Measuring the net circular polarization of the stochastic gravitational wave background with interferometers
Valerie Domcke, Juan Garcia-Bellido, Marco Peloso, Mauro Pieroni, Angelo Ricciardone, Lorenzo Sorbo, Gianmassimo Tasinato
DDESY 19-161
Measuring the net circular polarization of thestochastic gravitational wave background with interferometers
Valerie Domcke a , Juan Garc´ıa-Bellido b , Marco Peloso c,d , Mauro Pieroni b,e ,Angelo Ricciardone c , Lorenzo Sorbo f , Gianmassimo Tasinato ga Deutsches Elektronen Synchrotron (DESY), 22607 Hamburg, Germany b Instituto de F´ısica Te´orica UAM/CSIC, Nicol´as Cabrera 13, Universidad Aut´onoma deMadrid, Cantoblanco 28049 Madrid, Spain c INFN, Sezione di Padova, via Marzolo 8, I-35131, Padova, Italy d Dipartimento di Fisica e Astronomia “G. Galilei”, Universit`a degli Studi di Padova,via Marzolo 8, I-35131, Padova, Italy e Theoretical Physics, Blackett Laboratory, Imperial College, London, SW7 2AZ,United Kingdom f Amherst Center for Fundamental Interactions, Department of Physics, University ofMassachusetts, Amherst, MA 01003, USA g Department of Physics, Swansea University, Swansea, SA2 8PP, United Kingdom
Abstract
Parity violating interactions in the early Universe can source a stochastic gravitationalwave background (SGWB) with a net circular polarization. In this paper, we study possibleways to search for circular polarization of the SGWB with interferometers. Planar detectorsare unable to measure the net circular polarization of an isotropic SGWB. We discuss thepossibility of using the dipolar anisotropy kinematically induced by the motion of the solarsystem with respect to the cosmic reference frame to measure the net circular polarizationof the SGWB with planar detectors. We apply this approach to LISA, re-assessing previousanalyses by means of a more detailed computation and using the most recent instrumentspecifications, and to the Einstein Telescope (ET), estimating for the first time its sensitivityto circular polarization. We find that both LISA and ET, despite operating at differentfrequencies, could detect net circular polarization with a signal-to-noise ratio of order one in aSGWB with amplitude h Ω GW (cid:39) − . We also investigate the case of a network of groundbased detectors. We present fully analytical, covariant formulas for the detector overlapfunctions in the presence of circular polarization. Our formulas do not rely on particularchoices of reference frame, and can be applied to interferometers with arbitrary angles amongtheir arms. a r X i v : . [ a s t r o - ph . C O ] O c t Introduction
A direct detection of the SGWB represents a major future target of gravitational wave (GW)experiments working at interferometer scales. The characterization of the SGWB properties, andthe corresponding detection strategies, are essential for distinguishing between a cosmologicaland an astrophysical origin of the signal. See e.g. [1–6] for comprehensive reviews on theoreticaland experimental aspects of the physics of SGWBs. Among the properties that can characterizea SGWB is an intrinsic circular polarization, associated with an asymmetry in the amplitude ofGWs of left and right polarizations.The astrophysical SGWB is a combination of several independent signals from uncorrelatedsources. Therefore, we do not expect the astrophysical SGWB to carry a net polarization. Onthe other hand, cosmological SGWBs can be produced coherently (for example, the SGWBfrom inflation): if this coherence is accompanied by interactions that violate parity, then a cos-mological SGWB with net circular polarization can be generated. In fact, a sizable degree ofpolarization can be generated in well-motivated models of inflation with spontaneous parity vio-lation, manifesting itself e.g. in Chern-Simons couplings between the inflaton φ and curvature (as φ R ˜ R , [7–10]) or gauge fields (as φ F ˜ F , see e.g. [11–16]). Such a scenario, and its consequencesfor CMB polarization experiments, is the subject of active research, see e.g. [17–20] for reviews.Interestingly, recent numerical analysis [21] show that post-inflationary physics associated withmagnetohydrodynamic turbulence, in the presence of helical initial magnetic fields, can also giverise to net circular polarization of a SGWB potentially detectable with LISA. In this work, we willstudy the prospects for detecting a net circular polarization in the SGWB in GW interferometryexperiments. A positive detection would provide a smoking gun for parity violating effects andfor a cosmological origin of the SGWB signal.It has been proven [22–24] that parity violating effects in an isotropic SGWB can not bedetected by correlating a system of coplanar detectors. A planar interferometer responds in thesame way to a left-handed GW of wave vector (cid:126)k and to a right-handed GW of the same amplitudeand of wave vector (cid:126)k p , obtained from (cid:126)k by changing sign of the component of (cid:126)k perpendicularto the plane of the detector. In particular, this is the case for LISA and ET, which are planarinstruments. A way out of this argument is provided by an anisotropic SGWB [25, 26], since inthis case the GW arriving from the direction (cid:126)k have a different amplitude than those from the (cid:126)k p direction. Moreover, this problem is not present when one correlates signals from differentGW detectors which do not lie on the same plane [22, 23, 27], as is the case for a network ofground-based interferometers.In this work, we start from the consideration that a SGWB that is (statistically) isotropic inone frame O is not (statistically) isotropic in any other frame that is boosted with respect to O .This is true for any stochastic background, and this is for example the origin of the CMB dipole,which is induced kinematically by the motion of the solar system frame with respect to the cosmicreference frame. The latter is defined to be the one in which the CMB is statistically isotropic,and it is the rest frame of the cosmic fluid. It is reasonable to assume that this is also the framein which the SGWB is isotropic. The fact that measurements of parity odd SGWB anisotropiesallows the detection of circular polarization was already noticed and developed in [25, 26]. Inthe present work, we present a more detailed computation for the LISA instrument, discussing In fact, the analysis described in this paper will allow us to test this hypothesis, if the SGWB has a net circularpolarization.
2n full extent the properties of the instrument response functions under parity symmetry in thepresence of a dipolar anisotropy, and clarifying the relation between these properties and theSGWB circular polarization. Using the most up-to-date LISA instrument specifications, andtaking into account the full frequency band of the instrument, we re-assess the evaluation ofthe magnitude of the signal-to-noise ratio associated with measurements of the SGWB circularpolarization, obtaining a result about one order of magnitude greater that that of [25].This analysis can be readily extended to the the proposed ground-based Einstein Telescope(ET). A single third-generation telescope of this type features a planar configuration similarto that of LISA. Using also in this case the kinematically induced dipole, we estimate for thefirst time the signal-to-noise ratio (SNR) for this measurement at ET. We find that both LISAand ET, despite operating at different frequencies, could detect net circular polarization with asignal-to-noise ratio of order one in a SGWB with amplitude h Ω GW (cid:39) − .We then consider correlations of ground-based interferometers. In this case, as mentionedabove, a net circular polarization can already be measured from the SGWB monopole (namely,from its statistically isotropic component), since a network of two or more detectors is genericallynot coplanar. Such an analysis was already performed in [27] for the second and third generationground-based interferometers. While in [27] a numerical evaluation of the parity-dependentoverlap functions was employed, in this paper we compute, for the first time, the full analytic formof these functions (the overlap functions for parity even backgrounds were computed analyticallyin [30]). We present ‘covariant’ analytic formulas for overlap functions describing correlationsamong ground based interferometers in the small antenna limit (which applies to all existingground-based interferometers), also including the kinematically induced dipolar anisotropy. Ourexpressions are valid for any amount of polarization of the SGWB (namely, we provide separateformulas for the left-handed and the right-handed GW), they do not rely on any special choicesof frame (this is why we call them covariant), and they hold for arbitrary detector shape (namely,they are not limited to interferometers with orthogonal arms). While the angular integralsnecessary to obtain the overlap functions can be also computed numerically [27], evaluating theanalytic formulae given here is significantly faster, and we hope that it might speed up suchanalyses.The structure of the paper is the following: in Section 2 we compute the GW two-pointfunction for a detector which is boosted with respect to a frame in which the SGWB is isotropic;in Section 3 we present the dipole response functions for measuring the net circular polarization ofthe SGWB with LISA. We turn to ground-based detectors in Section 4, considering both the caseof cross-correlations among a network of (not coplanar) interferometers, for which already themonopole overlap function is sensitive to chirality, as well as the proposed Einstein Telescope,which can measure chirality upon taking into account the kinematic dipole. We conclude in ET will be a ground-based interferometer with a triangular shape, like LISA, with the difference that thearm length is L = 10 km. It will be an observatory of the third generation aiming to reach a sensitivity for GWsignals emitted by astrophysical and cosmological sources about a factor of 10 better than the advanced detectorscurrently operating. It will be formed by three detectors, each in turn composed of two interferometers (xylophoneconfiguration) [28, 29]. To detect chirality, we need to measure P L and P R separately, so at least three interferometers are needed.Two interferometers are enough if one assumes as an input the spectral form of the signal, as in this case themeasurements at different frequencies can be combined together [27]. For planar interferometers such as LISAand ET, the needed plurality of measurements is guaranteed by the different time-delay-interferometers at theirvertices. Let us assume that there exists a frame in which the SGWB is (statistically) isotropic. It isnatural to associate this frame to the cosmological frame, in which the CMB is isotropic. Thepeculiar motion of the solar system in this frame will kinematically make the observed SGWBanisotropic, as is this the case for the CMB, where it is found that our local system is movingwith speed v = 1 . × − in a direction ( φ E , θ E ) = (172 ◦ , − ◦ ) in ecliptic coordinates (seee.g. [31]). The possibility to detect a kinematically-induced dipolar anisotropy with ground basedexperiments was first quantitatively explored in [32], and more recently re-assessed in [33] for thespace-based experiment DECIGO. In this Section, we derive general formulas describing how adipolar anisotropy is induced on an otherwise isotropic SGWB. In Section 3, we use these resultsto study how such dipolar anisotropy can enable the detection of the net circular polarization ofa SGWB with the LISA instrument.We compute the GW two-point function seen by an observer who is moving with a constantvelocity (cid:126)v with respect to a frame in which the SGWB is isotropic. The motion with velocity (cid:126)v ofthe observer generates a dipole in the observed GW power spectrum at order v , a quadrupole atorder v and so on. Under the assumption that v (cid:28) O ( v ).We start the computation by considering a frame { t, (cid:126)x } in which the SGWB is isotropic.In this frame, we decompose the tensor field into modes of definite circular polarization, with λ = ± h ij ( t, (cid:126)x ) = (cid:90) d k e − πi(cid:126)k · (cid:126)x (cid:88) λ e ij,λ (ˆ k ) h λ ( t, (cid:126)k ) , (1)where the GW polarization operators in the chiral basis e ab,λ (ˆ k ) are introduced in AppendixA. The mode momentum-space operators of definite helicity satisfy the condition h λ ( t, (cid:126)k ) = h λ ( t, − (cid:126)k ) ∗ which, together with the property (A3), ensures that the expression (1) is real. Thisexpression satisfies the wave equation for a massless particle, which is solved by h λ ( t, (cid:126)k ) = A λ(cid:126)k cos(2 πk t ) + B λ(cid:126)k sin(2 πk t ) , (2)where A λ(cid:126)k = ( A λ − (cid:126)k ) ∗ and B λ(cid:126)k = ( B λ − (cid:126)k ) ∗ are stochastic variables that obey (cid:104) A λ(cid:126)k A λ (cid:48) (cid:126)k (cid:48) (cid:105) = (cid:104) B λ(cid:126)k B λ (cid:48) (cid:126)k (cid:48) (cid:105) = P λ ( k )4 πk δ λλ (cid:48) δ ( (cid:126)k + (cid:126)k (cid:48) ) , (cid:104) A λ(cid:126)k B λ (cid:48) (cid:126)k (cid:48) (cid:105) = 0 , (3)where P λ ( k ) is the GW helicity- λ power spectrum, depending only on the absolute value k due tostatistical isotropy. We note that, with our 2 π convention, k = | (cid:126)k | is the frequency of the mode.4oreover, we haveno net circular polarization ⇔ P R ( k ) = P L ( k ) ⇔ (cid:88) λ λ P λ ( k ) = 0 . (4)Equations (3) derive from the requirement that the equal time correlator takes the time-independentform (cid:104) h λ ( t, (cid:126)k ) h λ (cid:48) ( t, (cid:126)k (cid:48) ) (cid:105) ≡ P λ ( k )4 πk δ λλ (cid:48) δ ( (cid:126)k + (cid:126)k (cid:48) ) . (5)The gravitational wave correlator at arbitrary times then reads (cid:104) h ij ( (cid:126)x, t ) h i (cid:48) j (cid:48) ( (cid:126)x (cid:48) , t (cid:48) ) (cid:105) = (cid:88) σ (cid:90) d k πk e − πi(cid:126)k · ( (cid:126)x − (cid:126)x (cid:48) ) e ij,σ (ˆ k ) e i (cid:48) j (cid:48) ,σ ( − ˆ k ) P σ ( k ) cos(2 πk ( t − t (cid:48) ))= 12 (cid:88) σ (cid:90) d k πk e − πi(cid:126)k · ( (cid:126)x − (cid:126)x (cid:48) )+2 πik ( t − t (cid:48) ) e ij,σ (ˆ k ) e i (cid:48) j (cid:48) ,σ ( − ˆ k ) P σ ( k )+ 12 (cid:88) σ (cid:90) d k πk e − πi(cid:126)k · ( (cid:126)x − (cid:126)x (cid:48) ) − πik ( t − t (cid:48) ) e ij,σ (ˆ k ) e i (cid:48) j (cid:48) ,σ ( − ˆ k ) P σ ( k ) . (6)We now perform a boost to a frame { τ, (cid:126)y } that is moving with constant velocity (cid:126)v , directedalong the first coordinate, with respect to the { t, (cid:126)x } frame t = γ ( τ − v y ) , x = γ ( y − v τ ) , x = y , x = y , (7)where γ ≡ / √ − v . Being a rank-2 tensor, h ij transforms as h ij ( x , x , x , t ) = h ab ( γ ( y − v τ ) , y , y , γ ( τ − v y )) ∂y a ∂x i ∂y b ∂x j (cid:39) h ij ( γ ( y − v τ ) , y , y , γ ( τ − v y )) + O (cid:0) v (cid:1) . (8)Let us perform this transformation on the decomposition (6). To preserve the same plane wavestructure of the phase in the decomposition, we simultaneously perform a change in the integrationvariable, which can be also thought of as a boost on the momenta, with opposite signs of theboost parameter depending on whether we are in the negative (second line of eq. (6), (cid:126)k (cid:55)→ (cid:126)q ) orpositive (third line of eq. (6), (cid:126)k (cid:55)→ (cid:126)p ) frequency component of the unequal-time correlator,second line of eq . (6) third line of eq . (6) k = γ ( q − v q ) k = q k = q k = γ ( q − v q ) , k = γ ( p + v p ) k = p k = p k = γ ( p + v p ) (9)with q ≡ | (cid:126)q | and p ≡ | (cid:126)p | . Therefore, the unequal time correlator in the boosted frame can bewritten as (cid:104) h ij ( (cid:126)y, τ ) h i (cid:48) j (cid:48) ( (cid:126)y (cid:48) , τ (cid:48) ) (cid:105) = 12 (cid:88) σ (cid:90) d k πk e − πi(cid:126)q · ( (cid:126)y − (cid:126)y (cid:48) )+2 πiq ( τ − τ (cid:48) ) e ij,σ (ˆ k ) e i (cid:48) j (cid:48) ,σ ( − ˆ k ) P σ ( k ) Here we are considering the present-day SGWB, evaluated at times relevant for the detection. When consideringcosmological time scales (e.g. when comparing with the primordial power spectrum), the expansion of the Universemust be taken into account, encoded in the cosmic transfer function . (cid:88) σ (cid:90) d k πk e − πi(cid:126)p · ( (cid:126)y − (cid:126)y (cid:48) ) − πip ( τ − τ (cid:48) ) e ij,σ (ˆ k ) e i (cid:48) j (cid:48) ,σ ( − ˆ k ) P σ ( k ) , (10)where the dependence on the velocity (cid:126)v is hidden in the relation between the variables (cid:126)q , (cid:126)p and (cid:126)k . In the following, we perform explicit computations only on the first term on the right handside of eq. (10), since the second one is obtained from the first one with the replacements (cid:126)q → (cid:126)p , (cid:126)v → − (cid:126)v , τ ↔ τ (cid:48) . We obtain the correlator for the variables of definite helicity in momentumspace (cid:104) h λ ( (cid:126)l, τ ) h λ (cid:48) ( (cid:126)l (cid:48) , τ (cid:48) ) (cid:105) ≡ e ij,λ ( − ˆ l ) e i (cid:48) j (cid:48) ,λ (cid:48) ( − ˆ l (cid:48) ) (cid:90) d y d y (cid:48) e πi(cid:126)l · (cid:126)y +2 πi(cid:126)l (cid:48) · (cid:126)y (cid:48) (cid:104) h ij ( (cid:126)y, τ ) h i (cid:48) j (cid:48) ( (cid:126)y (cid:48) , τ (cid:48) ) (cid:105) = δ ( (cid:126)l + (cid:126)l (cid:48) ) e ij,λ ( − ˆ l ) e i (cid:48) j (cid:48) ,λ (cid:48) (ˆ l ) (cid:34) (cid:88) σ (cid:90) d k π k e πiq ( τ − τ (cid:48) ) δ ( (cid:126)q − (cid:126)l ) e ij,σ (ˆ k ) e i (cid:48) j (cid:48) ,σ ( − ˆ k ) P σ ( k )+ ( (cid:126)q → (cid:126)p, (cid:126)v → − (cid:126)v, τ ↔ τ (cid:48) ) (cid:35) . (11)Our task is then to eliminate (cid:126)k from the last equation, expressing it in terms of (cid:126)q only.Firstly, from d k = γ (1 − ˆ q · (cid:126)v ) d q and k = γ ( q − (cid:126)q · (cid:126)v ), we obtain d kk = (1 + 2 ˆ q · (cid:126)v ) d qq + O (cid:0) v (cid:1) . (12)Secondly, we decompose the product of the two polarization operators in eq. (11) in terms offour 1-index quantities e i,λ (see eq. (A1 )) and we use the identity (A4), that we can express as afunction of ˆ q using the relation ˆ k = ˆ q − (cid:126)v + ˆ q (ˆ q · (cid:126)v ) + O (cid:0) v (cid:1) , with ˆ q = ˆ l as a consequence of theDirac delta in eq. (11). Using these relations and the property l i e ij,λ ( − ˆ l ) = 0, we find that, tofirst order in (cid:126)v , the part of eq. (11) that depends on the polarization operators does not receiveany correction at linear order in v : e ij,λ ( − ˆ l ) e i (cid:48) j (cid:48) ,λ (cid:48) (ˆ l ) e ij,σ (ˆ k ) e i (cid:48) j (cid:48) ,σ ( − ˆ k ) δ ( (cid:126)q − (cid:126)l ) = e ij,λ ( − ˆ l ) e i (cid:48) j (cid:48) ,λ (cid:48) (ˆ l ) e ij,σ (ˆ l ) e i (cid:48) j (cid:48) ,σ ( − ˆ l ) + O ( v )= δ λσ δ λσ (cid:48) + O ( v ) . (13)Finally we expand P λ ( k ) = P λ ( γ ( q − (cid:126)q · (cid:126)v )) = P λ ( q ) − ( (cid:126)q · (cid:126)v ) P λ (cid:48) ( q ) + O ( v ).Using these results, and accounting for both terms in the second line of eq. (10), we finallyobtain the correlator in the boosted frame (cid:104) h λ ( (cid:126)l, τ ) h λ (cid:48) ( (cid:126)l (cid:48) , τ (cid:48) ) (cid:105) = δ λλ (cid:48) δ (3) ( (cid:126)l + (cid:126)l (cid:48) )4 π l (cid:40) P λ ( l ) cos[2 πl ( τ − τ (cid:48) )]+ i (ˆ l · (cid:126)v ) (cid:104) P λ ( l ) − l P λ (cid:48) ( l ) (cid:105) sin[2 πl ( τ − τ (cid:48) )] (cid:41) + O ( v ) . (14)It is worth noting that the dipole contribution vanishes in the equal-time case. This is because (cid:104) h λ ( (cid:126)l, τ ) h λ ( (cid:126)l (cid:48) , τ (cid:48) ) (cid:105) = (cid:104) h λ ( (cid:126)l (cid:48) , τ (cid:48) ) h λ ( (cid:126)l, τ ) (cid:105) , which implies that the correlator is invariant under (cid:126)l ↔ (cid:126)l (cid:48) in the equal time case. 6 Measuring the SGWB net circular polarization with LISA
We now discuss how the kinematically induced dipolar anisotropy can be used to measure thenet circular polarization of SGWBs with the planar interferometer LISA. This was first studiedin [25,26], where it was noticed that a measurement of parity odd SGWB anisotropies can be usedto detect parity violating effects in gravitational interactions. Those works focus on the smallfrequency limit of the detector response functions, and make use of the properties of the detectorin such regime, as discussed in [34, 35]. In our work, we first systematically discuss, in Section3.1, the general properties of the instrument response functions under parity symmetry, clarifyingthe relation between these properties and measurements of circular polarization. In Section 3.2,using the most up-to-date LISA instrument specifications and performing an analysis over thefull LISA frequency band, we re-assess the evaluation of the signal-to-noise ratio associated withmeasurements of the SGWB circular polarization.
The space-based laser interferometer LISA [36] will be a constellation of three satellites placedat the vertices (here placed at the positions { (cid:126)x , (cid:126)x , (cid:126)x } ) of an (approximate) equilateral trianglewith side length L = 2 . i -th interferometer is obtained byintegrating along the photon geodesic taking into account the perturbation of the metric due tothe gravitational wave. The result can be expressed as a convolution of the gravitational wavewith the response function Q i containing the geometry of the detector [6, 37, 38], σ i ( t ) ≡ δtt = δt L = (cid:88) λ (cid:90) d k h λ ( (cid:126)k, t − L ) e ab,λ (ˆ k ) Q iab ( (cid:126)x i , (cid:126)k ; { ˆ U j } ) , (15)with Q iab ( (cid:126)x i , (cid:126)k ; { ˆ U j } ) = 14 e − πi(cid:126)k · (cid:126)x i (cid:104) T ( kL, ˆ k · ˆ U i ) ˆ U ai ˆ U bi − T ( kL, − ˆ k · ˆ U i +2 ) ˆ U ai +2 ˆ U bi +2 (cid:105) , (16)where ˆ U i ≡ (cid:126)x i +1 − (cid:126)x i L is the unit vector in the direction of the arm that goes from the satellite (cid:126)x i to the satellite (cid:126)x i +1 . All indices { i, i + 1 , . . . } in eq. (16) are understood to be modulo 3.The detector transfer function T is given by T ( kL, ˆ k · ˆ U i ) ≡ e − π i k L [1+ˆ k · ˆ U i ] sinc (cid:104) π k L (cid:16) − ˆ k · ˆ U i (cid:17)(cid:105) + e π i k L [1 − ˆ k · ˆ U i ] sinc (cid:104) π k L (cid:16) k · ˆ U i (cid:17)(cid:105) , (17)which reduces to T (cid:39) kL (cid:28)
1. 7erforming linear combinations of the interferometers (cid:126)x i we can construct the Time DelayInterferometry (TDI) LISA channels { A, E, T } [39]Σ A ≡
13 (2 σ X − σ Y − σ Z ) , Σ E ≡ √ σ Z − σ Y ) , Σ T ≡
13 ( σ X + σ Y + σ Z ) . (18)For an isotropic background, we can exploit the symmetry under the exchange of the verticesof the equilateral triangle to see that all self correlators among σ X , σ Y , σ Z are equal to eachother, as are all cross correlations. This in particular implies (cid:104) Σ A Σ A (cid:105) = (cid:104) Σ E Σ E (cid:105) , while the crosscorrelations among Σ A , Σ E and Σ T vanish. As we will see explicitly below, these statements donot apply to anisotropic components of the SGWB.The signal induced by a passing gravitational wave in the channels O = { A, E, T } isΣ O ( t ) = (cid:88) λ (cid:90) d k h λ ( (cid:126)k, t − L ) e ab,λ (ˆ k ) Q Oab ( (cid:126)k ; { ˆ x j } ) , (19)with Q Oab ( (cid:126)k ; { ˆ x j } ) = (cid:80) i c Oi Q iab ( (cid:126)x i , (cid:126)k ; { ˆ U j } ), where the matrix c is given by c = − − − √ √
313 13 13 . (20)For more details on the derivation and notation, see Ref. [38]. Combining eq. (14) and (19) yields the two-point correlation function in the time domain, (cid:10) Σ O ( t )Σ O (cid:48) ( t (cid:48) ) (cid:11) = 14 (cid:88) λ (cid:90) dkk (cid:104) M λOO (cid:48) ( k ) P λ ( k ) cos (cid:2) πk ( t − t (cid:48) ) (cid:3) + v D λOO (cid:48) (cid:0) P λ ( k ) − kP (cid:48) λ ( k ) (cid:1) sin (cid:2) πk ( t − t (cid:48) ) (cid:3)(cid:105) , (21)where we have introduced the monopole and dipole response functions M λOO (cid:48) ( k ) ≡ (cid:90) d Ω ˆ k π e ab,λ (ˆ k ) e a (cid:48) b (cid:48) ,λ ( − ˆ k ) Q Oab ( (cid:126)k ) Q O (cid:48) a (cid:48) b (cid:48) ( − (cid:126)k ) , (22) D λOO (cid:48) ( k, ˆ v · ˆ n ) ≡ i (cid:90) d Ω ˆ k π e ab,λ (ˆ k ) e a (cid:48) b (cid:48) ,λ ( − ˆ k ) Q Oab ( (cid:126)k ) Q O (cid:48) a (cid:48) b (cid:48) ( − (cid:126)k ) ˆ k · ˆ v , (23)where ˆ n is the normal to the plane of LISA, that, for definiteness, we take it to be orientedupwards for an observer for whom the vertices labeled as (cid:126)x , (cid:126)x , (cid:126)x follow one another in theanti-clockwise direction.The two response functions satisfy the following properties1. M λOO (cid:48) and D λOO (cid:48) are real,2. M λOO (cid:48) does not depend on the orientation of the detector; D λOO (cid:48) depends on the directionof the detector only through the cosine of the angle between ˆ n and ˆ v ,8. M λOO (cid:48) → M λOO (cid:48) , D λOO (cid:48) → −D λOO (cid:48) if (cid:126)v → − (cid:126)v ,4. M ROO (cid:48) ( k ) = M LOO (cid:48) ( k ) , D ROO (cid:48) ( k, ˆ v · ˆ n ) = −D LOO (cid:48) ( k, ˆ v · ˆ n ),which we now prove.The first property immediately follows from the fact that Q Oab ( − (cid:126)k ) = ( Q Oab ( (cid:126)k )) ∗ , and identi-cally for the GW polarization operators.The second property is a consequence of statistical isotropy of the monopole, and of thestatistical isotropy of the dipole under rotations that preserve the direction of (cid:126)v . Let us verifythat the above relations ensure these properties. We start by noting that the transfer function T depends on ˆ k only through ˆ k · ˆ U i . The argument in the exponential pre-factor in Q i can beexpressed as 2 πik ˆ k · ( (cid:126)x + ( (cid:126)x i − (cid:126)x )) with (cid:126)x denoting the center of the equilateral triangle formedby the three satellites. The factor exp(2 πik ˆ k · (cid:126)x ) is thus universal to all Q i and drops out in thedipole response function due to the property Q Oab ( − (cid:126)k ) = ( Q Oab ( (cid:126)k )) ∗ . The remaining factor canalso be written as a scalar product between ˆ k and the direction of the LISA arms. For instance,for i = 1, we have ˆ k · ( (cid:126)x − (cid:126)x ) = ˆ k · (cid:18) (cid:126)x − (cid:126)x + (cid:126)x + (cid:126)x (cid:19) = ˆ k · (cid:16) ˆ U − ˆ U (cid:17) . (24)and analogously for i = 2 ,
3. Therefore, Q Oab ( (cid:126)k ) Q O (cid:48) a (cid:48) b (cid:48) ( − (cid:126)k ) = function of ˆ k · ˆ U , ˆ k · ˆ U , and of ˆ k · ˆ U . (25)As a consequence, any rotation of the LISA instrument (that for this discussion we consider as arigid equilateral triangle) can be “compensated” by a rotation of ˆ k . The rotation of ˆ k does notchange the monopole response function (22), as this is just the integration variable. It follows thatevery orientation of the instrument results in the same value for the monopole response function.In the case of the dipole response function, any change of the orientation of the instrument can be“compensated” by a rotation of ˆ k and of (cid:126)v , (since also the last factor must be unchanged). Again,since ˆ k is simply an internal variable, it follows that the dipole response function does not changeif we rotate both the instrument and (cid:126)v . If we now consider a rotation around the direction of (cid:126)v ,we then see that the dipole response function (23) is unchanged for rotations of the instrumentthat do not change the angle between (cid:126)v and the normal to the plane of the instrument. Therefore,it depends on the orientation of the instrument and of the dipole only through the product ˆ v · ˆ n .More specifically, if we consider a coordinate system in which ˆ n is directed along the z − axis, wesee that the last factor in eq. (23) factorizes a cosine of this angle.The third property follows immediately from the properties that we just proved, and fromthe definition of the response functions.The fourth property will be essential for our aim of measuring the SGWB circular polarization.To prove it, let us consider a mirror transformation with respect to the plane of the detector.Under this transformation, the component of a vector along ˆ n (that we denote as ⊥ ) changes sign,while the component of the vector on the plane of the detector (that we denote as // ) remainsinvariant. Therefore, the product Q Oab ( (cid:126)k ) Q O (cid:48) a (cid:48) b (cid:48) ( − (cid:126)k ) is invariant under this symmetry, due to (25).As seen from eq. (16), only the components of e ab,λ along the plane of the detector contributeto the response functions. One can verify by direct inspection (by using the explicit form ofeq. (A2)) that these components are unchanged if we perform this mirror transformation and9 AA R = M AA L = M EE R = M EE L - M OO σ ( k ) Figure 1:
Monopole response function. It vanishes in the AE cross-correlation channel while isidentical in the AA and EE auto-correlation channels and is insensitive to the chirality of theSGWB. we simultaneously change the GW chirality. Namely, e ab,λ Q iab (cid:0) k // , k ⊥ (cid:1) = e ab, − λ Q iab (cid:0) k // , − k ⊥ (cid:1) ,as we already proved in [38]. Under the mirror transformation, v ⊥ changes sign. Therefore, theintegrand of the monopole response function is unchanged if we perform this mirror symmetry,and we flip the two helicities, while the integrand of the dipole response function changes signunder the same transformations. The change of (cid:126)k can be then “undone” by a change of theintegration variable. This implies that the monopole response function is invariant when we flipthe two helicities, while the dipole response function changes sign.Having proved the above properties, let us now consider a re-labeling of two satellites, say (cid:126)x ↔ (cid:126)x . We see from the definitions (18) that the Σ A measurement is invariant under thisre-labeling, while Σ E changes sign. Therefore, the self-correlators (cid:104) Σ A Σ A (cid:105) and (cid:104) Σ E Σ E (cid:105) are evenunder the re-labeling, while the cross-correlator (cid:104) Σ A Σ E (cid:105) is odd. The re-labeling has the effectof inverting the direction of the normal to the plane of the instrument, as we have defined itbelow eq. (23). Due to the property (2.) demonstrated above, the monopole response function isinvariant under this inversion, while the dipole response function changes sign. Therefore M λAE = 0 , D λAA = D λEE = 0 . (26)These relations can be immediately verified by a direct evaluation of eqs. (22) and (23). InFigs. 1 and 2 we depict the monopole response functions for the AA and EE channel as well asthe dipole response function for the AE channel. We recall (property (4.) above) that the dipoleresponse function is odd under a flip of helicity, λ (cid:55)→ − λ , again reflecting that the dipole responsefunction is parity odd. In particular, due the summation over helicity, the total two-point function (cid:104) Σ A Σ E (cid:105) will only be non-zero if the stochastic background is chiral, i.e. if P λ ( k ) (cid:54) = P − λ ( k ).An important consequence of this is that one should be careful in assuming that a nonvanishingvalue for (cid:104) Σ A Σ E (cid:105) would be due only to noise. As we proved above, this cross-correlator vanishes Similarly, re-labeling of the tensor indices in eq. (23) while simultaneously flipping ˆ k (cid:55)→ − ˆ k yields D λAE = −D λEA .Consequently, since (cid:104) Σ A ( t )Σ E ( t (cid:48) ) (cid:105) = (cid:104) Σ E ( t (cid:48) )Σ A ( t ) (cid:105) , we conclude that the dipole contribution to (cid:104) Σ A ( t )Σ E ( t (cid:48) ) (cid:105) must be odd under the exchange t ↔ t (cid:48) , as reflected by the sine function in eq. (21). On the contrary, theauto-correlations (cid:104) Σ A ( t )Σ A ( t (cid:48) ) (cid:105) and (cid:104) Σ E ( t )Σ E ( t (cid:48) ) (cid:105) trivially have to be even under t ↔ t (cid:48) . AE R = - D AE L - - - | D AE R ( k , v ) | / c o s ( α ) Figure 2:
Absolute value of the dipole response function. The dashed (solid) line indicates positive(negative) values for D RAE = −D LAE . The angle α denotes the angle between the orientation ofthe dipole ˆ v and the plane of the detector. in presence of the monopole only, and one might be tempted to use any non-zero result as atoll for noise characterization. We have shown that this quantity is actually non-vanishing if theSGWB has a net polarization. As discussed above, the dipole response function (23) depends only on the angle between thedipole and the normal vector of the detector plane, ˆ v · ˆ n . The directional sensitivity of theintegrand of eq. (23) is more involved, encoding the geometrical sensitivity of the detector todifferent sky regions, the so-called antenna pattern. The antenna pattern of the monopole re-sponse function shows that GW interferometers are most sensitive to GWs arriving orthogonallyto the detector plane (see e.g. [37]). In Fig. 3 we depict the corresponding dipole antenna pat-tern, taking into account that, due to the motion of the LISA-plane around the sun, the effectivedipole will receive an annual modulation. See Section 3.2 for more details about the LISA orbitparametrization.These antenna patterns give allow for a qualitative understanding of the resolution of GWdetectors to higher order parity odd anisotropies. Moreover, as we will discuss in Sec. 3.2, theexpected annual modulation of the dipole response function can be used to optimize the signal-to-noise ratio of this measurement. This is in particular true if the SGWB dipole coincides withthe (known) dipole of the CMB. In the small frequency limit, k L (cid:28)
1, we can Taylor-expand the integrands of eqs. (22) and (23),and then perform the integrals numerically. We obtain M λAA ( k ) = M λEE ( k ) = 310 − π k L + O (cid:0) k L (cid:1) , ★◆◆ ★★ ◆◆ ★★ ◆◆ ★★ ◆◆★★◆◆ ★★◆◆ ★★◆◆ ★★◆◆ Figure 3:
Evolution of the dipole antenna pattern in ecliptic coordinates induced by the satelliterotation. The plots show the real part of the integrand of D RAE for f = 10 − Hz and every . months. The contour lines are at . , . , . , − . , − . , − . (red to blue). The greenstar denotes the direction of the dipole (assumed to coincide with the CMB dipole), and the browndot the direction of the LISA normal. M λT T ( k ) = π k L + O (cid:0) k L (cid:1) , D λAE ( k ) = λ ˆ v · ˆ n (cid:20) − π k L + O (cid:0) k L (cid:1)(cid:21) . (27)Obviously, these analytical expressions for the small kL expressions satisfy all the propertiesof the correlators discussed in the previous subsection. We note that the M λT T correlator vanishesat small frequencies. For this reason this channel is sometimes denoted as the “null-channel”,and it is expected to provide useful information for noise characterization [39].For LISA, with 2 πkL = 0 . (cid:18) k − Hz (cid:19) (cid:18) L . × km (cid:19) , (28)this Taylor expansion is only a good approximation for the lower part of the frequency band. Inthe following, and in particular in Sec. 3.2, we will work with the full response functions, therebyextending the work of Ref. [25]. On the other hand, when we turn to the Einstein Telescope inSec. 4.2, the small frequency limit will be fully sufficient. Performing a Fourier transform on eq. (21) yields the two-point function in the frequency domain, (cid:10) Σ O ( f )Σ O (cid:48) ( f (cid:48) ) (cid:11) = 14 (cid:88) σ (cid:90) dkk (cid:20) M σOO (cid:48) ( k ) P σ ( k ) (cid:90) dt (cid:90) dt (cid:48) e − πi ( tf + t (cid:48) f (cid:48) ) cos (cid:2) πk ( t − t (cid:48) ) (cid:3) v D σOO (cid:48) (cid:0) P σ ( k ) − kP (cid:48) σ ( k ) (cid:1) (cid:90) dt (cid:90) dt (cid:48) e − πi ( tf + t (cid:48) f (cid:48) ) sin (cid:2) πk ( t − t (cid:48) ) (cid:3)(cid:21) . (29)Here f and f (cid:48) can take both positive and negative values and the integration boundaries of thetime-integrals are ¯ t − ∆ T / ≤ t, t (cid:48) ≤ ¯ t + ∆ T / t denotes a reference time and ∆ T thetypical length of the data streams in the time domain which for LISA is expected to be O (10days). Since T is much longer than the inverse of the frequency range LISA is more sensitive to(which is of the order of hours), we will set ∆ T → ∞ from now on. We thus obtain (cid:10) Σ O ( f )Σ O (cid:48) ( f (cid:48) ) (cid:11) = 14 (cid:88) σ (cid:90) ∞ dk k {M σOO (cid:48) ( k ) P σ ( k ) (cid:2) δ ( − f + k ) δ ( f (cid:48) + k ) + δ ( f + k ) δ ( − f (cid:48) + k ) (cid:3)(cid:90) dk k − iv D σOO (cid:48) ( k ) (cid:0) P σ ( k ) − kP (cid:48) σ ( k ) (cid:1) (cid:2) δ ( − f + k ) δ ( f (cid:48) + k ) − δ ( f + k ) δ ( − f (cid:48) + k ) (cid:3)(cid:9) . (30)We note that the dipole contribution is odd under f ↔ f (cid:48) , indicating that the correspondingcontribution to the two-point function must vanish for O = O (cid:48) . This is an immediate consequenceof the sine function in eq. (14) which indicates that the dipole contribution has support only atunequal times, t (cid:54) = t (cid:48) . Let ˜ s O ( f ) be the signal registered by LISA in the O = { A, E } channels, in frequency space. Thesignal will be the sum of a physical signal Σ O ( f ) = δt ( f ) / L and of a noise ˜ n O ( f ):˜ s O ( f ) = Σ O ( f ) + ˜ n O ( f ) . (31)We define a frequency-dependent estimatorˆ F ( f , f ) ≡ W AE ( f , f ) ˜ s A ( f ) ˜ s E ( f ) , (32)where the filter function W AE ( f , f ) satisfies the reality condition W AE ( f , f ) ∗ = W AE ( − f , − f ).This implies that the frequency integrated estimator ˆ F ≡ (cid:82) df df W AE ( f , f ) ˜ s A ( f ) ˜ s E ( f ) isreal, and has expectation value (cid:104) ˆ F (cid:105) = (cid:90) df df W AE ( f , f ) (cid:104) ˜ s A ( f ) ˜ s E ( f ) (cid:105) = i (cid:90) ∞−∞ df W AE ( f , − f ) S s ( f ) , (33)where in the last step we have defined the AE correlator as (cid:104) ˜ s A ( f ) ˜ s E ( f ) (cid:105) = (cid:10) Σ A ( f )Σ E ( f (cid:48) ) (cid:11) = i δ ( f + f ) S s ( f ) , (34)with S s ( f ) real, and where we have assumed that the noises in the A and in the E channelsare uncorrelated. Note that since M σAE = 0, only the second line of eq. (30) contributes to thisexpression, that implies that S s ( − f ) = − S s ( f ).We next compute the variance of ˆ F assuming that the signal is noise dominated, with (cid:104) n O ( f ) n O (cid:48) ( f ) (cid:105) = δ OO (cid:48) P n,O ( f ) δ ( f + f ), so that (cid:104) ˆ F (cid:105) = (cid:90) ∞−∞ df df df df W AE ( f , f ) W AE ( f , f ) (cid:104) ˜ n A ( f ) ˜ n E ( f )˜ n A ( f ) ˜ n E ( f ) (cid:105) (cid:90) ∞−∞ df df W AE ( f , f ) W AE ( f , f ) ∗ P n,A ( f ) P n,E ( f ) . (35)The signal-to-noise ratio (SNR) is then given by (cid:104) ˆ F (cid:105) / (cid:113) (cid:104) ˆ F (cid:105) .To determine the filter function W AE ( f , f ) we define a noise-weighted scalar product infrequency space as ( A, B ) = (cid:90) ∞−∞ df df A ( f , f ) B ( f , f ) ∗ P n,A ( f ) P n,E ( f ) , (36)so that the SNR SNR = (cid:16) W AE , − iδ ( f + f ) S s ( f ) P n,A ( f ) P n,E ( f ) (cid:17)(cid:112) ( W AE , W AE ) , (37)is maximized for W AE ( f , f ) ∝ − i δ ( f + f ) S s ( f ) P nA ( f ) P nE ( f ) . For this optimal estimator, the SNRis thus given bySNR = (cid:20) T (cid:90) ∞−∞ df S s ( f ) P nA ( f ) P nE ( − f ) (cid:21) / = (cid:20) T (cid:90) ∞ df S s ( | f | ) P nA ( | f | ) P nE ( | f | ) (cid:21) / , (38)where T is the total duration of the measurement.Next, we write explicitly S s ( f ) using the response function D λAE ( k ) = λ D ( kL ) cos α , wherethe function D ( x ) is plotted in Figure 2. The quantity D λAE ( k ) (and, consequently, the expectationvalue for the signal S s ( f )) depends on time through the angle α between the direction of themotion of the solar system and the normal to LISA’s plane that rotates as the detector orbits theSun. As a consequence we will write the signal from now on as S s ( f, T ) ∝ cos α ( T ) and, whencomputing the SNR, we will replace the factor T in eq. (38) with an integral over dT , assumingthat the typical timescale on which Fourier transforms are computed is much shorter than themonth-long timescale on which α ( T ) changes significantly.We thus obtain S s ( f, T ) = 3 v H π f D ( | f | L ) (cid:32)(cid:88) λ λ Ω λGW (cid:33) cos α ( T ) , (39)where we have used the relation P λ ( f ) = H π f Ω λGW , and where we assume, to have a measureof the reach of this observable, that Ω λGW is does not depend on frequency within the LISAbandwidth, which implies that the quantity 2 P λ ( | f | ) − | f | P λ (cid:48) ( | f | ) that appears in eq. (30) equals H π f Ω λGW . It is worth reminding here that P λ ( f ) is the gravitational wave power spectrumevaluated at the time of detection, which is different from the primordial gravitational wavepower spectrum, and is related to it by the transfer function (see footnote 4).In order to determine the noise spectral functions P n,A ( f ) = P n,E ( f ) ≡ P n ( f ) we use theformulae given in [40, 41], that give f P n ( f ) (cid:39) × − (cid:18) ff ∗ (cid:19) (cid:32) − (cid:18) f ∗ f (cid:19) (cid:33)
14 2 . × − (1 + cos ( f /f ∗ )) (cid:18) f ∗ f (cid:19) (cid:32) × − (cid:18) f ∗ f (cid:19) (cid:33) (cid:32) (cid:18) ff ∗ (cid:19) (cid:33) , (40)where f ∗ = (2 πL ) − (cid:39) .
02 Hz.The final expression for the signal-to-noise ratio, for a scale invariant Ω λGW , is thusSNR = 9 H π v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) λ λ Ω λGW (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) (cid:90) dT cos α ( T ) (cid:90) ∞ dff D ( f L ) ( f P n ( f )) (cid:21) / (cid:39) . × v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) λ λ Ω λGW h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:34)(cid:90) T cos α ( x ) dx (cid:35) / . (41)Next, we have to estimate the integral (cid:82) T cos α ( x ) dx . LISA will be orbiting the Sun with itsnormal vector at 30 o with respect to the ecliptic plane, pointing south [36]. Placing the eclipticon the xy plane, and approximating that the orbit of the Earth with a circle, the unit vectornormal to LISA’s plane has components n = (cid:32) √
32 cos (cid:18) π t (cid:19) , √
32 sin (cid:18) π t (cid:19) , − (cid:33) . (42)Parametrizing the velocity vector as v = v (cos θ v sin φ v , cos θ v cos φ v , sin θ v ), we havecos α = n · v = √
32 sin (cid:18) π t φ v (cid:19) cos θ v − sin θ v . (43)The integral of cos α over 1 year gives the result (cid:20)(cid:90) cos α ( x ) dx (cid:21) / = (cid:112) θ v )4 , (44)that, depending on the value of cos θ v , ranges between . .
61. The value of the integral overthe total time T of observation, which appears in eq (41), can then be found multiplying theresult of eq (44) by (cid:112) T / (1 year).Thus, approximating (cid:104)(cid:82) cos α ( x ) dx (cid:105) / (cid:39) .
5, the total SNR turns out to be given approxi-mately by SNR (cid:39) (cid:16) v − (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) (cid:80) λ λ Ω λGW h . · − (cid:12)(cid:12)(cid:12)(cid:12) (cid:115) T . (45)This is one order of magnitude larger than the estimate obtained in [25].For definiteness, given that we use a different notation, we present in Appendix B a detailedcomparison among our computation and Seto’s results of [25]. On the other hand, we stress thatfor our analysis we use the most up-to-date LISA instrument specifications, and more completeformulas valid for the entire frequency band of the interferometer.15 Measuring the SGWB net circular polarization with ground-based interferometers
We now apply the formulas and techniques of the previous section to the case of ground-basedinterferometers. We develop fully analytical, ‘covariant’ formulas for overlap functions, describingcorrelations among ground based interferometers in the small antenna limit (condition (46) below) . Our formulas include the possibility that the SGWB is circularly polarized, do not relyon special choices of frame (this is why we call them covariant), and apply to any detectorshape (not limited to interferometers with orthogonal arms). When correlating distinct groundbased interferometers, it is well known that the SGWB monopole is already sensitive to circularpolarization (see e.g. [22, 23, 27]). We demonstrate this fact in terms of our analytic formulas,discuss the most convenient detector locations for maximizing sensitivity to circular polarization,and also include the kinematically induced dipole in our analysis. In the final part of this sectionwe turn to the future ground-based Einstein Telescope. A single instrument of this type willbe planar, and hence measuring the chirality of the SGWB requires taking into accoung thekinematic dipole, as in the analysis for LISA.Our starting point is given by relations (22) and (23), which apply also to pairs of ground-based interferometers (we actually choose a different overall normalization, as we discuss below).In these cases, the fact that the peak sensitivity of these detectors is at a frequency which issmall compared to their inverse arm length, results in a crucial simplification, allowing us toobtain fully analytical expressions for the overlap functions. Covariant, analytical formulas forthe unpolarized overlap function to the SGWB monopole M R ( k ) + M L ( k ) can already be foundin the literature [30, 42]. Here for the first time we provide covariant, analytic expressions forthe λ − dependent terms (contrary to LISA, these terms do not generally vanish, since pairs ofdetectors located in different locations on the Earth are generally not coplanar). Moreover, forthe first time we provide a covariant, analytic expressions for the overlap function to the SGWBdipole.For ground-based detectors, the crucial simplification arises from the fact that their sensitivityregion satisfies the “short arm condition” (referred to as “small kL limit” in Sec. 3.1.3)2 πk L (cid:39) . k
100 Hz L (cid:28) , (46)where we have normalized the frequency k to the region of best sensitivity for the existing andforthcoming detectors, and where we recall that the arms of the two LIGO sites are L = 4km long, while those of Virgo and KAGRA are L = 3 km long. In this limit, the quantity T indroduced in eq. (17) evaluates to T →
2. Using this value, eq. (15) assumes the simpler form σ i ( t ) ≡ δtt = D abi (cid:88) λ (cid:90) d k h λ (cid:16) (cid:126)k, t − L (cid:17) e ab,λ (ˆ k ) e − πi(cid:126)k · (cid:126)x i , D abi ≡ ˆ U ai ˆ U bi − ˆ V ai ˆ V bi , (47)where now ˆ U i and ˆ V i are the orientations of the arms of the i − th detector, that start from thecommon point located at (cid:126)x i . In the following, we refer to this point as to the “position of thedetector” for brevity. The vectors (cid:126)x i , ˆ U i , and ˆ V i for the two LIGO detectors, for Virgo, and forKAGRA are given in Appendix C. As customary in the literature, we call overlap functions the response functions for GW experiments thatcorrelate distinct detectors. (cid:10) Σ i ( t )Σ j ( t (cid:48) ) (cid:11) = (cid:88) λ (cid:90) dkk (cid:104) M λij ( k ) P λ ( k ) cos (cid:2) πk ( t − t (cid:48) ) (cid:3) − v D λij (cid:0) P λ ( k ) − kP (cid:48) λ ( k ) (cid:1) sin (cid:2) πk ( t − t (cid:48) ) (cid:3)(cid:105) , (48)with M λij ( k ) = D abi D cdj (cid:90) d Ω k π e − πi(cid:126)k · ( (cid:126)x i − (cid:126)x j ) e ab,λ (ˆ k ) e cd,λ ( − ˆ k ) D λij ( k, ˆ v ) = i D abi D cdj (cid:90) d Ω k π e − πi(cid:126)k · ( (cid:126)x i − (cid:126)x j ) e ab,λ (ˆ k ) e cd,λ ( − ˆ k ) ˆ k · ˆ v . (49)In Appendix D we compute these expression analytically. Parameterizing the positions of thedifferent detectors as κ ≡ πk | (cid:126)x i − (cid:126)x j | , ˆ s ij ≡ (cid:126)x j − (cid:126)x i | (cid:126)x i − (cid:126)x j | , (50)and introducing the functions f A ( κ ) ≡ j ( κ )2 κ + 1 − κ κ j ( κ ) , f B ( κ ) ≡ j ( κ ) κ − − κ κ j ( κ ) ,f C ( κ ) ≡ − j ( κ )4 κ + 35 − κ κ j ( κ ) ,f D ( κ ) ≡ j ( κ )2 − j ( κ )2 κ , f E ( κ ) ≡ − j ( κ )2 + 5 j ( κ )2 κ , (51)(where j (cid:96) are spherical Bessel functions) the overlap function for the SGWB monopole is M λij ( k ) = f A ( κ ) tr [ D i D j ] + f B ( κ ) ( D i ˆ s ij ) a ( D j ˆ s ij ) a + f C ( κ ) ( D i ˆ s ij ˆ s ij ) ( D j ˆ s ij ˆ s ij )+ λ f D ( κ ) [ D i D j ] ab (cid:15) abc ˆ s cij + λ f E ( κ ) ( D i ˆ s ij ) a ( D j ˆ s ij ) b (cid:15) abc ˆ s cij , (52)while that to the SGWB dipole is D λij ( k, ˆ v ) = f (cid:48) A ( κ ) ˆ v e ˆ s e ( D i D j ) aa + (cid:20) f (cid:48) B ( κ ) − f B ( κ ) κ (cid:21) ˆ v e ˆ s e ( D i ˆ s b ) a ( D j ˆ s ) a + f B ( κ ) κ [( D i ˆ v ) a ( D j ˆ s ) a + ( D i ˆ s ) a ( D j ˆ v ) a ]+ (cid:20) f (cid:48) C ( κ ) − f C ( κ ) κ (cid:21) ˆ v e ˆ s e ( D i ˆ s ˆ s ) ( D j ˆ s ˆ s ) + 2 f C ( κ ) κ [( D i ˆ s ˆ v ) ( D j ˆ s ˆ s ) + ( D i ˆ s ˆ s ) ( D j ˆ s ˆ v )]+ λ (cid:20) f (cid:48) D ( κ ) − f D ( κ ) κ (cid:21) ˆ v e ˆ s e ( D i D j ) ab (cid:15) abc ˆ s c + λ f D ( κ ) κ ( D i D j ) ab (cid:15) abc ˆ v c + λ (cid:20) f (cid:48) E ( κ ) − f E ( κ ) κ (cid:21) ˆ v e ˆ s e ( D i ˆ s ) a ( D j ˆ s ) b (cid:15) abc ˆ s c + λ f E ( κ ) κ (cid:110)(cid:104) ( D i ˆ v ) a ( D j ˆ s ) b + ( D i ˆ s ) a ( D j ˆ v ) b (cid:105) (cid:15) abc ˆ s c + ( D i ˆ s ) a ( D j ˆ s ) b (cid:15) abc ˆ v c (cid:111) . (53) We use a different normalization for the overlap function for ground-based interferometers with respect to theone used for LISA in Sec. 3, to respect the literature.
17n these expressions, we have used the combinations( D i ˆ v ) a ≡ D abi ˆ v b , ( D i ˆ v ˆ s ) ≡ D abi ˆ v b ˆ s a , ( D i D j ) ab ≡ D aci D cbi , . . . (54)As we mentioned, these expressions are valid in the regime in which the product between thefrequency and the arm lengths is much smaller than one, but do not assume that the productbetween the frequency and the separation distance between the two detectors, is also small(namely, κ does not need to be (cid:28) κ → M λij = ( D i D j ) aa , lim κ → D λij (ˆ v ) = 2 λ ( D i D j ) ab (cid:15) abc ˆ v c . (55)The analytic expressions (52) and (53) can be readily evaluated for any pair of detectors. InTable 1 in Appendix C we provide the explicit expressions for the vectors (cid:126)s i , ˆ U i , ˆ V i for the twoLIGO, the Virgo, and the KAGRA detectors. As an example, in Figure 4 we show the overlapfunctions for the pair of LIGO detectors (first row) and for the Virgo-KAGRA pair (secondrow). The figure confirms the correctness of the covariant, analytical expressions (52) and (53),obtained using both the analytical expressions given above and numerical evaluations. We haveverified that the agreement between the analytic and numerical results persists for other genericdirections of ˆ v , beyond the particular choice in Fig. 4. For the case of the monopole, equivalentformulas, but not covariant since they make use of a particular reference frame, can be foundin [22, 23]. Our general results identify clearly the ‘parity-violating’ contributions proportional tothe Levi-Civita tensor (cid:15) abc , and do not make any hypothesis on the shape of the detector (whosearms can form angles different than 90 degrees). M The last two contributions to the monopole overlap function (52), proportional to f D and f E ,distinguish between the two different GW polarizations and depend on the separation betweeninterferometers as well as their orientations. We note that they vanish in the limit of coincidentinstruments (see eq. (55)) or when the detector arms are oriented such that the quantity D aci D bdj is symmetric under the a ↔ b exchange. The former condition can be easily understood: bymeasuring the GW at one location, one cannot determine how its profile changes as it propagates,and hence left- and right-handed GWs cannot be distinguished. A geometrical interpretation forthe latter condition will be given below.To obtain a more explicit expression for the overlap function, we place and orient the detectorsat the following coordinates (this choice can always be done with no loss of generality),ˆ x = (1 , , , ˆ U = (0 , sin α, cos α ) , ˆ V = (0 , cos α, − sin α ) , ˆ x = (cos φ, sin φ, , ˆ U = ( − sin φ sin β, cos φ sin β, cos β ) , ˆ V = ( − sin φ cos β, cos φ cos β, − sin β ) , (56)where 0 ≤ φ ≤ π , and 0 ≤ α, β ≤ π . The angles α and β give the orientation of the ˆ U − arm interms of the angle from the north toward the east direction (where these directions are expressed18 ight chiralityLeft chirality - - - - - f [ Hz ] ℳ L H - LL Right chiralityLeft chirality - - - - - f [ Hz ] L H - LL Right chiralityLeft chirality - - f [ Hz ] ℳ V - K Right chiralityLeft chirality - - f [ Hz ] V - K Figure 4:
First row: monopole and dipole overlap functions for the LIGO Hanford (LH) andLIGO Livingston (LL) pair. Second row: monopole and dipole overlap functions for the Virgo(V) and KAGRA (K) pair. In the dipole case, ˆ v = (0 , , (in the coordinate system introduced inAppendix C) has been chosen for illustrative purposes. The solid lines are the analytic expressions(52) and (53). The dotted black lines at small frequency are the asymptotic values (55). The dotsare obtained from a numerical evaluation. at the location of each detector). With this choice, the unit vector going from the first to thesecond detector is ˆ s = 1 (cid:112) − cos φ ) ( − φ, sin φ, , (57)and the λ − dependent terms in the monopole overlap function (52) give rise to∆ M ≡ M + ij − M − ij = κ ( − φ ) j ( κ ) + (cid:2) (cid:0) − κ (cid:1) + (cid:0) κ (cid:1) cos φ (cid:3) j ( κ )24 κ sin (cid:18) φ (cid:19) sin [2 ( α + β )] , (58)where we note that φ is the angle (centered in the center of the Earth) between the two detectors,while α and β express, respectively, the orientations of the U − arm of the two detectors. We noticethat eq. (58) always vanishes when φ = 0 (the two detectors are coplanar) and when the sum( α + β ) is equal to zero or π/
2. If this condition occurs, indeed, the combination D aci D bdj issymmetric in the indexes ( a, b ): as we have discussed above, this implies null sensitivity toparity violating effects. This result can also interpreted geometrically as follows. If α = − β , thesystem of detectors is symmetric about the plane through the maximal circle on Earth that passes19igure 5: The location of all existing detectors on Earth, together with a LIGO-India detector inMaharashtra, and a hypothetical optimal-for-chiral-SGWB detector in Perth. We also show theantipodes of the LIGO-Livingston detector (green dot), which is not far from the Perth detector.We note that the Figure shows the point of view of an observer at a specific location in space,who sees less than half of the Earth. Lighter lines (red dots) are used to indicate continents(interferometers) that are not seen by this observer. halfway between the two detectors. As a consequence, a right-handed gravitational wave comingfrom one side of this plane is indistinguishable from a left-handed one coming from the oppositedirection, so that the system, after selecting the isotropic monopole contribution, is insensitive tochirality. This argument is analogous to, and generalizes, that given in [24], where it was shownthat coplanar detectors are insensitive to chirality (in that case, the symmetry plane coincidedwith the plane of the two detectors).In particular, if the detectors are located at the antipodes ( φ = π ), the absolute value ofeq. (58) is maximized and reduces to∆ M antipodes = − κ j ( κ ) + (cid:0) − κ (cid:1) j ( κ )6 κ sin [2 ( α + β )] . (59)In what comes next, using our formulas we discuss more quantitatively the best choices of locationfor antipodal ground based detectors in order to detect parity violating effects in the SGWB.Similar considerations can also be found in [22, 23]. Choice of Earth location for optimal detection of a chiral SGWB
If we search for the antipodes of the four known detectors (Hanford, Livingston, Virgo, KAGRA),we see that all of them fall in the Ocean (Pacific, Atlantic and Indian). The antipode of LIGO-Livingston (L) falls in the Indian Ocean near Australia. The closest large city to it is Perth20P). Let us compute the optimal overlap function for this pair of detectors. Recall that, in ourcoordinate system, defined in App. C, LIGO-Livingston (L) is located at (cid:126)x L = R ( − . , − . , . , (60)with R denoting the radius of the earth, and its arms are directed alongˆ u L = ( − . , − . , − . , ˆ v L = (0 . , − . , − . . (61)Moreover, in our coordinate system, P is located at (cid:126)x P = R ( − . , . , − . , (62)which gives a distance s = | (cid:126)x p − (cid:126)x L | (cid:39) . R ⇒ κ (cid:39) f
24 Hz . (63)Therefore the two detectors are nearly opposite, as can be seen in Fig. 5.We now place the arm ˆ u P at the angle α from the north direction towards east (from thepoint of view of an observer at P), while ˆ v P is at the angle π + α . We then haveˆ u P = cos α ( − . , . , . α ( − . , − . , , ˆ v P = − sin α ( − . , . , . α ( − . , − . , . (64)The difference in the overlap function ∆ M for the L-P pair gives∆ M = κ (cid:0) − . κ (cid:1) cos κ + (cid:0) − . κ (cid:1) sin κκ [ − .
22 cos (2 α ) + 1 . α )] . (65)We then have for:antipodes , α best = π ⇒ ∆ M = 1 . κ (cid:0) − κ / (cid:1) cos κ + (cid:0) − κ / (cid:1) sin κκ with κ = f . , L − P , α best = 2 . ⇒ ∆ M = 1 . κ (cid:0) − . κ (cid:1) cos κ + (cid:0) − . κ (cid:1) sin κκ with κ = f
24 Hz . (66)At small frequencies κ (cid:28) M antipodes ( α best ) (cid:39) − f
177 Hz , ∆ M L − P ( α best ) (cid:39) − f
191 Hz . (67)Consequently, an additional GW detector close to Perth, Australia, rotated clockwise by 2 . M + and M − can be utilized to distinguish a chiral from a non-chiral SGWB. This analysis21 ntipodesLL - P - - - f [ Hz ] Δ ℳ Figure 6:
The function ∆ M , sensitive to parity violation (difference of the overlap functions ofopposite chirality, see eq. (59) ) of two ideal detectors at the antipodes, and of LIGO-Livingstonwith a detector at Perth, Australia. By expanding the κ dependent part of eq. (59) for large κ , wefind that the zeros of this function occur at the frequencies f (cid:39) πd (cid:0) + n (cid:1) , where d is the diameterof the Earth and n is an integer number. By comparing with the figure, one can see that thisrelation works well already at n = 1 . (using numerically evaluated response functions) was performed in Ref. [27] for the Hanford andLivingston LIGO, VIRGO and KAGRA detectors and for a power-law signal. In the specificcase of a frequency-independent SGWB, it was found that maximal chirality can be detected orexcluded for an amplitude up to Ω GW (cid:38) − . It would be interesting to extend this analysis toinclude an antipodal detector with the optimal orientation α best , but this is beyond the scope ofthe present paper. The Einstein Telescope is a proposal for a ground-based interferometer with a triangular shapewith arm length L = 10 km. It will be an observatory of the third generation aiming to reacha sensitivity for GW signals emitted by astrophysical and cosmological sources about a factorof ten better than the currently operating ground based detectors. It will be formed by threedetectors, each in turn composed of two interferometers (xylophone configuration) [28, 29]. Thetriangular planar configuration of ET, similar to LISA, allows to use the same approach developedin Section 3 to compute the SNR for measuring the circular polarization. For the computationwe use eq. (41), where we consider the noise power spectrum P ET n ( f ) for a third-generationgravitational wave interferometer [43]. The expression for the SNR, for a scale invariant Ω λGW ,in this case isSNR ET = 3 H π v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) λ λ Ω λGW (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) (cid:90) dT cos α ( T ) (cid:90) ∞ dff D ( f L ) ( f P ET n ( f )) (cid:21) / ≈ . × v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) λ λ Ω λGW h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:34)(cid:90) T cos α ( x ) dx (cid:35) / . (68)where for the dipole response function D ( f L ) we have used the value at small frequency givenby eq. (27). 22omparing the sensitivity to circular polarization for operating ground-based detectors, de-rived in Ref. [27], with ET, we note that the improved sensitivity of the Einstein Telescope, inparticular at low frequencies, enables to out-perform the current LIGO configuration, taking intoaccount the expected magnitude of the kinematic dipole of v ∼ − . This nicely demonstratesthe important interplay between detector sensitivity, location and co-planarity for ground-baseddetectors. With two copies of the Einstein Telescope (or of the Cosmic Explorer [44]), one couldof course benefit from increased sensitivity and the elimination of the dipole factor v since themonopole is already sensitive to chirality. The detection of the SGWB is a major goal for GW interferometers, which is expected to beachieved in the coming years. On the other hand, the amplitude and properties of the cosmologicalSGWB are highly model dependent. Any detection or constraint on this cosmological SGWB willcontain valuable information about the early Universe. In this situation, it is crucial to extractand characterize all properties of any SGWB detected. In this paper we focus on the ability ofground- and space-based detectors to measure the net polarization of the SGWB, which couldbe a smoking gun for parity violating interactions in the early Universe.For an isotropic SGWB, a system of coplanar detectors is insensitive to the polarization ofthe SGWB [22–24]. Making the symmetries of the response functions of ground- and space-baseddetectors explicit, we provide a transparent demonstration of this result as well as of the twopossibilities to circumvent it: (i) for planar detectors (such as LISA or ET), we make use ofthe kinematic dipolar anisotropy induced by the motion of the solar system with respect to thecosmic rest frame [25, 32, 33] and (ii) for a network of ground-based detector, the curvature ofthe earth breaks co-planarity [22, 23, 27]. In the present work we reconsider previous results bytaking into account the full response functions and noise curves in the entire frequency band(for planar detectors). Moreover, we provide fully analytical and covariant expressions for the(parity-sensitive) response functions of a ground-based detector network.We find that LISA and ET, despite operating at very different frequencies, will have a similarsensitivity to a scale-invariant SGWB, and could detect an O (1) net polarization in a SGWBwith a magnitude of Ω GW h (cid:39) − with an SNR of order one. We emphasize that these twoinstruments should be seen as complementary probes, since the SGWB may vary significantlybetween the LISA and ET frequency bands. For both LISA and ET, the auto-correlation channelsare blind to chirality and the entire sensitivity stems from cross-correlating the two TDI channels.For a network of ground-based detectors we provide fully covariant analytical expressions forthe monopole and dipole response functions. It is much more rapid to evaluate these analyticexpressions than to compute numerically the angular integrals that are needed to obtain theresponse functions numerically, and therefore we hope that these analytic relations can be usedto speed up future studies of the SGWB polarization. Since the sensitivity to net polarization ofthe (dominant) monopole contribution to the SGWB arises from the departure from co-planarity,the detector location and orientation plays a crucial role.In summary, in this paper we studied a specific feature that can contribute to the characteri-zation of the SGWB: the possibility for measuring a circular polarization degree of a gravitationalwave background through a dipolar modulation induced by the motion of the reference frame withrespect to the cosmic frame. This could help single out specific cosmological mechanisms charac-23erized by violation of parity in the early universe, and it is therefore an interesting observationaltarget for current and future interferometers. As future work, it would be interesting to applythis approach to other GW detectors, which can be sensitive to the circular polarization usingthis method. For example, the proposed Japanese space-based GW observatory DECIGO [45],or to astrometric GW observations that aim to reveal effects induced by a SGWB using datafrom the Gaia mission [46]. Acknowledgments
We warmly thank the LISA Cosmology Research Group for many stimulating and insightful dis-cussions on various aspects of the SGWB. V.D. acknowledges funding by the Deutsche Forschungs-gemeinschaft under Germany’s Excellence Strategy - EXC 2121 “Quantum Universe” - 390833306.JGB acknowledges support from the Research Project FPA2015-68048-03-3P(MINECO-FEDER)and the Centro de Excelencia Severo Ochoa Program SEV-2016-0597. M.Pi. acknowledges thesupport of the Spanish MINECOs “Centro de Excelencia Severo Ochoa” Programme under grantSEV-2016-059. This project has received funding from the European Unions Horizon 2020 re-search and innovation programme under the Marie Sk(cid:32)lodowska-Curie grant agreement No 713366.The work of L.S. is partially supported by the US-NSF grant PHY-1820675. G.T. is partiallysupported by STFC grant ST/P00055X/1.
A GW polarization operators in the chiral basis
We follow the standard definition of the GW polarization operators, that we summarized in thework [38]. It is straightforward to show that the opertors can be also introduced as e ab,λ (cid:16) ˆ k (cid:17) = e a,λ (cid:16) ˆ k (cid:17) e b,λ (cid:16) ˆ k (cid:17) ≡ ˆ u a (cid:16) ˆ k (cid:17) + iλ ˆ v a (cid:16) ˆ k (cid:17) √ u b (cid:16) ˆ k (cid:17) + iλ ˆ v b (cid:16) ˆ k (cid:17) √ , (A1)where we recall that λ = +1 (respectively, λ = −
1) correspond to the right-handed (respectively,the left-handed) helicity, and whereˆ u (cid:16) ˆ k (cid:17) ≡ ˆ k × ˆ e z | ˆ k × ˆ e z | , ˆ v (cid:16) ˆ k (cid:17) ≡ ˆ k × ˆ u (cid:16) ˆ k (cid:17) = (cid:16) ˆ k · ˆ e z (cid:17) ˆ k − ˆ e z | ˆ k × ˆ e z | , (A2)where ˆ e z is the unit vector along the third-axis.It immediately follows that e ∗ ab,λ (ˆ k ) = e ab,λ ( − ˆ k ) = e ab, − λ (ˆ k ) , e ∗ ab,λ (ˆ k ) e ab,λ (cid:48) (ˆ k ) = δ λλ (cid:48) . (A3)Moreover, one can verify by direct inspection that e i,λ (ˆ k ) e i (cid:48) ,λ ( − ˆ k ) = − (cid:16) δ ii (cid:48) − ˆ k i ˆ k i (cid:48) − iλ (cid:15) ii (cid:48) j ˆ k j (cid:17) . (A4) Another basis that is often chosen for the polarization operators is the { + , ×} basis, related to the chiral basisby e ab,λ = e (+) ab + λie ( × ) ab √ . e ab,λ (cid:16) ˆ k (cid:17) e cd,λ (cid:16) − ˆ k (cid:17) = 14 (cid:104) δ ac − ˆ k a ˆ k c − iλ(cid:15) ace ˆ k e (cid:105) (cid:104) δ bd − ˆ k b ˆ k d − iλ(cid:15) bdf ˆ k f (cid:105) = 14 (cid:104)(cid:16) δ ac − ˆ k a ˆ k c (cid:17) (cid:16) δ bd − ˆ k b ˆ k d (cid:17) + (cid:16) δ ad − ˆ k a ˆ k d (cid:17) (cid:16) δ bc − ˆ k b ˆ k c (cid:17) − (cid:16) δ ab − ˆ k a ˆ k b (cid:17) (cid:16) δ cd − ˆ k c ˆ k d (cid:17)(cid:105) − iλ (cid:104)(cid:16) δ ac − ˆ k a ˆ k c (cid:17) (cid:15) bdf ˆ k f + (cid:16) δ bd − ˆ k b ˆ k d (cid:17) (cid:15) ace ˆ k e (cid:105) . (A5) B Comparison with previous computation
This Appendix provides a detailed comparison betweeen our results of Section 3.2 and the findingsof Seto in [25] for the magnitude of the signal-to-noise ratio associated with measurements of theSGWB circular polarization with LISA. The comparison is made difficult by the different notationused in the two works. Our aim is to carry on all the steps that allow us to re-write the resultsof [25] using the notation implemented in our paper. Our conclusion will be that our findings forthe magnitude of the signal-to-noise ratio is a factor of 10 larger than [25]. We use a superscript ( S ) to denote quantities in Seto’s work [25].We start from our decomposition for tensor fluctuations, h ij ( (cid:126)x, t ) = (cid:90) d ke − πi(cid:126)k · (cid:126)x (cid:88) P e ij,P (ˆ k ) (cid:104) e πikt h P ( k, ˆ k ) + e − πikt h ∗ P ( k, − ˆ k ) (cid:105) , (B1)where we decompose in P = + , × polarizations instead of L, R . In [25] the notation is h ij ( (cid:126)x, t ) = (cid:90) ∞−∞ df d ˆ n e πif ( t − ˆ n · x ) (cid:88) P ( S ) e ij,P (ˆ n ) ( S ) h P ( f, ˆ n ) , (B2)where ( S ) e ij,P (ˆ n ) = √ e ij,P (ˆ n ).To proceed with our comparison, we separate our expressions for h ij into (we use ˆ n = ˆ k ) h ij ( (cid:126)x, t ) = (cid:90) ∞ k dk d ˆ ne − πik ˆ n · (cid:126)x (cid:88) P e ij,P (ˆ n ) e πikt h P ( k, ˆ k )+ (cid:90) ∞ k dk d ˆ ne − πik ˆ n · (cid:126)x (cid:88) P e ij,P (ˆ n ) e − πikt h ∗ P ( k, − ˆ k ) , = (cid:90) ∞ f df d ˆ ne − πif ˆ n · (cid:126)x (cid:88) P e ij,P (ˆ n ) e πift h P ( f, ˆ n )+ (cid:90) −∞ f df d ˆ ne πif ˆ n · (cid:126)x (cid:88) P e ij,P (ˆ n ) e πift h ∗ P ( − f, − ˆ n ) , = (cid:90) ∞ f df d ˆ ne − πif ˆ n · (cid:126)x (cid:88) P e ij,P (ˆ n ) e πift h P ( f, ˆ n )+ (cid:90) −∞ f df d ˆ ne − πif ˆ n · (cid:126)x (cid:88) P e ij,P ( − ˆ n ) e πift h ∗ P ( − f, ˆ n ) , (B3)25here in the last step we have changed ˆ n → − ˆ n in the second integral.To compare with [25], we can make the identification √ ( S ) h P ( f, ˆ n ) = f (cid:26) h P ( f, ˆ n ) f > − P h ∗ P ( − f, ˆ n ) f < − P is +1 for P = + and − P = × . In order to prove this fact, we follow [25], andwrite ˆ n = (sin θ cos φ, sin θ sin φ, cos θ ) , (B5)and e θ = ∂ θ ˆ n = (cos θ cos φ, cos θ sin φ, − sin θ ) ,e φ = ∂ φ ˆ n = ( − sin θ sin φ, sin θ cos φ, . (B6)We have the relations e + = e θ e θ − e φ e φ , e × = e θ e φ + e φ e θ . On the other hand, ˆ n → − ˆ n isequivalent to θ → π − θ , φ → φ + π . This means that e θ → (( − cos θ ) ( − cos φ ) , ( − cos θ ) ( − sin φ ) , − sin θ ) = e θ ,e φ → ( − sin θ ( − sin φ ) , sin θ ( − cos φ ) ,
0) = − e φ , (B7)which then implies e + → e + , but e × → − e × . The work [25] defines i (cid:104) ( S ) h + ( f, ˆ n ) ( S ) h ∗× ( f (cid:48) , ˆ n (cid:48) ) − ( S ) h ∗ + ( f, ˆ n ) ( S ) h × ( f (cid:48) , ˆ n (cid:48) ) (cid:105) = δ ( f − f (cid:48) ) δ (ˆ n − ˆ n (cid:48) )4 π V ( f, ˆ n ) . (B8)We now translate this expression in our notation i (cid:104) h + ( f, ˆ n ) h ∗× ( f (cid:48) , ˆ n (cid:48) ) − h ∗ + ( f, ˆ n ) h × ( f (cid:48) , ˆ n (cid:48) ) (cid:105) = 2 δ ( f − f (cid:48) ) f δ (ˆ n − ˆ n (cid:48) )2 π V ( f, ˆ n ) f > ,i (cid:104) h ∗ + ( − f, ˆ n ) h × ( − f (cid:48) , ˆ n (cid:48) ) − h + ( − f, ˆ n ) h ∗× ( − f (cid:48) , ˆ n (cid:48) ) (cid:105) = − δ ( f − f (cid:48) ) f δ (ˆ n − ˆ n (cid:48) )2 π V ( f, ˆ n ) f < , (B9)where these two expressions are actually one the complex conjugate of the other.We now proceed to compute expressions in terms h L,R modes. We have h + = 1 √ h R + h L ) , h × = i √ h R − h L ) , (B10)so that i (cid:0) h + h ∗× − h ∗ + h × (cid:1) = | h R | − | h L | . (B11)The two point function found in eq. (14) reads (cid:104) h σ ( (cid:126)k, τ ) h σ (cid:48) ( (cid:126)k (cid:48) , τ (cid:48) ) (cid:105) = δ σσ (cid:48) δ (3) ( (cid:126)k + (cid:126)k (cid:48) )4 π k × (cid:110) P σ ( k ) cos[2 πk ( τ − τ (cid:48) )] − i (ˆ k · (cid:126)v ) (cid:2) P σ ( k ) − k P σ (cid:48) ( k ) (cid:3) sin[2 πk ( τ − τ (cid:48) )] (cid:111) , h σ ( (cid:126)k, τ ) = e πikt h σ ( k, ˆ k ) + e − πikt h ∗ σ ( k, − ˆ k ) , (B13)Then the LHS of equation (B12) rewrites (cid:104) (cid:104) e πikτ h σ ( k, ˆ k ) + e − πikτ h ∗ σ ( k, − ˆ k ) (cid:105) (cid:104) e πik (cid:48) τ (cid:48) h σ ( k (cid:48) , ˆ k (cid:48) ) + e − πik (cid:48) τ (cid:48) h ∗ σ ( k (cid:48) , − ˆ k (cid:48) ) (cid:105) (cid:105) = (cid:104) e πi ( kτ + k (cid:48) τ (cid:48) ) h σ ( k, ˆ k ) h σ ( k (cid:48) , ˆ k (cid:48) ) + e − πi ( kτ + k (cid:48) τ (cid:48) ) h ∗ σ ( k, − ˆ k ) h ∗ σ ( k (cid:48) , − ˆ k (cid:48) )+ e πi ( kτ − k (cid:48) τ (cid:48) ) h σ ( k, ˆ k ) h ∗ σ ( k (cid:48) , − ˆ k (cid:48) ) + e − πi ( kτ − k (cid:48) τ (cid:48) ) h ∗ σ ( k, − ˆ k ) h σ ( k (cid:48) , ˆ k (cid:48) ) (cid:105) . (B14)Now we note that this quantity must be a linear combination of sin[2 πk ( τ − τ (cid:48) )] and cos[2 πk ( τ − τ (cid:48) )]. This implies that (cid:104) h σ ( k, ˆ k ) h σ ( k (cid:48) , ˆ k (cid:48) ) (cid:105) = (cid:104) h ∗ σ ( k, − ˆ k ) h ∗ σ ( k (cid:48) , − ˆ k (cid:48) ) (cid:105) = 0 , (B15)since these terms multiply cosines and sines of 2 πk ( τ + τ (cid:48) ), and (cid:104) cos[2 πi ( kτ − k (cid:48) τ (cid:48) )] (cid:104) h σ ( k, ˆ k ) h ∗ σ ( k (cid:48) , − ˆ k (cid:48) ) + h ∗ σ ( k, − ˆ k ) h σ ( k (cid:48) , ˆ k (cid:48) ) (cid:105) + i sin[2 πi ( kτ − k (cid:48) τ (cid:48) )] (cid:104) h σ ( k, ˆ k ) h ∗ σ ( k (cid:48) , − ˆ k (cid:48) ) − h ∗ σ ( k, − ˆ k ) h σ ( k (cid:48) , ˆ k (cid:48) ) (cid:105) (cid:105) = δ (3) ( (cid:126)k + (cid:126)k (cid:48) )4 π k (cid:110) P σ ( k ) cos[2 πk ( τ − τ (cid:48) )] − i (ˆ k · v ) (cid:2) P σ ( k ) − k P σ (cid:48) ( k ) (cid:3) sin[2 πk ( τ − τ (cid:48) )] (cid:111) (B16)where we note in passing that the quantity in the second square bracket does not vanish.By comparing the time dependent parts we thus get the time-independent correlators (cid:104) h σ ( k, ˆ k ) h ∗ σ ( k (cid:48) , − ˆ k (cid:48) ) + h ∗ σ ( k, − ˆ k ) h σ ( k (cid:48) , ˆ k (cid:48) ) (cid:105) = δ (3) ( (cid:126)k + (cid:126)k (cid:48) )4 π k P σ ( k ) (cid:104) h σ ( k, ˆ k ) h ∗ σ ( k (cid:48) , − ˆ k (cid:48) ) − h ∗ σ ( k, − ˆ k ) h σ ( k (cid:48) , ˆ k (cid:48) ) (cid:105) = − δ (3) ( (cid:126)k + (cid:126)k (cid:48) )4 π k (ˆ k · (cid:126)v ) (cid:2) P σ ( k ) − k P σ (cid:48) ( k ) (cid:3) . (B17)We can then conclude that (cid:104) h σ ( k, ˆ k ) h ∗ σ ( k (cid:48) , ˆ k (cid:48) ) (cid:105) = δ (3) ( (cid:126)k − (cid:126)k (cid:48) )8 π k (cid:110) P σ ( k ) − (ˆ k · (cid:126)v ) (cid:2) P σ ( k ) − k P σ (cid:48) ( k ) (cid:3) (cid:111) . (B18)Let now compute (cid:104)| h R | − | h L | (cid:105) as (cid:104) h R ( k, ˆ k ) h ∗ R ( k (cid:48) , ˆ k (cid:48) ) − h L ( k, ˆ k ) h ∗ L ( k (cid:48) , ˆ k (cid:48) ) (cid:105) = δ (3) ( (cid:126)k − (cid:126)k (cid:48) )8 π k (cid:110) ∆ P σ ( k ) − (ˆ k · (cid:126)v ) (cid:2) P σ ( k ) − k ∆ P σ (cid:48) ( k ) (cid:3) (cid:111) (B19)(where ∆ P σ = P R − P L ), and compare with [25]. First, δ ( (cid:126)k − (cid:126)k (cid:48) ) = δ ( k − k (cid:48) ) k δ (2) (ˆ n − ˆ n (cid:48) ) where wedefine δ (2) (ˆ n − ˆ n (cid:48) ) = δ ( θ − θ (cid:48) ) δ ( φ − φ (cid:48) )sin θ (cid:48) , (B20)27o that (cid:90) d k δ (3) ( (cid:126)k − (cid:126)k (cid:48) ) = (cid:90) dk sin θ dθ dφ δ ( k − k (cid:48) ) δ (2) (ˆ n − ˆ n (cid:48) ) . (B21)Then we can write (cid:104) h R ( k, ˆ n ) h ∗ R ( k (cid:48) , ˆ n (cid:48) ) − h L ( k, ˆ n ) h ∗ L ( k (cid:48) , ˆ n (cid:48) ) (cid:105) = δ ( k − k (cid:48) ) δ (2) (ˆ n − ˆ n (cid:48) )8 π k × (cid:110) ∆ P σ ( k ) − (ˆ n · (cid:126)v ) (cid:2) P σ ( k ) − k ∆ P σ (cid:48) ( k ) (cid:3) (cid:111) . We can compare with eq. (B9) that can be written as (cid:104) h R ( f, ˆ n ) h ∗ R ( f (cid:48) , ˆ n (cid:48) ) − h ∗ L ( f, ˆ n ) h L ( f (cid:48) , ˆ n (cid:48) ) (cid:105) = δ ( f − f (cid:48) ) f δ (ˆ n − ˆ n (cid:48) ) π V ( f, ˆ n ) f > V ( f, ˆ n ) = 18 f (cid:110) ∆ P σ ( | f | ) − (ˆ n · (cid:126)v ) (cid:2) P σ ( | f | ) − | f | ∆ P σ (cid:48) ( | f | ) (cid:3) (cid:111) . (B23)Decomposing the quantity V in spherical harmonics, by choosing the direction of (cid:126)v as the z axis, we find V = 18 f ∆ P σ ( | f | ) ,V = − v f (cid:2) P σ ( | f | ) − | f | ∆ P σ (cid:48) ( | f | ) (cid:3) . (B24)The work [25] defines the quantity p that in our case, where we consider only the dipole contri-bution, reads p = | V | I t . (B25)Here I t is the total intensity, that in our regime is well approximated by | I | , while [25] defines I as 12 (cid:104) ( S ) h + ( f, ˆ n ) ( S ) h ∗ + ( f (cid:48) , ˆ n (cid:48) ) + ( S ) h × ( f, ˆ n ) ( S ) h ∗× ( f (cid:48) , ˆ n (cid:48) ) (cid:105) = δ ( f − f (cid:48) ) δ (ˆ n − ˆ n (cid:48) )4 π I ( f ) , = f (cid:104) h R ( f, ˆ n ) h ∗ R ( f (cid:48) , ˆ n (cid:48) ) + h L ( f, ˆ n ) h ∗ L ( f (cid:48) , ˆ n (cid:48) ) (cid:105) (cid:39) f δ ( k − k (cid:48) ) δ (ˆ n − ˆ n (cid:48) )8 π k (cid:88) σ P σ ( k ) , (B26)where we only keep the monopole contribution from eq. (B18). So we get I ( f ) = 18 f (cid:88) σ P σ ( | f | ) , (B27)and finally p = v | P σ ( | f | ) − | f | ∆ P σ (cid:48) ( | f | ) | (cid:80) σ P σ ( | f | ) . (B28)28hen using P σ ∝ Ω σ /f with Ω σ = constant, we obtain p = 4 v | (cid:80) λ λ Ω λ | (cid:80) λ Ω λ . (B29)The final result of [25], using (cid:80) λ Ω λ = Ω GW , can then be re-expressed as ( S ) SNR = 4 × v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) λ λ Ω λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:115) T . (B30)To compare with our findings, we rewrite our result of eq. (45) asSNR (cid:39) . × v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) λ λ Ω λ h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:115) T (cid:39) . × v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) λ λ Ω λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:115) T , (B31)To conclude, our result of eq. (B31) is a factor of 10 larger than the result of [25] in eq (B30). C Location and orientation of existing and forthcoming ground-based interferometers
We take the Earth to be a perfect sphere of radius R = 6 . · km. We consider a Cartesiancoordinate system with the origin located at the center of the Earth, and with the z − axis goingin the direction of the North Pole. The x − axis goes in the direction of the point connectingthe Earth Equator (latitude 0) and the Greenwich Meridian (longitude 0). The y − axis is thendetermined by ˆ e y = ˆ e z × ˆ e x , and it is directed toward the point on the Equator at 90 ◦ E longitude.In Table 1 we provide the Cartesian coordinates (in the system that we have just defined) fora set of three unit vectors for each detector. The first unit vector is the position (cid:126)x i of the i − thdetector divided by R (in practice, it is the unit-vector starting from the center of the Earthand pointing toward the center of the detector; by center we mean the point common to the twoarms). The other two unit-vectors are the directions of each arm of the detector. Therefore,they are unit-vectors starting from the center of the interferometer, and lying on a plane that istangent to the Earth at the point (cid:126)x i (ignoring the curvature of the Earth on the scales of theinterferometer arms).Ref. [47] provides the location (latitude and longitude) of the LIGO Hanford, the LIGOLivingston, and the Virgo detectors, together with the direction that the arms of these detectorsform with the North-South and East-West directions at that point on Earth. The same valuesfor the KAGRA detector can be found in Ref. [48]. The values in Table 1 are obtained from thesedata using basic trigonometry, and they are more convenient for us, since the unit-vectors in theTable can be directly employed in our computations of Section 4. D Analytics for ground-based interferometer overlap functions
In this Appendix we derive the analytic results (52) and (53) given in the main text. Let us startfrom the monopole response function (49), that we rewrite as M λij ( k ) = D abi D cdj × Γ ab,cd,λ ( κ, ˆ s ij ) , Γ Mab,cd,λ ( κ, ˆ s ) ≡ (cid:90) d Ω k π e iκ ˆ k · ˆ s e ab,λ (ˆ k ) e cd,λ ( − ˆ k ) . (D1)29IGO Hanford Central location {− . , − . , . } First Arm {− . , . , . } Second Arm {− . , . , − . } LIGO Livingston Central location {− . , − . , . } First Arm {− . , − . , − . } Second Arm { . , − . , − . } Virgo Central location { . , . , . } First Arm {− . , . , . } Second Arm {− . , − . , . } KAGRA Central location {− . , . , . } First Arm {− . , − . , . } Second Arm { . , − . , . } Table 1:
Cartesian coordinates of the unit-vectors specifying the positions of the interferometersand the direction of their arms, in the coordinate system described in this Appendix. For eachdetector, the distinction between the “first” and “second” arm is purely arbitrary, and plays norelevance in any computation.
The function Γ must be a rank 4 tensor, that is (separately) symmetric under the a ↔ b and the c ↔ d interchange, as well as under ab ↔ cd . These symmetries enforce the structureΓ Mabcd,λ ( κ, ˆ s ) = A λ ( κ ) δ ab δ cd + B λ ( κ ) ( δ ac δ bd + δ ad δ bc )+ C λ ( κ ) ( δ ab ˆ s c ˆ s d + δ cd ˆ s a ˆ s b )+ D λ ( κ ) ( δ ac ˆ s b ˆ s d + δ ad ˆ s b ˆ s c + δ bc ˆ s a ˆ s d + δ bd ˆ s a ˆ s c )+ E λ ( κ ) ˆ s a ˆ s b ˆ s c ˆ s d + F λ ( κ ) ( δ ac (cid:15) bde ˆ s e + δ ad (cid:15) bce ˆ s e + δ bc (cid:15) ade ˆ s e + δ bd (cid:15) ace ˆ s e )+ G λ ( κ ) (ˆ s a ˆ s c (cid:15) bde ˆ s e + ˆ s a ˆ s d (cid:15) bce ˆ s e + ˆ s b ˆ s c (cid:15) ade ˆ s e + ˆ s b ˆ s d (cid:15) ace ˆ s e ) , (D2)where our goal is to find the scalar functions A λ ( κ ) , . . . , G λ ( κ ). To obtain these functions, weconsider a set of independent contractions under which the angular integral of eq. (D1) becomesthe integral of a scalar quantity, that can be immediately performed. Specifically, we performthe following contractions on the left hand side of eq. (D2), p λ ( κ ) ≡ Γ abcd,λ δ ab δ cd = 0 ,q λ ( κ ) ≡ Γ abcd,λ ( δ ac δ bd + δ ad δ bc ) = 2 j ( κ ) ,r λ ( κ ) ≡ Γ abcd,λ ( δ ab ˆ s c ˆ s d + δ cd ˆ s a ˆ s b ) = 0 , ˆ s λ ( κ ) ≡ Γ abcd,λ ( δ ac ˆ s b ˆ s d + δ ad ˆ s b ˆ s c + δ bc ˆ s a ˆ s d + δ bd ˆ s a ˆ s c ) = 4 κ j ( κ ) ,t λ ( κ ) ≡ Γ abcd,λ ˆ s a ˆ s b ˆ s c ˆ s d = 2 κ j ( κ ) ,w λ ( κ ) ≡ Γ abcd,λ ( δ ac (cid:15) bde ˆ s e + δ ad (cid:15) bce ˆ s e + δ bc (cid:15) ade ˆ s e + δ bd (cid:15) ace ˆ s e ) = 4 λj ( κ ) ,z λ ( κ ) ≡ Γ abcd,λ (ˆ s a ˆ s c (cid:15) bde ˆ s e + ˆ s a ˆ s d (cid:15) bce ˆ s e + ˆ s b ˆ s c (cid:15) ade ˆ s e + ˆ s b ˆ s d (cid:15) ace ˆ s e ) = 4 λκ j ( κ ) , (D3)where we have also given the result of the integration in terms of spherical Bessel functions.Performing the same contractions on the right hand side of eq. (D2), and equating the results to30he expressions that we have just found, we then obtain the system A λ B λ C λ D λ E λ F λ G λ = p λ q λ r λ s λ t λ w λ z λ . (D4)This linear system is solved by A λ = − j ( κ )4 κ + 1 + κ κ j ( κ ) , B λ = j ( κ )4 κ + 1 − κ κ j ( κ ) ,C λ = j ( κ )4 κ − κ κ j ( κ ) , D λ = j ( κ )4 κ − − κ κ j ( κ ) ,E λ = − j ( κ )4 κ + 35 − κ κ j ( κ ) ,F λ = λ (cid:20) j ( κ )8 − j ( κ )8 κ (cid:21) , G λ = λ (cid:20) − j ( κ )8 + 5 j ( κ )8 κ (cid:21) . (D5)We have thus fully obtain the analytic expression for (D2). Once we contract with the detectorfunctions D abi D abj , the terms proportional to A ( κ ) and C ( κ ) do not contribute to the responsefunction (D1) due to the fact that these operators are traceless. The remaining terms give rise tothe expression (52), upon the relabelling 2 B λ → f A , D λ → f B , E λ → f C , F λ → λ f D , G λ → λ f E .Let us now move to the dipole response function (49), that we rewrite as D λij ( k ) = D abi D cdj × Γ Dab,cd,λ ( κ, ˆ s ij ) , Γ Dab,cd,λ ( κ, ˆ s, ˆ v ) ≡ i ˆ v · (cid:90) d Ω k π e iκ ˆ k · ˆ s e ab,λ (ˆ k ) e cd,λ ( − ˆ k ) . (D6)A direct comparison between eqs. (D1) and (D6) shows that the function Γ D can be obtainedfrom a derivative of the function Γ M that we just computedΓ Dab,cd,λ ( κ, ˆ s, ˆ v ) = (cid:20) κ ˆ v i ∂∂s i ˜Γ Mab,cd,λ ( κ, (cid:126)s ) (cid:21) (cid:12)(cid:12)(cid:12) s =1 (D7)Before differentiating, we need to promote the expression in eq. (D2) to be a function of avector (cid:126)s of arbitrary magnitude. This can be immediately done from the result we obtained bynoting (from the definition in eq. (D1)) that the magnitude can be absorbed in κ . Therefore, forexample, the term proportional to C becomes˜Γ Mab,cd,λ ( κ, (cid:126)s ) = · · · + C λ ( κ s ) (cid:18) δ ab (cid:126)s c (cid:126)s d + δ cd (cid:126)s a (cid:126)s b s (cid:19) + . . . . (D8)Taking this into account, inserting the result (D2) - (D5) in eq. (D7) leads to the analyticexpression for Γ Dab,cd,λ ( κ, ˆ s, ˆ v ). This expression, once contracted with D abi D cdj leads to the result(53). 31 eferences [1] B. Allen, “The Stochastic gravity wave background: Sources and detection,” in Relativisticgravitation and gravitational radiation. Proceedings, School of Physics, Les Houches,France, September 26-October 6, 1995 , pp. 373–417. 1996. arXiv:gr-qc/9604033[gr-qc] .[2] C. Caprini and D. G. Figueroa, “Cosmological Backgrounds of Gravitational Waves,”
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