=0$ (i.e., at late time) for the two models.

Figure 2

The time evolution of $|\frac{{\omega }_k}{2}[|{\mu }_k{|}^2+\frac{a^2}{{\omega }_k^2}|{\dot{\mu }}_k{|}^2]-\frac{1}{2}|$ for two representative cases for the kinetically coupled irrupton as extracted from the numerical solution for the choices of parameters $M=2.5$, $\epsilon =0.6$ and $k=2.3\times {10}^{-15}$ (left plot) and $M=4.0$, $\epsilon =0.6$ and $k=4.3\times {10}^{-5}$ (right plot). If ${\omega }_k^2>0$, $\frac{{\omega }_k}{2}[|{\mu }_k{|}^2+\frac{a^2}{{\omega }_k^2}|{\dot{\mu }}_k{|}^2]-\frac{1}{2}$ is real and equal to $|{\beta }_k{|}^2$. The evolution during the tachyonic phase is presented for demonstrative purposes to indicate the rapid increase during this period; we caution, however, that there is no fashion in which it can be interpreted as a particle number during the tachyonic phase. At both early- and late-times, the background space-time evolution is sufficiently adiabatic with respect to this mode that $|{\beta }_k{|}^2$ can indeed be interpreted as the occupation number for mode $\mathit{k}=k\widehat{\mathit{k}}$; note that the late-time behavior (i.e., after the end of inflation) shows that $|{\beta }_k{|}^2$ undergoes damped oscillation about a constant nonzero value, indicating particle production has occurred. The times $t_0^k$, $t_{\mathrm{HC}}$, $t_{*}$ and $t_{\mathrm{EI}}$ indicate, respectively, the times when the initial conditions Eq. (44) were imposed, when the mode crosses the Hubble radius ($k=aH$), when $\nu ={\nu }_{*}$, and when inflation ends ($\ddot{a}<0$).

Figure 3

Comoving particle density spectra $n_k^{\mathrm{c}}$ as a function of comoving $k$ (in units of $m$) for a minimally coupled scalar field with constant mass $M$ (as annotated; also in units of $m$) as discussed in Sec. 2a, but using the chaotic inflation model to describe the evolution of the background (i.e., the $\epsilon \to \infty$ limit of the kinetically coupled irrupton of Sec. 2c). Different curves are for various heavy particle masses $M$. For comparison, the thin (red) dashed line indicates the scaling of $n_k^{\mathrm{c}}$ with $k$ if the scaling were $k^3$ (constant $|{\beta }_k{|}^2$). The (blue) circles on the lines for $M=0.2$, 1, and 2 are sampled from spectra in Ref. [34]. The vertical dashed line denotes $k=a_{\mathrm{EI}}{H}_{\mathrm{EI}}$. See Table 1 for a summary of which numerical methods were applied to obtain these results.

Figure 4

Comoving particle density spectra $n_k^{\mathrm{c}}$ for the potentially coupled irrupton as a function of comoving $k$ (in units of $m$) for various parameter choices $M_g\equiv g{M}_{\mathrm{Pl}}=1.22\times {10}^{6}g$ (as annotated; in units of $m$). The choice of $M_g$ is indicated either by the black numbers annotating the lines, or by the numbers in the legend labeling the differently-styled (colored) lines. These latter cases are plotted differently to aid the reader visually, and also because these large-$M_g$ spectra show qualitatively different behavior, which is discussed in the text. The thin (red) dashed lines illustrate an $n_k^{\mathrm{c}}$ scaling proportional to $k^3$ (constant $|{\beta }_k{|}^2$). The vertical dashed lines denote $k=a_{*}{H}_{*}$ and $k=a_{\mathrm{EI}}{H}_{\mathrm{EI}}$, where $*$ denotes the values at the instant where $\varphi ={\varphi }_{*}$. See Table 1 for a summary of which numerical methods were applied to obtain these results.

Figure 5

Comoving particle density spectra $n_k^{\mathrm{c}}$ (left plot) and the late-time Bogoliubov coefficient (i.e., mode-$\mathit{k}$ occupancy number) $|{\beta }_k{|}^2$ (right plot) for the kinetically coupled irrupton for as a function of comoving $k$ (in units of $m$) for various heavy-particle masses $M$ (as annotated; also in units of $m$) with fixed $\epsilon =0.6$. We choose to present $|{\beta }_k{|}^2$ rather than $n_k^{\mathrm{c}}$ in the right plot for greater clarity in these very-large-$M$ cases. In the left plot, the thin (red) dashed lines illustrate a scaling proportional to $k^3$ (constant $|{\beta }_k{|}^2$). The vertical dashed lines denote $k=a_{*}{H}_{*}$ and $k=a_{\mathrm{EI}}{H}_{\mathrm{EI}}$, where $*$ denotes the values at the instant where $\varphi ={\varphi }_{*}$. See Table 1 for a summary of which numerical methods were applied to obtain these results, and the meaning of the open circles and solid lines in the right plot.

Figure 6

Comoving particle density spectra $n_k^{\mathrm{c}}$ as a function of comoving $k$ (in units of $m$) for the kinetically coupled irrupton, for various choices of $\epsilon$ with fixed $M=2$ (upper plot) and $M=4$ (lower-left plot). The thin (red) dashed lines illustrate a scaling proportional to $k^3$ (constant $|{\beta }_k{|}^2$). The (blue) circles on the line for $\epsilon \to \infty$ in the upper plot are sampled from a spectrum in Ref. [34]. The inset in the lower-left plot shows detail near the peak in the spectra. Also shown for the case of $M=4$ are the values of $|{\beta }_k{|}^2$ (lower-right plot). The vertical dashed lines denote $k=a_{*}{H}_{*}$ and $k=a_{\mathrm{EI}}{H}_{\mathrm{EI}}$, where $*$ denotes the values at the instant where $\varphi ={\varphi }_{*}$. See Table 1 for a summary of which numerical methods were applied to obtain these results, and the meaning of the open circles and solid lines in the lower plots.

Figure 7

The oscillation frequency ${\omega }_k^2/a^2$ and its three contributions for the kinetically coupled irrupton [see Eq. (31)] plotted as a function of cosmic time. The open circle in each plot indicates the point where two of the contributions to ${\omega }_k^2/a^2$, namely $M_{\mathrm{eff}}^2$ and $|a^{\prime \prime }/a^3|$, are approximately equal. The point where $k^2/a^2$ and $M_{\mathrm{eff}}^2$ first intersect is denoted by the open square. The value of $k$ at the “elbow” break-point, $k_{*}$, is the value of $k$ for which $k_{*}^2/a^2\approx |a^{\prime \prime }/a^3|\approx M_{\mathrm{eff}}^2$; i.e., the open circle and the square coincide. From the figure we see that this occurs for $k=k_{*}=2.5\times {10}^{-7}$. For $k\ll k_{*}$ (illustrated by the case $k={10}^{-14}$) the onset of the tachyonic phase is determined by $M_{\mathrm{eff}}^2$ and independent of $k$. Since the duration of the tachyonic phase is independent of $k$, $|{\beta }_k{|}^2$ should be independent of $k$, and $n_k^{\mathrm{c}}\propto k^3$. For $k\gg k_{*}$ (illustrated by the case $k={10}^{-2}$) the onset of the tachyonic phase is determined by $k^2/a^2$ and the duration of the tachyonic phase is reduced, which implies $|{\beta }_k{|}^2$ will decrease with $k$ and $n_k^{\mathrm{c}}$ will grow more slowly than $k^3$. It is clear that for even larger $k$, there may be no tachyonic phase at all, which exponentially suppresses the particle number produced. Note that larger values of $\epsilon$ flatten $M_{\mathrm{eff}}^2$, giving a longer maximum duration of the tachyonic phase, while larger values of the $M$ move the minimum of $M_{\mathrm{eff}}^2$ upward, which suppresses particle production by reducing the duration of the tachyonic phase (which argument holds until such a phase ceases to exist; see Fig. 8 for further consideration of this case).

Figure 8

As for Fig. 7, except this series of plots of the various contributions to ${\omega }_k^2/a^2$ for varying $k$ is shown for a case where no tachyonic phase is present. Nevertheless, the minimum value of ${\omega }_k^2/a^2$ still becomes $k$ independent at small $k$, which leads to a $n_k^{\mathrm{c}}\sim k^3$ infrared behavior in the particle spectrum. As $k$ becomes very large (lower plot), the minimum value of ${\omega }_k^2/a^2$ is clearly shifted upward, which leads to the large-$k$ suppression in the spectra.

Figure 9

The envelope $\pm (1/2)|{\omega }_k^{\prime }/{\omega }_k^2|$ (grey) bounding the real part of the rapidly oscillating integrand $(1/2)({\omega }_k^{\prime }/{\omega }_k^2)e^{-2i\mathrm{\Phi }}$ in Eq. (59) (black) for the first iterate ${{\beta }_k}^{(1)}$, for a variety of values of $k$ for the case $M=4$ and $\epsilon =0.6$ for the kinetically coupled irrupton, plotted as a function of the phase deviation $\mathrm{\Delta }\mathrm{\Phi }$ from when ${\omega }_k^2/a^2$ is minimized. The imaginary part of the integrand shows similar behavior; we omit it for clarity. We also show $|{{\beta }_k}^{(1)}(\mathrm{\Phi }){|}^2$ per Eq. (59) (right-scale on each axis; red dotted line) for illustrative purposes to indicate the impact of the shape of the envelope on the evolution of $|{\beta }_k{|}^2$. HC denotes Hubble-radius crossing; the mode with $k=1$ is always sub-Hubble-radius sized. The relevant comparisons we intend the reader to make from this series of plots are between plots A and B and between plot C and plots A or D; in particular, we caution the reader that the approximate order-of-magnitude equality of the value of $|{\beta }_k{|}^2$ at large positive $\mathrm{\Delta }\mathrm{\Phi }$ of $|{\beta }_k{|}^2$ in plot D (large $k$), and the values of the same quantity in plot A and B (small $k$), is a coincidental consequence of the values of $k$ we have chosen to display (see right plot of Fig. 5), so no deep significance should be attached to that approximate equality.

Figure 10

The maximum values attained during inflation of the adiabaticity parameter $|{\omega }_k^{\prime \prime }/{\omega }_k^3|$ (the behavior of the other adiabaticity parameter $|{\omega }_k^{\prime }/{\omega }_k^2{|}^2$ is similar) in the potentially coupled irrupton model at representative fixed $k$ (see legend) as a function of $M_g$ (in units of $m$). The black circles indicate the relevant values $M_g=\{17,60,2.1\times {10}^2,7.2\times {10}^2,1.2\times {10}^3,3.7\times {10}^3,1.2\times {10}^4\}$ (see Fig. 4). The divergences evident at small $M_g$ for some of the smaller-$k$ curves indicate that these modes allow for a tachyonic phase. The minimum of each curve is indicated by a grey diamond.

Figure 11

The present-day relic mass-density of stable irruptons for the kinetically coupled model (upper plot) and the potentially coupled model (lower plot) in units of ${({\mathrm{\Omega }}_{\mathrm{DM}}{h}^2)}_{\text{Planck}}\times (T_{\mathrm{RH}}/{10}^9\mathrm{GeV})\times (m/{10}^{13}\mathrm{GeV}{)}^2$ where ${({\mathrm{\Omega }}_{\mathrm{DM}}{h}^2)}_{\text{Planck}}=0.1186$ [41], as a function of the late-time effective mass $M_{\mathrm{eff}}^{\infty }$. Crosses (black) represent points we have explicitly sampled in our numerical computations. In the upper plot, solid (grey) lines join points at constant $M/m$ (from top to bottom $M=0.2$, 0.4, 0.6, 0.8, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 3.75, 4.0, 4.25, 4.5, 4.75, 5.0, 6.0, 7.0), and dashed (grey) lines join points at constant $\epsilon$ (from left to right $\epsilon =\infty$, 1.0, 0.8, 0.6, 0.5, 0.4, 0.35, 0.3, 0.27, 0.25), where interpolation between sampled points has been performed. The (green) square points in the upper plot are models where it was necessary to cut the spectrum off in the infrared to obtain a finite value; while the dash-dot (red) line is the constant-$M$ result taken from Kuzmin and Tkachev [34], which our $\epsilon \to \infty$ results recover well except at small $M$; see footnote (). In the lower plot, the solid (grey) line is a (log-log) cubic spline interpolant between the sampled points while the dashed (red) line is the result of our analytical expression, Eq. (b10). We show the ratio of the analytical to numerical result in the inset, including also numerically sampled points up to the mass $M_{\mathrm{eff}}^{\infty }/m=2.9\times {10}^4$ which are not shown on the main axes as they lie exactly on the $\mathrm{\Omega }{h}^2\propto M_g^{5/2}$ extrapolation line.