Kinematic Anisotropies in PTA Observations: Analytical Toolkit

TL;DR

The paper develops an analytical toolkit to model kinematic anisotropies in gravitational-wave backgrounds as observed by pulsar timing arrays, using a Doppler-boosted framework and a systematic $\beta$ expansion of the overlap reduction function (ORF). It provides corrected analytic expressions for the monopole, dipole, and especially the quadrupole contributions to cross- and auto-correlations, showing that the quadrupole term differs from previous work and is relevant for nonzero pulsar-term and finite-distance effects. The authors demonstrate how the kinematic dipole reshapes the ORF and assess deviations from the Hellings–Downs curve for realistic PTA configurations, including a 67-pulsar NANOGrav-like ensemble, with implications for upper bounds on the observer’s velocity and for forecasts in Bayesian analyses. The work validates the formalism through numerical checks and highlights the importance of auto-correlation terms in PTA analyses, thereby enabling more accurate discrimination of intrinsic versus kinematic anisotropies in the GWB.

Abstract

The reported evidence for an isotropic gravitational-wave background (GWB) from pulsar timing array (PTA) collaborations has motivated searches for extrinsic and intrinsic anisotropies. Kinematic anisotropies may arise as a consequence of a boosted observer moving with respect to the frame in which the GWB appears isotropic. In this work, we present an analytical toolbox to describe the effects of kinematic anisotropies on the overlap reduction function. Our analytical results differ from previous findings at the quadrupole order and are detailed in three appendices. For the first time, we also derive the corresponding auto-correlation using two approaches, taking the pulsar distances to be infinite or finite, respectively. Our formulas can be used in forecasts or Bayesian analysis pipelines.

Figures (4)

• Figure S1: Comparison of the solutions for the quadrupole contribution from Ref. Cruz:2024svc and our analysis (adjusting for normalization). The solid lines indicate a system in which the pulsar direction vectors lie in the plane transverse to the velocity vector, while the dashed plots are for the case of $\alpha_1 = \eta = \pi /4$ (angles for $\hat{\boldsymbol{v}}$; see Eq. \ref{['eq:vinspherical']}). The black stars indicate the result considered for the numerical solution in Eq. \ref{['eq:smallNumericalInOrthogonal']} and the vanishing solution for the same configuration in Eq. \ref{['eq:tasinatoGamma2Result']}.
• Figure S2: Comparison of the kinematic quadrupole contribution to the inter-pulsar cross-correlation as a function of the pulsar separation angle $\xi$. Left panel: Solution to Eq. \ref{['eq:integralGamma2']} computed via our numerical integration (grey), compared with that reported in Refs. Tasinato:2023zcgCruz:2024svc (teal). Right panel: Solution to Eq. \ref{['eq:integralGamma2']} computed via our numerical integration (grey), compared with our analytical solution (blue). The direction of the observer's motion has been set to the direction of the CMB dipole as inferred by Planck Planck:2018nkj.
• Figure S3: Location of the 67 pulsars that were used by the NANOGrav collaboration in NANOGrav:2023gor, shown here in galactic coordinates, and plotted using a Mollweide projection of the celestial sphere. The direction of the velocity vector $\hat{\boldsymbol{v}}$ as determined by the CMB dipole and its antipodal point are depicted by a red triangle (galactic coordinates $(l,b) = (264^{\circ},48^{\circ})$Planck:2018nkj) and a blue triangle, respectively. Pulsars within an angular separation of $40^\circ$ from the direction $\pm\hat{\boldsymbol{v}}$ are shown in a darker tone. (Pulsar positions taken from Ref. Manchester:2004bp.)
• Figure S4: Plot showing the ORF $\Gamma_{ab}$ as a function of the pulsar-pair angular separation $\xi$, assuming a power-law GWB intensity with constant spectral index $n_I = -7/3$ ($\alpha_I=0$) as defined in Eq. \ref{['eq:spectralParameter']}. The direction $\hat{\boldsymbol{v}}$ is chosen to align with that inferred from the CMB dipole. For each pulsar pair formed from the 67 pulsars in Fig. \ref{['fig:pulsarMapping']}, we show the corresponding value of the ORF as orange and purple dots for $\beta =0.1$ and $\beta = 0.3$, respectively. For pulsar pairs with both pulsars with an angular separation of at most $40^\circ$ from $\pm\hat{\boldsymbol{v}}$, we show the dots in dark orange and dark purple, respectively. For reference, we also plot the Hellings--Downs correlation ($\beta =0$) in black.