Supermassive black hole binary environments: Effects on the scaling laws and time to detection for the stochastic background

One of the primary gravitational wave (GW) sources for pulsar timing arrays (PTAs) is the stochastic background formed by supermassive black holes binaries (SMBHBs). In this paper, we investigate how the environments of SMBHBs effect the sensitivity of PTAs by deriving scaling laws for the signal-to-noise ratio (SNR) of the optimal cross-correlation statistic. The presence of gas and stars around SMBHBs accelerates the merger at large distances, depleting the GW stochastic background at low frequencies. We show that environmental interactions may delay detection by a few years or more, depending on the PTA configuration and the frequency at which the dynamical evolution transitions from being dominated by environmental effects to GW dominated.

Figure 1

The power spectral density for a power-law GW spectrum with $\gamma =13/3$ (solid blue line) and a broken-power-law spectrum (dashed red line). The top row shows spectra with $\kappa =10/3$ (stellar scattering), while the bottom row shows spectra with $\kappa =7/3$ (gas-driven dynamics). We also plot two different values of $f_b$: $f_b={10}^{-8}\mathrm{Hz}=1/(3\mathrm{yr})$ (left column) and $f_b=3\times {10}^{-9}\mathrm{Hz}=1/(10\mathrm{yr})$ (right column). The dotted horizontal lines indicated the power spectral density of the white noise for $\sigma =1\mu \mathrm{s}$ (top line), $\sigma =200\mathrm{ns}$ (middle line), and $\sigma =100\mathrm{ns}$ (bottom line). The dashed vertical line indicates the bend frequency $f_b$.

Figure 2

Timing RMS at which the GW power in the lowest frequency bin is equal to the white noise level, i.e., the transition from the weak-signal regime to the intermediate-signal regime, as a function of observation time. We compare a power-law spectrum (solid black line), a broken-power-law spectrum with $f_b={10}^{-8}\mathrm{Hz}=1/(3\mathrm{yr})$ (dashed blue line), and a broken-power-law spectrum with $f_b=3\times {10}^{-9}\mathrm{Hz}=1/(10\mathrm{yr})$ (dotted-dashed red line). On the left we plot a GW spectrum with $\kappa =10/3$ (stellar scattering), and on the right we plot a spectrum with $\kappa =7/3$ (gas-driven dynamics). The top row shows a GW spectrum with $A_{\mathrm{gw}}={10}^{-15}$, while the bottom row shows a GW spectrum with $A_{\mathrm{gw}}=5\times {10}^{-16}$. In all plots, $\gamma =13/3$, and the observing cadence is $c=20{\mathrm{yr}}^{-1}$. The horizontal dotted lines indicate timing RMS of $1\mu \mathrm{s}$, 200 ns, and 100 ns.

Figure 3

A schematic representation of the two possible scenarios in the weak-signal regime when observing a broken-power-law spectrum. Our PTA is sensitive to frequencies $f\in [f_L,f_H]$. The high-frequency cutoff is set by the observing cadence, $f_H=c/2$ where $c=1/\mathrm{\Delta }t$. The low-frequency cutoff is set by the observation time, $f_L=1/T$. In Case A, $f_L>f_b$, and so we only observe the steep portion of the spectrum. In Case B, $f_L<f_b$, which means we are sensitive to both the steep and shallow portions of the spectrum.

Figure 4

A schematic representation of the three possible scenarios in the intermediate-signal regime when observing a broken-power-law spectrum. Our PTA is sensitive to frequencies $f\in [f_L,f_H]$. The high-frequency cutoff is set by the observing cadence, $f_H=c/2$, where $c=1/\mathrm{\Delta }t$. The low-frequency cutoff is set by the observation time, $f_L=1/T$. For Cases A and B, $f_{*}>f_b$, so the GW signal intersects the white noise in the steep portion of the spectrum. In Case A, the PTA is not yet sensitive to the shallower portion of the spectrum (i.e., $f_L>f_b$), while in Case B the PTA is sensitive to both regions of the spectrum. For Case C, $f_{*}<f_b$, so only the shallow portion of the spectrum is above the white noise. Assuming infinite observation time, all PTAs will end up in either Case B or Case C.

Figure 5

The SNR of a PTA consisting of 20 pulsars observed with a cadence of $c=20{\mathrm{yr}}^{-1}$. The left panel shows an array with timing RMS of $1\mu \mathrm{s}$, and the right shows an array with timing RMS of $200\mathrm{ns}$. The stochastic background is a power law with $A_{\mathrm{gw}}={10}^{-15}$ and $\gamma =13/3$. The solid red line is the result of computing the S/N in the time domain. The dashed line shows the weak-signal regime scaling law given in Eq. (4.2), and the dashed-dotted line shows the intermediate-signal regime scaling law given in Eq. (4.22). At early times the SNR scales as $T^{\gamma }$, while at late times the SNR scales as $T^{1/2}$. The dotted horizontal line indicates an SNR of 3. The array with $\sigma =1\mu \mathrm{s}$ reaches an SNR of 3 after about 19 years, while the array with $\sigma =200\mathrm{ns}$ reaches it after about 9 years.

Figure 6

The SNR of a PTA consisting of 20 pulsars observed with a cadence of $c=20{\mathrm{yr}}^{-1}$ with a timing RMS of $1\mu \mathrm{s}$ (left) and $200\mathrm{ns}$ (right). The GW background has $A_{\mathrm{gw}}={10}^{-15}$, $\gamma =13/3$, and $f_b={10}^{-8}\mathrm{Hz}$. In the left panel $\kappa =10/3$, and in the right $\kappa =7/3$. The solid red line is the result of computing the S/N in the time domain. The dotted horizontal line indicates an SNR of 3. At early times the SNR of the PTA with $\sigma =1\mu \mathrm{s}$ is described by the weak-signal scaling law in Eq. (4.2) and scales like $T^{\gamma }$, as shown by the dashed line. Once it is sensitive to the turnover in the spectrum, it is described by Eq. (4.13) and scales like $T^{\gamma -\kappa }$, indicated by the dashed-dotted line. This PTA does not enter the intermediate-signal regime until more than 250 years of observation, and the stochastic background is undetectable. The PTA with $\sigma =200\mathrm{ns}$ begins in the weak-signal regime [Eq. (4.2); dashed line]. The turnover in the spectrum delays the transition into the intermediate-signal regime until after about 6 years of observation, after which the SNR scales like $T^{1/2}$ [Eq. (4.36); dashed-dotted line]. The stochastic background is detectable after about 14 years, compared to 9 years for the same PTA configuration observing a spectrum with no bend.

Figure 7

The SNR of a PTA consisting of 20 pulsars observed with a cadence of $c=20{\mathrm{yr}}^{-1}$ with timing RMS of $1\mu \mathrm{s}$ (left) and $200\mathrm{ns}$ (right). The GW background has $A_{\mathrm{gw}}={10}^{-15}$, $\gamma =13/3$, $\kappa =10/3$, and $f_b=3\times {10}^{-9}\mathrm{Hz}$. The solid red line is the result of computing the S/N in the time domain. The dashed line shows the weak-field scaling law given in Eq. (4.2), and the dashed-dotted line shows the intermediate-field scaling law given in Eq. (4.31). At early times the SNR scales as $T^{\gamma }$, while for late times the SNR scales as $T^{1/2}$. Since the turnover in the spectrum is at low frequency, there is very little difference between these cases and observing a power-law spectrum (Fig. 5). For the PTA with $\sigma =1\mu \mathrm{s}$, detection is delayed by a few years; for the PTA with $\sigma =200\mathrm{ns}$, detection is delayed by less than a year.