Forecasted Detection Limits on the (Dark) Matter Density in Supermassive Black Hole Binaries for LISA

Abstract

Supermassive black hole binaries (SMBHBs) are among the most powerful known sources of gravitational waves (GWs). Accordingly, these systems could dominate GW emission in the micro- and millihertz frequency range. Within this domain, SMBHs evolve rapidly and merge with each other. Dynamical friction from stars and gas at the centers of galaxies typically helps to bring together two SMBHs when they are at relatively far separations ($\approx$ kpc $-$ 100 pc), but becomes less efficient at smaller separations. However, dark matter (DM) spikes around SMBHs could enhance dynamical friction at close separations and, thus, shorten the evolution times. In this paper, we simulate the effects of DM spikes on GW signals in the micro- to millihertz frequency range and confirm that the GW signals from SMBHBs with DM spikes can be clearly distinguished from those without any additional matter. Making use of the projected sensitivity curve of the Laser Interferometer Space Antenna (LISA), we forecast upper limits for the (dark) matter density for given future SMBHB observations. We then compare these thresholds with the theoretical density profiles expected for self-interacting dark matter (SIDM) spikes.

Figures (11)

• Figure 1: Radial density profiles for CDM ($\alpha = 7/3$) and SIDM spikes with different slopes, assuming $m_\mathrm{BH} = 10^7\,\mathrm{M_\odot}$ and $\sigma = 200\,\mathrm{km/s}$. The vertical dashed gray line indicates the influence radius $r_h$.
• Figure 2: Comparison of the temporal evolution of two different equal-mass SMBHBs (left) and their GW spectra (right) for CDM (blue) and SIDM (orange) spikes with $\alpha = 7/4$. The case where the binary systems lose energy only through GW emission is represented by the pink lines (vacuum). Solid lines: For $m_1 = m_2 = 10^5\,\mathrm{M_{\odot}}$. Dashed lines: For $m_1 = m_2 = 10^7\,\mathrm{M_{\odot}}$. Left: Temporal evolution of the separation $r$ for circular orbits. Note that, in our setup, the initial separation is not fixed but grows with the mass of the SMBHB, resulting in an increase in the time until coalescence, even though the mass increases by two orders of magnitude. Right: GW spectra. The solid black line represents the projected sensitivity curve of LISA Robson_2019. Note: For SIDM spikes with slopes smaller than $7/4$, the corresponding curves would lie closer to those for the vacuum case.
• Figure 3: Temporal energy loss as a function of the observed GW frequency $f_\mathrm{obs}$ for the SMBHB with $m_1 = m_2 = 10^5\,\mathrm{M_\odot}$ and CDM (blue) or SIDM spikes with $\alpha = 7/4$ (orange). Solid lines: For dynamical friction ($i = \mathrm{DF}$). Dashed lines: For GW emission ($i = \mathrm{GW}$). Note: For $m_1 = m_2 = 10^7\,\mathrm{M_\odot}$ or SIDM spikes with smaller $\alpha$, the transition would occur at lower $f_\mathrm{obs}$.
• Figure 4: Dephasing as a function of $f_\mathrm{obs}$ for SMBHs surrounded by CDM (blue) or SIDM spikes with $\alpha = 7/4$ (orange). Solid lines: For $m_1 = m_2 = 10^5\,\mathrm{M_{\odot}}$. Dashed lines: For $m_1 = m_2 = 10^7\,\mathrm{M_{\odot}}$. Note: For SIDM spikes with $\alpha < 7/4$, the dephasing effect would be weaker across the full frequency range.
• Figure 5: Evolution of the eccentricity $e$ depending on the semi-major axis $a$ for different initial eccentricities $e_0 > 0$. For the left plot, CDM spikes were used, while for the right plot, SIDM with $\alpha = 7/4$ spikes were considered. Solid lines: For $m_1 = m_2 = 10^5\,\mathrm{M_{\odot}}$. Dots: For $m_1 = m_2 = 10^7\,\mathrm{M_{\odot}}$. The results are the same for both considered equal-mass SMBHBs (see text for explanation). Note: For the vacuum case, the two SMBHBs only lose energy through the emission of GWs. This effect circularizes the orbits throughout the entire evolution. Additionally, the smaller the SIDM spike slope $\alpha$, the weaker the eccentrification of the orbits due to dynamical friction.