Effect of small interpulsar distances in stochastic gravitational wave background searches with pulsar timing arrays

One of the primary objectives for pulsar timing arrays (PTAs) is to detect a stochastic background generated by the incoherent superposition of gravitational waves (GWs), in particular from the cosmic population of supermassive black hole binaries. Current stochastic background searches assume that pulsars in a PTA are separated from each other and the Earth by many GW wavelengths. As more millisecond pulsars are discovered and added to PTAs, some may be separated by only a few radiation wavelengths or less, resulting in correlated GW phase changes between close pulsars in the array. Here we investigate how PTA overlap reduction functions (ORFs), up to quadrupole order, are affected by these additional correlated phase changes, and how they are in turn affected by relaxing the assumption that all pulsars are equidistant from the solar system barycenter. We find that in the low-frequency GW background limit of $f\sim {10}^{-9}\mathrm{Hz}$, and for pulsars at varying distances from the Earth, these additional correlations only affect the ORFs by a few percent for pulsar pairs at large angular separations, as expected. However, when nearby (order 100 pc) pulsars are separated by less than a few degrees, the correlated phase changes can introduce variations of a few tens of percent in the magnitude of the isotropic ORF, and much larger fractional differences in the anisotropic ORFs—up to 188 in the $m=0$, $l=2$ ORF for equidistant pulsars separated by 3°. In fact, the magnitude of most of the anisotropic ORFs is largest at small, but nonzero, pulsar separations. Finally, we write down a small angle approximation for the correlated phase changes which can easily be implemented in search pipelines, and for completeness, examine the behavior of the ORFs for pulsars which lie at a radiation wavelength from the Earth.

Figure 1

The “computational” frame: pulsar $a$ is on the $z$ axis at a distance $L_a$ from the origin (solar system barycenter), and pulsar $b$ is in the $x$-$z$ plane at a distance $L_b$ from the origin making an angle $\zeta$ with pulsar $a$. $\widehat{\mathrm{\Omega }}$ is the direction of GW propagation with principal axes $\widehat{m}$ and $\widehat{n}$ such that $\widehat{m}\times \widehat{n}=\widehat{\mathrm{\Omega }}$. The polar and azimuthal angles are given by $\theta$ and $\varphi$, respectively.

Figure 2

The Earth-term-only overlap reduction functions for isotropic, (a); dipole, (b); and quadrupole, (c) GW energy density distributions [27]. In the chosen reference frame, ${\mathrm{\Gamma }}_l^{-m}(\zeta )=(-1{)}^m{\mathrm{\Gamma }}_l^m(\zeta )$. The legends are as follows: in Fig. 2, the $m=0$ curve is the solid (blue) curve, $m=1$ is the dashed (green) curve, and $m=-1$ is the dashed-dotted (red) curve. In Fig. 2, $m=0$ is the solid (blue) curve, $m=1$ is the dashed (green) curve, $m=-1$ is the dotted (cyan) curve, and $m=\pm 2$ is the dashed-dotted (red) curve. Features of these ORFs are discussed in Sec. 3a.

Figure 3

Geometry of pulsar in the “strong pulsar-term regime.” Here $L_x$ is the distance to pulsar $x$ from the SSB, and $\zeta$ is the angular separation of the pulsars. The dimensionless product $f{L}_x$ is the number of gravitational radiation wavelengths from the SSB to each pulsar. The geometry indicates two possible movements: pulsar $b$ is moved azimuthally by $\zeta f{L}_a$ radiation wavelengths from $a$, or $b$ is moved radially away from the origin by $f{L}_b-f{L}_a$.

Figure 4

The effect of pulsar distance variations on the magnitude of the isotropic and dipole overlap reduction functions, with $f{L}_a=10$ fixed. Panels on the left-hand side are truncated at 40°, as pulsar-term oscillations rapidly converge to zero at large angular separations. (a) Magnitude of the isotropic overlap reduction function. (b) Strong pulsar term regime for ${}^{(ab)}{\mathrm{\Gamma }}_0^0(fL,\zeta )$. (c) Magnitude of the dipole ${}^{(ab)}{\mathrm{\Gamma }}_1^0(fL,\zeta )$ ORF. (d) Strong pulsar term regime for ${}^{(ab)}{\mathrm{\Gamma }}_1^0(fL,\zeta )$. (e) Magnitude of the dipole ${}^{(ab)}{\mathrm{\Gamma }}_1^1(fL,\zeta )$ ORF. (f) Strong pulsar term regime for ${}^{(ab)}{\mathrm{\Gamma }}_1^1(fL,\zeta )$.

Figure 5

The effect of pulsar distance variations on the magnitude of the quadrupole overlap reduction functions, with $f{L}_a=10$ fixed. Panels on the left-hand side are truncated at 40°, as pulsar-term oscillations rapidly converge to zero. Note that the maximum value of these ORFs is achieved for small, but nonzero, pulsar separations. (a) Magnitude of the quadrupole ${}^{(ab)}{\mathrm{\Gamma }}_2^0(fL,\zeta )$ ORF. (b) Strong pulsar term regime for ${}^{(ab)}{\mathrm{\Gamma }}_2^0(fL,\zeta )$. (c) Magnitude of the quadrupole ${}^{(ab)}{\mathrm{\Gamma }}_2^1(fL,\zeta )$ ORF. (d) Strong pulsar term regime for ${}^{(ab)}{\mathrm{\Gamma }}_2^1(fL,\zeta )$. (e) Magnitude of the quadrupole ${}^{(ab)}{\mathrm{\Gamma }}_2^2(fL,\zeta )$ ORF. (f) Strong pulsar term regime for ${}^{(ab)}{\mathrm{\Gamma }}_2^2(fL,\zeta )$.

Figure 6

The small angle approximation of the isotropic ORF compared to the full ORF. Moving from right to left: the solid curve is the full ORF, and the dashed one is the approximation given by Eq. (36) for $fL=10$ (online blue), $fL=51.2$ (green), and $fL=100$ (magenta). The approximation appears to hold for $\zeta \le 2.3°$ for $fL\ge 10$, Eq. (37). Afterward the ORF reverts to the Earth-term-only solution, which appears flat due to the small range of angles considered.

Figure 7

In all panels ${{}^{(ab)}\mathrm{\Gamma }}_0^0(fL,\zeta )$ is the solid curve, ${{}^{(ab)}\mathrm{\Gamma }}_1^0(fL,\zeta )$ is the dashed curve (blue), and ${{}^{(ab)}\mathrm{\Gamma }}_2^0(fL,\zeta )$ is the dotted curve (red). (a) The behavior of the pulsar term only when $L_a=L_b=L$ and $fL=1$ for the aforementioned ORFs. This is found by subtracting the Earth-term solution from the numerically integrated ORF. (b) The imaginary part only of ${{}^{(ab)}\mathrm{\Gamma }}_0^0(fL,\zeta )$, ${{}^{(ab)}\mathrm{\Gamma }}_1^0(fL,\zeta )$, and ${{}^{(ab)}\mathrm{\Gamma }}_2^0(fL,\zeta )$ when $fL=1$. As there is no imaginary part in the computational frame where the Earth term is calculated, we cannot display the difference as is done in panel (a). Note that these imaginary values are only a factor of a few smaller than their real counterparts, with the exception of ${{}^{(ab)}\mathrm{\Gamma }}_0^0(fL,\zeta )$, where the imaginary part is zero. Moreover, they do not quickly converge to zero as in previous cases for $fL\ge 10$. (c) Magnitude of the overlap reduction functions.

Figure 8

(a) The energy density distribution for $Y_2^0(\theta ,\varphi )$. The red and blue regions are positive and negative, respectively. (b) The Earth (green, equivalently the SSB) is at the center with the two pulsars above. The magnitude of the ${{}^{(ab)}\mathrm{\Gamma }}_2^0(fL,\zeta )$ ORF is enhanced by small $0<\delta fL\lesssim 1$ pulsar $b$ displacements, over the $\delta fL=0$ case. The arrow shows the direction of a GW propagating with incoming angle $\theta$. The lighter shaded regions of the diagram show the regions of the sky from which the signal will contribute positively to the ORF. The darker shaded regions will contribute negatively to the ORF, though their size depends on $\delta fL$. The brackets indicate the [sign of the pulsar-term correlation, sign of the background energy density].