Maximum Redshift of Gravitational Wave Merger Events

Future generations of gravitational wave detectors will have the sensitivity to detect gravitational wave events at redshifts far beyond any detectable electromagnetic sources. We show that if the observed event rate is greater than one event per year at redshifts $z\ge 40$, then the probability distribution of primordial density fluctuations must be significantly non-Gaussian or the events originate from primordial black holes. The nature of the excess events can be determined from the redshift distribution of the merger rate.

Figure 1

Halo mass functions at $z=20$, 30, 40, and 50. At each redshift the multiple gray lines correspond to the mass functions derived in Refs. [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. The red line corresponds to the Press-Schecter mass function [31], while the blue line corresponds to the Sheth, Mo, and Tormen mass function [30]. At each redshift, the range of mass function values is bounded roughly by these two analytic mass functions. The dashed blue lines correspond to the Sheth, Mo, and Tormen mass function with a correction [32] for a cosmology with a non-Gaussianity parameter $f_{\mathrm{NL}}=43$ [33].

Figure 2

Red contours depict the quantity ${\log }_{10}[⟨\epsilon (M,z)⟩{\dot{M}}_g/M_{\odot }{\mathrm{yr}}^{-1}]$, i.e., the logarithm of the mass accretion rate of gas that makes black holes in halos of mass $M$ at redshift $z$. The values of the contours are from ${10}^{-1}-{10}^{-10}$ in factors of 10 from left to right. Thick contours correspond to the simulated gas accretion rate of Ref. [40], thin contours correspond to the analytic prediction of Ref. [43]. The dashed gray curves show the number of standard deviations that correspond to the fluctuations of the power spectrum that give rise to halos of mass $M$ at $z$. Black lines correspond to the minimum halo mass for molecular hydrogen cooling. The thick black solid line corresponds to the simulation results of Ref. [44] in the case where streaming velocities of gas are included in the calculation of gas cooling, while the thin solid black line corresponds to the minimum mass when relative motion between gas and dark matter is not considered (see Ref. [44]). By redshift $z\approx 40$ the rate of infalling gas decreases dramatically, while at the same time the minimum mass of a halo that can harbor star formation corresponds to extremely rare density peaks.

Figure 3

The number of gravitational wave events of $m_{\mathrm{BH}}=30M_{\odot }$ black hole pairs originating from redshifts greater than $z$ [Eq. (1)] as a function of redshift. The blue curve corresponds to the upper limit on the halo mass function [30], a low value of $M_{\min }$ (i.e., ignoring the effects of a relative speed between dark matter and baryons [44]) and a high value of gas accretion [40]. The dashed blue curve makes the same assumptions as above, but with a modified mass function that includes a correction corresponding to non-Gaussianity with $f_{\mathrm{NL}}=43$ [33]. The red curve assumes the lower limit on the mass function [31], a large minimum mass (assuming relative velocities between baryons and dark matter [44]) and a low gas accretion rate [39]. The shaded area represents everything in between these two extreme cases. The two vertical lines correspond to the $5\sigma$ and $10\sigma$ sensitivity to $m_{\mathrm{BH}}=30M_{\odot }$ black hole pairs with the future gravitational wave detector, Cosmic Explorer [8].