Hayama, K.

Kawamura, S.

Nishizawa, A.

Sakagami, M.

Taruya, A.

English

MPG.PuRe

Searching for nontensorial polarizations of stochastic gravitational waves with laser interferometers

A. Nishizawa et al. 1Searching for nontensorial polarizations of stochasticgravitational waves with laser interferometersAtsushi Nishizawa1(a), Atsushi Taruya(b);(c), Kazuhiro Hayama(d);(e), Seiji Kawamura(d),Masa-aki Sakagami(f)(a)Division of Theoretical Astronomy, National Astronomical Observatory of Japan, Mitaka, Tokyo181-8588(b)Research Center for the Early Universe, School of Science, The University of Tokyo, Tokyo 113-0033(c)Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba277-8568(d)TAMA Project, National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588(e)Albert-Einstein-Institut (Max-Planck-Institut fur Gravitationsphysik), Callinstrae 38, D-30167Hannover, Germany(f)Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501AbstractIn general relativity, a gravitational wave (GW) has two polarization modes, while inmodied gravity, the GW is allowed to have additional polarizations. Thus, the obser-vation of the GW polarizations can be utilized for the test of gravity theories. In thisarticle, we investigated the mode-separability and detectability of additional polariza-tion modes of gravitational waves, particularly focusing on a stochastic gravitational-wave background, with laser-interferometric detectors based on the ground and space.As a result, we found that the additional polarization modes can be succcesslly sep-arated and detected.1 IntroductionGeneral relativity (GR) has been strictly tested in the solar system [1, 2], however, has not been stronglyconstrained at a cosmological scale and in a strong eld regime. If the gravity theory is deviated fromGR, it gives rise to various observational signatures. The properties of a gravitational wave (GW) are alsoaltered in the propagation speed, waveforms, and polarization modes. In GR, a GW has two polarizationmodes (plus and cross modes), while in a general metric theory of gravitation, the GW is allowed tohave, at most, six polarizations [1, 3]. Such additional polarizations appear in modied gravity andextra-dimensional theories, corresponding to extra degrees of freedom in the theories. Therefore, theobservation of the GW polarizations can be utilized for the test of the gravity theory.Currently, there are a few observational constraints on the additional polarization modes of GWs. Forthe scalar GWs, the observed orbital-period derivative of PSR B1913+16 agrees well with the predictedvalues of GR, conservatively, at a level of 1% error [2], indicating that the contribution of scalar GWs tothe energy loss is less than 1%. On the other hand, a null result in a search for a stochastic gravitational-wave background (GWB) by LIGO [4] has given an upper limit on an energy density, h20gw . 3:610 6.No detection can also be applied to non-Einsteinian polarizations, though a factor of the upper limit wouldbe corrected, depending on a detector response.In this article, we focus on the stochastic GWB here and investigate the separability of the polarizationmodes of the GWB with laser-interferometric detecters.2 GW polarizations and cross-correlation analysisUsing the unit vectors m^ and n^ perpendicular to the unit vector pointing at the GW propagation direction^ and to each other, the polarization tensors for p = +;; b; `; x; and y called plus, cross, breathing,1Email address: atsushi.nishizawa@nao.ac.jp2 Searching for nontensorial polarizations of stochastic GWslongitudinal, vector-x, and vector-y modes, respectively, are dened by [1, 3]: e+ = m^  m^   n^  n^,e = m^ n^+ n^ m^, eb = m^ m^+ n^ n^, e` =p2 ^ ^, ex = m^ ^+ ^ m^, ey = n^ ^+ ^ n^.Each polarization mode is orthogonal to one another and is normalized so that epijeijp0 = 2pp0 ; p; p0 =+;; b; `; x; and y. The angular response function of the I-th interferometer is given by contraction ofthe polarization tensors with the detector tensors: F pI (^)  DI : ep(^) and DI  [u^ u^  v^  v^] =2.The unit vectors u^ and v^ are directed to each detector arm. Note that the above expression is validonly when the arm length of the detector, L, is much smaller than the wavelength of GWs, g, in theobservational frequency band we consider. This condition holds for both ground-based detectors andspace-based detector such as DECICO.We assume that a stochastic GWB is (i) isotropic, (ii) independently polarized (not correlated be-tween polarizations), (iii) stationary, and (iv) Gaussian. Conventionally, the amplitude of GWB foreach polarization is characterized by pgw(f)  (dpgw=d ln f)=c, where c = 3H20=8G and H0 =100h0 km sec 1Mpc 1 [5{8]. Then, we dene the GWB energy density in tensor, vector, and scalarpolarization modes as Tgw  +gw + gw, Vgw  xgw + ygw, Sgw  bgw + `gw = bgw(1 + ). Here weassume +gw = gw and xgw = ygw. For the scalar mode, we introduced a model-dependent parameter,(f)  `gw(f)=bgw(f).To distinguish a stochastic GWB from detector random noise, one needs to correlate detector's signals[5{8]. We assume that the amplitude of GWB is much smaller than detector noise. In the cross-correlationanalysis between I-th and J-th detectors, a GW signal can be written as =3H20202Tobs sin2 Z 1 1df jf j 3 ~Q(f)Tgw(f)TIJ (f) + Vgw(f)VIJ(f) + (f)Sgw(f)SIJ (f);where Tobs is observation time, ~Q(f) is a lter function, which weight the correlation signal so thatsignal-to-noise ratio (SNR) is maximized. The parameter dened by (f)  [1 + 2(f)]=3[1+ (f)] takesthe value in the range 1=3    2=3, depending on the ratio of the energy density in the longitudinalmode to the breathing mode. The prefactor, sin2  = 1   (u^  v^)2, comes from the detector tensor withnon-orthogonal detector arms. The sensitivity to the GWB with each polarization can be characterizedby so-called overlap reduction functions (ORF) [9]MIJ(f) 1sin2 ZS2d^4e2if^ ~X=cRMIJ ;with RTIJ(^)  (5=2) (F+I F+J + FI FJ ), RVIJ(^)  (5=2) (F xI F xJ + F yI F yJ ), and RSIJ (^)  [15=(1 +2)] (F bIF bJ + F `IF `J). The subscript M denotes M = T; V; S, and ~X  ~XI   ~XJ .3 Mode-separation and SNRThe three polarization modes, in principle, can be separated by linearly combining more than threeindependent correlation signals from detector pairs. In general case with arbitrarily large number Npairof the correlation signal, an SNR by optimally combining the correlation signals is given by [10]SNRM =9H204022TobsZ 10df(Mgw(f))2 detF(f)f6FM (f)1=2; (1)F(f) =0@ FTT FTV FTSFTV FV V FV SFTS FV S FSS1A ; FMM 0(f) =XiMi (f)M 0i (f)Ni(f) ;where M and M 0 denote polarization modes, M;M 0 = T; V; S. The quantity FM is the determinant ofthe submatrix, which is constructed by removing the M 's elements from F. The subscript i designates adetector pair (for I-th and J-th detector pair, i = IJ), and Ni(f) is dened as, say, N12(f)  P1(f)P2(f).The analytical t of the noise power spectrum of a single interferometer is given byP (f) ="6:4 10 51f1Hz 4+ 1:7 10 48 + 5:8 10 50f1Hz2#Hz 1 :A. Nishizawa et al. 3Figure 1: Four clusters, A, A', B, and C, sharingthe orbit, whose radius is 1AU.Figure 2: Detectable h20gw (h20gw for the scalarmode) after the mode separation with four clustersof DECIGO.for DECIGO andP (f) =8>>>><>>>>:10 44f10Hz 4+ 10 47:25f100Hz 1:7Hz 1 for 10Hz  f  240Hz ;10 46f1000Hz3Hz 1 for 240Hz  f  3000Hz ;1 otherwise :for advanced LIGO. The mode separability signicantly aects the SNR via detF in the integrand inEq. (1). To successfully separate the modes, the condition, detF 6= 0, is necessary and leads to twoconditions that have to be satised for the detector conguration: (i) the detectors have to be located ata distance more than one wavelength of the GW, (ii) detector pairs do not geometrically degenerated. Ifone of the two conditions fails, detF  0 suppresses the SNR.In the calculations below, we will assume that Mgw(f) has a at spectrum, i.e. frequency-independent,and the observation time is Tobs = 3yr. We set the detection threshold to SNR = 5, then it leads to thedetectable h20gw (h20gw for the scalar mode).3.1 Ground-based interferometersWe perform the polarization mode separation with three detectors (minimum set needed to separatemodes) among the advanced interferometers on the Earth (at  100Hz) such as AIGO, advanced LIGOat Hanford and Livingston, advanced VIRGO, and LCGT. Here we assume that all interferometershave the same noise spectrum as that of advanced LIGO, i.e. PI(f) = P (f). The SNR calculation isstraightforward because these interferometers are located on the Earth at the distance more than onewavelength of a GW at  100Hz. Therefore, there is no problem concerning the mode degeneracy andthe geometrical degeneracy of the detectors. According to [9], the set of three advanced detectors issensitive to the GWB of h20gw  10 9   10 8 for each polarization. This sensitivity is almost the sameas that without the mode separation.3.2 Space-based interferometersDECIGO in the current conceptual design [11] is composed of four clusters, orbiting at 1AU fromthe Sun, as shown in Fig. 1. Each cluster has three spacecrafts, which form three Fabry-Perot cavitywith the armlength 103 km. By measuring the relative distance between a pair of the spacecrafts, threeinterferometer's signals are obtained in a cluster. The correlation signals that we use for the SNR4 Searching for nontensorial polarizations of stochastic GWscalculation are those constructed with interferometers of interclusters, because in a cluster, one cannotobtain the correlation signal sensitive enough to a GWB at low frequencies [10].The detector conguration is shown in Fig. 1. For mathematical details of this conguration, see [10].Since the detector separation can be approximated as the distance between the guiding centers of clusters,the distances between the clusters D  j~Xj are unchanged during their orbital motion and are givenby DAB() = 2R0j sin(=2)j for the AB link, DAC() = DAB() for AC link and DBC() = 2R0j sinjfor the BC link. One should be noted that the only parameter in this conguration is .In this detector conguration, the detector separation D is typically of the order of 1 AU. This meansthat the ORF starts to oscillate and rapidly decay above the characteristic frequency, fc  c=(2D) 10 3Hz. Thus, the large detector separation considerably degrades the sensitivity to the GWB in 0:1 1Hz band, however, since the ORF of each polarization mode oscillates dierently in the band, the modeseparability is pretty good.The SNR as a function of  is calculated with, in total, 54 correlation signals (AA', AB, AC, A'B,A'C, BC 9 links = 54). The result is shown in Fig. 2. At   120, the sensitivity degrades due to thesymmetry of the detector conguration, in other words, some correlation signals are degenerated. As approach 0 and 180, the detector sensitivity peaks, since the clusters A and B (or C), and B and Care closely located, respectively. However, such a conguration considerably loses the angular resolutionto point GW sources. Thus, an optimal angle would be  = 60, which leads to h20Tgw = 2:2  10 14,h20Vgw = 1:110 14, and h20Sgw = 1:910 14. These sensitivities should be compared with those whenthe polarization modes are not separated. For two clusters that are colocated and coaligned, e.g. clustersA and A', the sensitivity is h20gw = 7:1 10 17. The mode separation degrades the sensitivity by a fewhundred times. However, the important point here is that the non-Einsteinian-polarization search doesnot impair the cross-correlation sensitivity to a GWB with the colocated and coaligned clusters at all,though the mode is not separated.4 ConclusionThe GW polarizations can be utilized as a novel and accurate test of gravity. We showed that theground-based advanced interferometers and the proposed space-based detectors such as DECIGO andBBO can successfully separate and probe the GWB with the non-Einsteinian polarization modes. TheGWB search with the GW detectors is complemental to the searches at much dierent frequencies: CMBand pulsar timing. If the non-Einstein polarizations would be detected, it implies that GR should beextended.References[1] C. M. Will, Theory and experiment in gravitational physics, Cambridge University Press (1993).[2] C. M. Will, Living Rev. Relativity 9, 3 (2006).[3] D. M. Eardley, D. L. Lee. A. P. Lightman, R. V. Wagoner, and C. M. Will, Phys. Rev. Lett. 30, 884(1973).[4] The LIGO Scientic Collaboration and The Virgo Collaboration, Nature 460, 990 (2009).[5] N. Christensen, Phys. Rev. D 46, 5250 (1992);[6] E. E. Flanagan, Phys. Rev. D 48, 2389 (1993);[7] B. Allen and J. D. Romano, Phys. Rev. D 59, 102001 (1999);[8] M. Maggiore, Phys. Rep. 331, 283 (2000).[9] A. Nishizawa, A. Taruya, K. Hayama, K. Kawamura, and M. Sakagami, Phys. Rev. D 79, 082002(2009).[10] A. Nishizawa, A. Taruya, and S. Kawamura, arXiv:0911.0525 (2009).[11] S. Sato et al., Journal of Physics: Conference Series 154, 012040 (2009).

Searching for nontensorial polarizations of stochastic gravitational waves with laser interferometers

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