Probing Formation of Double Neutron Star Binaries around 1mHz with LISA Lucy O. McNeilland Naoki Seto Department of Physics Kyoto University Kyoto 606-8502 Japan

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Probing Formation of Double Neutron Star Binaries around 1mHz with LISA

Lucy O. McNeill∗and Naoki Seto

Department of Physics, Kyoto University, Kyoto 606-8502, Japan

(Dated: October 11, 2022)

We propose a novel method to examine whether Galactic double neutron star binaries are formed

in the LISA band. In our method, we assign an eﬀective time fraction τto each double neutron

star binary detected by LISA. This fraction is given as a function of the observed orbital period

and eccentricity and should be uniformly distributed in the absence of in-band binary formation.

Applying statistical techniques such as the Kolmogorov-Smirnov test to the actual list of τ, we can

inspect the signature of the in-band binary formation. We discuss the prospects of this method,

paying close attention to the available sample number of Galactic double neutron star binaries

around 1mHz.

I. INTRODUCTION

Double neutron star binaries (hereafter DNSBs) are

bountiful astrophysical targets. They have been detected

as radio pulsars [1, 2], and the current sample has orbital

periods Pfrom 1.9 hours [3] to six weeks [4]. This sam-

ple contains around 20 DNSBs, and searches might be in-

complete at the faint end of the luminosity function (also

limited by the beaming fraction). Shorter period (P <

1 hour) DNSBs ought to exist in the Galaxy as well.

However, due to Doppler smearing and shorter merger

timescales, their detection would be more diﬃcult than

the longer period ones [5].

The Laser Interferometer Space Antenna (LISA) is

planned in the 2030’s [6] and is sensitive to gravita-

tional waves (GWs) around 0.1-100mHz. It will detect

all Galactic DNSBs in the frequency range f&1.5mHz

(corresponding to the orbital period P= 2/f .20 min),

unlike the longer-period radio sample. Observationally

motivated estimates [7] suggest that at least dozens of

DNSBs exist in the Galaxy at f&1.5mHz. Numerical

galactic modelling [8, 9] predicts that LISA will detect

altogether from a few to upto hundreds of DNSBs.

DNSB formation depends on the complex interplay

between many astrophysical processes. In the general

picture, the binary must survive two supernova explo-

sions. Preceding these supernovae, various mechanisms

have been theorized with respect to mass loss/exchange

in the binary [10–13] after hydrogen burning. These pro-

cesses all play a role in determining the separation at

formation, if the binary survives.

However, the related eﬃciencies and rates in popula-

tions are not well established (see [14] for a review). In

particular, it’s unclear whether there is a critical mini-

mum orbital period (or maximum orbital frequency) for

isolated DNSB formation.

Dynamical encounters in star clusters is an alternative

pathway for short period DNSB formation, though their

contribution to the LISA sample is estimated to be small

[15, 16]. Also, for the dynamical scenario it will be diﬃ-

∗email: mcneill@tap.scphys.kyoto-u.ac.jp

cult to solidly estimate the distribution function for the

orbital periods of the generated DNSBs.

In this paper, we are interested in the possibility of

DNSB formation speciﬁcally at f&1mHz. We hereafter

call this channel as “in-band” (mHz) DNSB formation, or

simply “injections”. Considering the aforementioned the-

oretical uncertainties, it will be fruitful to observationally

examine the in-band formation in a model independent

manner.

We thus develop a statistical method to examine the

in-band formation with LISA (see also [17] for forma-

tion between the LISA band and the lower frequencies

already probed by the radio sample). Recently, several

studies have proposed to statistically deal with multiple

LISA sources in the Galaxy. Among others, a large num-

ber (∼104) of white dwarf binaries (WDBs) will be a

powerful data set for various astronomical analyses (e.g.,

probing the Galactic structure [18, 19]). In this context,

one of the authors suggested to measure the ﬂuxes of the

Galactic WDBs in frequency space [20]. He pointed out

that the measurement will enable us to follow the col-

lective evolution of the WDBs, resulting in mergers or

stable mass transfers.

One might imagine that we can get some information

about the in-band formation by similarly measuring the

DNSB ﬂux with LISA at various frequencies. Unfortu-

nately, LISA will detect much fewer DNSBs than WDBs,

and the small number statistics will severely limit the

ﬂux approach for DNSBs. On the other hand, unlike

WDBs, DNSBs can be well regarded as point particle

systems in the LISA band, and their long-term orbital

evolution from GW emission can be predicted quite ac-

curately [1]. Considering these pros and cons of DNSBs,

we newly introduce the eﬀective time fraction τ, corre-

sponding to the fraction of total time that each binary

has spent in the mHz band. Without in-band forma-

tion, the fraction τshould be uniformly distributed. By

analysing the observed list of τ, we can examine potential

in-band formation through its deviation from a uniform

distribution.

The basic assumption of our study is that the mHz

DNSB population is in the “steady state” [21]. This is

a reasonable assumption, since a DNSB passes through

the mHz band in the timescale of Myr but the Galactic

arXiv:2210.04407v1 [astro-ph.HE] 10 Oct 2022

DNSB merger rate will change in the Hubble timescale

[22, 23].

This paper is organized as follows. In section II, we

roughly estimate how many Galactic DNSBs are likely

to be detected by LISA. In section III we study the long-

term orbital evolution of DNSBs and deﬁne the eﬀective

time fraction τ, as a function of the GW frequency and

the orbital eccentricity. Then, in section IV, using sta-

tistical tools such as the Kolmogorov-Smirnov test, we

discuss how well we can probe in-band formation with

LISA. In section V, we mention potential extensions of

this study. We summarize our ﬁndings in section VI.

II. GALACTIC DOUBLE NEUTRON STAR

BINARIES

A. Expected number in the LISA band

We ﬁrst estimate the merger rate RMW of DNSBs

in our Galaxy (Milky Way). As a basic observa-

tional input, we use the comoving merger rate R=

660+1040

−530 Gpc−3yr−1from a recent report by the LVK col-

laboration [24] (Multi source model).

To relate the two rates Rand RMW, we apply the

traditional argument based on the eﬀective number den-

sity of Milky Way equivalent galaxies [15, 25], and put

RMW =LB,MWR/LB. Here LBis the B-band lumi-

nosity per comoving volume and LB,MW is the B-band

luminosity of our Galaxy. Using their typical values, we

obtain

RMW =6.0×10−5yr−1R

660Gpc−3yr−1

×LB,MW

9×109L LB

1017LGpc−3−1

.(1)

We should notice that the Galactic merger rate RMW still

has large uncertainties (at least a factor of three).

Next, we roughly estimate the total number of Galactic

DNSBs in the LISA band. In this paper, we use the

notation fspeciﬁcally for the second harmonic frequency

(given by f= 2forb with the orbital frequency forb). Due

to radiation reaction, the GW frequency fevolves as [26]

dt=96π8/3G5/3f11/3M5/3

5c5(1 −e2)7/21 + 73

24e2+37

96e4(2)

with the orbital eccentricity eand the chirp mass Mof

the binary [26]. At the same time, eccentricity evolves

according to

dt=−304eπ8/3G5/3f8/3M5/3

15c5(1 −e2)5/21 + 121

304e2.(3)

These equations are a good approximation for DNSBs in

the LISA band since the relativistic corrections are small

(except for 1 −e1).

Simply assuming (i) the steady state condition for the

Galactic DNSB population at f&1mHz, and (ii) no bi-

nary formation there, we have the frequency distribution

df=RMW df

dt−1

∝RMWf−11/3M−5/3.(4)

Here we ignored the eccentricity dependence of ˙

f. After

the frequency integral, we obtain the cumulative number

N(> f) = 30 M

1.2M−5/3f

1.5mHz−8/3

×RMW

6.0×10−5yr−1.

(5)

Note that the chirp mass distribution of known DNSBs

is narrow and centred around M= 1.2M[27].

In fact, we will later relax the assumption (ii). But

the above result will serve as a rough guide, except for

extreme model settings.

B. Gravitational wave observations

1. Identiﬁcation of binary neutron stars

Next, we consider a DNSB at distance D, with chirp

mass M, eccentricity eand gravitational wave frequency

f. When the binary is approximated as monochromatic

(f= constant), the dimensionless gravitational wave

strain amplitude from the binary in the second orbital

harmonic is

h2=8G5/3M5/3π2/3f2/3

51/2Dc41−5

2e2+35

24e4+O(e6)

(6)

[28–30]. This expression is obtained using the strain am-

plitude in the nth orbital harmonic hn∝g(n, e)1/2/n,

with g(n, e) given by Equation (20) in [31]. For binaries

upto e≤0.2, the correction for the eccentricity is less

than ∼10 percent.

In a gravitational wave detector with a sensitivity

curve Sn(f), the angle averaged signal-to-noise ratio ¯ρ2

over an observing time Tis given by

¯ρ2=h2

Sn(f)1/2√T . (7)

We put the noise curve of LISA by Sn(f) = Sd(f) +

Sc(f) with the detector noise Sd(f) and astrophysical

foreground confusion noise Sc(f) [28], where the latter is

a function of T[32].

Now we will calculate some estimates related to gravi-

tational wave detection of Galactic DNSBs. Taking a cir-

cular binary with the chirp mass 1.2M, conservatively

located at 20kpc, Equations (6) and (7) are used to ob-

tain ¯ρ2= 7.8 and 452 for f= 1.5 mHz and 50mHz re-

spectively for an observation time T= 4 years. If the

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ProbingFormationofDoubleNeutronStarBinariesaround1mHzwithLISALucyO.McNeillandNaokiSetoDepartmentofPhysics,KyotoUniversity,Kyoto606-8502,Japan(Dated:October11,2022)WeproposeanovelmethodtoexaminewhetherGalacticdoubleneutronstarbinariesareformedintheLISAband.Inourmethod,weassignaneectivetimefraction...

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Probing Formation of Double Neutron Star Binaries around 1mHz with LISA Lucy O. McNeilland Naoki Seto Department of Physics Kyoto University Kyoto 606-8502 Japan

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