Post-Newtonian-accurate pulsar timing array signals induced by inspiralling eccentric binaries accuracy computational cost and single-pulsar search Abhimanyu Susobhanan

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Post-Newtonian-accurate pulsar timing array signals induced by inspiralling

eccentric binaries: accuracy, computational cost, and single-pulsar search

Abhimanyu Susobhanan∗

Center for Gravitation, Cosmology, and Astrophysics,

University of Wisconsin-Milwaukee, Milwaukee, WI 53211, USA

(Dated: June 29, 2023)

Pulsar Timing Array (PTA) experiments are expected to be sensitive to gravitational waves

(GWs) emitted by individual supermassive black hole binaries (SMBHBs) inspiralling along

eccentric orbits. We compare the computational cost of diﬀerent methods of computing

the PTA signals induced by relativistic eccentric SMBHBs, namely approximate analytic

expressions, Fourier series expansion, post-circular expansion, and numerical integration.

We show that the fastest method for evaluating PTA signals is by using the approximate

analytic expressions, which provides up to ∼50 times improvement in computational speed

over the alternative methods. We investigate the accuracy of the approximate analytic

expressions by employing a mismatch metric valid for PTA signals. We show that this

method is accurate within the region of the binary parameter space that is of interest to

PTA experiments. We introduce a spline-based method to further accelerate the PTA signal

evaluations for narrowband PTA datasets. The eﬃcient methods for computing the eccentric

SMBHB-induced PTA signals were implemented in the GWecc.jl package and can be readily

accessed from the popular ENTERPRISE package to search for such signals in PTA datasets.

Further, we simplify the eccentric SMBHB PTA signal expression for the case of a single-

pulsar search and demonstrate our computationally eﬃcient methods by performing a single-

pulsar search in the 12.5-year NANOGrav narrowband dataset of PSR J1909−3744 using

the simpliﬁed expression. These results will be crucial for searching for eccentric SMBHBs

in large PTA datasets.

I. INTRODUCTION

Routine detections of gravitational waves (GWs) from stellar mass compact binary merger

events by ground-based GW detectors such as the advanced LIGO and the advanced Virgo have

ushered in the era of GW astronomy [1–3]. Pulsar Timing Arrays [PTAs: 4,5] are experiments that

aim to detect GWs in the nanohertz frequency range by routinely monitoring ensembles of millisec-

ond pulsars using some of the world’s most sensitive radio telescopes. Ongoing PTA campaigns

∗abhimanyu.susobhanan@nanograv.org

arXiv:2210.11454v3 [gr-qc] 27 Jun 2023

include the North American Nanohertz Observatory for Gravitational waves [NANOGrav: 6], the

European Pulsar Timing Array [EPTA: 7], the Parkes Pulsar Timing Array [PPTA: 8], the Indian

Pulsar Timing Array [InPTA: 9], the MeerKAT Pulsar Timing Array [MPTA: 10], and the Chinese

Pulsar Timing Array [CPTA: 11]. The International Pulsar Timing Array consortium [IPTA: 12]

aims to combine data and resources from diﬀerent PTA campaigns to accelerate the discovery of

nanohertz GWs and improve the prospects of post-discovery science. PTA experiments have grown

in sensitivity over the years, and are expected to achieve their ﬁrst detection in the near future

[13–16].

The most prominent sources of nanohertz GWs are expected to be supermassive black hole

binaries (SMBHBs), usually hosted by active galactic nuclei (AGNs) [17]. The ﬁrst PTA detection

is expected to be that of a stochastic GW background [GWB: 18] formed via the incoherent addition

of GWs emitted by an ensemble of unresolved SMBHBs, followed by the detection of individual

SMBHBs that stand out above the GWB [19–21]. Several promising SMBHB candidates have been

identiﬁed through electromagnetic observations of AGNs [e.g. 22–25], and PTA experiments have

already put increasingly stringent constraints on the presence of SMBHB signals in their datasets

[26–30].

SMBHBs are believed to form through galaxy mergers, where the central black holes of the

merging galaxies sink to the center of the merger remnant, eventually forming a bound system [31].

Such binary systems shrink due to energy and angular momentum exchange with the surrounding

stars and gas until the orbital evolution is dominated by GW emission [32], and can retain signiﬁcant

eccentricities as they enter the PTA frequency band [e.g. 33,34]. The SMBHB candidate OJ 287

is believed to host a binary system with eccentricity ∼0.6 [23]. It is therefore desirable to search

for signals induced by eccentric SMBHB systems in PTA datasets.

The PTA responses to GW signals (known as PTA signals) due to inspiralling eccentric SMBHB

systems were modeled by Refs. [26,35,36] (see Section II for details). This usually involves

modeling the relativistic motion of the binary system using the post-Newtonian (PN) formalism,

where general relativistic corrections to Newtonian dynamics are expressed in powers of (v/c)2∼

GM/(c2r), and where Mis the total mass of the binary, ris the relative separation, and vis

the relative speed [37]. The GW strain amplitudes in the two orthogonal polarizations h+,×can

then be expressed in terms of the polar coordinates rand ϕin the orbital plane and their time

derivatives. Finally, the PTA signal R(t) involves the time integrals of h+,×(t).

A signiﬁcant challenge in searching for such signals in PTA datasets is posed by the cost of

computing the PTA signal given a set of pulse times of arrival (TOAs) of an ensemble of pulsars.

(See Refs [35,38] for estimates of the sensitivity degradation experienced when using the computa-

tionally inexpensive circular PTA signals [26] to search for eccentric sources.) This computational

cost is mainly incurred in two stages: (a) solving the orbital evolution, and (b) computing the

PTA signal as a function of the orbital variables. Refs [36,39] provided an analytic solution to the

quadrupolar-order orbital evolution equations governing the motion of inspiralling non-spinning ec-

centric binaries, thereby mitigating the computational cost of solving the orbital evolution of such

systems. A few diﬀerent approaches have been presented in the literature for computing the PTA

signals given the orbital variables as a function of time, including an approximate analytic integral

[26], a Fourier series expansion [35], a post-circular expansion, and numerical integration [36]. In

this work, we compare the computational cost of these methods and investigate the accuracy of

the most eﬃcient method. We then introduce a new spline-based method for further improving

the computational performance of PTA signal evaluations by leveraging the properties of real PTA

datasets. We have implemented the optimal methods for computing the PTA signals due to eccen-

tric SMBHBs in the GWecc.jl package, and this can be used readily with the popular ENTERPRISE

package [40] for searching PTA datasets. Further, to demonstrate our methods, we perform a

single-pulsar search for eccentric SMBHB signals in the NANOGrav 12.5-year narrowband dataset

of PSR J1909−3744 [6].

This paper is arranged as follows. In Section II, we introduce the PTA signals induced by

inspiralling eccentric binaries, and the GW phasing approach used for their computation. We

compare the computational cost of diﬀerent approaches for evaluating the PTA signals given the

orbital evolution as a function of time in III. We investigate the accuracy of the most computation-

ally eﬃcient PTA signal computation method in IV. In Section V, we introduce a new spline-based

method to further reduce the computational cost of evaluating PTA signals by exploiting the struc-

ture of typical PTA datasets. We provide a brief description of the GWecc.jl package in Section

VI. In Section VII, we demonstrate our methods by performing a single-pulsar search for eccentric

SMBHB signals using the NANOGrav 12.5-year narrowband data of PSR J1909−3744 as a proof

of concept. Finally, we summarize our results in Section VIII.

II. THE PTA SIGNAL MODEL FOR ECCENTRIC BINARIES

A. The gravitational waveform and the PTA signal

We begin by brieﬂy describing the PN-accurate PTA signal model for inspiralling eccentric

binaries. GWs traveling across the line of sight to a pulsar induce modulations on the observed

TOAs of its pulses. These modulations are given by [26,41,42]

R(tE) = ZtE

t0h(t′

E)−h(t′

E−∆p)dt′

E,(1)

where his the GW strain (precisely deﬁned below), the time variables tEand t′

Eare measured in

the solar system barycenter (SSB) frame, ∆pis a geometric time delay given by

∆p=Dp

c(1 −cos µ),(2)

Dpis the pulsar distance, µis the angle between the lines of sight to the pulsar and the GW source,

cis the speed of light, and t0is an arbitrary ﬁducial time. The coordinate time tEmeasured in the

SSB frame relates to the coordinate time tmeasured in the GW source frame via the cosmological

redshift as

tE−t0= (1 + z)(t−t0).(3)

Despite the above distinction between the coordinate times measured in the SSB frame (tE) and

the GW source frame (t), the redshift zis not measurable for an individual SMBHB source due to

the fact that it is covariant with the orbital frequency and the total mass of the source [see, e.g.

43]. Therefore, we drop the subscript Edenoting the SSB frame with the understanding that the

frequencies and masses appearing in the equations hereafter have been scaled by redshift.

The dimensionless GW strain h(t) is given by

h(t) = hF+F×i



cos 2ψ−sin 2ψ

sin 2ψcos 2ψ





h+(t)

h×(t)

,(4)

where h+,×(t) are the two GW polarization amplitudes, F+,×are the antenna pattern functions

that depend on the sky locations of the pulsar and the GW source (see, e.g., Ref. [44] for explicit

expressions), and ψis the GW polarization angle. Hence, R(t) can be expressed in terms of

functions s+,×(t) deﬁned as

s+,×(t) = Zt

h+,×(t′)dt′,(5)

such that

R(t) = hF+F×i



cos 2ψ−sin 2ψ

sin 2ψcos 2ψ





s+(t)−s+(t−∆p)

s×(t)−s×(t−∆p)

.(6)

The s+,×(t) and s+,×(t−∆p) contributions are known as the Earth term and the pulsar term

respectively.

The quadrupolar-order h+,×expressions valid for binary systems inspiralling along eccentric

orbits are given by [36]

h+=H"c2

ι+ 1 2e2

t−χ2+χ−2

(1 −χ)2cos(2ϕ)−2p1−e2

tξ

(1 −χ)2sin(2ϕ)!+s2

(1 −χ)#,(7a)

h×=H2cι"2p1−e2

tξ

(1 −χ)2cos(2ϕ) + 2e2

t−χ2+χ−2

(1 −χ)2sin(2ϕ)#,(7b)

where H=GMη

DLc2x,ηis the symmetric mass ratio, DLis the luminosity distance to the binary,

x=GM(1 + k)n/c32/3is a dimensionless PN parameter, nis the mean motion of the orbit, k

is the relativistic advance of periapsis per orbit, etis the time eccentricity, cι= cos ι,sι= sin ι,

ιis the orbital inclination, χ=etcos u,ξ=etsin u,uis the true anomaly, and ϕis the angular

coordinate in the orbital plane also known as the orbital phase. The variables ϕand ucan be

obtained as functions of time using the gravitational wave phasing approach [45], and this is what

we discuss in the next subsection.

B. The GW phasing formalism for computing the orbital motion

The conservative dynamics of the binary system can be expressed using the PN-accurate quasi-

Keplerian parametrization [46,47]. We begin by deﬁning the mean anomaly las

l(t) = Zt

n(t′)dt′.(8)

The eccentric anomaly ucan be written implicitly as a function of lusing the PN-accurate Kepler

equation [47,48]

l=u−etsin u+Ft(u),(9)

where etis known as the time eccentricity and Ft(u) is a periodic function of u. Although this

transcendental equation admits an analytic Fourier series solution [48], it is usually solved numeri-

cally in practice due to the computationally expensive nature of the analytic solution. An eﬃcient

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Post-Newtonian-accuratepulsartimingarraysignalsinducedbyinspirallingeccentricbinaries:accuracy,computationalcost,andsingle-pulsarsearchAbhimanyuSusobhanan∗CenterforGravitation,Cosmology,andAstrophysics,UniversityofWisconsin-Milwaukee,Milwaukee,WI53211,USA(Dated:June29,2023)PulsarTimingArray(PTA)expe...

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Post-Newtonian-accurate pulsar timing array signals induced by inspiralling eccentric binaries accuracy computational cost and single-pulsar search Abhimanyu Susobhanan

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