Gravitationally Produced Decaying Dark Matter

• Gravitationally produced decaying dark matter is characterized by a freeze-in mechanism via Planck-suppressed gravitational interactions that produce DM out of equilibrium.
• Relic abundance and mass constraints are determined by integrating the Boltzmann equation, tightly linking the DM mass window to the maximum post-reheating temperature.
• Indirect detection through high-frequency gravitational waves and graviton-to-photon conversion offers unique astrophysical probes of these minimally interacting dark sectors.

Gravitationally produced decaying dark matter denotes a class of scenarios in which dark matter (DM) is generated exclusively via gravitational interactions in the early universe and subsequently undergoes slow decay through processes mediated by gravity—most notably into gravitons or lighter gravitational-sector states. Such scenarios, devoid of non-gravitational couplings to the Standard Model (SM) at tree level, lead to characteristic predictions for DM mass ranges, decay signatures, and cosmological imprints. The landscape encompasses both minimal effective field theory constructions as well as higher-dimensional (Kaluza-Klein) extensions involving towers of gravitationally coupled states.

1. Gravitational Production Mechanisms

In the absence of sizable non-gravitational interactions, DM production occurs through "freeze-in," rather than thermal equilibration. For a real scalar DM field (here denoted $X$), the interaction with SM fields $\phi$ is provided only by Einstein gravity:

$$\mathcal{L} = \sqrt{-g} \left[ \frac{1}{16\pi G} R + \frac{1}{2}g^{\mu\nu}(\partial_\mu \phi \partial_\nu \phi + \partial_\mu X \partial_\nu X) - V(\phi,X) \right]$$

where $V$ is the scalar potential with no direct $X$–SM couplings beyond gravity (Tang et al., 2016).

The leading production process is $\phi\phi \to XX$ via single-graviton $s$-channel exchange. The thermally averaged cross section scales as

$$\langle \sigma v \rangle \sim \kappa^4 T^2 = \frac{T^2}{M_{\text{Pl}}^4}$$

with $\kappa = M_{\text{Pl}}^{-1}$, $T$ the temperature. Since $n_\phi\langle \sigma v \rangle \ll H$ throughout, $X$ remains out of equilibrium and is populated through freeze-in.

In the context of multi-dimensional models, such as the "Dark Dimension" scenario, the gravitational production rate encompasses the sum over accessible Kaluza-Klein (KK) modes. For a compact extra dimension of length $l_5$, the density of modes accessible at temperature $T$ is $N(T) \sim T l_5$, and the production rate per unit volume becomes

$\Gamma_{\text{prod}}(T) \sim \frac{T^6 l_5}{M_4^4}$

where $M_4$ is the reduced 4D Planck mass (Obied et al., 2023).

2. Relic Abundance and Mass Constraints

The final comoving dark matter yield $Y = n_X/s$ resulting from gravitational freeze-in is determined by integrating the Boltzmann equation; the dominant production occurs near the highest available temperature $T_{\max}$ after reheating. Matching the resulting relic abundance to the present $\Omega_{\text{DM}}$ places tight constraints on $m_X$ and the allowed $T_{\max}$.

For the pure scalar scenario: $$1\, \text{TeV} \lesssim m_X \lesssim 10^{11}\, \text{GeV}$$ is required for $X$ to account for the observed DM abundance, given $$10^{-7}\,M_{\text{Pl}} \lesssim T_{\max} \lesssim 10^{-4}\,M_{\text{Pl}}$$ (Tang et al., 2016).

In Peebles–Vilenkin quintessential inflation, gravitational reheating fixes DM masses to

$$1 \times 10^{16}\,\text{GeV} \lesssim m_{Y} \lesssim 4.1 \times 10^{17}\,\text{GeV}$$

with a low reheat temperature constrained by BBN and GW overproduction (1904.02393).

For KK graviton towers,

$$m_n = \frac{n}{l_5},\quad l_5 \sim 1-10\, \mu\text{m}$$

limits are set by cosmological, astrophysical, and fifth-force experiments (Obied et al., 2023).

3. Effective Operators, Decay Channels, and Lifetimes

At tree level, gravitationally produced DM is cosmologically stable in the absence of symmetry-violating operators. However, gravitational radiative corrections (one-loop and higher) inevitably induce higher- and lower-dimensional operators, resulting in slow decay (Tang et al., 2016).

The dominant decay channel is generally into two gravitons: $$\mathcal{L}_{\text{eff}} \supset \frac{1}{A} X R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} \implies \Gamma_{X \to hh} = \frac{m_X^3}{4\pi A^2 M_{\text{Pl}}^2}$$ with $A$ a UV-dependent mass scale typically $A \sim M_{\text{Pl}}$ (Dunsky et al., 24 Mar 2025). For scalar DM with a cubic self-coupling $\mu_X$,

$$\Gamma_X^h \simeq \frac{m_X}{32\pi} \left( \frac{\mu_X m_X^2}{16\pi^2 M_{\text{Pl}}^2} \right)^2$$

and the requirement $\tau_X \gtrsim t_U \sim 4.3\times10^{17}$ s yields $m_X \lesssim 10^{7}$ GeV for $\mu_X \sim 1$.

Decays to SM states (e.g., $X \to \gamma\gamma$, $X \to ZZ, WW, gg$, etc.) appear only at two-loop order and are further suppressed: $$\Gamma_X^{\text{SM}} \sim \left( \frac{m_X^2}{16\pi^2 M_{\text{Pl}}^2} \right)^2 \Gamma_X^h$$ rendering such channels unobservable with current sensitivities (Tang et al., 2016).

In the DD scenario, the decay of heavy KK modes into lighter modes is induced by inhomogeneities in the compact dimension. The aggregate effect is a "cascade" that redshifts the average DM mass as $\langle m(t)\rangle \propto t^{-2/7}$ (Obied et al., 2023).

4. Astrophysical and Cosmological Signatures

Since direct detection of weakly interacting gravitational DM or its decay products is challenging, indirect signatures become vital. The decay $X \to 2h$ yields a relic flux of high-frequency gravitational waves. In SM decay channels, the predicted fluxes are suppressed by factors $<10^{-48}$ relative to the graviton channel for $m_X < 10^7$ GeV, far below any realistic detection threshold (Tang et al., 2016).

Crucially, gravitons produced from DM decays can convert into photons via the Gertsenshtein effect within cosmic filaments hosting large-scale magnetic fields. The net conversion probability per domain is

$$P_{h\to\gamma}(z) \simeq \frac{B(z)^2 \lambda_{\text{coh}}^2}{M_{\text{Pl}}^2}$$

with $B(z)\sim 60$–$250$ nG, $\lambda_{\text{coh}}\sim 1$–$4$ Mpc. The cumulative photon flux is then compared against the isotropic gamma-ray background (IGRB) as measured by Fermi-LAT to constrain the DM decay lifetime (Dunsky et al., 24 Mar 2025).

A key result is that for $m_{\text{DM}} \sim 1$ TeV and $B_0 = 250$ nG,

$\tau_{\text{DM}} \gtrsim 10^{26}\, \text{s}$

with stronger limits at higher mass or for larger $B_0$. The Advanced Particle-astrophysics Telescope (APT) is projected to improve these bounds by up to an order of magnitude (Dunsky et al., 24 Mar 2025).

In the dark dimension scenario, the transfer of kinetic energy to daughter KK gravitons induces a non-thermal velocity dispersion that suppresses structure formation on small scales, providing a unique cosmological signature. The CMB anisotropy and large-scale structure data (Planck, BAO, KiDS-1000) constrain the present-day kick velocity to $v_{\text{today}} \leq 2.2 \times 10^{-4}$ c (95% CL), which restricts the size of the extra dimension to $l_5 \sim 1-10\,\mu$m (Obied et al., 2023).

5. Model Implementations and Phenomenological Variants

Pure Scalar Gravitational DM (EFT):

• Real scalar $X$ field, tree-level isolation except via gravity.
• Dominant production via gravitational freeze-in.
• Decays primarily into gravitons once self-coupling $\mu_X$ is introduced.
• Range: $$1\,\text{TeV} \lesssim m_X \lesssim 10^{11}\,\text{GeV}$$.
• Cosmologically stable in the absence of explicit symmetry breaking; otherwise, decay rates set by loop-induced operators and suppressed by Planck mass to high powers (Tang et al., 2016).

Quintessential Inflation Models:

• Gravitational production during nonadiabatic breaks at the end of inflation (e.g., Peebles–Vilenkin).
• Superheavy $X$/stable $Y$ species: $m_Y \sim 10^{16}$–$10^{17}$ GeV (1904.02393).
• $Y$ behaves as DM; $X$'s decay reheats the universe under strong BBN and GW constraints.

Dark Dimension (KK Gravitons):

• SM confined to brane; gravity propagates in five-dimensional bulk with compact length $l_5$.
• Tower of spin-2 KK gravitons as DM candidates.
• Cascade decay chain populates hierarchy of mass states, with mass and velocity distributions evolving in time.
• Constraints link extra dimension’s size and laboratory fifth-force bounds: $l_5 \sim 1$–$10\,\mu$m (Obied et al., 2023).

6. Observational Constraints and Future Tests

Empirical probes of gravitationally produced decaying DM hinge on indirect detection. For two-graviton decay, the only realistic signature is the population of high-energy gravitons, which produce secondary photons in cosmic magnetic fields through the Gertsenshtein effect. The measured IGRB thus enables the first constraints on purely gravitational DM decay (see Table):

$m_{\text{DM}}$ (GeV)	Fermi-LAT limit $\tau_{\text{DM}}$ ($B_0$=60 nG, s)	Fermi-LAT limit $\tau_{\text{DM}}$ ($B_0$=250 nG, s)
0.1	$3 \times 10^{22}$	$3 \times 10^{23}$
1	$1 \times 10^{23}$	$1 \times 10^{24}$
$10^3$	$1 \times 10^{25}$	$1 \times 10^{26}$
$10^8$	$1 \times 10^{27}$	$1 \times 10^{28}$

Future experiments (e.g. APT) are anticipated to improve these sensitivity bounds by approximately an order of magnitude for $m_{\text{DM}} \gtrsim 100$ GeV, accessing lifetimes up to $\sim 10^{29}$ s (Dunsky et al., 24 Mar 2025).

In the context of multi-graviton dark sectors, laboratory tests of gravity at sub-millimeter scales complement cosmological bounds on the kick velocity and decaying DM signatures (Obied et al., 2023).

7. Theoretical and Empirical Significance

Gravitationally produced decaying dark matter models are highly predictive, as the production and decay mechanisms are fixed by Planck-suppressed interactions and early-universe cosmology. The predicted mass windows are disjoint from conventional WIMP regimes and motivate searches for ultra-heavy DM and high-frequency gravitational waves. A unique, irreducible signal is the extragalactic photon background induced by graviton-to-photon conversion, setting the first indirect detection limits on scenarios where DM decays only gravitationally (Dunsky et al., 24 Mar 2025).

Combined cosmological, astrophysical, and laboratory signatures offer multi-pronged tests of both minimal Planckian DM and more complex gravitational dark sectors (e.g., KK graviton towers), rendering these models increasingly accessible to present and near-future experiments.