Analytical Estimates of Gravitational Wave Background Anisotropies from Shot Noise and Large-Scale Structure in Pulsar Timing Arrays

Abstract

An important next step for pulsar timing arrays (PTAs) is to measure anisotropies in the gravitational wave background (GWB) at $\sim$ nano-Hz frequencies. We calculate the expected GWB anisotropies using empirically calibrated models for the merger rates of supermassive black hole binaries (SMBHBs). The anisotropies reflect both shot-noise in the discrete SMBHB populations while also tracing, in part, the large-scale structure (LSS) of the universe. The shot-noise term is sensitive to the high-mass end of the merging SMBH mass function, depends somewhat on the low-redshift tail of the merger distribution, and is a strong function of observing frequency. The precise frequency dependence provides a test of SMBHB residence times. In our models, the mean shot-noise anisotropy typically lies close to or above the broad frequency-band NANOGrav upper limits. Consequently, near-future PTA data, and potentially re-analyses of existing measurements using frequency-dependent shot-noise anisotropy templates, should be capable of detecting this signal or placing meaningful constraints on SMBHB merger models. A full interpretation, however, will require modeling the probability distribution of shot-noise amplitudes rather than relying solely on ensemble-averaged predictions. The LSS-induced anisotropies are at least two to three orders of magnitude smaller. Although the LSS contribution contains valuable information regarding the redshift distribution and clustering bias of the merging SMBHBs, detecting this component will be challenging.

Figures (6)

• Figure 1: Model predictions for the GWB anisotropy power spectrum and current observational bounds. The downward-pointing arrows show current 3-$\sigma$ upper bounds on the anisotropy signal from the NANOGrav collaboration based on a broad frequency-band analysis. The blue solid line gives the LM shot-noise model prediction at a frequency of $f=1/10 \, {\rm yr}^{-1}$, while the blue dashed line shows the LM shot-noise model at $f=1/3 \, {\rm yr}^{-1}$. The orange lines show the same for the SZQ model. The quantity $D_{\ell, h^2}$ is plotted (see text), and so shot-noise scales as $\propto \ell^2$ here. The green and red solid lines show the LSS anisotropy power spectrum in the LM and SZQ models, respectively, with the solid lines adopting $\langle b_{\rm BH}(z) \rangle = 1$ and dashed lines taking $\langle b_{\rm BH}(z) \rangle=5$. Each of the shot-noise and LSS models assumes $z_{\rm min}=0.05$ and adopts a maximum mass of $M_{\rm BH, max}=10^{10.5} M_\odot$. As we discuss below, the expected shot-noise signals are typically close to, or exceed, the NANOGrav bounds. Although a rigorous comparison between the models and the NANOGrav bounds is beyond the scope of this work, the results here suggest that shot-noise detections may be possible in the near term. The LSS signal is much smaller than the shot-noise one and the current observational bounds.
• Figure 2: A comparison between the broad frequency-band NANOGrav constraint and the frequency dependence of the shot-noise models (left panel). The blue lines give the shot-noise power spectrum in the LM model, while the orange lines are for SZQ, each assuming a narrow logarithmic bin in frequency. Each model takes $z_{\rm min} = 0.05$, while various values of $M_{\rm BH, max}$ are adopted, as indicated in the legend. The green downward-pointing arrow shows the $3-\sigma$ upper bound from NANOGrav. The line indicates the approximate frequency range spanned in this broadband measurement, although we caution that the frequency weighting is non-uniform. In each model, the shot-noise is a strongly increasing function of frequency owing to the decreasing residence time of SMBHB mergers towards high frequency. Future NANOGrav analyses adopting a frequency-dependent shot-noise template may help detect or constrain the shot-noise signal, providing further tests of our local SMBH census and of merger residence time models. The results also reveal sensitivity to the high-mass tail of the SMBH mass function, especially in SZQ. The right panel shows the $1-\sigma$ spread in shot-noise model predictions given the SMBH mass function parameter uncertainties, for a few different values of $M_{\rm BH, max}$. Each case adopts $f=0.1 \, {\rm yr}^{-1}$ (denoted by the vertical dashed line in the left panel). Note that, especially when $C_{\ell>0,h^2}^{\rm SN}/4\pi \gtrsim 1$, we expect a large sample variance around the mean shot-noise signal.
• Figure 3: Contribution to the mean GWB ($\langle h^2 \rangle$, solid lines) and shot-noise induced anisotropy ($\langle h^4 \rangle$, dashed lines) per logarithmic mass bin (left panel), without adopting an $M_{\rm BH,max}$, and per redshift bin (right panel). All curves are normalized such that they integrate to unity. The blue lines are for the LM model, and the orange lines are for SZQ. The shot-noise anisotropy is weighted towards higher mass and lower redshift than the mean GWB signal, and so is sensitive to the extreme high-mass end of the SMBH function and to the distribution of nearby sources. The $\langle h^4 \rangle$ curve peaks at $z=z_{\rm min} =0.05$, reaching a redshift kernel value of 7.9 for LM and 16.8 for SZQ, exceeding the y-axis range shown.
• Figure 4: Left panel: the minimum redshift-dependence of anisotropies from both shot-noise and LSS, normalized to their values at $z_{\rm min}=0.05$. The blue and orange lines show, respectively, the LM and SZQ shot-noise power spectrum models at $f=1/10 \, {\rm yr}^{-1}$, while the green and red curves show the corresponding LM and SZQ LSS predictions at $\ell=10$. The shot-noise power spectra formally diverge as $z \rightarrow 0$, and so there is some sensitivity to the precise choice of $z_{\rm min}$ assumed. Right panel: the sensitivity of the shot-noise anisotropies to the redshift distribution power-law index, $\gamma$. For the LM model (here with $M_{\rm BH,max}=10^{10.5} M_\odot$) and $\gamma > 1$, the results converge towards $z_{\rm min} \rightarrow 0$, while the shot-noise amplitude drops with increasing $\gamma$.
• Figure 5: Similar to Fig. \ref{['fig:SN-signals-f']} but with varying redshift distribution power-law index, $\gamma$, for the SMBH merger rates in the LM model. The original LM and SZQ models are also shown for comparison (blue and orange lines), where the redshift integral in Eq. \ref{['eq:h4zmin']} is cut off at $z_{\rm min}=0.05$. The modified LM models with $\gamma=1.5$ and $2.0$ are integrated to redshift 0. All models assume $M_{\mathrm{BH,max}}=10^{10.5} M_\odot$.