Draft version January 1, 2020 Typeset using LATEX twocolumn style in AASTeX62 Detecting gravitational scattering of interstellar objects using pulsar timing Ross J. Jennings,1 James M. Cordes,1, 2 and Shami Chatterjee1, 2 1Department of Astronomy, Cornell University, Ithaca, NY 14853, USA 2Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, NY 14853, USA ABSTRACT Gravitational scattering events, in which the path of an interstellar object is deflected by a pulsar or the solar system, give rise to reflex motion which can potentially be detected using pulsar timing. We determine the form of the timing signal expected from a gravitational scattering event, which is ramp-like and resembles the signal produced by a glitch or a gravitational wave burst with memory (BWM), and investigate the prospects for detecting such a signal using a pulsar timing array. The level of timing precision currently achieved for some millisecond pulsars makes it possible to detect −10 objects as small as 10 M , less than the mass of the dwarf planet Ceres, at impact parameters as large as 1 au. The signals produced by gravitational scattering could provide independent constraints on models of dark matter involving asteroid-mass objects or subhalos, and should be considered as potential false positives in searches for BWMs. 1. INTRODUCTION planets (Charnoz & Morbidelli 2003). Some such bodies If a compact astrophysical object were to pass near have even been directly observed: In particular, the as- enough to a pulsar or to the solar system to interact teroid 1I/`Oumuamua (Meech et al. 2017) and the comet gravitationally, it would slightly alter the motion of the 2I/Borisov (Guzik et al. 2019) have both been identified pulsar relative to the solar system barycenter, poten- as interstellar in origin. To produce a detectable timing tially producing a detectable effect on the pulse times signal, an asteroid-like ISO would have to be a few hun- of arrival (TOAs). The North American Nanohertz Ob- dred kilometers in diameter and pass within a few au of servatory for Gravitational Waves (NANOGrav; Arzou- a pulsar. This is significantly larger than either `Oumua- manian et al. 2018a) and other pulsar timing arrays mua or Borisov, but smaller than the largest objects in (PTAs; Desvignes et al. 2016; Hobbs 2013) have col- the asteroid belt. lected hundreds of pulsar-years of high-precision (sub- Some theories of dark matter predict that at least microsecond) pulsar timing data as part of their ef- a fraction of it is composed of massive ISOs, such as forts to detect gravitational waves. The various regional primordial black holes (PBHs) or subhalos. As a re- PTAs collaborate to form the international pulsar tim- sult, several searches for ISOs have been carried out ing array (IPTA; Hobbs et al. 2010), which occasion- with the goal of understanding the nature of dark mat- ally releases combined data sets (e.g. Verbiest et al. ter. For the most part, these have involved microlensing 2016; Perera et al. 2019). The qualities of PTAs that surveys, which are sensitive to any object with mass make them well suited for detecting gravitational waves along the line of sight to a star. The pioneering MA- { namely, the precision of arrival time measurements and CHO (Alcock et al. 2001), EROS (Tisserand et al. 2007), the long spans of time over which they may be collected and OGLE (Wyrzykowski et al. 2011) surveys largely { also make them highly sensitive to perturbations of ruled out the possibility that the majority of dark mat- −8 arXiv:1910.08608v2 [astro-ph.HE] 27 Dec 2019 this type. ter could consist of objects between 10 and 100 M , An interstellar object (ISO) gives rise to a gravita- but did not significantly constrain objects smaller than −8 tional scattering signal in pulsar timing data if it passes 10 M (approximately 20 times the mass of Ceres). close enough to a pulsar or to the solar system. One im- Subsequently, attention has focused on primordial black −11 −8 portant category of ISOs consists of free-floating planets holes with masses between 10 and 10 M as dark and smaller asteroid- and comet-like bodies. Almost cer- matter candidates (Carr et al. 2016). The strongest lim- tainly, a large number of these bodies exist in interstellar its on PBHs and other relatively compact objects in this space, since the planet formation process is thought to mass range come from a recent microlensing survey of result in the ejection of large numbers of rocky bod- the Andromeda galaxy using the Hyper Suprime Cam ies, ranging in size from planetesimals to fully-formed (HSC) on the 8.2-meter Subaru telescope (Niikura et al. 2 2019). The implications of these constraints for the ex- shapes with the template pulse shape using a Fourier- pected event rate are discussed further in Section6. domain matched filtering algorithm (Taylor 1992). Gen- A PBH or similar dark matter constituent would in- erally, averages with N 1 pulses are used, with typical teract with a pulsar in a manner indistinguishable from values of N being between 105 and 106. a baryonic ISO of equal mass. If they constituted a sig- The minimum uncertainty in estimating an arrival nificant fraction of dark matter by mass, PBHs would time comes from noise in the pulse profile measurement also be much more numerous than baryonic ISOs of the introduced by the receiver (radiometer noise). Comput- same mass. Because of this, pulsar timing has been pro- ing TOAs using a matched filtering algorithm minimizes posed as a means of detecting PBHs. Some proposals this contribution to the arrival time error for a given have focused on the Shapiro delay signal caused by ob- level of radiometer noise. In the absence of other sources jects along the line of sight to the pulsar (Siegel et al. of error, the uncertainty in arrival times computed using 2007; Clark et al. 2016), but others (Seto & Cooray 2007; matched filtering is given by (Cordes 2013; Lam et al. Kashiyama & Seto 2012; Kashiyama & Oguri 2018; Dror 2016): p et al. 2019) have discussed the Doppler signal, which is PWeff σMF = p : (1) more closely related to the timing signal described here. S Nφ Detectable encounters with ISOs are likely to be rare, Here P is the pulse period, S is the signal-to-noise ratio but examining a large volume of data makes their ob- (SNR), Nφ is the number of samples in pulse phase, and servation more likely. In this respect, it is important Weff is an effective pulse width, given by to distinguish between encounters with a pulsar and en- Z P −1 counters with the solar system. By analogy with the 0 2 Weff = U (t) dt ; (2) terminology used for gravitational wave signals, we call 0 the former \pulsar-term" scattering events and the lat- where U (t) is the pulse shape (normalized to unit max- ter \Earth-term" events. If many pulsars are observed, imum). the two scenarios give complementary sensitivities | In practice, TOA errors are larger than equation (1) since Earth-term events affect all pulsars, it is possible would predict, because other effects contribute. These to detect weaker events by cross-correlating the signals, include pulse jitter from the motion of emission regions but objects are more likely to pass near one of the pul- in pulsar magnetospheres; dispersion and scintillation, sars in a PTA simply because there are more pulsars. caused by propagation of the signal through the ionized This means that Earth-term events are more detectable interstellar medium; and spin noise, caused by interac- for large populations of very small objects, but pulsar- tions between the neutron star crust and its magneto- term events are more detectable for small populations sphere and superfluid interior. Of these, pulse jitter is of larger objects. uncorrelated in time, but scintillation, dispersion mea- In what follows, we calculate the shape of the timing sure variations, and spin noise are generally correlated, signal expected from a gravitational scattering event in producing gradual but random drifts in pulse times of ar- detail, and use the results to assess the circumstances rival. Spin noise has a \red" power spectrum with most under which such events may be detectable. In Sec- of the power concentrated at low frequencies. By con- tion2, we discuss the factors which limit the precision trast, radiometer and jitter noise have a \white" power of pulsar timing measurements. In Section3, we derive spectrum, contributing approximately equal power at all the expected timing signal. In Section4, we discuss the frequencies. conditions under which pulsar-term events may be de- Broadly speaking, pulsars can be divided into two cat- tected. Section5 describes the corresponding conditions egories: canonical pulsars (CPs), which are relatively for Earth-term events. In Section6, we review methods young and tend to have periods of order one second of estimating the number density of potential perturbers and surface magnetic fields of order 1012 gauss; and mil- in the galaxy, and speculate as to the frequency of de- lisecond pulsars (MSPs), which are much older and are tectable events. Finally, in Section7, we summarize our thought to have been spun up by accreting matter from results and draw conclusions. a companion. MSPs have significantly shorter periods (a few milliseconds) and weaker magnetic fields (of order 2. PULSAR TIMING PRECISION 108 gauss) than canonical pulsars. Additionally, MSPs Contemporary pulsar timing methods rely on the fact spin down much more slowly and have lower levels of that every pulsar has a characteristic average pulse spin noise (by a factor of around 106) than CPs (Shan- shape which is stable on time scales of decades.