The PTA Hellings and Downs correlation unmasked by symmetries

The Hellings and Downs correlation curve describes the correlation of the timing residuals from pairs of pulsars as a function of their angular separation on the sky and is a smoking-gun signature for the detection of an isotropic stochastic background of gravitational waves. We show that it can be easily obtained from realizing that Lorentz transformations are conformal transformations on the celestial sphere and from the conformal properties of the two-point correlation of the timing residuals. This result allows several generalizations, e.g. the calculation of the three-point correlator of the time residuals and the inclusion of additional polarization modes (vector and/or scalar) arising in alternative theories of gravity.


Introduction
The recent Pulsar Timing Arrays (PTA) collaborations, NANOGrav [1,2], EPTA [3][4][5], PPTA [6][7][8] and CPTA [9], have recently released data showing evidence for the presence of a stochastic background of Gravitational Waves (GWs).Pulsars are high-quality clocks generating pulses in the radio frequencies band which arrive at Earth with regular and predictable arrival times.The latter are slightly perturbed by the presence of the GWs which induce correlated perturbations between different radio pulsars, called "timing residuals".
The correlations are predicted to follow the so-called "Hellings and Downs" (HD) correlation curve [10] which describe the expected correlation in the timing residuals from a pair of pulsars as a function of their angular separation on the sky.It is a prediction of general relativity for an unpolarized and isotropic GW background (see the recent Ref. [11] for a nice set of comments about the HD correlation).
The goal of this paper is to show that the HD correlation function can be obtained by very general symmetry arguments.The key point is the realization that 1. Lorentz transformations between inertial observers are conformal motions of the surface of a unit two-sphere; 2. such Lorentz transformations can be interpreted as conformal transformations on the celestial sphere; 3. under such transformations the HD correlation transforms in a well-defined way with a welldefined conformal weight.This allows one to obtain the HD correlation with no explicit and cumbersome calculation.The symmetry arguments are also suitable to generalize the HD correlation to the case in which three or more pulsars are correlated amd to include additional polarization modes (vector and/or scalar) in theories of gravity different from general relativity.
The paper is organized as follows.In section 2 we describe the Lorentz group and show how Lorentz transformations are seen as conformal transformation in the celestial sphere.In section 3 we obtain the generic two-point correlator on the celestial sphere and show in section 4 how the HD correlation function is recovered.Section 5 extends our computation of the overlap function to the so-called short-arm detectors.Section 6 contains our conclusions.The paper contains also three Appendices where some technical details are added.

The Lorentz group, inertial observers and the celestial sphere
Let us first review the notion of the celestial sphere of an inertial observer in Minkowski spacetime [12][13][14].The knowledgeable reader can skip this section.The celestial sphere is an imaginary sphere of practically infinite radius centered at the position of an inertial observer.We wish to show now that Lorentz transformations between different inertial observers correspond to conformal transformations of the celestial sphere.First, we recall the stereographic projection and projective coordinates of the two-dimensional sphere S 2 .The latter is defined as ( The points (0, 0, 1) and (0, 0, −1) are the north (N ) and the south (S) poles of the sphere.The we find that Eq. (2.2) is written as where is the standard metric on the unit S 2 .Since the metric on the S 2 is written as (2.9) Note in the above construction, the north pole is mapped nowhere.However, by extending C to C ∪ {∞}, the north pole is mapped to itself and therefore, we can identify the whole of S 2 with C∪{∞}.Therefore, complex analytic automorphisms f : S 2 → S 2 are represented by homomorphic mappings g : C∪{∞} → C∪{∞}.In particular, it is easy to find that and therefore, complex analytic automorphisms of S 2 are conformal transformations.In fact, it can be shown that the conformal orientation-preserving automorphisms of S 2 form a group isomorphic to the group of all complex analytic automorphisms of S 2 .The latter are generated by the composition of inversion and the linear transformation and therefore, complex analytic automorphism of S 2 are the Möbius transformations which form a group isomorphic to SL(2, C)/Z 2 .Thus, the group of Möbius transformations is the group of all complex analytic automorphisms of On the other hand, it is known that the proper orthochronous Lorentz transformations O ↑ + (1, 3) is isomorphic to the Möbius group SL(2, C)/Z 2 [12,13].A convenient set of generators of the latter are where σ i (i = 1, 2, 3) are the Pauli matrices.The generators L i , K i satisfy the commutation relations of the SL(2, C) algebra (2.17)

One concludes that
Lorentz transformations between inertial observers are conformal motions of the two-sphere.

Lorentz transformations and the celestial sphere
The sphere described above is actually a spatial sphere, that is a sphere at radius r 2 = x 2 + y 2 + z 2 .
However, we are interested in the sphere that an observer looks at, that is the celestial sphere.This is not a spatial sphere defined by r =constant, since ingoing, radial light-rays emitted by the sphere need a time r to arrive from the sphere to the observer at the origin r = 0, (we have set the speed of light to unity).Instead, one can parametrize the emission time of radial ingoing light rays by the value of the advanced time coordinate u = t + r.In other words, we specify events in spacetime by using the Bondi coordinates (u, r, ζ).A sphere at constant r > 0 and constant u in Bondi coordinates, is the sphere seen by an observer at r = 0 and time u.The celestial sphere at time u is then the sphere located at an infinite distance r → ∞, and at a fixed u, and it is the sphere formed by all directions towards which an observer sitting at r = 0 may look.
In terms of the Bondi coordinates, the metric of the flat Minkowski spacetime is written as where ds 2 2 in the metric on the unit S 2 given in Eq. (2.9).Isometries of Minkowski spacetime are just Lorentz transformations, which act linearly on the flat Cartesian coordinates x µ .On the other hand, Lorentz tranformations act highly non-linear on the Bondi coordinates.However, they have a simpler form at large r and in particular, one finds that under the Lorentz transformations given in Eq. (2.17), the Bondi coordinates transform as (u, r, ζ) → (u ′ , r ′ , ζ ′ ) where ) ) and Eq. (2.19c) is clearly, to leading order, the standard Möbius transformations on S 2 and therefore, Lorentz transformations are conformal transformations of the celestial sphere.
In particular, by using Eq.(2.17), we find that the matrix which is nothing else than a Lorentz boost with rapidity along the z-direction.In Eq. (2.23), the unit vector n turns out to be and therefore, Lorentz boosts corresponds to the following transformations of the Bondi coordinates where is the Doppler shift factor.It is easy to verify that Eq.(2.25) gives rise to the aberration formula for outgoing light rays (2.27) Then by using Eq.(2.27), we find that under a Lorentz boost in the n direction, the metric of the celestial sphere transforms as (2.28) 3 The two-and three-point Correlators on the Celestial Sphere The goal of this section is two explain how a generic two-point correlator can be written on the celestial sphere.
The projective coordinate ζ in C∪{∞} can be parametrized in terms of the unit vector n as so that and the metric on the S 2 becomes Notice that since the S 2 is formed by light rays which satisfy the unit vector can be expressed as Clearly, n transforms as a vector under space rotations, whereas, under Lorentz boosts, which act linearly on the Minkowski coordinates x µ , it is easy to find, using the transformation properties of (x 0 , ⃗ x) that n i transforms as For an infinitesimal Lorentz boost generated by K, using Eq.(2.26) we get It is not difficult to verify that indeed, Lorentz boosts that transform the unit vector n as in Eq.
(3.7), act like conformal transformations on S 2 since Let us now consider the two-point correlator ) at equal time t and for points ⃗ x 1 and ⃗ x 2 both on the S 2 .We will demand invariance under the Lorentz group.Rotational invariance leads to dependence on the distance where is the geodesic invariant distance between points on S 2 .Therefore, we should have where w is the scaling dimension of C 2 since Lorentz boosts are actually conformal rescalings of the S 2 .In particular, we find that under Lorentz boosts, σ transforms as or, in infinitesimal form, In addition, if the scaling dimension of C 2 is w, we will have that Therefore, the change in the two-point correlator C 2 under Lorentz boosts is Hence, we find that Invariance under Lorentz boosts (δ K C 2 = 0) then leads to and therefore, In particular, for w = 0 we get (as a limit of Eq. (3.19)) where α 2 and β 2 are constants.4 Hellings and Downs from Lorentz invariance or conformal symmetry on the celestial sphere We now come to the HD correlation function.The goal of this section is to demonstrate that it can be understood in terms of Lorentz transformations or conformal transformations on the celestial sphere.Let us consider metric perturbations which, for the most general GW is a superposition of (transverse-traceless) plane waves, i.e., where A = +, × are the polarizations of the GWs, n their direction of propagation and f their frequency.The polarization tensors e A ij ( n) are given by where the vectors n, u and w are explicitly given by n = (− sin θ cos ϕ, − sin θ cos ϕ, − cos θ), u = (sin ϕ, − cos ϕ, 0), and depicted in Fig. 2. In addition, we will need later the ensemble average of a stationary, isotropic and unpolarized, stochastic GW background, which is characterized by where S(f ) is the spectral density of the stochastic background.
Let us now recall how the periodicity of the pulses of a pulsar is affected by the passing GW.If the pulsar is at position ⃗ x a = d a n a , then the change in the periodicity (time residual) due to a GW h T T ij (t, ⃗ x) as seen by an observer at ⃗ x = 0 is The term in the parenthesis represents the difference between the metric perturbation which arrive at the position of the observer at ⃗ x = 0 and the position of the pulsar at ⃗ x a .As it is standard in any similar consideration, we will ignore the pulsar metric perturbation [15] and write instead A stochastic GW background can be constructed by the superposition of plane GW of all possible frequencies from all possible directions.Therefore, the change in the periodicity of the pulsar due to a stochastic GW background can be expressed as where Eq. (4.1) has been used.Then by using Eq.(4.4), we find that the correlator of time residuals ⟨z a z b ⟩ of two pulsars at directions n a and n b is given by where we have defined the overlap function We are interested in the transformation properties of the correlator Γ ab in Eq.(4.9) under Lorentz transformations.We have seen that three-dimensional rotational invariance requires that the correlator is a function of σ ab only that is a function of the angular separation of the pulsars.It remains to determine the transformation properties of Γ ab under Lorentz boosts.We will consider a Lorentz boost of velocity ⃗ v in the direction of the GW, i.e., ⃗ v = n.Notice that this particular choice does not spoil the generality of the following analysis, since we need to integrate over all directions n to account for a stochastic GW background.This is equivalent to keeping n fixed and boosting in all possible directions.To determine the transformation law of Γ ab , we recall Eqs.(3.7), (3.11) and (3.13) so that and 12) The polarization tensors e A ij are invariant under Lorentz boosts along the direction of propagation.Indeed, in this case, ( u, w) are invariant and so are the polarization tensors.In the Newman-Penrose formalism, ( u, w) span the (m α , m a ) plane, which is invariant under Lorentz boosts in the (ℓ a , n a ) directions.Thus, using Eq.(4.11) we find that Under a Lorentz boost, the volume element on the S 2 in Eq. (4.9) transforms as The amplitude of the GW is also invariant as follows from the transformation of the Weyl scalar Ψ 4 under Lorentz boosts [16] Ψ and ω = 2πf transforms as we get that the amplitudes h A are invariant under Lorentz boosts.Therefore, for a stationary, isotropic and unpolarized stochastic GW background, the ensemble average in Eq. (4.4) is invariant or, in other words, respectively, we get that the spectral density of the stochastic background transforms as and therefore, since df transforms as we find that df S(f ) is invariant This can also be deduced from the relation Since the left-hand side of Eq. (4.24) is invariant, so is the right-hand side.
Putting all together, it is easy to verify that overlap function transforms under Lorentz boosts as However, this transformation has fixed points since there are Lorentz boosts that leave Γ ab invariant.
Indeed, since for pulsars located at opposite directions, n a and n b = − n a , a Lorentz boost along their direction (so Then, the reduced overlap function Γ − ab = Γ ab − Γ v transforms as in Eq. (4.25) without fixed points and therefore,

Recovering the Hellings and Downs correlation
Our goal now is to determine the constants α, β and Γ v of Eq. ( 4) and to recover the HD correlation function.For this we need two conditions: 1. PTA detectors operate in the "long-arm" (or short-wavelength) limit in which the distance between the pulsars and the earth is much longer than the wavelengths of the GWs; this induces a well-defined relation of the correlators for pulsars in the same and in opposite directions; 2. the vanishing of the integral of the two-point correlator over the whole celestial sphere basically due to the isotropy of the stochastic GW background and its traceless and transverse properties.Knowing that the overall normalization, that is the coefficient α is arbitrary [11], setting α = 1, standard form of the HD curve is recovered

Going beyond the Hellings and Downs correlation: the three-point correlator
Conformal symmetry on the celestial sphere can also specify (up to a multiplicative constant) the three-point correlator.Indeed, applying similar considerations as in the previous section, the three- ) is again, due to rotational symmetry, a function of the geodesic invariant distances σ ij = |⃗ x i − ⃗ x j | 2 /4, (i, j = 1, 2, 3) so that C 3 = C 3 (σ 12 , σ 23 , σ 13 ).Let us further assume that under Lorentz boosts (conformal transformations), the three-point correlator C 3 of fields of dimension w transforms as Then, since invariance under Lorentz boosts (δ K C 3 = 0) should hold for any n i and any ⃗ v, one is led to the equations which are solved by Again, for w = 0, the three-point correlators C 3 (σ 12 , σ 23 , σ 13 , w) = C 3 (σ, w), turns out to be where α 3 and β 3 are constants.To proceed, let us assume that there exists some sort of non-Gaussianity in the stochastic gravitational wave background.In this case, the three-point correlator of time residuals C abc = ⟨z a z b z c ⟩ will be given by

.41)
We are interested in the transformation properties of the above correlator under Lorentz boosts, that is conformal transformations of the celestial sphere.By rotational symmetry, the correlator in Eq.
(4.41) will depend on the invariants σ ab , σ bc and σ ac .By using Eqs.(4.12), (4.17) and (4.22), it can easily be verified that the three-point correlator transforms under Lorentz boosts as where In the last equality in Eq. (4.43), the invariance of C abc under n i → n j (i, j = 1, 2, 3) has been used.
As in the case of the two-point correlator, by defining and taking v = n 1 , we find that Ω abc = 0.In addition, it can be easily verified that C abc transforms like a three-point correlator for fields of dimension w = 0, and therefore, according to Eq. ( 4.40), it will be given by This specifies the three-point correlator of the time residuals to be It should be noted that both the two-and three-point correlators have the form of the corresponding correlators of a logarithmic conformal field theory as discussed in Appendix A.
The Eq. (4.46) is our generic result, where the coefficients depend on the non-Gaussian nature of the tensor modes.However, even if the GW source signal is intrinsically non-Gaussian and with a non-vanishing three-point correlator, one has to account for the fact that the stochastic signal measured by the PTA collaborations is a sum of the superposition of GWs from a large number of independent sources.This suppresses the non-Gaussianity of the time residuals, since it destroys the phase coherence needed to have a nonvanishing non-Gaussianity.Indeed, consider that the GW propagates in a perturbed universe and travel for cosmological distances.Each GW will pick up a Shapiro time delay from the time of emission t e δt = 2 where Ψ is the gravitational potential along the trajectory and t e the emission time.When interfering with the light from the pulsars, the GW has therefore acquired a phase shift of the order of f δt compared to the propagation in a homogeneous universe.This would not a problem were the phase shift the same for all the GW measurements.However, if they vary, the average correlation of waves at three points would pick up a factor of exp(i which gives an exponential suppression of the three-point correlator of the time residuals when evaluated at the age of the universe [18,19], unless one considers flattened triangle configurations in frequency space [20].Eventually, one might also hope to escape the suppression if the GWs are produced within the same structure, e.g. by clustered binaries of supermassive black holes.
5 Recovering the overlap function for an array of "short-arm" LIGO-like detectors The Hellings and Downs correlation is obtained in the so-called long-arm approximation, when the wavelengths of the GWs are much smaller than the distance between the earth and the pulsars.In the opposite limit, the so-called short-arm, in which the wavelengths are much larger than the arms of the detectors, e.g., LIGO and Virgo, the overlap function is also easily recovered by our symmetry arguments as follows.
The time residual is written in the short-arm approximation as and therefore, the correlator ⟨z a z b ⟩ turns out to be in this case where the overlap function is given by Repeating the steps of section 4, we define Γ f − ab = Γ f ab − Γ f v , which transforms under Lorentz boost as where the transformation of Eq.(3.8) has been used.Then, by using the fact that, since Γ f ab is quadratic in n a , it is invariant under n a → − n a or, equivalently, under σ ab → 1 − σ ab , we get which is solved by Implementing the σ ab → 1 − σ ab symmetry of the overlap function we find (5.8) In addition, the identity leads to the condition as the polarization tensors are traceless.Using Eq. (5.8) in Eq. (5.10) we find that Γ f v = −α/3.Therefore, the final expression for the overlap function in the short-arm approximation turns out to be with P 2 the ℓ = 2 Legendre polynomial P ℓ .It provides the same value at opposite angles.The coefficient α is an arbitrary parameter, which can be set to 3 to reproduce the value reported in the literature [11].

Conclusions
We have shown that the HD correlation function can be recovered by simple symmetry arguments once it is realized that Lorentz symmetries act as conformal symmetries on the celestial sphere and that the HD correlation function has well-defined conformal transformation properties.
We believe that this result is useful to describe variations of the standard HD function, for instance when there are additional polarization modes (vector and/or scalar) that may arise in alternative-gravity theories [21,22] for which a different angular correlation functions are expected.
Anisotropy, linear, or circular polarization in the stochastic GW background [23] gives rise to extra structure in the two-point correlation function and cannot be written simply in terms of the angular separation of the two pulsars.One relevant question we would like to investigate is if a deformation of the conformal field theory on the celestial sphere can help in describing these cases too.
Then, it turns out that the two-point correlators C 2 (0) = Γ ab (σ) C Proof that the integral of the HD correlation function over the celestial sphere vanishes In order to prove Eq. (4.33), let us fix n b = (0, 0, 1) and let n a = (sin θ cos ϕ, sin θ sin ϕ, cos θ).Then from the definition of the two-point correlator in Eq. (4.9) we get

Figure 1 .
Figure 1.The stereographic projection and projective coordinates of the two-dimensional sphere S 2 .
(x, y) plane at z = 0 can be identified with C by defining ζ = x + iy.The stereographic projection ζ : S 2 \{N } → C is then the map that associates to every point (X, Y, Z) ∈ S 2 \{N } the intersection ζ of the line which connects the north pole N with (x, y, z = 0).In other words, we have

Figure 2 .
Figure 2. The GW source is in the − n direction and the GW propagates towards the origin.

First, as shown 1 0
in Appendix B, by using the fact that the two-point correlator of two pulsars separated by π (σ ab = 1) is half that for 0 angular separation (σ = 0), we get that Γ ab = α σ ab ln σ ab + β σ ab − 2β.(4.32)Secondly, as shown in Appendix C, one can prove that the integral of the two-point function over the whole celestial sphere should vanish and therefore d 2 Ω 2 Γ ab = −4π dσ ab α σ ab ln σ ab + β σ ab − 2β = π(α + 6β)

d 2 Ω 2 Γ ab = d 2 n a n i a n j a 1 −2 n a n i a n j a 1 − n • n a = d 2 n a n i a n j a 1 −n a n i a n j a 1 −
n • n a × A=+,× ϵ A ij ( n) × (terms independent of n a ).(C.1)Now we want to evaluate the integral of the right-hand side of Eq. (C.1), which due to symmetry should be written asd s n • n a s=1 = A(s)n i n j + B(s)δ ij s=1 , (C.2)where A, B are constants.Since the above expression is contracted with the transverse and traceless polarization tensors ϵ A ij ( n) we have that in generalA(s)n i n j + B(s)δ ij ϵ A ij ( n) n • n a A=+,× ϵ A ij ( n) = 0 or d 2 Ω 2 Γ ab = 0,(C.4)asclaimed.Let us note that we have inserted the parameter s ≥ 1 in Eq. (C.2) in order to avoid the pole at n • n a = 1.We could equally use a +iϵ shift in the denominator.Both ways lead to the vanishing of the integral in Eq. (C.4).
.15) Due to the aforementioned isomorphism, one can prove that boosts in the z-direction in Minkowski spacetime by a hyperbolic angle ξ generated by K 3 , corresponds to the conformal (Möbius) transformation ζ → e ξ ζ.In analogy, the generators K 1 , K 2 of SL(2, C) generate boosts in the x and −y directions.Similarly, L 1 , L 2 and L 3 , generate rotations around the −x, y and −z directions,