Anisotropies in the stochastic gravitational-wave background:Formalism and the cosmic string case

# Anisotropies in the stochastic gravitational-wave background: Formalism and the cosmic string case

## Abstract

We develop a powerful analytical formalism for calculating the energy density of the stochastic gravitational wave background, including a full description of its anisotropies. This is completely general, and can be applied to any astrophysical or cosmological source. As an example, we apply these tools to the case of a network of Nambu-Goto cosmic strings. We find that the angular spectrum of the anisotropies is relatively insensitive to the choice of model for the string network, but very sensitive to the value of the string tension .

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## I Introduction

The direct detection of gravitational waves (GW) from binary black hole mergers Abbott et al. (2016a, b, 2017a, 2017b, 2017c) and from a binary neutron star merger Abbott et al. (2017d) has opened a new window to the Universe. Gravitational waves offer a powerful tool for understanding the early stages of the Universe, particularly the prerecombination era that is inaccessible to conventional (electromagnetic) astronomy. Apart from the events so far detected by the LIGO and Virgo collaborations, we expect many more which are too distant to be individually detected. These quieter events, produced by many weak, independent and unresolved sources, constitute the stochastic GW background (SGWB). A variety of sources may lead to a SGWB, such as compact binary mergers, cosmic strings Vachaspati and Vilenkin (1985); Sakellariadou (1990) or phase transitions in the early Universe Binetruy et al. (2012), while at much higher redshifts one expects a contribution from a cosmological background due to a mechanism such as inflation.

Gravitational wave sources with an inhomogeneous spatial distribution lead to a SGWB characterized by preferred directions, and hence anisotropies. The main contribution to such an anisotropic background comes from astrophysical sources (such as compact binaries) that follow the local distribution of matter. The finiteness of the GW sources and the nature of the spacetime along the line of propagation of GWs will also contribute to anisotropies in the SGWB. The aim of this work is to develop a formalism for anisotropies in the SGWB of any astrophysical or cosmological source, and then apply it to the case of GWs sourced by cosmic string networks.

Our study is divided into two parts. In Sec. II, we follow the formalism presented in Ref. Cusin et al. (2017), which we develop further in order to derive a general expression for anisotropies in the SGWB, written in a form consistent with the usual GW literature. In addition, we derive a simple condition for the SGWB to be a Gaussian random field (GRF), and make a clear distinction between background and foreground sources in order to calculate the background in an unbiased way. We compute the kinematic dipole, which must be subtracted since it interferes with the anisotropy statistics. Finally, we show how to relate our results to future observational work. In Sec. III, we apply this formalism to the case of cosmic string networks. In particular, we study gravitational waves emitted from cusps, kinks and kink-kink collisions for three analytic models of Nambu-Goto string networks Vilenkin and Shellard (2000); Blanco-Pillado et al. (2014); Lorenz et al. (2010).

## Ii General formalism

Consider a Friedman-Lemaître-Robertson-Walker (FLRW) spacetime with scalar perturbations,

 \dds2=a2\qty[−\qty(1+2ψ)\ddη2+\qty(1−2ϕ)\dd\vb∗x\vdot\dd\vb∗x], (1)

where is the scale factor, denotes conformal time and , are the two Bardeen potentials, decomposed as , respectively. Using units with , setting , and keeping only linear order perturbations, the energy density of GWs with observed frequency arriving from a solid angle centered on the direction , is given in Ref. Cusin et al. (2017) as

 \dd[3]ρgw\ddνo\dd[2]σo\qty(νo,\vu∗eo)=14\uppi∫ηo0\ddηa4∫\dd\vb∗ζ¯nLs[1+δn−3\qty(Ψo+Πo)+4\qty(Ψ+Π)+\vu∗eo\vdot\qty(3\vb∗vo−2\vb∗v)+6∫ηoη\ddη′\pdvΨη′], (2)

with “s” and “o” subscripts indicating quantities evaluated at the GW source and at the observer, respectively, and with the integral along the line of sight, . Note that stands for the peculiar 3-velocity of the cosmic fluid. Here, is the gravitational luminosity at emitted frequency of a source with parameters , with the emitted frequency given in terms of the observed frequency by

 νs=νoa[1+Ψo+Πo−Ψ−Π+\vu∗eo\vdot\qty(\vb∗v−\vb∗vo)−2∫ηoη\ddη′\pdvΨη′]. (3)

We also define as the source number density—per physical volume, not comoving volume—with homogeneous background value . The number density inhomogeneities are expressed in terms of the density contrast

 δn\qty(η,\vu∗eo,\vb∗ζ)≡n−¯n¯n, (4)

so that .

In order to express Eq. (2) in a form consistent with the SGWB literature, we change from linear to logarithmic frequency, and normalize with respect to the critical density , giving the density parameter,

 Ωgw\qty(νo,\vu∗eo)≡1ρc\dd[3]ρgw\dd(lnνo)\dd[2]σo=8\uppiGνo3H2o\dd[3]ρgw\ddνo\dd[2]σo. (5)

Thus, using the above definition, the dimensionless quantity expressing the intensity of a stochastic background of gravitational waves, in the context of a FLRW universe with scalar perturbations, is

 Ωgw\qty(νo,\vu∗eo)=2Gνo3H2o∫ηo0\ddηa4∫\dd\vb∗ζ¯nLs[1+δn−3\qty(Ψo+Πo)+4\qty(Ψ+Π)+\vu∗eo\vdot\qty(3\vb∗vo−2\vb∗v)+6∫ηoη\ddη′\pdvΨη′]. (6)

We decompose this in terms of the isotropic monopole term and the GW energy density contrast , giving

 Ωgw≡¯Ωgw\qty(1+δgw). (7)

This definition implicitly takes the average of over the celestial sphere as zero, so we must choose a gauge in which the spatial average of the cosmological potentials is also zero (the spatial average of the density contrast is zero by definition). Note that corresponds to the average GW flux at frequency per unit solid angle, so that the total flux at this frequency is . This factor of must be taken into account when comparing our results with isotropic models of the SGWB, as the latter are usually expressed in terms of the total flux.

### ii.1 Relating strain and luminosity

The gravitational luminosity of any astrophysical or cosmological source that emits a series of GW signals can be decomposed as

 Ls\qty(νs,\vb∗ζ)=\dvEsνsR\qty(\vb∗ζ), (8)

where is the total energy lost from the system due to each signal, and is the rate at which the signals are emitted (i.e. the product is the signal rate per unit physical volume, per unit source time).2 We compute as a function of the GW strain by integrating the solid angle over a spherical surface of radius centered on the source, where is large enough to use linearized general relativity on a Minkowski background, but small enough to neglect cosmological effects. We thus obtain

 Es =132\uppiG∫S2\dd[2]σsr\mathrlap2s∫+∞−∞\ddts\pdvhTTijts\pdvhTTijts =132\uppiG∫S2\dd[2]σsr\mathrlap2s∫+∞−∞\ddts∑A=+,×(\pdvhAts)2,

where is the strain in the transverse traceless (TT) gauge, with “plus” and “cross” mode amplitudes , and are the coördinates of the local Minkowski metric Maggiore (2007). Writing the strain in terms of its Fourier transform ,

and using (since is real), we find

 Es=\uppi4G∫S2\dd[2]σsr\mathrlap2s∫+∞0\ddνsν2s∑A=+,×\qty|~hA\qty(νs)|2,

which we have written in terms of positive frequencies only (i.e. this is a one-sided spectrum). Since in what follows we are not interested in polarization effects, we can simplify the above expression by defining the total strain magnitude

 ~h≡√|~h+|2+|~h×|22. (9)

Rewriting in terms of and using the definition in Eq. (8), the luminosity spectrum of a single source is therefore given by

 Ls\qty(νs,\vb∗ζ)=\uppiν2sR\qty(\vb∗ζ)2G∫S2\dd[2]σsr\mathrlap2s~h2. (10)

Using Eq. (6) and Eq. (10), the density parameter is given, to linear order, by

 (11)

where we have used Eq. (3) to convert the source-frame frequency in Eq. (10) to the corresponding observer-frame frequency. Note that this changes the linear perturbation terms—in particular, there is no net contribution from the source’s peculiar motion, only from that of the observer.

### ii.2 Gaussian and non-Gaussian backgrounds

Analyzing the anisotropic statistics of the SGWB is greatly simplified if is a Gaussian random field (GRF). In particular, Wick’s theorem tells us that if the field is Gaussian, we can fully characterize its anisotropies in terms of the mean and the two-point correlation function (2PCF), as defined in Sec. II.4. It is also convenient from the point of view of GW data analysis if the detector strain associated with the SGWB is a Gaussian process, as this gives a simple likelihood function for the strain Romano and Cornish (2017). Current LIGO/Virgo searches for the SGWB exploit this fact, and use pipelines optimized for Gaussian backgrounds, though we note that search methods for non-Gaussian backgrounds do exist (see e.g. Ref. Smith and Thrane (2018)).

However, one must be careful when speaking of a “Gaussian background”, as it is not clear a priori that being a Gaussian process implies that is a GRF, or vice versa. In this section we use a simple model of a background composed of independent GW bursts to derive sufficiency conditions for Gaussianity of each of the relevant quantities: first, we reproduce the standard condition that gives a Gaussian strain ; then we find a different condition that makes the isotropic energy density Gaussian; and finally we extend this to find a condition for the energy density as a function of sky location to be a GRF, given some angular resolution .

#### A simple model of an incoherent SGWB

Suppose we observe the SGWB over a time interval . It can then be written in terms of a discrete set of frequencies , where . Let us focus on the signal in a single frequency bin centered on (with width set by the observation time). Since we are considering a background composed of many independent transient bursts, we write the complex GW strain at this frequency as the sum of bursts

 h\qty(t)=N∑i=1hi\qty(t),hi\qty(t)≡{Aiei\qty(2\uppiνt+ϕi)if ti≤t≤ti+\upDeltat,0else, (12)

where the (real) amplitudes , times of arrival , and initial phases of the bursts are all random variables. We take the amplitudes as being independent and identically distributed (i.i.d.) according to some unknown probability distribution that depends on the frequency bin. The times of arrival are distributed according to a Poisson process with rate parameter (also dependent on frequency bin), while the phases are uniformly distributed on .

When we speak of “burst signals”, we mean signals whose duration is “short” in some sense. We can quantify this by saying that a burst lasts for no more than wavelengths in each frequency bin, so that its duration in a given frequency bin can be taken as . This is a good approximation for most burstlike signals (e.g. supernovae Buonanno et al. (2005) or cosmic string cusps and kinks Damour and Vilenkin (2001)). Specific GW sources will have different signal durations, but for the sources mentioned above, and for more general transient sources, this approximation is accurate to within an order of magnitude. This is not the case for GW signals from compact binary coalescences, where the duration in some small frequency interval is roughly during the inspiral phase (where is the chirp mass) Maggiore (2007). For a discussion of the Gaussianity of the stochastic background in this case, see Ref. Jenkins et al. (2018).

#### Conditions for h\qty(t) to be Gaussian

It is well known that for to be a Gaussian process, it is sufficient for the duty cycle to be much greater than unity. We define this quantity below and give a brief justification of this condition, using the simple model described above.

At any time , the observed GW strain due to the SGWB is the superposition of all the bursts with arrival times up to before the time , as each burst has a duration of . This means that is the sum of some number of i.i.d. random variables, and in the limit where this number is large is Gaussian by the central limit theorem.

Since the times of arrival are given by a Poisson process with rate , the total number of bursts will tend to in the limit where . (The expected number of bursts will always be . However, there will be random fluctuations around this value, which vanish only when .) For each of these bursts, there is a probability of roughly that they will arrive at a time between and , so the expected number of bursts contributing to the strain at time is . By the law of large numbers, the number of contributing bursts therefore converges to in the limit where . So for , the number of bursts in-band at time converges to . This quantity is called the duty cycle, . In order to ensure that is Gaussian, it is therefore sufficient to take and . In these limits, the fluctuations in the number of signals with respect to time are small, so if the signal is Gaussian at some time then it is Gaussian for the whole observing period .

For reasons discussed below, we usually only consider frequencies , so the limit implies that . We are therefore left with a single sufficiency condition for Gaussianity:

 Λ≫1⟹∀t∈[0,T],h\qty(t) is% Gaussian. (13)

When “Gaussian” GW backgrounds are discussed in the literature, this is usually what is meant. For studying anisotropies in the background, however, it is the density parameter that is important, rather than the strain.

#### Conditions for ¯Ωgw to be Gaussian

As we hinted at before, being Gaussian at frequency is not the same as the isotropic energy density being Gaussian. To see this, we express explicitly using

 ¯Ωgw=ν\upDeltaν148\uppiH2o\ev˙h˙h∗. (14)

Here is the frequency resolution. The factor of is equivalent to the derivative used in the continuum case . The angle brackets represent an average over many periods, as this is required to have a well-defined notion of “energy” for a GW. It is only possible to perform this average if we observe the SGWB for many periods, so we must have . Assuming this is the case, we use the decomposition Eq. (12) to find

 ¯Ωgw=\uppiν212H2o\qty[N∑i=1A2i+∑coincidentpairs {i,j}Bij], (15)

where . As well as the contribution due to the energy of each individual burst (the first sum in the expression above), we also have a contribution from cross-terms between coincident bursts (the second sum), whose times of arrival are within of each other. There are pairs of bursts, and probability of any pair of bursts overlapping is roughly , so by the law of large numbers, the number of coinciding pairs converges to when . As before, taking ensures that . The random variables and are i.i.d. for all bursts and for all coincident pairs respectively, so the central limit theorem guarantees that is Gaussian if and .

Putting all this together, we find that the conditions

 RT≫1,RT\qty(RT−1)/νT≫1, (16)

are sufficient for to be Gaussian at frequency . With some rearranging, we see that the second condition implies the first. Rewriting in terms of the duty cycle, we have

 νT≫1Λ+1Λ2⟹¯Ωgw\qty(ν) is Gaussian, (17)

where we only consider frequencies .

We see that is always Gaussian if . This means that being Gaussian implies that is Gaussian, but note that the converse does not hold. In fact, no matter how non-Gaussian is (i.e. no matter how small the duty cycle is), it is in principal possible to make Gaussian by increasing the observation time (the required observation time will depend entirely upon the duty cycle of the sources considered).

#### Conditions for Ωgw\qty(ν,\vu∗e) to be a GRF

The discussion thus far has been about the isotropic GW energy density, . To extend this to the angular distribution of this energy as a field on the sky, we divide the sphere into some number of pixels of equal size , and let be the energy density in GWs arriving from the pixel. If the background is statistically isotropic, then the probability of a given burst arriving from one particular pixel is . If the number of bursts is large, then the number arriving from the -th pixel converges to . Referring back to our discussion about Eq. (15), we see that for the number of bursts in a given pixel converges to , and the number of coincident pairs of bursts converges to . For the total energy in that pixel from each burst and from cross-terms to be Gaussian, it is therefore sufficient to have

 RT≫1,1νTRTNpix\qty(RTNpix−1)≫1. (18)

If this is the case, then all the pixels are Gaussian, and the SGWB is a Gaussian random field. As before, the second condition above implies the first, so we simplify to find that

 νT≫NpixΛ+N2pixΛ2⟹Ωgw\qty(ν,\vu∗eo) is a GRF, (19)

where we emphasize once more that we are only interested in frequencies . We can also eliminate in favour of the angular resolution to write this as

 νT≫4\uppiΛ\updeltaσ+16\uppi2Λ2\qty(\updeltaσ)2⟹Ωgw\qty(ν,\vu∗eo) is a GRF. (20)

In practice, we expect that it is only necessary for the LHS to be an order of magnitude larger than the RHS.

Equation (20) could potentially be a useful guide for the future observing strategies of advanced GW detectors. Given an estimate of the duty cycle of a particular background source in a given frequency bin, and given the angular resolution of the detector network, Eq. (20) specifies the requisite observing time to ensure that the field is Gaussian. (Note that this time need not correspond to one unbroken observing period; it will likely be necessary to combine many separate observing runs.) In principle, any background can be made to satisfy the criterion Eq. (20) at any angular resolution by increasing , so our treatment in Sec. II.4 and II.5 assumes that this criterion is met. In practice, it may be desirable to measure the integral of over some frequency interval much larger than the frequency bin size , as this would increase the integrated GW power and therefore make Gaussianity more achievable for shorter observing times.

We emphasize once again that Eq. (20) is only relevant for a stochastic background composed of GW bursts that decay after wavelengths in-band, such that their duration can be approximated by . In particular, it does not apply to the astrophysical background from compact binaries, due to the assumption about the GW burst duration—this case is addressed in Ref. Jenkins et al. (2018). It also does not apply to a background from continuous sources (or very long transients, lasting longer than the observation time). However, this case is somewhat simpler as there is a fixed number of continuous signals , whose distribution amongst the pixels does not vary with time. By a very similar argument to that given above, having continuous sources will ensure is Gaussian (by the central limit theorem), and having and ensures that is a GRF (as the number of signals and number of overlapping signals per pixel are then large enough for the central limit theorem to apply).

### ii.3 Separating background from foreground

Equation (11) includes all of the GW sources considered as part of the stochastic background. However, to calculate the SGWB in an unbiased way, one must be careful not to include any loud, rare, individually resolvable signals that make up the foreground 3—this was pointed out for the case of cosmic strings in Ref. Damour and Vilenkin (2001).

There has been some debate in the literature over what constitutes a “resolvable” signal. Arguably the most thorough approach is to decide this on a signal-by-signal basis with Bayesian model selection, as in Ref. Cornish and Romano (2015). For our purposes however, it is sufficient to distinguish between the two using the duty cycle . As above, this is defined as the average number of overlapping signals at frequency experienced by the observer Regimbau et al. (2012). For foreground signals we have , as the majority of the observation time contains no such signals (equivalently, the typical interval between these signals arriving is much greater than their duration). For the SGWB we have , as this background consists of a large number of superimposed signals (equivalently, the interval between signals that are part of the background is much shorter than their duration). We stress that this is a detector-independent (and therefore more general) way of defining what we mean by the “stochastic background”. There will be many GWs that are not resolved by the detector network but which have , and therefore could in principle be resolved by an idealized zero-noise detector; these might reasonably be described as “background signals”, but here we consider them part of the foreground.

Let denote the duty cycle for observed signals that are emitted from conformal time onward—i.e. the conformal time at emission obeys . Then we define the SGWB as all of the signals emitted at times , where is defined by

 Λ\qty(νo,η∗)≡1. (21)

We are thus excluding nearby sources whose combined duty cycle is less than unity, meaning that, on average, they do not overlap in time. The SGWB is what remains: a continuous signal composed of many objects at large distances . If for a given frequency there is no solution to the above equation, then we let ; this means that there are not enough sources at this frequency to constitute a background. Since we compute the duty cycle as an average quantity, we take as being the same in all directions on the sky.

Note that the duty cycle used in Sec. II.2 includes all signals that are part of the background, which is equal to the total duty cycle of all (background and foreground) signals, , minus the duty cycle of foreground signals, . When checking the Gaussianity of the background, the appropriate to use in Eq. (20) is therefore .

In order to calculate , we write

 Λ\qty(νo,η∗)=\upDeltat∫\dd\vb∗ζ∫ηoη∗\dd[3]V\qty(η)fo¯nR, (22)

where we define as the fraction of the emitted signals that are observable at frequency (this accounts for e.g. beaming effects and cutoffs in the frequency spectrum of the signal). So, is just the rate of arrival of observable signals originating at , multiplied by their duration . In principle we should allow to depend on and , but for burst signals such as those we consider in Sec. III we can make the simple assertion that  Regimbau et al. (2012).

We can write the physical volume element in our perturbed FLRW metric as

 \dd[3]V=\dd[2]σo\ddηa3r2\qty(1+Ψ+Π+\vu∗eo\vdot\vb∗v), (23)

where is the comoving distance measure, written in terms of the conformal time as

 r\qty(η)≡∫ηoη\ddη\qty(1+2Ψ). (24)

Integrating over solid angle averages out the cosmological perturbations, and hence the cutoff time is found by solving the integral equation

 4\uppiνo∫\dd\vb∗ζ∫ηoη∗\ddηa3\qty(ηo−η)2fo¯nR=1. (25)

We therefore modify the conformal time integral in our previously found linear-order expression Eq. (11) and get

 Ωgw\qty(νo,\vu∗eo)=\uppiν3o3H2o∫η∗0\ddηa2∫\dd\vb∗ζ¯nR[1+δn−Ψo−Πo+2\qty(Ψ+Π)+\vu∗eo\vdot\vb∗vo+2∫ηoη\ddη′\pdvΨη′]×∫S2\dd[2]σsr\mathrlap2s~h2. (26)

This expression Eq. (26) is the main result of our analysis; it can be used for any astrophysical or cosmological source of anisotropies in the stochastic background of gravitational waves.

### ii.4 Characterizing the anisotropies

We will initially focus on the anisotropy due to the source density contrast , and therefore neglect most of the cosmological perturbations. The only other term we include is the peculiar motion of the observer , as this introduces a “kinematic dipole” that interferes with the anisotropy statistics. In the case of the cosmic microwave background (CMB), this dipole is roughly 100 times greater than the “true” cosmological fluctuations we are interested in, so it is usually subtracted from the raw data before calculating any statistics. We will do the same for the SGWB.

There are two possible approaches to this: either measure the observed kinematic dipole of the SGWB directly at each frequency and subtract it, or use CMB data to measure the direction of the dipole, and use the formalism discussed above to generate a theoretical prediction for its magnitude. Since SGWB measurements are likely to be much less precise than CMB measurements in both overall magnitude and angular resolution for the foreseeable future, the latter seems to us the best approach.

Thus, setting everywhere and everywhere except at the observer, we have

 Ωgw\qty(νo,\vu∗eo)=\uppiν3o3H2o∫η∗0\ddηa2∫\dd\vb∗ζ¯nR×\qty(1+δn+\vu∗eo\vdot\vb∗vo)∫S2\dd[2]σsr\mathrlap2s~h2, (27)

with the emitted frequency given by

 νs=νoa\qty(1−\vu∗eo\vdot\vb∗vo). (28)

We thus see that the observer’s peculiar motion causes a Doppler shift in the observed frequencies for each source, which will vary in importance depending on the cosmological redshifts of the sources. This means that the magnitude of kinematic dipole will depend on the waveform and distance of every source that contributes to the SGWB, making the required calculation more complicated than that for the CMB dipole. We sketch here how to calculate the size of the dipole, with a more concrete treatment for the cosmic string case given in Sec. III.2.

As we are working only to linear order, we define

 x\qty(\vu∗eo)≡1+\vu∗eo\vdot\vb∗vo (29)

and express all modifications due to the kinematic dipole as powers of . This depends only on , and is therefore unaffected by the integrals over and . With reference to Eq. (7), we see that the averaged isotropic background value (monopole) is given by

 ¯Ωgw\qty(νo)≡14\uppi∫S2\dd[2]σoΩgw\qty(νo,\vu∗eo)=Ωgw∣∣x=1,δn=0 (30)

with the anisotropies described by the SGWB energy density contrast,

 δgw(νo,\vu∗eo)≡Ωgw−¯Ωgw¯Ωgw. (31)

The quantity we are interested in is the density contrast due to the source distribution alone, with the kinematic dipole subtracted. This is defined as

 δ\qtygw(s)(νo,\vu∗eo)≡δgw∣∣x=1=Ωgw∣∣x=1−¯Ωgw¯Ωgw (32)

where “s” stands for “source”. We can compute the linear-order correction due to the kinematic dipole with a Taylor expansion around ,

 Ωgw =Ωgw∣∣x=1+\vu∗eo\vdot\vb∗vo\pdvΩgwx∣∣x=1 =¯Ωgw\qty(1+δ\qtygw(s))+\vu∗eo\vdot\vb∗vo\pdvΩgwx∣∣x=1,δn=0,

where the latter equality holds because is second order. We therefore find

 δgw=δ\qtygw(s)+D\vu∗eo\vdot\vu∗vo,D≡vo¯Ω−1gw\pdvΩgwx∣∣x=1,δn=0, (33)

where , , and is a frequency-dependent coefficient describing the size of the kinematic dipole, which depends on the GW waveforms and spatial distribution of the sources. Note that this approach is only valid if ; otherwise we must go beyond the linear expansion.

Now we are able to study , either directly or in terms of its statistics. One particularly useful statistical descriptor is the two-point correlation function (2PCF), defined as the second moment of the density contrast,

 Cgw\qty(θo,νo)≡\evδ\qtygw(s)\qty(νo,\vu∗eo)δ\qtygw(s)(νo,\vu∗e\mathrlap′o), (34)

where , and the angle brackets denote an averaging over all pairs of directions , whose angle of separation is . The first moment (i.e. mean) vanishes by definition, and if the background is a GRF (as discussed in II.2) then all higher moments either vanish or are expressed in terms of the second moment by Wick’s theorem. The 2PCF therefore uniquely characterizes the anisotropies in the Gaussian part of the background. It is common practice (particularly in the CMB literature) to perform a multipole expansion of the 2PCF,

 Cgw\qty(θo,νo)=∞∑ℓ=02ℓ+14\uppiCℓ\qty(νo)Pℓ(cosθo), (35)

where denotes the Legendre polynomial. The anisotropies are then described in terms of the components, which are given by

 Cℓ\qty(νo)≡2\uppi∫+1−1\dd\qty(cosθo)Cgw\qty(θo,νo)Pℓ\qty(cosθo). (36)

The quantity is roughly the contribution to the variance of per logarithmic bin in , as can be seen by considering

 var\qty(δ(s)gw)=∑ℓ2ℓ+14\uppiCℓ≈∫\dd\qty(lnℓ)ℓ\qty(ℓ+1)2\uppiCℓ.

Defined in this way, the 2PCF excludes the kinematic dipole. The effects of including this on the components are described in the Appendix.

### ii.5 Estimating the 2PCF from observations

The decomposition of the 2PCF described above is not the only way of describing the SGWB anisotropies. Another convenient tool is the spherical harmonic decomposition of itself,

 Ωgw(νo,\vu∗eo)=∞∑ℓ=0+ℓ∑m=−ℓΩℓm(νo)Yℓm(\vu∗eo), (37)

where are the Laplace spherical harmonics, and

 Ωℓm(νo)≡∫S2\dd[2]σoΩgw(νo,\vu∗eo)Y\mathrlap∗ℓm(\vu∗eo). (38)

We can perform the same decomposition for ,

 δgw(νo,\vu∗eo)=∞∑ℓ=0+ℓ∑m=−ℓωℓm(νo)Yℓm(\vu∗eo),ωℓm(νo)≡∫S2\dd[2]σoδgw(νo,\vu∗eo)Y\mathrlap∗ℓm(\vu∗eo), (39)

with the components given in terms of the ’s by

 ωℓm=¯Ω−1gwΩℓm−√4\uppiδℓ0δm0. (40)

Here we have used the orthogonality condition for the spherical harmonics

 ∫S2\dd[2]σoYℓm\qty(\vu∗eo)Y\mathrlap∗ℓ′m′\qty(\vu∗eo)=δℓℓ′δmm′, (41)

and the fact that .

Since we are interested in the ’s of the source anisotropies , we want to remove the kinematic dipole from Eq. (40). Doing so inevitably involves a particular choice of coördinates . For simplicity, we take the direction of the kinematic dipole as the direction, so that

 δ(s)gw=δgw−Dcosθo. (42)

The dipole is then proportional to , so performing the decomposition,

 δ(s)gw(νo,\vu∗eo)=∞∑ℓ=0+ℓ∑m=−ℓω(s)ℓm(νo)Yℓm(\vu∗eo),ω(s)ℓm(νo)≡∫S2\dd[2]σoδ(s)gw(νo,\vu∗eo)Y\mathrlap∗ℓm(\vu∗eo), (43)

we see that Eq. (40) becomes

 ω(s)ℓm=¯Ω−1gwΩℓm−√4\uppiδℓ0δm0−√4\uppi3Dδℓ1δm0. (44)

The relationship between these spherical harmonic decompositions and the components can be found by writing

 Cgw ≡\evδ(s)gw(\vu∗eo)δ(s)gw(\vu∗e\mathrlap′o) =∞∑ℓ=0∞∑ℓ′=0+ℓ∑m=−ℓ+ℓ′∑m′=−ℓ′\evω(s)ℓmω(s)∗ℓ′m′Yℓm(\vu∗eo)Y\mathrlap∗ℓ′m′(\vu∗e\mathrlap′o) =∞∑ℓ=02ℓ+14\uppiCℓPℓ(\vu∗eo\vdot\vu∗e\mathrlap′o).

We require the RHS above to be invariant under rotations of the sphere, which implies that is proportional to . Using the addition theorem for spherical harmonics,

 +ℓ∑m=−ℓYℓm(\vu∗eo)Y\mathrlap∗ℓm(\vu∗e\mathrlap′o)=2ℓ+14\uppiPℓ(\vu∗eo\vdot\vu∗e\mathrlap′o), (45)

we therefore see that

 \evω(s)ℓmω(s)∗ℓ′m′=Cℓδℓℓ′δmm′, (46)

and thus

 Cℓ=12ℓ+1+ℓ∑m=−ℓ\evω(s)ℓmω(s)∗ℓm (47)

which directly relates the ’s to the ’s 4. Note that the angle brackets here indicate an ensemble average over random realizations of the field.

This expression shows that the components contain more information about each random realization of the SGWB than the ’s do. There is an averaging process (the angle brackets) that takes us from the ’s to the ’s (or, equivalently, from to ), so there must be many possible configurations of the field that all correspond to the same ’s but give different ’s. This means that we cannot invert the above equation and reconstruct in terms of the ’s alone.

With a view towards future observational work, we can relate the and components to the GW strain measured by the observer. This is given by

 (48)

where , are polarization tensors and , are the Fourier components of the background Maggiore (2000). The signal is often characterized by the quadratic expectation value of these Fourier components. For a SGWB that is unpolarized, Gaussian, and stationary (but still anisotropic), these expectation values can be written as Romano and Cornish (2017)