Gravitational Production of Massive Spin-2 Particles During Reheating

We study the minimal gravitational portal for a massive spin-2 dark matter candidate $X_{\mu\nu}$ produced during perturbative reheating. The dark sector couples to the visible sector only via gravity, and we analyze two unavoidable channels: (i) inflaton condensate annihilation, $\phi+\phi \to X+X$, and (ii) thermal scatterings, ${\rm SM}+{\rm SM}\to X+X$, both mediated by graviton exchange. Working in the Fierz-Pauli framework for a free massive spin-2 field of mass $m_2$, we derive the graviton-mediated amplitudes and perform a full helicity decomposition of the final state. The relic abundance is obtained analytically in terms of $m_2$ and the reheating temperature $T_{\rm RH}$. In the light mass regime $m_2 \ll m_\phi$ (with $m_\phi$ the inflaton mass during oscillations), production is overwhelmingly dominated by the longitudinal (helicity-0) mode: the $2\to2$ cross section is parametrically enhanced, scaling as $\sim (m_\phi/m_2)^4$, and yields efficient dark matter production despite purely gravitational couplings. Compared to lower-spin cases (spin-$0$, $1/2$, $1$, and $3/2$), massive spin-$2$ production is substantially more efficient for the same reheating history. Over most of the parameter space the inflaton condensate channel dominates the yield, while the thermal contribution is negligible. Avoiding overproduction typically requires either a relatively low $T_{\rm RH}$ or a spin-$2$ mass near threshold, $m_2 \lesssim m_\phi$. This places the spin-$2$ portal on similar footing to other higher spins in reheating scenarios, while emphasizing the central role of the helicity-$0$ mode and the reheating history in setting the dark matter density.