Gravitational atoms

Particles in a yet unexplored dark sector with sufficiently large mass and small gauge coupling may form purely gravitational atoms (quantum gravitational bound states) with a rich phenomenology. In particular, we investigate the possibility of having an observable signal of gravitational waves or ultrahigh-energy cosmic rays from the decay of gravitational atoms. We show that, if ordinary Einstein gravity holds up to the Planck scale, then within the Lambda-cold dark matter model ($\mathrm{\Lambda }\mathrm{CDM}$), the frequency of the gravitational wave signal produced by the decays is always higher than ${10}^{13}\mathrm{Hz}$. An observable signal of gravitational waves with smaller frequency from such decays, in addition to probing near Planckian dark physics, would also imply a departure from Einstein gravity near the Planck scale or an early epoch of nonstandard cosmology. As an example, we consider an early universe cosmology with a matter-dominated phase, violating our assumption that the Universe is radiation dominated after reheating, which gives a signal in an interesting frequency range for near-Planckian bound states. We also show how gravitational atoms arise in the minimal Planckian interacting dark matter scenario and compute their gravitational wave signature.

Figure 1

Energy spectrum of gravitational waves emitted from bound state decay in the early Universe as a function of the observed frequency. The spectrum has the functional form $x^2{e}^{-x^2}$, with $x={\omega }_0/m_B$ and it is peaked at ${\overline{\omega }}_0$.

Figure 2

Peak frequency of the monochromatic gravitational wave signal produced by ground state (red) and 3d excited (blue) gravitational atoms as a function of the bound state mass with $T_{rh}\sim {10}^{-3}{m}_p$ (top) and $T_{rh}\sim {10}^{-8}{m}_p$ (bottom). The frequency is in hertz and the mass in Planck units. The mass range is limited by condition (13) coming from disruption of bound states due to Hubble expansion, $m_X\gtrsim T_{rh}^{2/3}{m}_p^{1/3}$. Even for $m_X$ as high as 0.5 $m_p$, the peak frequency is still at least a factor of $10^3$ larger than $T_0\sim 10\mathrm{GHz}$.

Figure 3

Gravitational wave density parameter per unit logarithmic energy ${\mathrm{\Omega }}_{\mathrm{GW}}({\omega }_0)$ as a function of the signal frequency ${\omega }_0$, measured in units of gigahertz, for $T_{rh}\sim {10}^{-3}{m}_p$ and $m_X\sim 0.1m_p$. We parametrize the initial abundance of atoms as $n_{B,i}=\alpha T_{rh}^3$, where $\alpha$ is roughly the abundance of atoms as a fraction of the thermal plasma number density. From top to bottom, we have $\alpha =1$ (purple), $\alpha ={10}^{-10}$ (blue), $\alpha ={10}^{-20}$ (orange), and $\alpha ={10}^{-30}$ (magenta). The red line represents the nucleosynthesis bound on the GW density parameter, ${\mathrm{\Omega }}_{\mathrm{GW}}<5\times {10}^{-6}$.

Figure 4

Gravitational wave density parameter per unit logarithmic energy ${\mathrm{\Omega }}_{\mathrm{GW}}({\omega }_0)$ as a function of the signal frequency ${\omega }_0$ (in units of gigahertz) in the PIDM scenario with $T_{rh}\sim {10}^{-3}{m}_p$ and $m_X\sim 0.01m_p$. The initial abundance of gravitational atoms is set by freeze-in, Eq. (33).

Figure 5

Peak frequency of the monochromatic gravitational wave signal produced by ground state (red) and 3d excited (blue) gravitational atoms as a function of the bound state mass with an extended period of early matter domination and $H_i\sim 5\times {10}^{-6}{m}_p$, saturating the experimental bound on tensor modes. The frequency is in hertz and the mass in Planck units. The mass range is limited by condition (13) coming from disruption of bound states due to Hubble expansion, $m_X\gtrsim 0.01m_p$. Due to the enhancement of the redshift factor during matter domination, the signal frequency is pushed down to an interesting range for extremely massive atoms, $m_X\gtrsim 0.1m_p$.

Figure 6

Gravitational wave density parameter per unit logarithmic energy ${\mathrm{\Omega }}_{\mathrm{GW}}({\omega }_0)$ as a function of the signal frequency ${\omega }_0$ (in units of hertz) in the nonminimally coupled PIDM scenario with $T_{rh}\sim {10}^{-3}{m}_p$, $m_X\sim 0.01m_p$, and ${\xi }_X\sim 100$. The spectrum is truncated at 1000 GHz, since this is the lowest possible frequency attainable in this scenario, corresponding to atoms decaying immediately after being produced.

Figure 7

Scalar to scalar bound state amplitude as a function of the hard cutoff $\mathrm{\Lambda }$. The amplitude converges in the nonrelativistic region ${\alpha }_G{m}_X<\mathrm{\Lambda }<m_X/{\alpha }_G$ before diverging in the relativistic region $\mathrm{\Lambda }>m_X/{\alpha }_G$.