Understanding the gravitational-wave Hellings and Downs curve for pulsar timing arrays in terms of sound and electromagnetic waves
UUnderstanding the gravitational-wave Hellings and Downs curve forpulsar timing arrays in terms of sound and electromagnetic waves
Fredrick A. Jenet ∗ University of Texas at Brownsville, Department of Physics and Astronomy,and Center for Advanced Radio Astronomy, Brownsville, TX 78520
Joseph D. Romano † University of Texas at Brownsville, Department of Physics and Astronomy,and Center for Gravitational-Wave Astronomy, Brownsville, TX 78520 (Dated: March 30, 2015)Searches for stochastic gravitational-wave backgrounds using pulsar timing arrays look for cor-relations in the timing residuals induced by the background across the pulsars in the array. Thecorrelation signature of an isotropic, unpolarized gravitational-wave background predicted by gen-eral relativity follows the so-called
Hellings and Downs curve, which is a relatively simple function ofthe angle between a pair of Earth-pulsar baselines. In this paper, we give a pedagogical discussion ofthe Hellings and Downs curve for pulsar timing arrays, considering simpler analogous scenarios in-volving sound and electromagnetic waves. We calculate Hellings-and-Downs-type functions for thesetwo scenarios and develop a framework suitable for doing more general correlation calculations.
I. INTRODUCTION
A pulsar is a rapidly-rotating neutron star that emitsa beam of electromagnetic radiation (usually in the formof radio waves) from its magnetic poles. If the beam ofradiation crosses our line of sight, we see a flash of ra-diation, similar to that of a lighthouse beacon. Theseflashes can be thought of as ticks of a giant astronomicalclock, whose regularity rivals that of the best human-made atomic clocks. By precisely monitoring the arrivaltimes of the pulses, astronomers can determine: (i) in-trinisic properties of the pulsar—e.g., its rotational pe-riod and whether it is spinning up or spinning down; (ii)extrinsic properties of the pulsar—e.g., whether it is ina binary system, and if so what are its orbital parame-ters; and (iii) properties of the intervening ‘stuff’ betweenus and the pulsar—e.g., the column density of electronsin the interstellar medium.
Indeed, it was the precisemonitoring (for over 30 years) of the pulses from binarypulsar PSR B1913+16 that has given us the most com-pelling evidence to date for the existence of gravitationalwaves. The measured decrease in the orbital period ofbinary pulsar PSR B1913+16 agrees precisely with thepredictions of general relativity for the energy loss dueto gravitational-wave emission. (See Fig. 1.) This was apath-breaking result, with the discovery of the binarypulsar being worthy of a Nobel Prize in Physics forJoseph Taylor and Russell Hulse in 1993.Monitoring the gravitational-wave-induced decay of abinary system, like PSR B1913+16, is one method for de-tecting gravitational waves. Another method is to lookfor the effect of gravitational waves on the radio pulsesthat propagate from a pulsar to a radio antenna on Earth.The basic idea is that when a gravitational wave transitsthe Earth-pulsar line of sight, it creates a perturbationin the intervening spatial metric, causing a change inthe propagation time of the radio pulses emitted by the pulsar. (This is the timing response of the Earth-pulsarbaseline to a gravitational wave.) One can then comparethe measured and predicted times of arrival (TOAs) ofthe pulses, using timing models that take into accountthe various intrinsic and extrinsic properties of the pul-sar. Since standard timing models factor in only deter-ministic influences on the arrival times of the pulses, thedifference between the measured and predicted TOAs willresult in a stream of timing residuals , which encode theinfluence of both deterministic and stochastic (i.e., ran-dom) gravitational waves as well as any other randomnoise processes on the measurement. If one has a setof radio pulsars, i.e., a pulsar timing array (PTA), onecan correlate the residuals across pairs of Earth-pulsarbaselines, leveraging the common influence of a back-ground of gravitational waves against unwanted, uncorre-lated noise. The key property of a PTA is that the signalfrom a stochastic gravitational-wave background will becorrelated across the baselines, while that from the othernoise processes will not. This is what makes a PTA func-tion as a galactic-scale, gravitational-wave detector. For an isotropic, unpolarized stochastic background ofquadrupole gravitational radiation composed of the plus(+) and cross ( × ) polarization modes predicted by gen-eral relativity, the expected correlated response of a pairof Earth-pulsar baselines to the background follows theso-called Hellings and Downs curve, named after the twoauthors who first calculated it in 1983. A plot of theHellings and Downs curve as a function of the angle be-tween a pair of baselines is shown in Fig. 2. Searchesfor stochastic gravitational-wave backgrounds using pul-sar timing arrays effectively compare the measured cor-relations with the expected values from the Hellings andDowns curve to determine whether or not a signal froman isotropic, unpolarized background is present (or ab-sent) in the data. Gravitational-wave backgrounds pre-dicted by alternative theories of gravity, which have dif-ferent polarization modes, or backgrounds that have an a r X i v : . [ g r- q c ] M a r FIG. 1: Decrease in the orbital period of binary pulsar PSR B1913+16. The measured data points and error bars agree withthe prediction of general relativity (parabola) for the rate of orbital decay due to gravitational-wave emission. anisotropic distribution of gravitational-wave energy onthe sky, will induce different correlation signaturesand must be searched for accordingly. To date no detec-tions have been made, but upper limits on the strengthof the background have been set that constrain certainmodels of gravitational-wave backgrounds produced bythe inspirals of binary supermassive black holes (SMBHs)in merging galaxies throughout the universe.Mathematically, the Hellings and Downs curve is thesky-averaged and polarization-averaged product of theresponse of a pair of Earth-pulsar baselines to a planewave propagating in a particular direction with either + or × polarization. It has the analytic form χ ( ζ ) = 12 − (cid:18) − cos ζ (cid:19) + 32 (cid:18) − cos ζ (cid:19) ln (cid:18) − cos ζ (cid:19) , (1)where ζ is the angle between two Earth-pulsarbaselines. (See Fig. 3 for the Earth-pulsar baselinegeometry.) The integration that one must do in or-der to obtain the above expression is non-trivial enoughthat Hellings and Downs originally used the symbolicmanipulation computer system MACSYMA to do thecalculation. It turns out that is also possible to eval-uate the integral by hand, using contour integration for
Angle ζ between Earth−pulsar baselines (degrees) E x pe c t ed c o rr e l a t i on χ ( ζ ) FIG. 2: Hellings and Downs curve for the expected correlated response of a pair of Earth-pulsar baselines to an isotropic,unpolarized stochastic gravitational-wave background, plotted as a function of the angle between the baselines, cf. Eq. (1). part of the integration (see, e.g., Appendix A). But forsome reason, perhaps related to the difficulty of analyt-ically evaluating the sky integral, students or beginningresearchers who are first introduced to the Hellings andDowns curve see it as a somewhat mysterious object, inti-mately connected to the realm of pulsar timing. Granted,the precise analytic form in (1) is specific to the responseof a pair of Earth-pulsar baselines to an isotropic, un-polarized stochastic gravitational-wave background, butHellings-and-Downs-type functions show up in any sce-nario where one is interested in the dependence of thecorrelated response of a pair of receivers on the geomet-rical configuration of the two receivers. The geometry re-lating the configuration of one receiver to another mightbe more complicated (or simpler) than that for the pul-sar timing case, but the basic idea of correlation acrossreceivers is exactly the same.The purpose of this paper is to emphasize thiscommonality, and to calculate Hellings-and-Downs-typefunctions for two simpler scenarios. Scenario 1 will befor a pair of receivers constructed from omni-directionalmicrophones responding to an isotropic stochastic soundfield. Scenario 2 will be for a pair of receivers con-structed from electric dipole antennas responding toan isotropic and unpolarized stochastic electromagneticfield. These two scenarios were chosen since the deriva-tion of the corresponding Hellings-and-Downs-type func-tions (cf. Eqs. (27) and (48)) and the evaluation of thenecessary sky-integral and polarization averaging (for theelectromagnetic-wave case) are relatively simple. But thesteps that one must go through to obtain these results areidentical to those for the gravitational-wave pulsar tim-ing Hellings and Downs function, even though the math-ematics needed to derive the relevant expression for thepulsar timing case (cf. Eq. (58)) is more involved. Hope-fully, after reading this paper, the reader will understand the pulsar timing Hellings and Downs curve in its propercontext, and appreciate that it is a special case of a gen-eral correlation calculation.The rest of the paper is organized as follows: In Sec-tion II, we describe a general mathematical formalism forworking with random fields, which we will use repeat-edly in the following sections. In Section III we applythis formalism to calculate a Hellings-and-Downs-typefunction for the case of omni-directional microphones inan isotropic stochastic sound field. Section IV extendsthe calculation to electric dipole antennas in an isotropicand unpolarized stochastic electromagnetic field, whichrequires us to deal with the polarization of the compo-nent waves. Finally, in Section V, we summarize the basicsteps needed to calculate Hellings-and-Downs-type func-tions in general, and then set-up up the calculation forthe actual pulsar timing Hellings and Downs curve, leav-ing the evaluation of the final integral to the motivatedreader. (We have included details of the calculation inAppendix A, in case the reader has difficulty completingthe calculation.)
II. RANDOM FIELDS AND EXPECTATIONVALUES
Probably the most important reason for calculatingHellings-and-Downs-type functions is to determine thecorrelation signature of a signal buried in noisy data.The situation is tricky when the signal is associated witha random field (e.g., for a stochatic gravitational-wavebackground), since then one is effectively trying to de-tect “noise in noise.” Fortunately, it turns out that thereis a way to surmount this problem. The key idea is thatalthough the signal associated with a random field is typ-ically indistinguishable from noise in a single detector or
Earth ≈ SSBpulsar 1pulsar 2x yz ζ u u Wednesday, February 4, 15
FIG. 3: Geometry for the calculation of the Hellings and Downs function for the correlated response of a pair of Earth-pulsarbaselines to an isotropic, unpolarized stochastic gravitational-wave background. The coordinate system is chosen so that theEarth is located at the origin and pulsar 1 is located on the z -axis, a distance D from the origin. Pulsar 2 is located in the xz -plane, a distance D from the origin. The two Earth-pulsar baselines point along the unit vectors ˆ u and ˆ u . The anglebetween the two baselines is denoted by ζ , and is given by cos ζ = ˆ u · ˆ u . (Actually, the origin of the coordinate system forthe calculation is the fixed solar system barycenter (SSB), and not the moving Earth. But since the the distance between theEarth and SSB (1 au) is much smaller than the typical distance to the two pulsars, D , ∼ × au, there is nopractical difference between the Earth-pulsar and SSB-pulsar baselines.) receiving system, it is correlated between pairs of de-tectors or receiving systems in ways that differ, in gen-eral, from instrumental or measurement noise. (In otherwords, by using multiple detectors, one can leverage thecommon influence of the background field against un-wanted, uncorrelated noise processes.) At each instant oftime, the measured correlation is simply the product ofthe output of two detectors. But since both the field andthe instrumental noise are random processes, the mea-sured correlation will fluctuate with time as dictated bythe statistical properties of the field and noise. By aver-aging the correlations over time, we obtain an estimate ofthe expected value of the correlation, which we can thencompare with predicted values assuming the presence (orabsence) of a signal. The purpose of this section is to de-velop the mathematical machinery that will allow us toperform these statistical correlation calculations.In the following three sections of the paper, we will beworking with fields (sound, electromagnetic, and gravi-tational fields) that are made up of waves propagatingin all different directions. These waves, having been pro-duced by a large number of independent and uncorrelatedsources, will have, in general, different frequencies, am-plitudes, and phases. (In the case of electromagnetic andgravitational waves, they will also have different polar-izations.) Such a superposition of waves is most conve- niently described statistically , in terms of a Fourier inte-gral whose Fourier coefficients are random variables. Thestatistical properties of the field will then be encodedin the statistical properities of the Fourier coefficients,which are much simpler to work with as we shall showbelow, cf. Eqs. (13) and (14).To illustrate these ideas as simply as possible, wewill do the calculations in this section for an arbitrary scalar field Φ( t, x ). Analogous calculations would also gothrough for vector and tensor fields (e.g., electromagneticand gravitational fields) with mostly just an increase innotational complexity, coming from the vector and ten-sor nature of these fields and their polarization proper-ties. Scalar fields are particularly simple since they aredescribed by a single real (or complex) number at eachpoint in space x , at each instant of time t . Sound waves,which we will discuss in detail in Section III, are an ex-ample of a scalar field. The Fourier integral for a scalarfield Φ( t, x ) has the form˜Φ( t, x ) = (cid:90) d k ˜ A ( k ) e i ( k · x − ωt ) , Φ( t, x ) = Re[ ˜Φ( t, x )] , (2)with ω/k = v , where v is the speed of wave propaga-tion and k = | k | . The relation ω/k = v is requiredfor e i ( k · x − ωt ) to be a solution of the wave equation.The Fourier coefficients ˜ A ( k ) are complex-valued randomvariables, and can be written as˜ A ( k ) = A ( k ) e iα ( k ) = a ( k ) + ib ( k ) , (3)where A , α , a , and b are all real-valued functions of k .The statistical properties of the field Φ( t, x ) are com-pletely determined by the joint probability distributions p n (Φ , t , x ; Φ , t , x ; · · · ; Φ n , t n , x n ) , n = 1 , , · · · (4)in terms of which one can calculate the expectation values (cid:104) Φ( t , x ) (cid:105) , (cid:104) Φ( t , x )Φ( t , x ) (cid:105) , etc . (5)For example, the expectation value of the field at spatiallocation x at time t is defined by (cid:104) Φ( t , x ) (cid:105) ≡ (cid:90) ∞−∞ dΦ Φ ( t , x ) p (Φ , t , x ) . (6)Equivalently, the expectation values can be defined interms of an ensemble average, e.g., (cid:104) Φ( t, x ) (cid:105) ≡ lim N →∞ N N (cid:88) i =1 Φ ( i ) ( t, x ) , (7)where Φ ( i ) ( t, x ) denotes a particular realization ofΦ( t, x ). The usefulness of knowing the expectation val-ues given in (5) is that such knowledge is equivalent toknowing the joint probability distributions (4) and hencethe complete statistical properties of the field. Theseexpectation values in turn are completely encoded in theexpectation values of the products of the Fourier coeffi-cients ˜ A ( k ).The simplest case, which is also the one we consider, isfor a multivariate Gaussian -distributed field, since knowl-edge of the quadratic expectation values is sufficient todetermine all higher-order moments. Without loss of gen-erality, we will work with the real random variables a ( k )and b ( k ), and assume that any non-zero constant valuehas been subtracted from the field: (cid:104) a ( k ) (cid:105) = 0 , (cid:104) b ( k ) (cid:105) = 0 . (8)We will also assume that the field Φ( x , t ) is stationary intime and spatially homogeneous —i.e., that the statisticalproperties of the field are unaffected by a change in eitherthe origin of time or the origin of spatial coordinates: t → t + t (cid:48) , x → x + x (cid:48) . (9) This means that the quadratic expectation values candepend only on the difference between these coordinates: (cid:104) Φ( t , x )Φ( t , x ) (cid:105) = C ( t − t , x − x ) . (10)Such behavior follows from (cid:104) a ( k ) a ( k (cid:48) ) (cid:105) = 12 B ( k ) δ ( k − k (cid:48) ) , (cid:104) b ( k ) b ( k (cid:48) ) (cid:105) = 12 B ( k ) δ ( k − k (cid:48) ) , (cid:104) a ( k ) b ( k (cid:48) ) (cid:105) = 0 , (11)with C ( t − t , x − x ) = 12 (cid:90) d k B ( k ) × cos [ k · ( x − x ) − ω ( t − t )] . (12)For readers interested in proving this last statement,write Φ as the sum Φ = ( ˜Φ + ˜Φ ∗ ) / a ( k ) and b ( k ). Given Eq. (11),Eq. (10) then follows with C ( t − t , x − x ) given byEq. (12).The physical meaning of the Dirac delta functions thatappear in the expectation values of Eq. (11) is that wavespropagating in different directions ˆ k and ˆ k (cid:48) and havingdifferent angular frequencies ω = kv and ω (cid:48) = k (cid:48) v are sta-tistically independent of one another. In other words, theexpected correlations are non-zero only for waves travel-ing in the same direction and having the same frequency.Using Eqs. (8) and (11), it is also straightforward to showthat the complex Fourier coefficients ˜ A ( k ) = a ( k ) + ib ( k )satisfy (cid:104) ˜ A ( k ) (cid:105) = 0 , (cid:104) ˜ A ∗ ( k ) (cid:105) = 0 , (13)and that (cid:104) ˜ A ( k ) ˜ A ( k (cid:48) ) (cid:105) = 0 , (cid:104) ˜ A ∗ ( k ) ˜ A ∗ ( k (cid:48) ) (cid:105) = 0 , (cid:104) ˜ A ( k ) ˜ A ∗ ( k (cid:48) ) (cid:105) = B ( k ) δ ( k − k (cid:48) ) . (14)These two sets of expectation values for the Fourier co-efficients ˜ A ( k ) are the main results of this section. Thevanishing of the first two expectation values in (14) imply (cid:10) Φ ( t, x ) (cid:11) = 14 (cid:68) ( ˜Φ( t, x ) + ˜Φ ∗ ( t, x ))( ˜Φ( t, x ) + ˜Φ ∗ ( t, x )) (cid:69) = 14 (cid:110) (cid:104) ˜Φ( t, x ) ˜Φ( t, x ) (cid:105) + (cid:104) ˜Φ ∗ ( t, x ) ˜Φ ∗ ( t, x ) (cid:105) + (cid:104) ˜Φ( t, x ) ˜Φ ∗ ( t, x ) (cid:105) + (cid:104) ˜Φ ∗ ( t, x ) ˜Φ( t, x ) (cid:105) (cid:111) = 12 (cid:68) ˜Φ( t, x ) ˜Φ ∗ ( t, x ) (cid:69) , (15)which we will use repeatedly in the following sections.As discussed at the start of this section, we are ulti-mately interested in calculating the expected correlation (cid:104) r ( t ) r ( t ) (cid:105) of the responses r ( t ), r ( t ) of two receiv-ing systems R , R to the field Φ( t, x ). It is this ex-pected correlation that we can compare against the ac-tual measured correlation, assuming that the other noiseprocesses are uncorrelated across different receiving sys-tems. The response of the receiving systems will be linearin the field, given by a convolution of R and R with Φ.Since Φ( t, x ) is a random field, r ( t ) and r ( t ) will berandom functions of time. In addition, the expectationvalue (cid:104) r ( t ) r ( t ) (cid:105) will depend only on the time differ-ence t − t , as a consequence of our assumption regard-ing stationarity of the field, cf. (10). Hence, the expectedcorrelation (cid:104) r ( t ) r ( t ) (cid:105) will be independent of time, andwe expect to be able to estimate this correlation by aver-aging together measurements made at different instantsof time: (cid:104) r ( t ) r ( t ) (cid:105) ≡ lim N →∞ N N (cid:88) i =1 r ( t i ) r ( t i ) . (16)Random processes for which this is true—i.e., for whichtime averages equal ensemble averages over different re-alizations of the field—are said to be ergodic .In what follows we will assume that all our randomprocesses are ergodic so that ensemble averages can bereplaced by time averages (and/or spatial averages) if de-sired. This will allow us to calculate expectation valuesby averaging over segments of a single realization, whichis usually all that we have in practice. Although ergod-icity is often a good assumption to make, it is impor-tant to note that not all stationary random processes areergodic. An example of a stationary random processthat is not ergodic is an ensemble of constant time-series x ( t ) = a , where the values of a are uniformly distributedbetween − (cid:104) x ( t ) (cid:105) = 0 forall t , but the time-average of a single realization equalsthe value of a for whichever time-series is drawn from theensemble. For simplicity of presentation in the remain-der of this paper, we will continue to treat the Fourierexpansion coefficients as random variables and calculateensemble averages of these quantities, rather than time (and/or spatial) averages of products of the plane wavecomponents e i ( k · x − ωt ) . III. SCENARIO 1: SOUND WAVES
The first scenario we consider involves sound. Math-ematically, sound waves in air are pressure deviations(relative to atmospheric pressure) that satisfy the 3-dimensional wave equation. If we denote the pressuredeviation at time t and spatial location x by p ( t, x ), then ∇ p − c s ∂ p∂t = 0 , (17)where ∇ denotes the Laplacian and c s denotes thespeed of sound in air (approximately 340 m / s at roomtemperature). The most general solution of the 3-dimensional wave equation is a superposition of planewaves: p ( t, x ) = (cid:90) d k A ( k ) cos( k · x − ωt + α ( k )) , (18)where the wave vector k and angular frequency ω arerelated by ω/k = c s in order that (17) be satisfied for each k . As discussed in Section II, it will be more convienentto work with the complex-valued solution˜ p ( t, x ) = (cid:90) d k ˜ A ( k ) e i ( k · x − ωt ) , ˜ A ( k ) = A ( k ) e iα ( k ) , (19)for which p ( t, x ) is the real part.For a stochastic sound field, the Fourier coefficients˜ A ( k ) are random variables. We will assume that thesecoefficients satisfy (13) and (14), with the additional re-quirement that the function B ( k ) be independent of di-rection ˆ k , which is appropriate for a statistically isotropic sound field. (This means there is no preferred directionof wave propagation at any point in the field.) As weshall now show, the function B ( k ) ≡ B ( k ) is simply re-lated to the power per unit frequency in the sound fieldintegrated over all directions. To prove this last claim wecalculate the mean-squared pressure deviations: (cid:104) p ( t, x ) (cid:105) = 12 (cid:104) ˜ p ( t, x )˜ p ∗ ( t, x ) (cid:105) = 12 (cid:28)(cid:90) d k ˜ A ( k ) e i ( k · x − ωt ) (cid:90) d k (cid:48) ˜ A ∗ ( k (cid:48) ) e − i ( k (cid:48) · x − ω (cid:48) t ) (cid:29) = 12 (cid:90) d k (cid:90) d k (cid:48) (cid:68) ˜ A ( k ) ˜ A ∗ ( k (cid:48) ) (cid:69) e i ( k − k (cid:48) ) · x e − i ( ω − ω (cid:48) ) t = 12 (cid:90) d k B ( k )= 12 (cid:90) ∞ k d k (cid:90) S d Ω ˆ k B ( k )= 2 π (cid:90) ∞ k d k B ( k ) . (20)Thus, if we write (cid:104) p ( t, x ) (cid:105) = (cid:90) ∞ d ω d (cid:104) p (cid:105) d ω , (21)then d (cid:104) p (cid:105) d ω = 2 πk c s B ( k ) (22)as claimed.To determine the acoustical analogue of the pulsar tim-ing Hellings and Downs function, we need to calculate theexpected correlation of the responses r ( t ) and r ( t ) oftwo receiving systems to an isotropic stochastic soundfield. A single receiving system will consist of a pair of omni-directional (i.e., isotropic) microphones that areseparated in space as shown in Fig. 4. For simplicity, wewill assume that the microphones are indentical and havea gain G that is independent of frequency. The response r ( t ) of receiving system 1, consisting of microphones A and B , is defined to be the real part of˜ r ( t ) = ˜ V A ( t ) − ˜ V B ( t ) , (23)where˜ V A ( t ) = G ˜ p ( x A , t ) , ˜ V B ( t ) = G ˜ p ( x B , t ) . (24)The response of receiving system 2, consisting of micro-phones A and C , is defined similarly,˜ r ( t ) = ˜ V A ( t ) − ˜ V C ( t ) , (25)with microphone C replacing microphone B . Note thatmicrophone A is common to both receiving systems, andthat we have taken the time of the measurement to be thesame at both microphones, which physically correspondsto running equal-length wires from each microphone toour receiving system.The expected value of the correlated response is then (cid:104) r ( t ) r ( t ) (cid:105) = 12 Re {(cid:104) ˜ r ( t )˜ r ∗ ( t ) (cid:105)} = 12 Re (cid:110)(cid:68)(cid:16) ˜ V A ( t ) − ˜ V B ( t ) (cid:17) (cid:16) ˜ V ∗ A ( t ) − ˜ V ∗ C ( t ) (cid:17)(cid:69)(cid:111) = 12 Re (cid:26) G (cid:90) d k (cid:90) d k (cid:48) (cid:68) ˜ A ( k ) ˜ A ∗ ( k (cid:48) ) (cid:69) e − i ( ω − ω (cid:48) ) t (cid:2) − e i k · x B (cid:3) (cid:104) − e − i k (cid:48) · x C (cid:105)(cid:27) = 12 Re (cid:26) G (cid:90) d k B ( k ) (cid:2) − e i k · x B (cid:3) (cid:2) − e − i k · x C (cid:3)(cid:27) = 12 Re (cid:26) G (cid:90) ∞ k d k B ( k ) (cid:90) S d Ω ˆ k (cid:104) − e i k · x B − e − i k · x C + e i k · ( x B − x C ) (cid:105)(cid:27) = (cid:90) ∞ d ω d (cid:104) p (cid:105) d ω Γ ( ω ) , (26)where the correlation functionΓ ( ω ) = Re (cid:26) G π (cid:90) S d Ω ˆ k (cid:104) − e i k · x B − e − i k · x C + e i k · ( x B − x C ) (cid:105)(cid:27) . (27) microphone Amicrophone Bmicrophone Cx yz ζ u u Wednesday, February 4, 15
FIG. 4: Geometry for the calculation of the Hellings-and-Downs-type function for a pair of receivers constructed from omni-directional microphones responding to an isotropic stochastic sound field. Receiving system 1 is constructed from microphones A and B , and points along the unit vector ˆ u . Receiving system 2 is constructed from microphones A and C , and points alongthe unit vector ˆ u . The coordinate system is chosen so that microphone A , which is common to both recieving systems, islocated at the origin. Microphone B is located on the z -axis, a distance D B from the origin, while microphone C is locatedin the xz -plane, a distance D C from the origin. The angle between the two receiving systems is denoted by ζ , and is given bycos ζ = ˆ u · ˆ u . (Note the similarity of this figure and Fig. 3.) The integrals of the exponentials over all directions ˆ k areof the form (cid:82) S d Ω ˆ k e i k · x , where x is a fixed vector. Suchan integral is most easily evaluated in a frame in whichthe z -axis is directed along x . In this frame, (cid:90) S d Ω ˆ k e i k · x = (cid:90) π d φ (cid:90) − d(cos θ ) e ikD cos θ = 2 π ikD (cid:0) e ikD − e − ikD (cid:1) = 4 π sinc( kD ) , (28)where D = | x | and sinc( x ) ≡ sin x/x . Since the sincfunction rapidly approaches zero for x (cid:29) kD (cid:29)
1, or equivalently,provided D (cid:29) /k = c s /ω . This is called the short-wavelength approximation . For audible sound, which hasfrequencies f ≡ ω/ (2 π ) in the range ∼
20 Hz to ∼
20 kHz,this condition becomes D (cid:29) c s ω = 340 m / s2 π ·
20 Hz = 2 . . (29)So assuming that the individual microphones are sepa-rated by more than this amount, we have χ ( ζ ) ≡ Γ ( ω ) (cid:39) G . (30) kD s i n c ( k D ) FIG. 5: Plot of sinc( kD ) versus kD . In other words, the Hellings and Downs function for anisotropic stochastic sound field is simply a constant, inde-pendent of the angle between the two receiving systems.The expected correlation is thus (cid:104) r r (cid:105) (cid:39) G (cid:104) p (cid:105) , (31)which is the mean power in the sound field multiplied bya constant G . This result is to be expected for omni-directional microphones in an isotropic stochastic soundfield.Although this was a somewhat long calculation to ob-tain an answer that, in retrospect, did not require any calculation, the formalism developed here can be appliedwith rather minor modifications to handle more compli-cated scenarios as we shall see below. IV. SCENARIO 2: ELECTROMAGNETICWAVES
The second example we consider involves electromag-netic waves. Similar to sound, electromagnetic waves aresolutions to a 3-dimensional wave equation, but with thespeed of light c = 2 . × replacing the speed of sound c s : ∇ E − c ∂ E ∂t = 0 , ∇ B − c ∂ B ∂t = 0 . (32)The most general solution to the wave equation for theelectric and magnetic fields is given by a sum of planewaves similar to that in Eq. (2),˜ E ( t, x ) = (cid:90) d k (cid:110) ˜ E ( k )ˆ (cid:15) (ˆ k ) + ˜ E ( k )ˆ (cid:15) (ˆ k ) (cid:111) e i ( k · x − ωt ) , ˜ B ( t, x ) = (cid:90) d k ˆ k c × (cid:110) ˜ E ( k )ˆ (cid:15) (ˆ k ) + ˜ E ( k )ˆ (cid:15) (ˆ k ) (cid:111) e i ( k · x − ωt ) , (33)with E ( t, x ) = Re[ ˜ E ( t, x )] , B ( t, x ) = Re[ ˜ B ( t, x )] , (34)and ω/k = c . In the above expressions, ˆ (cid:15) α (ˆ k ) ( α =1 ,
2) are two unit polarization vectors, orthogonal to oneanother and to the direction of propagation:ˆ (cid:15) α (ˆ k ) · ˆ (cid:15) β (ˆ k ) = δ αβ , ˆ k · ˆ (cid:15) α (ˆ k ) = 0 . (35)Note that there is freedom to rotate the polarization vec-tors in the plane orthogonal to ˆ k . For simplicity, we will chooseˆ (cid:15) (ˆ k ) = cos θ cos φ ˆ x + cos θ sin φ ˆ y − sin θ ˆ z = ˆ θ , ˆ (cid:15) (ˆ k ) = − sin φ ˆ x + cos φ ˆ y = ˆ φ , (36)whenever ˆ k points in the direction given by the standardangular coordinates ( θ, φ ) on the sphere:ˆ k = sin θ cos φ ˆ x + sin θ sin φ ˆ y + cos θ ˆ z . (37)Since the receiving systems that we shall consider beloware constructed from electric dipole antennas, which re-spond only to the electric part of the field, we will ignorethe magnetic field for the remainder of this discussion.For a stochastic field, the Fourier coefficients arecomplex-valued random variables. We will assume thatthey have expectation values (cf. (13) and (14)): (cid:104) ˜ E α ( k ) (cid:105) = 0 , (cid:104) ˜ E ∗ α ( k ) (cid:105) = 0 , (38)and (cid:104) ˜ E α ( k ) ˜ E β ( k (cid:48) ) (cid:105) = 0 , (cid:104) ˜ E ∗ α ( k ) ˜ E ∗ β ( k (cid:48) ) (cid:105) = 0 , (cid:104) ˜ E α ( k ) ˜ E ∗ β ( k (cid:48) ) (cid:105) = δ ( k − k (cid:48) ) P αβ ( k ) . (39)As before, the Dirac delta function ensures that the radi-ation propagating in different directions and having dif-ferent angular frequencies are statistically independent ofone another. If the field is also statistically isotropic andunpolarized, then the polarization tensor P αβ ( k ) will beproportional to the identity matrix δ αβ , with a propor-tionality constant independent of direction on the sky: P αβ ( k ) = P ( k ) δ αβ . (40)Similar to the case for sound, the function P ( k ) turnsout to be simply related to the power per unit frequencyin the electric field when summed over both polarizationmodes and integrated over all directions. To see this wecalculate mean-squared electric field: (cid:104) E ( t, x ) (cid:105) = 12 (cid:104) ˜ E ( t, x ) · ˜ E ∗ ( t, x ) (cid:105) = 12 (cid:42)(cid:90) d k (cid:88) α =1 , ˜ E α ( k ) ˆ (cid:15) α (ˆ k ) e i ( k · x − ωt ) · (cid:90) d k (cid:48) (cid:88) β =1 , ˜ E ∗ β ( k (cid:48) ) ˆ (cid:15) β (ˆ k (cid:48) ) e − i ( k (cid:48) · x − ω (cid:48) t ) (cid:43) = 12 (cid:90) d k (cid:90) d k (cid:48) (cid:88) α =1 , (cid:88) β =1 , (cid:68) ˜ E α ( k ) ˜ E ∗ β ( k (cid:48) ) (cid:69) ˆ (cid:15) α (ˆ k ) · ˆ (cid:15) β (ˆ k (cid:48) ) e i ( k − k (cid:48) ) · x e − i ( ω − ω (cid:48) ) t = 12 (cid:90) d k P ( k ) (cid:88) α =1 , ˆ (cid:15) α (ˆ k ) · ˆ (cid:15) α (ˆ k )= 4 π (cid:90) ∞ k d k P ( k ) = (cid:90) ∞ d ω d (cid:104) E (cid:105) d ω , (41)0for which d (cid:104) E (cid:105) d ω = 4 πk c P ( k ) (42)as claimed. Note that this has the same form as that forsound, cf. (22), with the speed of light c replacing thespeed of sound c s , and the extra factor of two comingfrom the summation over the two (assumed statisticallyequivalent) polarization modes for the electromagneticfield.To determine the electromagnetic analogue of the pul-sar timing Hellings and Downs function, we need to cal-culate the expected correlation of the responses r ( t ) and r ( t ) of two receiving systems to an isotropic, unpolar-ized stochastic electromagnetic field. A single receivingsystem will consist of a pair of electric dipole antennasthat are separated in space as shown in Fig. 6. For sim-plicity, we will assume that the electric dipole antennasare identical and short relative to the wavelengths thatmake up the electric field. The response r ( t ) of receivingsystem 1, consisting of electric dipole antennas A and B , is defined to be the real part of˜ r ( t ) = ˜ V A ( t ) − ˜ V B ( t ) , (43)where˜ V A ( t ) = ˆ u · ˜ E ( x A , t ) , ˜ V B ( t ) = ˆ u · ˜ E ( x B , t ) . (44)The response r ( t ) of receiving system 2, consisting ofelectric dipole antennas A (cid:48) and C , is defined similarly˜ r ( t ) = ˜ V A (cid:48) ( t ) − ˜ V C ( t ) , (45)where˜ V A (cid:48) ( t ) = ˆ u · ˜ E ( x A , t ) , ˜ V C ( t ) = ˆ u · ˜ E ( x C , t ) . (46)Note that ˜ V A (cid:48) ( t ) differs from ˜ V A ( t ) since the dipole an-tenna for A (cid:48) points along ˆ u , while that for A pointsalong ˆ u .The expected value of the correlated response is then (cid:104) r ( t ) r ( t ) (cid:105) = 12 Re {(cid:104) ˜ r ( t )˜ r ∗ ( t ) (cid:105)} = 12 Re (cid:110)(cid:68)(cid:16) ˜ V A ( t ) − ˜ V B ( t ) (cid:17) (cid:16) ˜ V ∗ A (cid:48) ( t ) − ˜ V ∗ C ( t ) (cid:17)(cid:69)(cid:111) = 12 Re (cid:40) (cid:90) d k (cid:90) d k (cid:48) (cid:88) α =1 , (cid:88) β =1 , (cid:68) ˜ E α ( k ) ˜ E ∗ β ( k (cid:48) ) (cid:69) ˆ u · ˆ (cid:15) α (ˆ k ) ˆ u · ˆ (cid:15) β (ˆ k (cid:48) ) × e − i ( ω − ω (cid:48) ) t (cid:2) − e i k · x B (cid:3) (cid:104) − e − i k (cid:48) · x C (cid:105) (cid:41) = 12 Re (cid:40)(cid:90) d k P ( k ) (cid:88) α =1 , (ˆ u · ˆ (cid:15) α (ˆ k ))(ˆ u · ˆ (cid:15) α (ˆ k )) (cid:2) − e i k · x B (cid:3) (cid:2) − e − i k · x C (cid:3)(cid:41) = 12 Re (cid:40)(cid:90) ∞ k d k P ( k ) (cid:90) S dΩ ˆ k (cid:88) α =1 , (ˆ u · ˆ (cid:15) α (ˆ k ))(ˆ u · ˆ (cid:15) α (ˆ k )) (cid:2) − e i k · x B (cid:3) (cid:2) − e − i k · x C (cid:3)(cid:41) = (cid:90) ∞ d ω d (cid:104) E (cid:105) d ω Γ ( ω ) , (47)where Γ ( ω ) = Re (cid:40) π (cid:90) S dΩ ˆ k (cid:88) α =1 , (ˆ u · ˆ (cid:15) α (ˆ k ))(ˆ u · ˆ (cid:15) α (ˆ k )) (cid:2) − e i k · x B (cid:3) (cid:2) − e − i k · x C (cid:3)(cid:41) . (48)If we ignore the contribution of the integrals involving e i k · x B , e − i k · x C , and e i k · ( x B − x C ) , assuming as we did forsound that we are working in the short-wavelength ap- proximation, thenΓ ( ω ) (cid:39) π (cid:90) S d Ω ˆ k (cid:88) α =1 , (ˆ u · ˆ (cid:15) α (ˆ k ))(ˆ u · ˆ (cid:15) α (ˆ k )) , (49)which is the sky-averaged and polarization-averagedproduct of the inner products of ˆ u and ˆ u with the po-1 antenna Aantenna Bantenna Cx yz ζ u u antenna A ′ Wednesday, February 4, 15
FIG. 6: Geometry for the calculation of the Hellings-and-Downs-type function for a pair of receivers constructed from electricdipole antennas responding to an isotropic, unpolarized stochastic electromagnetic field. Receiving system 1 is constructedfrom dipole antennas A and B , which are both directed along ˆ u , which points from A to B . Receiving system 2 is constructedfrom dipole antennas A (cid:48) and C , which are both directed along ˆ u , which points from A (cid:48) to C . The coordinate system is chosenso that the two dipole antennas A and A (cid:48) are located at the origin. Dipole antenna B is located on the z -axis, a distance D B from the origin, while dipole antenna C is located in the xz -plane, a distance D C from the origin. The angle between the tworeceiving systems is denoted by ζ , and is given by cos ζ = ˆ u · ˆ u . (Again, note the similarity of this figure and Fig. 3.) larization vectors ˆ (cid:15) α (ˆ k ).The above integral for the correlation function Γ ( ω )can easily be evaluated in the coordinate system shownin Fig. 6. In these coordinates, ˆ u = ˆ z and ˆ u = sin ζ ˆ x +cos ζ ˆ z . Using the expressions for the polarization vectorsgiven in (36) it follows thatˆ u · ˆ (cid:15) (ˆ k ) = − sin θ , ˆ u · ˆ (cid:15) (ˆ k ) = 0 , ˆ u · ˆ (cid:15) (ˆ k ) = sin ζ cos θ cos φ − cos ζ sin θ , ˆ u · ˆ (cid:15) (ˆ k ) = − sin ζ sin φ , (50)for whichΓ ( ω ) (cid:39) π (cid:90) S d Ω ˆ k (cid:88) α =1 , (ˆ u · ˆ (cid:15) α (ˆ k ))(ˆ u · ˆ (cid:15) α (ˆ k ))= 18 π (cid:90) π d φ (cid:90) − d(cos θ ) ( − sin θ ) × (sin ζ cos θ cos φ − cos ζ sin θ )= 14 cos ζ (cid:90) − d x (1 − x ) = 13 cos ζ . (51)Thus χ ( ζ ) ≡ Γ ( ω ) (cid:39)
13 cos ζ (52) and (cid:104) r r (cid:105) (cid:39)
13 cos ζ (cid:104) E (cid:105) . (53)So the Hellings and Downs function for an isotropic, un-polarized stochastic electromagnetic field is simply pro-portional to the cosine of the angle between the two re-ceiving systems. V. SUMMARY AND DISCUSSION
In the preceeding two sections, we calculated Hellings-and-Downs-type functions χ ( ζ ) for two simple scenarios:(i) omni-directional microphones in an isotropic stochas-tic sound field, and (ii) electric dipole antennas in anisotropic, unpolarized stochastic electromagnetic field.The result for sound was trivial, χ ( ζ ) = const, and inretrospect did not even require a calculation. The resultfor the electromagnetic case was slightly more compli-cated, χ ( ζ ) = cos( ζ ), as we had to take account of thepolarization of the electromagnetic waves as well as thedirection of the electric dipole antennas. But the basicsteps that we went through to obtain the results werethe same in both cases, and, in fact, can be abstracted towork for receivers in a general field, which we will denotehere by Φ( t, x ):
21. Write down the most general expression for thefield in terms of a Fourier expansion. Let theFourier coefficients be random variables whose ex-pectation values encode the statistical properties ofthe field—e.g., isotropic, unpolarized, · · · .2. Using the expectation values of the Fourier coeffi-cients, calculate (cid:104) Φ ( t, x ) (cid:105) . Use this expression todetermine how the power in the field is distributedas a function of frequency (cid:104) Φ ( t, x ) (cid:105) = (cid:90) ∞ d ω d (cid:104) Φ (cid:105) d ω . (54)3. Write down the response r I ( t ) of receiver I to thefield Φ( t, x ). For a linear receiving system, the re-sponse will take the form of a convolution: r I ( t ) = ( R I ∗ Φ)( t )= (cid:90) d τ (cid:90) d y R I ( τ, y )Φ( t − τ, x I − y ) . (55)For the simple examples we considered in Sec-tions III and IV, R I ( τ, y ) was proportional to asum of a product of delta functions like δ ( τ ) δ ( y ),but that need not be the case in general.4. Using the expectation values of the Fourier coef-ficients, calculate the expected value of the corre-lated response (cid:104) r ( t ) r ( t ) (cid:105) for any pair of receivers.Use this expression to determine the correlation function Γ ( ω ) defined by (cid:104) r ( t ) r ( t ) (cid:105) = (cid:90) ∞ d ω d (cid:104) Φ (cid:105) d ω Γ ( ω ) . (56)5. For fixed frequency ω , the correlation Γ ( ω ) is,by definition, the value of the Hellings and Downsfunction evaluated for the relative configuration ofthe two receiving systems. For example, χ ( ζ ) = Γ ( ω ) (57)for the simple examples that we considered in Sec-tions III and IV, where ζ is the angle betweenthe two receiving systems, relative to an origin de-fined by the common microphone A for sound, orthe co-located dipole antennas A and A (cid:48) for theelectromagnetic field. For more complicated re-ceivers, such as ground-based laser interferometerslike LIGO, Virgo, etc., χ will be a function of sev-eral variables; the separation vector between thevertices of the two interferometers s ≡ x − x , aswell as the unit vectors ˆ u , ˆ v , and ˆ u , ˆ v , whichpoint along the arms of the two interferometers.The above five steps are generic and will work for anyscenario.We conclude this paper by stating without proof theexpression for the actual gravitational-wave pulsar timingHellings and Downs function: χ ( ζ ) ≡ π (cid:90) S d Ω ˆ k (cid:88) α =+ , × (cid:18) ˆ u ⊗ ˆ u k · ˆ u (cid:19) : (cid:15) α (ˆ k ) 12 (cid:18) ˆ u ⊗ ˆ u k · ˆ u (cid:19) : (cid:15) α (ˆ k ) , (58)whereˆ u I ⊗ ˆ u I : (cid:15) α (ˆ k ) ≡ (cid:88) a =1 3 (cid:88) b =1 u aI u bI (cid:15) α,ab (ˆ k ) , I = { , } (59)and (cid:15) + (ˆ k ) = ˆ θ ⊗ ˆ θ − ˆ φ ⊗ ˆ φ , (cid:15) × (ˆ k ) = ˆ θ ⊗ ˆ φ + ˆ φ ⊗ ˆ θ , (60)are the two gravitational-wave polarization tensors.(Here ˆ u and ˆ u are unit vectors pointing from Earthto the two pulsars, and ζ is the angle between ˆ u and ˆ u as shown in Fig. 3.) The extra factors of 1 / (1 + ˆ k · ˆ u )and 1 / (1 + ˆ k · ˆ u ) that appear in (58)—as comparedto the analogous electromagnetic expression (49)—comefrom the calculation of the timing residual response ofan Earth-pulsar baseline to the gravitational-wave field,when integrating the metric pertubations perturbations h ab ( t, x ) along the photon world-line from the pulsar to Earth. This is a non-trivial example of the convolu-tion described in Step 3 above, and the mathematicaldetails needed to derive the precise form of (58) are out-side the scope of this paper. (Readers who are interestedin seeing a derivation of (58) are encouraged to consultRef. 22.) But all in all, the pulsar timing Hellings andDowns function is just a sky-averaged and polarization-averaged product of two geometrical quantities, as is thecase for any Hellings-and-Downs-type function. It is nowjust a matter of doing the integrations, which we leave tothe motivated reader. The final result should be pro-portional to (1), which has been normalized by an overallmultiplicative factor of 3.3
Acknowledgments
The content of this paper was originally presented byFAJ during the student week of the 2011 IPTA sum-mer school at the University of West Virginia. JDR ac-knowledges support from NSF Awards PHY-1205585 andCREST HRD-1242090. We also thank the referees for nu-merous suggestions that have improved the presentationof the paper.
Appendix A: Details of the calculation for the pulsartiming Hellings and Downs function
Here we fill in some of the details of the integrationof the pulsar timing Hellings and Downs function (58), following the hints given in Endnote 23. The approachthat we follow is based on similar presentations found inthe appendices of Refs. 13,15,22.In the coordinate system shown in Fig. 3, the twopulsars are located in directions ˆ u = ˆ z and ˆ u =sin ζ ˆ x + cos ζ ˆ z , so thatˆ k · ˆ u = cos θ , ˆ k · ˆ u = cos ζ cos θ + sin ζ sin θ cos φ . (A1)Using the definition (60) of the gravitational-wave polar-ization tensors (cid:15) α (ˆ k ) with ˆ θ , ˆ φ defined in (36), it is fairlyeasy to show thatˆ u ⊗ ˆ u : (cid:15) + (ˆ k ) = sin θ , ˆ u ⊗ ˆ u : (cid:15) × (ˆ k ) = 0 , ˆ u ⊗ ˆ u : (cid:15) + (ˆ k ) = (sin ζ cos θ cos φ − cos ζ sin θ ) − sin ζ sin φ , ˆ u ⊗ ˆ u : (cid:15) × (ˆ k ) = − ζ cos θ cos φ − cos ζ sin θ ) sin ζ sin φ . (A2)The quantities F αI (ˆ k ) ≡ (cid:18) ˆ u I ⊗ ˆ u I k · ˆ u I (cid:19) : (cid:15) α (ˆ k ) , I = { , } , α = { + , ×} , (A3)which appear in (58) are then given by F +1 (ˆ k ) = 12 (1 − cos θ ) ,F × (ˆ k ) = 0 ,F +2 (ˆ k ) = 12 (cid:20) (1 − cos ζ cos θ − sin ζ sin θ cos φ ) − ζ sin φ ζ cos θ + sin ζ sin θ cos φ (cid:21) ,F × (ˆ k ) = − (cid:20) sin ζ cos θ sin(2 φ ) − sin(2 ζ ) sin θ sin φ ζ cos θ + sin ζ sin θ cos φ (cid:21) , (A4)where for F +2 (ˆ k ) we cancelled the denominator with part of the numerator to isolate the complicated φ -dependence.In this reference frame, the pulsar timing Hellings and Downs function (58) simplifies to χ ( ζ ) = 18 π (cid:90) S d Ω ˆ k F +1 (ˆ k ) F +2 (ˆ k ) = 116 π (cid:90) − d x (1 − x ) I ( x, ζ ) , (A5)where x ≡ cos θ , and I ( x, ζ ) ≡ (cid:90) π d φ F +2 (ˆ k )= 12 (cid:90) π d φ (cid:20) (1 − x cos ζ − (cid:112) − x sin ζ cos φ ) − ζ sin φ x cos ζ + √ − x sin ζ cos φ (cid:21) . (A6)The first part of the integral for I ( x, ζ ) is trivial: I ( x, ζ ) ≡ (cid:90) π d φ (1 − x cos ζ − (cid:112) − x sin ζ cos φ ) = π (1 − x cos ζ ) . (A7)The second part can be be evaluated using contour integration, which we illustrate below. Making the4usual substitutions z = e iφ , cos φ = ( z + z − ), etc.,we obtain I ( x, ζ ) ≡ − sin ζ (cid:90) π d φ sin φ x cos ζ + √ − x sin ζ cos φ = − sin ζ (cid:73) C d z f ( z ) , (A8)where f ( z ) = i ( z − z (cid:2) z (1 + x cos ζ ) + 2 √ − x sin ζ ( z + 1) (cid:3) (A9)and C is the unit circle in the complex z -plane. Thedenominator of f ( z ) can be factored using the quadraticformula for the expression in square brackets:4 z (1 + x cos ζ ) + 2 (cid:112) − x sin ζ ( z + 1)= 2 (cid:112) − x sin ζ ( z − z + )( z − z − ) , (A10)where z + ≡ − (cid:115)(cid:18) ∓ cos ζ ± cos ζ (cid:19) (cid:18) ∓ x ± x (cid:19) , z − ≡ z + . (A11)In the above expression, the top signs correspond to theregion − cos ζ ≤ x ≤ − ≤ x ≤ − cos ζ . One can show that for both of these regions, z + is inside the unit circle C (i.e., | z + | ≤ z − is outside the unit circle and does not contribute. Inaddition, z = 0 lies inside the unit circle and contributesto the contour integral as a pole of order two. Using theresidue theorem (cid:73) C f ( z ) d z = 2 πi (cid:88) i Res( f, z i ) , (A12)withRes( f, z + ) = lim z → z + { ( z − z + ) f ( z ) } = i ( z + − z − )2 √ − x sin ζ , Res( f,
0) = lim z → (cid:26) ddz (cid:2) z f ( z ) (cid:3)(cid:27) = i ( z + + z − )2 √ − x sin ζ , (A13)it follows that (cid:73) C f ( z ) d z = 2 π (1 ± x )(1 ± cos ζ ) , (A14)for which I ( x, ζ ) = π (1 − x cos ζ ) − π (1 ∓ cos ζ )(1 ± x ) . (A15)It is now a relatively simple matter to the evaluate theintegral over x to obtain χ ( ζ ): χ ( ζ ) = 116 (cid:40) (cid:90) − d x (1 − x )(1 − x cos ζ ) − ζ ) (cid:90) − cos ζ − d x − − cos ζ ) (cid:90) − cos ζ d x (1 − x )(1 + x ) (cid:41) = 116 (cid:40) ζ − ζ )(1 − cos ζ ) − − cos ζ ) (cid:20) (cid:18) − cos ζ (cid:19) − (1 + cos ζ ) (cid:21) (cid:41) = 18 + 124 cos ζ + 14 (1 − cos ζ ) ln (cid:18) − cos ζ (cid:19) = 16 − (cid:18) − cos ζ (cid:19) + 12 (cid:18) − cos ζ (cid:19) ln (cid:18) − cos ζ (cid:19) . (A16)Note that the above expression differs from (1) by anoverall normalization factor of 1 /
3. The normalizationused in (1) was chosen so that for zero angular separation, χ ( ζ ) | ζ =0 = 1 / ∗ Electronic address: [email protected] † Electronic address: [email protected] D. R. Lorimer and M. Kramer,
Handbook of Pulsar Astron-omy . Cambridge University Press, Cambridge, UK, Dec.2004. D. R. Lorimer, “Binary and Millisecond Pulsars,”
Living Reviews in Relativity , vol. 11 (8), pp. 1–90, Nov. 2008. I. H. Stairs, “Testing General Relativity with Pulsar Tim-ing,”
Living Reviews in Relativity , vol. 6 (5), pp. 1–49,Sept. 2003. J. M. Weisberg, D. J. Nice, and J. H. Taylor, “Tim-ing Measurements of the Relativistic Binary Pulsar PSR B1913+16,”
Astrophys. J. , vol. 722, pp. 1030–1034, Oct.2010. R. A. Hulse and J. H. Taylor, “Discovery of a pulsar in abinary system,”
Astrophys. J. , vol. 195, no. 2, pp. L51–L53, 1975. F. B. Estabrook and H. D. Wahlquist, “Response ofDoppler spacecraft tracking to gravitational radiation,”
General Relativity and Gravitation , vol. 6, pp. 439–447,Oct. 1975. M. V. Sazhin, “Opportunities for detecting ultralong grav-itational waves,”
Soviet Ast. , vol. 22, pp. 36–38, Feb. 1978. S. Detweiler, “Pulsar timing measurements and the searchfor gravitational waves,”
Astrophys. J. , vol. 234, pp. 1100–1104, Dec. 1979. A pulsar timing model predicts the arrival times of thepulses given values for the pulsar’s spin frequency, fre-quency derivative, location on the sky, proper motion withrespect to the solar system barycenter, its orbital param-eters if the pulsar is in a binary, etc. The values of theseparameters are typically determined by an iterative least-squares fitting procedure, which minimizes the root-mean-squared (rms) deviation of the resultant timing residuals.Systematic errors in the timing model parameters can usu-ally be identified by this iterative procedure, but unmod-elled processes in the timing model will lead to errors inthe timing residuals that cannot easily be removed. R. S. Foster and D. C. Backer, “Constructing a pulsartiming array,”
Astrophys. J. , vol. 361, pp. 300–308, Sept.1990. R. W. Hellings and G. S. Downs, “Upper limits on theistotropic gravitational radiation background from pulsartiming analysis,”
Astrophys. J. , vol. 265, pp. L39–L42,1983. K. J. Lee, F. A. Jenet, and R. H. Price, “Pulsar Timing asa Probe of Non-Einsteinian Polarizations of GravitationalWaves,”
Astrophys. J. , vol. 685, pp. 1304–1319, Oct. 2008. C. M. F. Mingarelli, T. Sidery, I. Mandel, and A. Vec-chio, “Characterizing gravitational wave stochastic back-ground anisotropy with pulsar timing arrays,”
Phys. Rev.D , vol. 88, p. 062005(17), Sept. 2013. S. R. Taylor and J. R. Gair, “Searching for anisotropicgravitational-wave backgrounds using pulsar timing ar-rays,”
Phys. Rev. D , vol. 88, p. 084001(25), Oct. 2013. J. Gair, J. D. Romano, S. Taylor, and C. M. F. Mingarelli,“Mapping gravitational-wave backgrounds using methodsfrom CMB analysis: Application to pulsar timing arrays,”
Phys. Rev. D , vol. 90, p. 082001(44), Oct. 2014. R. M. Shannon, V. Ravi, W. A. Coles, G. Hobbs, M. J.Keith, R. N. Manchester, J. S. B. Wyithe, M. Bailes,N. D. R. Bhat, S. Burke-Spolaor, J. Khoo, Y. Levin,S. Os(cid:32)lowski, J. M. Sarkissian, W. van Straten, J. P. W.Verbiest, and J.-B. Wang, “Gravitational-wave limits frompulsar timing constrain supermassive black hole evolu-tion,”
Science , vol. 342, no. 6156, pp. 334–337, 2013. We are assuming here that the two pulsars—even for thecase ζ = 0—are distinct (i.e., they do not occupy the samephysical location in space). If we consider the same pulsar,as would be the case for an autocorrelation calculation,then the right-hand-side of (1) should have an extra termequal to δ ( ζ ) / This statement is a generalization (to fields) of the math-ematical result that the Fourier transform of the proba-bility distribution p ( x ) for a random variable x (i.e., theso-called characteristic function of the random variable)can be written as a power series with coefficients givenby the expectation values (cid:104) x k (cid:105) for k = 1 , , · · · . Thus,the expectation values (cid:104) x k (cid:105) for k = 1 , , · · · completelydetermine the probability distribution p ( x ) and hence thestatistical properties of the random variable x . C. W. Helstrom,
Statistical Theory of Signal Detection .Pergamon Press, Oxford, United Kingdom, 1968. Recall: ∇ = (cid:16) ∂ ∂x + ∂ ∂y + ∂ ∂z (cid:17) in Cartesian coordinates( x, y, z ). The field might actually be a tensor field, like thegravitational-wave field h ab ( t, x ), and hence should havetensor indices in general. But for simplicity, we will ignorethat complication here. M. Anholm, S. Ballmer, J. D. E. Creighton, L. R. Price,and X. Siemens, “Optimal strategies for gravitational wavestochastic background searches in pulsar timing data,”
Phys. Rev. D , vol. 79, p. 084030(19), Apr. 2009. Hint: Work in the coordinate system shown in Fig. 3 withthe Earth at the origin and the two pulsars located alongthe z -axis and in the xz -plane, respectively. Evaluate (59)in this frame using (60) and the expressions for ˆ θ , ˆ φ givenin (36). Finally, use contour integration to do the integralover the azimuthal angle φ . It is a long calculation, butworth the effort. If you have trouble completing the calcu-lation, see Appendix A for more details. M. L. Boas,