# Pulsars and neutron stars/Searching for gravitational waves
## Introduction

Einstein's theory of general relativity predicts the existence of gravitational waves (GWs) - ripples in the fabric of space and time. Such waves exist - they were confirmed by observing the binary pulsar PSR B1913+16 (Hulse & Taylor 1975), but most astronomers consider this an "indirect detection" of GWs. Using bar detectors, space-craft tracking, ground-based interferometers and now, as discussed here, pulsars, scientists have been attempting to make a "direct detection" of GWs.
GWs are described by an amplitude, frequency (or wavelength) and polarization state. In contrast to electromagnetic waves, the polarizations of a GW are at 45 degrees (see animations to the right). GWs, according to general relativity, travel at the speed of light.
It was realised during the late 1970s that pulsar timing observations could be used to search for very low frequency ($f\sim 10^{-9}\rightarrow 10^{-7}{\rm {Hz}}$) GWs. The mathematical formalism was laid out by Sazhin (1979) and Detweiler (1979) who showed that (Detweiler 1979):
The strength of the induced timing residuals is described below in detail. However, the induced timing residuals are small. It is only with recent high-precision timing experiments with the world's largest radio telescopes that pulsar astronomers are close to detecting such waves. The most likely sources of GWs detectable by pulsar timing are supermassive binary black holes in the centre of merging galaxies. It is also possible that GWs may be detectable from the inflationary era of the Universe or from cosmic strings.
Observations of a single pulsar can be used to constrain the amplitude of GW signals, but as pulsar timing residuals already exhibit unexplained signatures (such as "timing noise") it would not be possible to unambiguously identify a GW signal from the timing residuals of a single pulsar. Foster & Backer (1990) introduced the concept of a "pulsar timing array" in which the search for GWs is carried out by timing a relatively large number of millisecond pulsars and looking for the GW evidence in all the data sets. Currently almost all of the major pulsar observatories carry out pulsar timing array research. The data are shared and processed as part of the International Pulsar Timing Array (IPTA), but, to date, no GW detection has been made.

## How gravitational waves induce pulsar timing residuals

A GW will induce a shift in the observed pulse frequency, $\delta \nu$, of:
${\frac {\delta \nu }{\nu }}=-H^{ij}\left[h_{ij}(t_{e},x_{e}^{i})-h_{ij}(t_{e}-D/c,x_{p}^{i})\right]$
where $H^{ij}$ is a geometrical term that depends upon the position of the GW source, the Earth and the pulsar, $h_{ij}(t,x)$ is the GW strain evaluated at time $t$ and position $x$. The strain evaluated at the current time at the Earth is known as the "Earth term". The strain evaluated at the pulsar at the time that the pulse was emitted is known as the "Pulsar term". Clearly, for multiple pulsars the Earth term will be the same, but the geometrical factor and the pulsar term will be different. The geometrical factor is given in Hobbs et al. (2010) and Lee et al. (2011).
The induced timing residuals for a given pulsar are:
$R(t)=-\int _{0}^{t}{\frac {\delta \nu }{\nu }}dt$
Assuming general relativity we can describe the GW in terms of two polarisation states and write the Earth term as:
$R_{\rm {Earth}}(t)=\int _{0}^{t}{\frac {P_{+}A_{+}(t)+P_{\times }A_{\times }(t)}{2(1-\gamma )}}dt$
where $\gamma$ is the GW-Earth-pulsar angle. For a non-evolving GW source, the $A_{+,\times }$ terms are given by
$A_{+,\times }=A_{+,\times }e^{i\omega _{g}t}$.
The pulsar term is the same as the Earth term apart from an extra phase.
It is likely that a background of many GWs will be present. We can therefore sum the equations above over many individual GW sources. The pulsar terms will still be uncorrelated, but the Earth terms will, for an isotropic, stochastic, unpolarised background, lead to a well-defined angular correlation. This was first derived by Hellings & Downs (1983) and is often called the "Hellings and Downs curve". It is given by
$c(\theta )={\frac {3}{2}}x\ln x-{\frac {x}{4}}+{\frac {1}{2}}+{\frac {1}{2}}\delta (x)$
where $x=[1-\cos \theta ]/2$ for angle $\theta$ on the sky between two pulsars. This is shown graphically in the Figure.
Pulsar timing array experiments are currently searching for the signature of either an individual GW source or for the GW background.

### More terminology

The terminology used to describe gravitational waves and the sensitivity of gravitational wave detectors is confusing. The table below provides some of the common terms:
| Symbol | Term | Synonymous terms | Description | Relationship |
| $h$ | rms strain amplitude | $h_{s}$, effective strain amplitude | The rms strain amplitude of a source averaged over orientation (or equivalently, sky position) and summed over polarisations | |
| $h_{c}$ | characteristic strain | dimensionless strain, strain | Expected strain amplitude per logarithmic frequency | |
| $S_{h}$ | strain spectral density | | For a stochastic, isotropic unpolarised gravitational wave background the spectral density is enough to describe the strain tensor at the Earth | $h_{c}(f)={\sqrt {fS_{h}(f)}}$ |
| $h_{n}$ | Characteristic noise strain | | Characteristic strain corresponding to the sensitivity of a detector | |
| $S_{n}$ | strain-equivalent noise spectral density | Strain noise amplitude spectral density, equivalent strain rms PSD | The sensitivity of various detectors are defined by the power spectral density of their noise | $h_{n}(f)={\sqrt {fS_{h}(f)}}$ |
| $A$ | gravitational wave background amplitude | | The characteristic strain for a background described by a power law can be defined by an amplitude, $A$ and spectral exponent $alpha$ | $h_{c}(f)=A\left({\frac {f}{f_{\rm {1yr}}}}\right)^{\alpha }$ |
| $\Omega _{GW}$ | | | Fractional energy density in gravitational waves per logarithmic frequency interval | $\Omega _{g}(f)={\frac {2\pi ^{2}}{3H_{0}^{2}}}f^{3}S_{h}(f)$ |
| $h_{rss}$ | Strain per root Hz | Strain-equivalent noise, root spectral density per root Hz, root-sum-square strain, strain sensitivity | Square root of either the strain spectral density or the strain-equivalent noise spectral density, depending on the context | $h_{rss}={\sqrt {S_{h}}}$, $h_{rss}={\sqrt {S_{n}}}$ |

## Single sources

Peters (1964) analysed the emission of GWs from two point masses and provided equations for the decay of the orbit caused by GWs and the lifetime of any given system. The GWs will be emitted at twice the orbital frequency. For an inspiralling circular binary of component masses $m_{1}$ and $m_{2}$ the GW strain amplitude is given by:
$h_{0}=2{\frac {(GM_{c})^{5/3}}{c^{4}}}{\frac {(\pi f)^{2/3}}{d_{L}}}$
where $d_{L}$ is the luminosity distance of the source and $M_{c}$ is the chirp mass:
$M_{c}={\frac {(m_{1}m_{2})^{3/5}}{(m_{1}+m_{2})^{1/5}}}$
An estimate of the induced timing residuals caused by a binary system can be obtained from Jenet et al. (2009):
$t\sim 10{\rm {ns}}\left({\frac {1{\rm {Gpc}}}{d}}\right)\left({\frac {M}{10^{9}{\rm {M}}_{\odot }}}\right)^{5/3}\left({\frac {10^{-7}{\rm {Hz}}}{f}}\right)^{1/3}$
where $d$ is the distance to the system which has a total mass of $M/(1+z)$ and emits GWs at frequency $f$.
For black holes in a circular orbit with a separation of 0.01pc then the time to coalescence will depend on the black hole masses. If both black holes have masses of $10^{10}{\rm {M}}_{\odot }$ then the time scale for merging will be only approximately 3 years. For such a system the frequency of the GWs detected on Earth will significantly change over typical data spans. For masses of $\sim 10^{9}{\rm {M}}_{\odot }$ the time scale is around 3000 years. The GW frequency observed at Earth will not significantly change during the span of observations, but the signal at the pulsar and at the Earth will have a different frequency. For masses of $10^{8}{\rm {M}}_{\odot }$ the merging time scale is $>10^{6}{\rm {yr}}$ and no evolution will be observed during the data span or in the light travel time to the pulsars.

### 3C66B

Sudou et al. (2003) presented evidence for a supermassive black hole binary system in the radio galaxy 3C 66B. They provided estimates of the system's total mass ($M=5\times 10^{10}{\rm {M}}_{\odot }$), the orbital period (~1 year) and the distance (~85 Mpc). Jenet et al. (2004) showed that such a system would be emitting GWs that would easily be detectable in the timing residuals of existing pulsars (they considered PSR B1857+09). As the induced timing residuals are not observed the system, as proposed by Sudou et al., was ruled out with high confidence. This was, arguably, the first major astrophysical result based on GW astronomy.

### General searches for individual sources of GWs

Lommen & Backer (2001) unsuccessfully searched for GW emission from Sagittarius A* which had been proposed to be a binary system. They also carried out the first search for GWs from other nearby galaxies. Yardley et al. (2010) used data from the Parkes telescope to calculate a sensitivity curve (as a function of the GW frequency) to such individual GW sources. Zhu et al. (2014) updated this work with longer Parkes data and obtained an upper limit on the GW strain amplitude of
$h_{0}<1.7\times 10^{-14}$ at 10 nHz.
This bound leads to constraints on the local merger rate density of supermassive binary black holes.

### Bursts

Sources of potentially detectable burst GW emission include 1) the formation of supermassive black holes, 2) highly eccentric supermassive black hole binaries, 3) close encounters of massive objects and 4) cosmic string cusps.

#### Bursts with memory

Seto (2009), van Haasteren & Levin (2010), Pshirkov, Baskaran & Postnov (2010), and Pollney & Reisswig (2011) have considered GW bursts "with memory". Such events lead to a permanent distortion in the spacetime metric. They can be caused during mergers of supermassive binary black holes. A GW memory event will lead to a step change in the pulse frequency of all pulsars. For a single pulsar the timing residuals will have the form of a glitch event (but with either sign) that does not decay. With a pulsar timing array, it is therefore possible to search for GW memory events passing the Earth by searching for glitch events that occur simultaneously in the timing of all of the pulsars in the array. (Bounds on the existence of such events can also be placed by limiting the size of glitches in an individual data set; this limits both the GW memory events at the Earth and at the pulsar).
The GW memory signal can be modelled as a step-function:
$h_{+}(t)=h^{\rm {mem}}\Theta (t-t_{0}),h_{x}(t)=0$
where $t_{0}$ is the time the GW memory signal reaches the observer on the Earth (the choice of the entire signal being in the plus polarisation state is discussed in Favata 2009). The function, $\Theta (t)$ is the Heaviside step function.
Cordes & Jenet (2012) provide a simple way to obtain an order-of-magnitude estimation of the signal strength:
$h^{\rm {mem}}\sim 5\times 10^{-16}(\mu /10^{8}{\rm {M}}_{\odot })(1{\rm {{Gpc}/D)}}$
where $D$ is the distance to the binary black hole system and $\mu$ is its reduced mass.
Searches for bursts with memory have been carried out by Arzoumanian et al. (2015) and Wang et al. (2015) and discussed by Madison et al. (2014). Wang et al. (2015) used Parkes observations to search (and limit) GW memory events passing the Earth. No detection was made, but, as shown in the paper, this was not unexpected. The conclusion in their paper is that it is unlikely that GW memory events will be detected in the near future. However, the predictions are uncertain and so it makes sense to keep looking - just in case!

## The gravitational wave background
### Definitions

A stochastic background of gravitational waves can be cosmological (e.g., from inflation, cosmic strings or phase transitions) or astrophysical (e.g., caused by coalescing massive black hole binary systems that result from mergers of their host galaxies).
The simplest definition of a GW background is in terms of its energy density per logarithmic frequency,
$E_{\rm {GW}}={\frac {d\rho _{\rm {GW}}}{d\log f}}=f{\frac {d\rho _{\rm {GW}}}{df}},$
as a fraction of the critical energy density of the Universe,
$\rho _{c}=3H_{0}^{2}c^{2}/(8\pi G).$
Here, $G$ is Newton's gravitational constant, $f$ is the received GW frequency at the Earth and $H_{0}=h\times (100)$km s$^{-1}{\rm {Mpc}}^{-1}$ is the Hubble constant. This `little $h$' is sometimes written as $h_{\rm {100}}$ or $h_{0}$. The fraction GW energy density, $\Omega _{\rm {GW}}$, is then:
$\Omega _{\rm {GW}}={\frac {8\pi G}{3H_{0}^{2}c^{2}}}{\frac {d\rho _{\rm {GW}}}{d\log f}}.$
As PTA measurements effectively constrain $E_{\rm {GW}}$ rather than $\Omega _{\rm {GW}}$, constraints on $\Omega _{\rm {GW}}$ can be written in terms of $\Omega _{\rm {GW}}h^{2}$, which is independent of the exact (uncertain) value of $H_{0}$.
Another useful definition of the gravitational wave background is the characteristic strain amplitude over a logarithmic frequency interval:
$h_{c}(f)={\sqrt {fS_{h}(f)}}.$
It is common to define:
$h_{c}(f)=A\left({\frac {f}{f_{1{\rm {yr}}}}}\right)^{\alpha }$
where $f_{1{\rm {yr}}}=1/(1{\rm {yr}})$. The spectral exponent $\alpha =-2/3$, $-1$ and$-7/6$ for likely GW backgrounds caused by coalescing black hole binaries, cosmic strings and relic GWs respectively. This gives:
$\Omega _{\rm {GW}}(f)={\frac {2}{3}}{\frac {\pi ^{2}}{H_{0}^{2}}}A^{2}f^{2\alpha +2}f_{1yr}^{-2\alpha }.$
Most of the earliest limits assumed that $\alpha =-1$ implying that $\Omega _{GW}$ has a flat spectrum. More recent limits have considered a wider range of $\alpha$ values.

### The supermassive binary black hole background

Standard models for galaxy formation and evolution are based on the hierarchical galaxy formation model. In this model, black holes exist at the centres of galaxies. The galaxies merge and the black holes coalesce. Burke-Spolaor (2011) provides a review of the literature which describes how a galaxy pair first becomes virially bound and then the massive black holes become centralised via dynamical friction. For a sufficiently close system, GWs will finally become the dominant mechanism for energy loss and, ultimately, the coalescence of the two black holes. The GWs emitted from a large number of such coalescing systems form the GW background that pulsar astronomers are searching for.
Even though plenty of observational evidence already exists for individual supermassive binary black holes and for merging galaxies, there is no direct evidence for massive binary black hole systems that are sufficiently close to emit strong GWs. Rodriguez et al. (2006) discovered a system in the radio galaxy 0402+379 (4C+37.11), which has a projected separation between the positions of the postulated black holes of just 7.3 pc. However, this is still too wide to be emitting detectable GWs.
The lack of known binary black hole systems that emit GWs is not too surprising. Standard survey techniques do not provide sufficient resolution to probe parsec-scales at the centres of distant galaxies and there is no known identifiable electromagnetic radiation signal that would lead to an unambiguous detection of a black hole system. It is therefore necessary to predict the GW background signal by attempting to model the black hole properties and their merger rates.

#### Modelling the GW background

Various papers have been written on predicting the GW background signal by calculating the coalescence rate of supermassive binary black holes of various masses at various redshifts (for instance, Rajagopal & Romani 1995; Jaffe & Backer 2003; Wyithe & Loeb 2003, Enoki et al. 2004, Sesana et al. 2008, Sesana & Vecchio 2010, Ravi et al. 2012, Sesana 2013, Ravi et al. 2014, Ravi et al. 2015). The majority of these papers assume that the black holes are in circular orbits and evolve purely from the GW emission.

#### Bounding the GW background

The most constraining limit to date on the background was published by Shannon et al. (2015). They provided an upper bound on A of:
$A<10^{-15}$ with 95% confidence.
Previous limits on this type of background are:
| Reference | Limit |
| Shannon et al. (2015) | $10^{-15}$ (95% confidence) |
| Lentati et al. (2015) | $3\times 10^{-15}$ (95% confidence) |
| Shannon et al. (2013) | $2.4\times 10^{-15}$ (95% confidence) |
| Demorest et al. (2013) | $7\times 10^{-15}$ (95% confidence) |
| Jenet et al. (2006) | $11\times 10^{-15}$ |

##### Implications of the bounds on the GW background
#### Towards detecting the GW background

Jenet et al. (2005) provided a method for detecting a stochastic background of gravitational waves. This work indicated that regular timing observations of 40 pulsars each with a timing accuracy of 100 ns will be able to make a direct detection of the predicted stochastic background from coalescing black holes within 5 years. With an improved prewhitening algorithm, or if the background is at the upper end of the predicted range, a significant detection should be possible with only 20 pulsars.

### Background from the inflationary era
## The timing arrays

Most large radio telescopes that can observe pulsars are currently part of a pulsar timing array (PTA) project. In 2004 the Parkes Pulsar Timing Array (PPTA) started observing enough pulsars with sufficient sensitivity to attempt a detection of GWs. The North American PTA (NANOGrav) and the European equivalent (EPTA) started soon after. Currently the following major PTAs exist:
* The European Pulsar Timing Array (EPTA; Janssen et al. 2008): combines data from five European telescopes (Effelsberg, Jodrell Bank, Nançay, Westerbork and Sardinia).
* The Parkes Pulsar Timing Array (PPTA; Manchester et al. 2013) uses observations from the Parkes radio telescope
* NANOGrav (Jenet et al. 2009) observes with the Arecibo and Green Bank telescopes.
The three PTAs share data and resources under the auspices of the International Pulsar Timing Array (IPTA) project. Details of the IPTA project are given in Manchester et al. (2013) and Hobbs et al. (2010).

## Frequently Asked Questions
### Gravitational waves versus gravity waves

For simplicity we often talk about "gravity waves". However, gravity waves are waves generated in a fluid when the force of gravity tries to restore equilibrium. These are not the same as the "gravitational waves" discussed here.