Superheavy scalar dark matter from gravitational particle production in $\alpha$-attractor models of inflation

We study the phenomenon of gravitational particle production as applied to a scalar spectator field in the context of $\alpha$-attractor inflation. Assuming that the scalar has a minimal coupling to gravity, we calculate the abundance of gravitationally produced particles as a function of the spectator’s mass $m_{\chi }$ and the inflaton’s $\alpha$ parameter. If the spectator is stable and sufficiently weakly coupled, such that it does not thermalize after reheating, then a population of spin-0 particles is predicted to survive in the Universe today, providing a candidate for dark matter. Inhomogeneities in the spatial distribution of dark matter correspond to an isocurvature component, which can be probed by measurements of the cosmic microwave background anisotropies. We calculate the dark-matter-photon isocurvature power spectrum and by comparing with the upper limits from Planck, we infer constraints on $m_{\chi }$ and $\alpha$. If the scalar spectator makes up all of the dark matter today, then for $\alpha =10$ and $T_{\mathrm{RH}}={10}^4\mathrm{GeV}$ we obtain $m_{\chi }>1.8\times {10}^{13}\mathrm{GeV}\approx 1.2m_{\varphi }$, where $m_{\varphi }$ is the inflaton’s mass.

Figure 1

Inflaton potentials for the $T$-model and $E$-model $\alpha$ attractors. Each panel shows several values of $\alpha$, and the value of $m_{\varphi }$ is chosen such that $A_s$ agrees with the value measured by Planck. There are three dots along each curve; the red dot indicates the value of $\varphi$ at the end of inflation, the magenta dot indicates 10 $e$-foldings before the end of inflation, and the blue dot indicates 60 $e$-foldings before the end of inflation.

Figure 2

Evolution of the spacetime background for a fiducial model of inflation. We show the $T$-model $\alpha$ attractor with $\alpha =1$ and $\mu =2.5\times {10}^{13}\mathrm{GeV}$, implying $m_{\varphi }=1.5\times {10}^{13}\mathrm{GeV}$, $H_{\mathrm{cmb}}=1.4\times {10}^{13}\mathrm{GeV}$, $H_e=5.9\times {10}^{12}\mathrm{GeV}$, and $N_{\mathrm{cmb}}=60$. The top panels show the FRW scale factor $a$ (top left) and the Hubble parameter $H$ (top right). Both bottom panels show the Ricci scalar $R$ with the bottom-right panel showing an enlargement to highlight oscillations near the end of inflation. A vertical black line indicates the end of inflation at $\eta ={\eta }_e=0$, and the top of the frame shows the number of $e$-foldings after the end of inflation $N(\eta )=\ln a(\eta )/a_e$.

Figure 3

Evolution of the scalar spectator’s mode functions near the end of inflation. All panels correspond to the $T$-model $\alpha$ attractor with $\alpha =0.1$ and $\mu =2.528\times {10}^{13}\mathrm{GeV}$ and a minimally coupled scalar spectator with mass $m_{\chi }=3m_{\varphi }$. From top to bottom the rows correspond to different modes with wave numbers $k/a_e{H}_e=0.01$, 0.1, 1, and 10. For each mode we show the scaled amplitude of the mode function, $a(\eta )m_{\chi }|{\chi }_{\mathit{k}}(\eta ){|}^2$, as a function of scaled conformal time $a_e{H}_e\eta$. Inflation ends at $\eta ={\eta }_e=0$. The solid-blue curve shows the numerical solution, and the dashed-blue curve shows the Bunch-Davies mode function (18), which sets the initial condition at early time.

Figure 4

The spectrum of spin-0 particles arising from gravitational particle production in an $\alpha$-attractor model of inflation. We show the comoving number spectrum $a^3{n}_k$ where $n_k\equiv kdn/dk$ in units of $a_e^3{H}_e^3$. All four panels correspond to the $T$-model $\alpha$ attractor with inflaton mass $m_{\varphi }\approx 6\times {10}^{-6}{M}_p$, and the value of $\alpha$ is varied across the panels from $\alpha =1$, 3, 10, to $\infty$. In each panel, the various curves show different values of the scalar spectator’s mass from $m_{\chi }=0.001m_{\varphi }$ to $3.16m_{\varphi }$.

Figure 5

The abundance of spin-0 particles arising from gravitational particle production in $\alpha$-attractor models of inflation. The left panel shows the comoving number density $a^{3}n$ in units of $a_e^3{H}_e^3$, and the right panel shows the dimensionless relic abundance $\mathrm{\Omega }{h}^2$. This calculation is performed for the $T$-model $\alpha$ attractor with $\mu =2.5\times {10}^{13}\mathrm{GeV}$ and $m_{\varphi }=1.5\times {10}^{13}\mathrm{GeV}$. We vary $\alpha$ and the spectator’s mass, $m_{\chi }$. The right panel also takes $T_{\mathrm{RH}}={10}^5\mathrm{GeV}$, and for models with a different reheating temperature, the relic abundance can be inferred from Eq. (23) and using the left panel.

Figure 6

The relic abundance of light ($m_{\chi }\ll H_e$) scalar spectator particles that arise from gravitational particle production in $\alpha$-attractor models of inflation. The relic abundance $\mathrm{\Omega }{h}^2$ is insensitive to $m_{\chi }$ for $m_{\chi }\ll H_e$, as seen in Fig. 5, and we take $m_{\chi }={10}^{-3}{H}_e$ here. We take $T_{\mathrm{RH}}={10}^2\mathrm{GeV}$, and for other reheating temperatures $\mathrm{\Omega }{h}^2\propto T_{\mathrm{RH}}$ as in Eq. (23). The blue and orange curves show the predicted relic abundance $\mathrm{\Omega }{h}^2$ for the $T$-model and $E$-model $\alpha$ attractors, respectively. For comparison we show the measured dark matter relic abundance ${\mathrm{\Omega }}_{\mathrm{dm}}{h}^2\simeq 0.12$ (red line) and the prediction of quadratic chaotic inflation (green-dashed line), which is the limit of the $T$- and $E$-model $\alpha$ attractors for $\alpha \to \infty$.

Figure 7

The predicted isocurvature power spectrum ${\mathrm{\Delta }}_{\mathcal{S}}^2(k)$. As the spectator mass is raised from $m_{\chi }/m_{\varphi }\ll 1$ to $m_{\chi }/m_{\varphi }\gtrsim 1$, the spectrum’s low-$k$ tail changes from a red spectral tilt (red-colored curves) to a blue spectral tilt (blue-colored curves). For this example we take the $T$-model $\alpha$ attractor with $m_{\varphi }\approx 6\times {10}^{-6}{M}_p$ and $\alpha =3$.

Figure 8

A slice of the parameter space with various constraints. We show the scalar spectator field’s mass $m_{\chi }$, the $T$-model $\alpha$ attractor’s parameter $\alpha$, and we fix $m_{\varphi }=6\times {10}^{-6}{M}_p$. The solid curves denote the minimum $m_{\chi }$ and maximum $\alpha$ at a given reheating temperature $T_{\mathrm{RH}}$ for which the predicted isocurvature is consistent with the upper bound on ${\mathrm{\Delta }}_{\mathcal{S}}^2(k_{\mathrm{cmb}})$ inferred by Planck. Along the dashed curves, the scalar spectator’s predicted relic abundance saturates the measured dark matter relic abundance $\mathrm{\Omega }{h}^2\simeq 0.12$, and the spectator is overproduced for smaller $m_{\chi }$ or larger $\alpha$ at a given $T_{\mathrm{RH}}$. The gray-dashed line denotes the maximum $\alpha$ for which Planck and BICEP/Keck’s constraint on the tensor-to-scalar ratio $r<0.064$ (95% C.L.) is satisfied at $N_{\mathrm{cmb}}=60$, and the limit strengthens to $\alpha \lesssim 20$ at $N_{\mathrm{cmb}}=50$. Note that the parameter space shown here is consistent with the limit in Eq. (10).