# Pulsar Timing and Its Application for Navigation and Gravitational Wave Detection

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## Abstract

Pulsars are natural cosmic clocks. On long timescales they rival the precision of terrestrial atomic clocks. Using a technique called pulsar timing, the exact measurement of pulse arrival times allows a number of applications, ranging from testing theories of gravity to detecting gravitational waves. Also an external reference system suitable for autonomous space navigation can be defined by pulsars, using them as natural navigation beacons, not unlike the use of GPS satellites for navigation on Earth. By comparing pulse arrival times measured on-board a spacecraft with predicted pulse arrivals at a reference location (e.g. the solar system barycenter), the spacecraft position can be determined autonomously and with high accuracy everywhere in the solar system and beyond. We describe the unique properties of pulsars that suggest that such a navigation system will certainly have its application in future astronautics. We also describe the on-going experiments to use the clock-like nature of pulsars to “construct” a galactic-sized gravitational wave detector for low-frequency (\(f_{GW}\sim 10^{-9} \text{--} 10^{-7}\) Hz) gravitational waves. We present the current status and provide an outlook for the future.

## Keywords

Pulsars Neutron stars High precision timing Atomic clocks Pulsar timing XNAV Pulsar-navigation Pulsar-based-navigation Autonomous spacecraft navigation Gravitational waves Gravitational wave detector## 1 Introduction

For millenia, keeping time was safely in the hands of astronomers, who watched the heavens to serve the societies’ needs for time measurements. They used the Earth’s rotation, the moon, the Sun and the stars to keep time, but, of course, they also exploited their observations to study and explore the Universe that we live in. Practical purposes, especially the need to navigate, turned the art of timekeeping into the hands of clockmakers and, hence, engineers and physicists. Today, world time is kept by a set of ultra-precise atomic clocks, and the sophistication and accuracy of these clocks is hugely impressive (see other contributions to this Topical Collection). The stability of these clocks is best on small timescales. When time stability is needed on long timescales, one can (and does) “hand over” time from clock to clock—or one can revert back to astronomical observations. Moreover, the fact that the Earth is moving on an elliptical orbit around the Sun, and hence at varying distances, means that every clock on Earth experiences a varying gravitational potential throughout the year. This leads to seasonal changes in clock rates that affect all clocks on Earth simultaneously. Astronomy can, hence, provide an “independent” time standard that is unaffected by effects on Earth or in the solar system. The key in providing such an astronomical time standard are objects called “pulsars.”. Pulsars are compact, highly-magnetized rotating neutron stars which act as “cosmic lighthouses” as they rotate, enabling a number of applications as precision tools.

This contribution describes pulsars, the technique of pulsar timing and some of the resulting applications. Coming full circle, these applications include navigation and also time keeping. Here we concentrate on the former (see Sect. 2) and applications in fundamental physics, especially the detection of low-frequency gravitational waves (see Sect. 3). We note, that at the end, despite attractive features of a pulsar timescale, pulsars will not be able to compete with the precision and practicality of the best atomic clocks on Earth. Nevertheless, it is the combination of pulsar clocks with terrestrial clocks that allow, via clock comparison experiments, to probe a wide range of physics—and the Universe that we live in, continuing the tradition of astronomers.

### 1.1 Pulsars

Most known pulsars have spin periods between 0.1 and 1.0 s with period derivatives of typically \(\dot{P}=10^{-15}\) s s^{−1}. Selection effects are only partly responsible for the limited number of pulsars known with very long periods, the longest known period being 8.5 s (Young et al. 1999). The dominant effect is due to the “death” of pulsars when their slow-down has reached a critical state. This state seems to depend on a combination of \(P\) and \(\dot{P}\) known as the *pulsar death-line*. The normal life of radio pulsars is limited to a few tens or hundreds million years or so.

The described evolution does not explain the over 200 pulsars in the lower left of the \(P\text{--}\dot{P}\)-diagram (Fig. 2). These pulsars have simultaneously small periods (few milliseconds) and small period derivatives (\(\dot{P}\le 10^{-18}\) s s^{−1}). Based on their evolutionary history and characteristic ages, these *millisecond pulsars* (MSPs) are believed to be much older than ordinary pulsars with ages up to \(\sim 10^{10}\) yr. MSPs evolve from pulsars with a binary companion. Once the binary companion evolves and overflows its Roche lobe, it transfers mass and thereby angular momentum (e.g. Alpar et al. 1982). In this process, previously “dead” pulsars are recycled to MSPs via an accreting X-ray binary phase. This has a number of observational consequences: (a) most normal pulsars do not develop into a MSP as they have long lost a possible companion during their violent birth event; (b) for surviving binary systems, X-ray binary pulsars represent the progenitor systems for MSPs; (c) the final spin period of recycled pulsars depends on the mass of the initial binary companion. A more massive companion evolves faster, limiting the duration of the accretion process; (d) the majority of MSPs have low-mass white-dwarf companions as the remnant of the binary star. These systems evolve from low-mass X-ray binary systems; (e) high-mass X-ray binary systems represent the progenitors for double neutron star systems (DNSs). DNSs are rare since these systems need to survive a second supernova explosion. The resulting MSP is only mildly recycled with a period of tens of millisecond. This picture explains the observation that \(\sim 80\)% of all MSPs are in a binary orbit while this is true for only less than 1% of the non-recycled population. For MSPs with a low-mass white dwarf companion the orbit is nearly circular. In case of DNSs, the orbit is affected by the unpredictable nature of the kick imparted onto the newly born neutron star in the asymmetric supernova explosion of the companion. If the system survives, the result is typically an eccentric orbit with an orbital period of a few hours. However, there is also evidence for the existence of low-kick supernova, producing DNSs with low eccentricity and relatively low-mass neutron star companions (e.g. Tauris et al. 2015).

Source with the largest estimated magnetic fields (\(\sim 10^{15}\) G), the so called *magnetars*, are located in the upper right corner of Fig. 2. Here, the observed luminosity appears to exceed the neutron stars’ spin-down energy loss, suggesting that magnetars in addition to the spin-down energy are powered by converting magnetic field energy (see e.g. Kaspi and Kramer 2016 for a comparison of magnetars to rotation powered radio pulsars). Only four magnetars have been detected as (transient) radio sources while all appear to be X-ray and gamma-ray sources. The long-term timing is not regular, so that applications as discussed below, are unlikely to be possible.

From timing measurements of binary MSPs (see Sect. 1.2), we can measure neutron star masses. Those are found in a range between about 1.2 and 2\(M_{\odot }\) (Özel and Freire 2016; Antoniadis et al. 2016) with the maximum mass (Antoniadis et al. 2013) ruling out the softest equation-of-sate (Demorest et al. 2010). The MSP mass distribution is strongly asymmetric. The diversity in spin and orbital properties of high-mass NSs suggests that this is most likely not a result of the recycling process, but rather reflects differences in the NS birth masses. The asymmetry is best accounted for by a bimodal distribution with a low mass component centered at \(1.39M_{\odot }\) and a high-mass component with a mean of \(1.81M_{\odot }\) (Antoniadis et al. 2016). These equations-of-state yield radii not too different from the very first calculations by Oppenheimer and Volkoff (1939), i.e. about 20 km in diameter, and are consistent with the blackbody emission radii determined from X-ray observations (Özel and Freire 2016). This makes neutron stars the most compact objects in the observable Universe.

### 1.2 Pulsar Timing: Pulsars as Clocks

*braking index*takes the value \(n=3\). If \(\nu \) and its derivatives are accurately known and if \(t_{0}\) coincides with the arrival of a pulse, all following pulses should appear at integer values of \(N\) — when observed in an inertial reference frame. However, our observing frame is not inertial, as we are using telescopes that are located on a rotating Earth orbiting the Sun. Therefore, we need to transfer the pulse times-of-arrival (TOAs) measured with the observatory clock (

*topocentric arrival times*) to the center of mass of the solar system as the best approximation to an inertial frame available. The transformation of a topocentric TOA to such

*barycentric arrival times*, \(t_{\mathrm{SSB}}\), is given by

#### 1.2.1 Clock and Frequency Corrections

The observatory time is typically maintained by local Hydrogen-maser clocks monitored by GPS signals. In a process involving a number of steps, clock corrections, \(t_{\mathrm{corr}}\), are retroactively applied to the arrival times in order to transfer them to a uniform atomic time (“Terrestrial Time”, or TT) that would be kept by an ideal atomic clock on the geoid. It is published retroactively by the *Bureau International des Poids et Mesures* (BIPM).

The free electrons in the interstellar medium interact with the propagation radio signal, causing a frequency dependent group velocity. Consequently, the frequency components of the broadband pulse signal emitted a higher radio frequency \(f_{\mathrm{obs}}\) arrive earlier than the corresponding low frequency components. Hence, due to this dispersion, the measured arrival time depends on the observing frequency, \(f_{\mathrm{obs}}\) and the dispersion measure (DM). The TOA is therefore corrected for a pulse arrival at an infinitely high frequency (last term in Eq. (4)). For the best pulsars, the limiting factor in timing precision is often “interstellar weather” that causes small changes in DM and hence time-varying drifts in the TOAs. In those cases, the above term is, for instance, extended to include time-derivatives of DM which can be measured using multi-frequency observations.

#### 1.2.2 Barycentric Corrections

*Roemer delay*, \(\Delta_{\mathrm{Roemer,} \odot }\), is the classical light-travel time between the phase center of the telescope and the solar system barycenter (SSB). Given a unit vector, \(\hat{s}\), pointing from the SSB to the position of the pulsar and the vector connecting the SSB to the observatory, \(\vec{r}\), we find:

The *Shapiro delay*, \(\Delta_{\mathrm{Shapiro,} \odot }\), is a relativistic correction that corrects for an extra delay due to the curvature of space-time in the solar system (Shapiro 1964). It is largest for a signal passing the Sun’s limb (\(\sim 120\) μs) while Jupiter can contribute as much as 200 ns. In principle one has to sum over all bodies in the solar system, but in practice only the Sun is usually taken into account.

The last term in Eq. (5), \(\Delta_{\mathrm{Einstein,} \odot }\), is called *Einstein delay* and it describes the combined effect of gravitational redshift and time dilation due to motion of the Earth and other bodies, taking into account the variation of an atomic clock on Earth in the varying gravitational potential as it follows its elliptical orbit around the Sun (Backer and Hellings 1986).

#### 1.2.3 Relative Motion & Shklovskii Effect

If the pulsar is moving relative to the SSB, the transverse component of the velocity, \(v_{t}\), can be measured as the vector \(\hat{s}\) in Eq. (7) changes with time. Present day timing precision is not sufficient to measure a radial motion although it is theoretically possible. This leaves Doppler corrections to observed periods, masses etc. undetermined. The situation changes if the pulsar has an optically detectable companion such as a white dwarf for which Doppler shifts can be measured from optical spectra.

*Shklovskii effect*, also known in classical astronomy as

*secular acceleration*. With the pulsar motion, the projected distance of the pulsar to the SSB is increasing, leading to an increase in any observed change of periodicity, such as pulsar spin-down or orbital decay. The observed pulse period derivative, for instance, is increased over the intrinsic value by

#### 1.2.4 Binary Pulsars

Equation (4)–(5) is used to transfer the measured TOAs to the SSB. If the pulsar has a binary companion, the light-travel time across the orbit and further relativistic effects need to be taken into account (see Eq. (6)). That adds additional orbital parameters to the set of timing parameters which have to be solved in the timing process (see below). In the simplest case, five Keplerian parameters need to be determined, i.e. orbital period, \(P_{\mathrm{b}}\); the projected semi-major axis of the orbit, \(x\equiv a \sin i\) where \(i\) is the (usually unknown) orbital inclination angle; the orbital eccentricity, \(e\); the longitude of periastron, \(\omega \); and the time of periastron passage, \(T_{0}\). For a number of binary systems this Keplerian description of the orbit is not sufficient and corrections need to be applied. These can be either time derivatives of Keplerian parameters or parameters describing completely new effects (e.g. those of a Shapiro delay due to curved spacetime near the companion). In any case, it is important to note that we do not have to assume a particular theory of gravity when measuring such relativistic corrections, called “post-Keplerian” (PK) parameters (Damour and Deruelle 1985; Damour and Deruelle 1986). Instead, we can take the observational values and compare them with predictions made within a framework of specific theories of gravity (Damour and Taylor 1992).

The first PK parameter, \(\dot{\omega }\), describes the relativistic advance of periastron in rad s^{−1}. It is the easiest to measure for orbits with non-zero eccentricities (note that \(\omega \) is only poorly defined for \(e\approx 0\) and so is \(\dot{\omega }\)). From a measurement of \(\dot{\omega }\), we obtain from Eq. (9) the total mass of the system, \((M_{p}+M_{c})\).

The orbital decay due to gravitational-wave damping is expressed by the (dimensionless) change in orbital period, \(\dot{P}_{\mathrm{b}}\). Any metric theory of gravity that embodies Lorentz-invariance in its field equations predicts gravitational radiation and, hence, \(\dot{P}_{ \rm b}\). If a theory satisfies the strong equivalence principle, like GR, gravitational *dipole* radiation is not expected, but *quadrupole* emission will be the lowest multipole term. In alternative theories, while the *inertial* dipole moment may remain uniform, the *gravitational wave* dipole moment may not, and dipole radiation may be predicted. The magnitude of this effect depends on the difference in gravitational binding energies, expressed by the difference in coupling constants to a scalar gravitational field.

The other two parameters, \(r\) and \(s\), are related to the Shapiro delay caused by the curvature of space-time due to the gravitational field of the companion. They are measurable, depending on timing precision, if the orbit is seen nearly edge-on.

#### 1.2.5 Obtaining a Timing Solution

Equations (2) to (6) contain the set of timing parameters that need to be determined. Many of them are not known a priori (or only with limited precision after the discovery of a pulsar) and need to be determined precisely in a least squares fit analysis of the measured TOAs. The parameters can be categorized into three groups: (a) *astrometric parameters* (i.e. position, proper motion, parallax contained in the Römer and Shapiro delay, respectively); (b) *spin parameters* (i.e. rotation frequency, \(\nu \), and higher derivatives); (c) *binary parameters*.

In order to obtain a phase-coherent solution that accounts for every single rotation of the pulsar between two observations, one starts off with a small set of TOAs that were obtained sufficiently close in time so that the accumulated uncertainties in the starting parameters do not exceed one pulse period. Gradually, the data set is expanded, maintaining coherence in phase. When successful, post-fit residuals expressed in pulse phase show a Gaussian distribution around zero with a root mean square that is comparable to the TOA uncertainties. Incorrect or incomplete timing models cause systematic structures in the post-fit residuals identifying the parameter that needs to be included or adjusted. The precision of the parameters improves with length of the data span and the frequency of observation, but also with orbital coverage in the case of binary pulsars. See, for instance, Verbiest et al. (2016) for examples and references therein for further details.

Deviations from a regular pulsar spin-down are collectively called “timing noise”. The origin of timing noise is usually identified with processes in the interior of neutron stars, observable as “glitches” (i.e. sudden increases in rotational frequency), state-changes in the pulsar magnetosphere causing a change in spin-down rate, or possibly variations in the observed pulse arrival time due to interstellar weather. The first two sources of timing noise are mostly observed in young pulsars, while recycled millisecond pulsars appear as much more stable clocks. All cases may severely affect the usage of individual sources, as we discuss in somewhat more detail in Sect. 2.2.

### 1.3 Pulsars as Tools

Pulsar combine a number of interesting properties: they are compact objects with strong gravitational fields; they are small enough to essentially act as test masses in binary systems; the act as natural precision clocks; their interior contains the most extreme dense matter in the observable Universe; they emit coherent radio emission that is up to 100% polarized; they are not only radio sources but pulsar emission is often seen across the whole electromagnetic spectrum. There are many more properties that one could list, but just those are sufficient to make pulsars an exciting and useful tool for probing a wide range of physics or for probing the interstellar or other surrounding media. They are especially useful for testing theories of gravity. This can be done either with pulsars as part of a binary system, or also as part of a Galactic-sized detector for low-frequency gravitational waves. We will expand on the latter in more detail below, but concentrate for a moment on binary pulsars.

Theory (class) | Method | Ref. |
---|---|---|

| ||

Jordan–Fierz–Brans–Dicke | Limits by PSR J1738+0333 and PSR J0348+0432, comparable to best Solar system test (Cassini) | (Freire et al. 2012) |

Quadratic scalar-tensor gravity | For \(\beta _{0} < -3\) and \(\beta _{0} > 0\) best limits from PSR-WD systems, in particular PSR J1738+0333 and PSR J0348+0432 | (Freire et al. 2012) |

Massive Brans–Dicke | For \(m_{\varphi }\sim 10^{-16}\) eV: PSR J1141−6545 | (Alsing et al. 2012) |

| ||

Einstein–Æther | Combination of pulsars (PSR J1141−6545, PSR J0348+0432, PSR J0737−3039, PSR J1738+0333) | (Yagi et al. 2014) |

Hořava gravity | Combination of pulsars (see above) | (Yagi et al. 2014) |

| ||

Bekenstein’s TeVeS | Excluded using Double Pulsar | (Kramer 2016) |

TeVeS-like theories | Excluded using PSR 1738+0333 | (Freire et al. 2012) |

The tests of relativistic gravity are complementary to what is achievable with detections of gravitational waves using ground-based detectors. The latter allow access to the highly-relativistic regime which is dominated by the effects of gravitational wave emission. However, these tests are currently limited in their precision. In comparison, the Double Pulsar can test the lowest order of the general relativity’s quadrupole 0 about three orders more precisely than possible with the brightest LIGO event so far (Kramer 2016). Higher order terms, in contrast, can only be tested with LIGO—at least until Pulsar Timing Arrays (see below) can make a detection. Therefore, binary pulsar experiments and direct detection of gravitational waves are highly complementary.

Examples of obtained relative precision using pulsar timing. Corresponding (example) references are cited in the last column

Clock properties: | ||

Spin frequency | ≲10 | (Verbiest et al. 2008) |

Orbital period | ≲10 | (Kramer 2016) |

Frequency stability | ≲10 | (Hobbs et al. 2012) |

Masses measurements: | ||

Neutron star | ≲10 | (Weisberg et al. 2010) |

White dwarf companion | ≲10 | (Hotan et al. 2006) |

Main sequence star companion | ≲10 | (Freire et al. 2011) |

Mass of Jupiter and moons | ≲10 | (Champion et al. 2010) |

Astrometry: | ||

Distance | ≲10 | (Verbiest et al. 2008) |

Proper motion | ≲10 | (Verbiest et al. 2008) |

Gravity tests: | ||

Test of general relativity | ≲10 | (Kramer 2016) |

Constancy of grav. constant, | ≲10 | (Freire et al. 2012) |

GW chara. strain Δ | ≲10 | (Verbiest et al. 2016) |

## 2 Autonomous Spacecraft Navigation Based on Pulsar Timing

Possible implementations of autonomous navigation systems were already discussed in the early days of space flight (Battin 1964). In principle, the orbit of a spacecraft can be determined by measuring angles between solar system bodies and astronomical objects; e.g., the angles between the Sun and two distant stars and a third angle between the Sun and a planet. However, because of the limited angular resolution of on-board star trackers and sun sensors, this method yields spacecraft positions with uncertainties that accumulate typically to several thousand kilometers. Alternatively, the navigation fix can be established by observing multiple solar system bodies: It is possible to autonomously triangulate the spacecraft position from images of asteroids taken against a background field of distant stars. This method was realized and flight-tested on NASA’s Deep-Space-1 mission between October 1998 and December 2001. The Autonomous Optical Navigation (AutoNav) system on-board Deep Space 1 provided the spacecraft orbit with \(1\sigma \) errors of \(\pm 250\) km and \(\pm 0.2\) m/s, respectively (Riedel et al. 2000).

An alternative and very appealing approach to autonomous spacecraft navigation is based on pulsar timing. The idea of using these celestial sources as a natural aid to navigation goes back to the 1970s when Downs (1974) investigated the idea of using pulsating radio sources for interplanetary navigation. He analyzed a method of position determination by comparing TOAs at the spacecraft with those at a reference location. Within the limitations of technology and pulsar data available at that time (a set of only 27 radio pulsars were considered), Downs showed that spacecraft position errors on the order of 1500 km could be obtained after 24 hours of signal integration. A possible improvement in precision by a factor of 10 was estimated if better (high-gain) radio antennas were available for the observations.

Chester and Butman (1981) adopted this idea and proposed to use X-ray pulsars, of which about one dozen were known at the time. They estimated that 24 hours of data collection from a small on-board X-ray detector with 0.1 m^{2} collecting area would yield a three-dimensional position accurate to about 150 km. Their analysis, though, was not based on detailed simulations or actual pulsar timing analyses; neither did it take into account the technological requirements or weight and power constraints for implementing such a navigation system.

These early studies on pulsar-based navigation estimated relatively large position and velocity errors so that this method was not considered to be an applicable alternative to the standard navigation schemes. However, pulsar astronomy has improved considerably over the last 40 years since these early proposals. Meanwhile, pulsars have been detected across the electromagnetic spectrum and their emission properties have been studied in great detail (see Sect. 1). Along with the recent advances in detector and telescope technology this motivates a general reconsideration of the idea of pulsar-based spacecraft navigation.

### 2.1 The Relevance of the Various Pulsar Types for Navigation

We already discussed the various types of pulsars, namely rotation-, accretion-, and magnetic-field-powered neutron stars in Sect. 1. The very complex spin behavior and often unpredictable evolution of rotation period in accretion powered pulsars manifest themselves in erratic changes between spin-up and spin-down as well as X-ray burst activities. This unsteady and non-coherent timing behavior disqualifies them as reference sources for navigation. Similar arguments for magnetars invalidates these sources also for the use in a pulsar-based spacecraft navigation system.

Concerning their application for navigation, the only pulsar class that really qualifies is that of rotation-powered ones. Here, the much higher timing stability of MSPs is of major importance for their use in a pulsar-based navigation system. Of the \(\sim 2500\) rotation-powered pulsars known today (Fig. 2), about 150 have been detected in the X-ray band (Becker 2009), and approximately \(1/3\) of them are MSPs. In the past \(30\text{--}40\) years many of them have been regularly timed with high precision especially in radio observations. Consequently, their ephemerides (RA, DEC, \(P\), \(\dot{P}\), binary orbit parameters, TOAs and absolute pulse phase for a given epoch, pulsar proper motion etc.) are known with very high accuracy (see Table 2). This is an essential requirement for using these celestial objects as navigation beacons, as it enables us to predict the TOAs of a pulsar for any location in the solar system and beyond.

### 2.2 Timing Irregularities

So far, we have neglected that young rotation-powered pulsars can show *glitches* in their spin-down behavior, i.e., abrupt increases of rotation frequency, often followed by an exponential relaxation toward the pre-glitch frequency (Espinoza et al. 2011; Yu et al. 2013). While this is often observed in young pulsars, it is very rarely in old and millisecond pulsars. Nevertheless, the glitch behavior of pulsars has to be taken into account by a pulsar-based navigation system. While it is possible to change the set of pulsars taken as reference when a glitch has been observed, the intrinsic *timing noise* is a factor which sets a hard limit for the overall accuracy possible for a pulsar-based spacecraft navigator. Timing noise is present in all pulsars, but the level varies between sources (Deng et al. 2012, 2013). Typical MSPs show timing residuals with an root mean square (RMS) at a level of \(\sim 0.1\text{--}1\) μs, which limits the position accuracy then to 30–300 m.

### 2.3 Principles of Pulsar-Based Navigation

In this scheme, a phase shift (6) with respect to the absolute pulse phase corresponds to a range difference \(\Delta x = c P (\Delta \phi + n)\) along the line of sight toward the observed pulsar. Here, \(\Delta \phi \) the phase shift and \(n=0, \pm 1, \pm 2,\ldots \) an integer that takes into account the periodicity of the observed pulses. If the phase shift is non-zero, the position and velocity of the spacecraft needs to be corrected accordingly and the next iteration step is taken (7). If the phase shift is zero, or falls below a certain threshold, the position and velocity used during the barycenter correction was correct (8) and corresponds to the actual orbit of the spacecraft.

Since the position of the spacecraft is deduced from the phase (or TOA) of a periodic signal, ambiguous solutions may occur. This problem can be solved by constraining the domain of possible solutions to a finite volume around an initial assumed position (Bernhardt et al. 2010, 2011), or by observing additional pulsars as illustrated in Fig. 6.

## 3 Gravitational Wave Detection Based on Pulsar Timing

### 3.1 Principles of Gravitational Wave Detection with Pulsar Timing

Pulsars are complex astrophysical objects, governed by largely unknown physics that might affect their intrinsic stability and the GW signal has to be disentangled from a plethora of other effects (see Lentati et al. 2015 for a comprehensive discussion). For example, a deterministic sinusoidal GW signal might be virtually indistinguishable from the effect of a binary companion. Therefore, the GW residual signature needs to be simultaneously detected in several MSPs. This is even more crucial if the GW signal has a stochastic nature. In this case, \(r_{a}(t)\) is just a realization of a stochastic process, at par with other stochastic noise sources. GW detection therefore requires the cross correlation of the TOAs from an ensemble of MSPs, forming a pulsar timing array (PTA, Foster and Backer 1990). Three main PTA collaborations are now active in the monitoring of several tens of MSPs: The European Pulsar Timing Array (EPTA, Desvignes et al. 2016), the Parkes Pulsar Timing Array (PPTA, Reardon et al. 2016) and the North American Nanohertz Observatory for Gravitational Waves (NANOGrav, Arzoumanian et al. 2015). The three collaborations are constantly improving their data, that they also share since 2010 under the aegis of the International Pulsar Timing Array (IPTA, Verbiest et al. 2016), aiming at the formation of a combined, more sensitive dataset.

Altogether, the three PTAs are timing order of fifty of the best MSPs at a weekly cadence (\(\Delta {t}\)) for a timespan \(T\) of several years (more then 20 for some cases), with a timing precision of few micro-seconds to few tens of nano-seconds. PTAs are therefore sensitive to GWs in the frequency range \(1/T< f<1/(2\Delta {t})\), corresponding to few to few-hundred nano-Hertz. Possible GW sources in this frequency range include cosmological stochastic backgrounds from inflation, phase transitions or cosmic strings (Lasky et al. 2016), but the loudest signals are expected from the cosmic population of inspiralling supermassive black hole binaries (SMBHBs) formed following galaxy mergers (Sesana et al. 2008).

### 3.2 Signals from Supermassive Black Hole Binaries

The signal is therefore deterministic and can be described as \(r_{a}(\overrightarrow{\lambda },t)\), where \(\overrightarrow{\lambda }\) is the vector of parameters specifying the SMBHB (including SMBH masses, binary sky location, inclination, frequency and initial phase). Note that \(r_{a}\) also depends on the sky location of the pulsar through the response function in Eq. (17) and on the distance to the pulsar that affects the phase and possibly the frequency of the pulsar term (see Babak et al. 2016 for a full description).

*at any time*a large population of SMBHB emitting GWs in the PTA band. Therefore, the superposition of many incoherent signals results in a stochastic GW background (GWB), with energy content described in terms of its GW energy density \(\rho_{\mathrm{gw}}\) per unit logarithmic frequency, divided by the critical energy density, \(\rho_{c}\), to close the Universe:

^{−1}Mpc

^{−1}is the Hubble expansion rate. In the limit of circular GW-driven SMBHBs the ‘characteristic strain’, \(h_{c}(f)\), associated with a GWB energy density \(\varOmega_{\mathrm{gw}}(f)\) is parametrized as a single power-law:

^{−1}. Finally, \(h_{c}\) is directly related to the observable quantity induced by a GWB in the timing residuals, the one-sided power spectral density, \(S(f)\), given by:

Therefore, in the case of a stochastic GWB, the signal is described by a stochastic red process with power spectral density given by Eq. (26). This can be described by a vector of two parameters \(\overrightarrow{\eta }=(A,\gamma )\), where \(\gamma =-13/3\) for circular GW-driven SMBHBs. We will see below that this power has a very specific correlations between pairs of pulsars, which is the distinctive signature that has to be searched in PTA data.

### 3.3 Current Status: Upper Limits from Pulsar Timing Arrays

Here \(n\) is the length of the vector \(\overrightarrow{\delta t} = \cup \delta {t}_{a}\) obtained by concatenating the individual pulsar TOA series \(\delta {t}_{a}\), \(m\) is the total number of parameters describing the timing model (see Sect. 1.2), and the matrix \(G\) is related to the design matrix (see van Haasteren and Levin 2013 for details). The variance-covariance matrix \(C\), in its more general version, contains contributions from the GWB and from the white and (in general) red noise: \(C = C_{gw}(\overrightarrow{ \eta }) + C_{wn} + C_{rn}\). We refer the reader to Lentati et al. (2015) for exact expressions of the noise variance-covariance matrix.

Despite of its complexity, Eq. (27) is conceptually not distant from the more familiar Eq. (15). In obtaining a timing solution one describes the ‘distance’ between the data and the model in terms of \(\chi^{2}\), and adopts as best solution the set of parameters minimizing this distance. The inverse of the \(\chi^{2}\) can be thought as a likelihood function, and the solution of the problem is the maximum likelihood point. Likewise, the multivariate Gaussian likelihood in Eq. (27) is a measure of how well the data is described by some parametric model for resolvable sources (\(\overrightarrow{ \lambda }\) parameters) and a stochastic GWB (\(\overrightarrow{\eta }\) parameters). Conversely, The timing model parameters—those obtained in Eq. (15)—are marginalized over using the design matrix \(G\). One can then find the set of model parameters that maximizes the likelihood, or compare the likelihoods for models with and without a specific GW source to decide whether the data favor the presence of a signal, etc. Note that, since the fingerprint of a stochastic GWB is hidden in the specific correlation pattern found in the residuals collected from an ensemble of sources, it is necessary to describe the correlation properties of the residual via the computation of the variance-covariance matrix \(C\). Although this implies a significant computational burden, it does not change the nature of the problem.

Likelihoods similar to Eq. (27) have been used to search for both (i) a deterministic single GW sources (e.g., particularly loud SMBHBs rising above the level of the stochastic GWB), or (ii) a stochastic GWB.

Here \(\zeta_{IJ}\) is the angle between the pulsars \(I\) and \(J\) on the sky and \(\varGamma (\zeta_{IJ})\) is the overlap reduction function, which represents the expected correlation between the TOAs given an isotropic stochastic GWB, and the \(\delta_{IJ}\) term accounts for the pulsar term for the autocorrelation.

^{−1}, as defined in Eq. (26). Current limits are \(A=3\times 10^{-15},1.5\times 10^{-15},10^{-15},1.7 \times 10^{-15}\) for EPTA (Lentati et al. 2015), NANOGrav (Arzoumanian et al. 2016), PPTA (Shannon et al. 2015) and IPTA (Verbiest et al. 2016) respectively. Note that, although the IPTA limit is not the most stringent one, it has not been obtained combining the most recent individual PTA datasets. Compared to the individual dataset used in the combination, the IPTA limit is a factor of \(\approx 2\) better, demonstrating the great potential of collaborating at a worldwide level. As shown in Fig. 9, these limits start to dig into the level expected by a cosmological population of SMBHBs, and a positive detection might be expected in the near future.

## 4 Summary and Future Prospects

Pulsars are unique astrophysical sources that combine a number of special properties, which converts them into effective cosmic laboratories for fundamental physics. After having provided the very first evidence for GWs (Taylor 1994), pulsars can also be used as GW detectors, probing a nHz-frequency range, where we expect the signals from supermassive binary black holes to occur.

In fact, following galaxy mergers, the SMBHs hosted at their centers pair together forming a binary system. Since galaxy mergers are observed to be frequent in the Universe, a large population of adiabatically inspiralling SMBHBs is expected to fill the nHz GW sky. Such low frequencies are inaccessible to ground and space based GW interferometers, but are in the reach of ongoing and forthcoming PTAs.

Current PTAs are already placing stringent limits to the maximum GW strain emitted by these systems, starting to pierce through the level predicted by current SMBH assembly models. As timing continues, PTA sensitivity naturally increases and progressively extends at lower frequencies (where the signal is expected to be the loudest). Moreover, new instrumentation will play a pivotal role in the coming years. MeerKAT in South Africa and FAST in China will provide even better timing precision on a larger ensemble of pulsars, and by the mid 20s’ we will be surveying thousands of MSPs throughout the galaxy with SKA.

With these premises, it is likely that nHz GW detection will be likely within the next decade, placing another milestone in our understanding of the cosmic evolution of the biggest compact objects in the Universe.

But the clock-like properties of pulsars, and their accessibility across the whole electromagnetic spectrum, also open the door for a wide rage of practical applications. Here we have especially discussed spacecraft navigation.

The knowledge of how to use stars, planets and stellar constellations for navigation was fundamental for mankind in discovering new continents and subduing living space in ancient times. It is fascinating to see how history repeats itself in that a special population of stars may play again a fundamental role in the future of mankind by providing a reference for navigating their spaceships through the Universe.

Autonomous spacecraft navigating with pulsars is feasible when using either phased-array radio antennas of at least 150 m^{2} antenna area or compact light-weighted X-ray telescopes and detectors, which are currently developed for the next generation of X-ray observatories.

Using the X-ray signals from MSPs, Becker et al. (2015) estimated that navigation would be possible with an accuracy of \(\pm 5\) km in the solar system and beyond. The uncertainty is dominated by the inaccuracy of the X-ray pulse profile templates that were used for the pulse peak fittings and pulse-TOA measurements. As those are known with much higher accuracy in the radio band, it is possible to increase the accuracy of pulsar navigation down to the meter scale by using radio signals from pulsars for navigation.

The disadvantage of radio observations in a navigation application, though, is the large size and mass of the phased-antenna array. The antenna area is inversely proportional to the square root of the integration time; i.e., the same signal quality can be obtained with a reduced antenna size by increasing the observation time. However, the observing time is limited by the Allen variance of the receiving system and, therefore, cannot become arbitrarily large. In addition, irradiation from the on-board electronics requires an efficient electromagnetic shielding to prevent signal feedback. This shielding will further increase the navigator weight in addition to the weight of the antenna.

The optimal choice of the observing band depends on the boundary conditions given by a specific mission. What power consumption and what navigator weight might be allowed for may determine the choice for a specific wave band.

In general, however, it is clear already today that this navigation technique will find its applications in future astronautics. The technique behind it is very simple and straightforward, and pulsars are available everywhere in the Galaxy. Today \(\approx 2500\) pulsars are known. With the next generation of radio observatories, like the SKA, it is expected to detect signals from about 20 000 to 30 000 pulsars (Smits et al. 2009).

Finally, pulsar-based navigation systems can operate autonomously. This is one of their most important advantages, and is interesting also for current space technologies; e.g., as augmentation of existing GPS/Galileo satellites. Future applications of this autonomous navigation technique might be on planetary exploration missions and on manned missions to Mars or beyond.

## Notes

### Acknowledgement

Open access funding provided by Max Planck Society. A.S. is supported by a University Research Fellowship of the Royal Society.

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