Supplement To: Search for Tensor, Vector, and Scalar Polarizations in the Stochastic Gravitational-Wave Background

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1 Supplement To: Search or Tensor, Vector, and Scalar Polarizations in the Stochastic GravitationalWave Background B. P. Abbott et al. (LIGO Scientiic Collaboration & Virgo Collaboration) This documents contains additional inormation and results supplemental to the material presented in Re. []. In particular, we illustrate the contributions o dierent requency bands to Advanced LIGO s stochastic search sensitivity, urther describe the hypothesis testing procedure used in Re. [], and discuss our treatment o calibration uncertainties. Finally, we show complete parameter estimation results under a variety o hypotheses or the polarization content o the stochastic gravitationalwave background. Sensitive Frequency Bands Although the stochastic search or nonstandard polarizations utilizes the ull 7 Hz requency band, dierent requency subbands contribute variously to our overall search sensitivity. To illustrate this, we can investigate the contribution rom each requency bin to a background s optimal signaltonoise ratio (SR), given by [] SR = H [ A π T γ A()Ω A () ] d. P ()P () (A) Up to additive constants, SR and O SIG are related by ln O SIG SR /. Using the measured O search sensitivity, Fig. illustrates the cumulative raction o the squaredsr o several representative hypothetical backgrounds, obtained by integrating Eq. (A) rom Hz up to a cuto requency. Results are shown or purely tensor (blue), vector (red), and scalarpolarized (green) backgrounds, with spectral indices α =,, and. As seen in Fig., the most sensitive requency band or a given background is highly dependent on the background s spectral index. For steeply negativelysloped backgrounds (α = ), the majority o the measured SR is obtained at very low requencies between Hz. Meanwhile, the Hz band is most sensitive to lat backgrounds, and high requencies above 7 Hz are most sensitive to steeply positivelysloped backgrounds. Although trends are generally independent o polarization, Fig. does show somewhat dierent behaviors or tensor, vector, and scalar modes. These dierences are due to the dierent overlap reduction unctions or each polarization sector. Model Construction Here, we briely summarize the construction o our Signal, Gaussian noise, onstandard polarization, and Tensorpolarization hypotheses; see Re. [] or urther details. Gaussian noise: We assume that no signal is present and the observed crosspower Ĉ() is Gaussian distributed about zero with variance given by Eqs. (9) and () o Re. []. Although Advanced LIGO instrumental noise is neither stationary nor Gaussian, searches or the stochastic background Cumulative SR Tensor Vector Scalar α = α = α = (Hz) FIG.. Cumulative squared signaltonoise ratios as a unction o requency or hypothetical backgrounds o tensor (blue), vector (red), scalar (green) polarizations with spectral indices α =,, and (solid, dashed, and dotdashed, respectively). The three α = curves lie nearly on top o one another, as do the three α = curves. The Advanced LIGO network is most sensitive to negativelysloped backgrounds at low requencies, while high requencies contribute the most sensitively to positivelysloped backgrounds. are nonetheless welldescribed by Gaussian statistics due to the large number o timesegments combined to orm the inal crosspower spectrum Ĉ() []. Signal: The Signal hypothesis is the union o seven subhypotheses, which together allow or each unique combination o tensor, vector, and scalar polarizations. The TVS subhypothesis, or example, assumes the simultaneous presence o all polarization modes, with a canonical energydensity spectrum o the orm: ( ) ( ) ( ) Ω TVS () = Ω T + Ω V + Ω S. (A) The TS subhypothesis, meanwhile, assumes only the existence o tensor and scalar modes: ( ) ( ) Ω TS () = Ω T + Ω S. (A) In this ashion, we can construct seven unique subhypotheses: {T,V,S,TV,TS,VS,TVS}. The union o these

2 seven possibilities is the Signal hypothesis. onstandard polarization () Analogous to the Signal hypothesis above, this is the union o the six subhypotheses {V,S,TV,TS,VS,TVS} containing nonstandard polarizations. Tensorpolarization () We assume the stochastic background is present and purelytensor polarized, with the energydensity spectrum ( ) Ω = Ω T ; (A) this hypothesis is identical the T signal subhypothesis above. Odds Ratios Here, we review the procedure or constructing odds O SIG between Signal and Gaussian noise hypotheses, and odds O between the and hypotheses. As above, see Re. [] or urther details. The odds between two hypotheses M and is the ratio o posterior probabilities or each hypothesis, given data d: O M = p(m d) p( d) π(m) = B M π( ), (A) where B M is the Bayes actor between the two hypotheses and π(m) and π( ) are the prior probabilities on M and, respectively. The ratio π(m)/π( ) is known as the prior odds. To obtain O SIG, we irst compute the Bayes actor B A between each Signal subhypothesis A {T, V, S,...} and the oise hypothesis. Because each subhypothesis is independent, O SIG is then just the sum O SIG A SIG A SIG O A B A π(a SIG)π(SIG), π() (A) where we have expanded π(a) = π(a SIG)π(SIG). We choose equal prior probabilities on the Signal and oise hypotheses, such that π(sig)/π() =. Within the Signal hypothesis, we assign equal probabilities to each subhypothesis, giving π(a SIG) = /7. The odds O is analogously given by O A B A π(a )π(). (A7) π() We set π()/π() = and again choose equal prior probabilities or each subhypotheses within, such that π(a ) = /. TABLE I. Bayes actors between each signal subhypothesis and the Gaussian noise hypothesis, as computed by Multiest. These Bayes actors are combined ollowing Eqs. (A) and (A7) to obtain odds O SIG between Signal and Gaussian noise hypotheses, and odds O between and hypotheses. ln B A T. V. S. TV. TS. VS. TVS.99 Our chosen prior odds between hypotheses are necessarily somewhat arbitrary, and dierent choices will yield dierent values o O SIG and O. For completeness, Table I provides the Bayes actors between each signal subhypothesis and Gaussian noise. These Bayes actors allow readers to recompute odds O SIG and O using dierent choices o prior odds. Calibration Uncertainty The strain measured by LIGOHanord and LIGO Livingston is not known perectly, but is subject to nonzero calibration uncertainty. For imperectly calibrated data, the crosspower measurements Ĉ() are not estimators o A γ A()Ω A (), but rather o λ A γ A()Ω A (), where λ is some multiplicative actor []. Perect calibration would yield λ =, but in general λ is unknown. We include the calibration actor λ as an additional parameter in Multiest, so that the likelihood unction becomes L(Ĉ() ΩA, α A, λ) (Ĉ() λ A γ A()Ω A (/ ) α A exp σ () ), (A) with Ĉ() and σ () given by Eqs. () and () o Re. [], respectively. We place a Gaussian prior on λ, centered at λ = : ( ) (λ ) π(λ) exp ɛ. (A9) The standard deviation ɛ encapsulates the amplitude calibration uncertainty. Within the 7 Hz requency band, LIGOHanord and LIGOLivingston have maximum estimated amplitude uncertainties o.% and.%, respectively []. These uncertainty estimates have been improved relative

3 to the the uncertainties previously adopted in Res. [7 9]. For our analysis, we take ɛ =.7, the quadrature sum o the Hanord and Livingston uncertainties [7]. All results are given ater marginalization over λ. In the above prescription we have made two simpliying assumptions. First, we have neglected phase calibration uncertainty, which is expected to be a subdominant source o uncertainty in the stochastic analysis [, ]. Secondly, although calibration uncertainties are requencydependent, or simplicity we adopt uniorm amplitude uncertainties across all requencies. Our quoted amplitude uncertainties are conservative, encompassing the largest calibration uncertainties in the stochastic sensitivity band [, 7]. Detailed Parameter Estimation Results In Re. [], we presented marginalized posteriors or the tensor, vector, and scalar background amplitudes under the TVS hypothesis. In Fig. we show the ull sixdimensional parameter estimation results obtained when choosing loguniorm amplitude priors. Diagonal subplots show marginalized posteriors on the amplitudes and slopes o each polarization, while the interior subplots show joint posteriors between each pair o parameters. The spectral index posteriors (not shown in Re. []) are largely consistent with our choice o prior, but indicate a slight bias against large positive spectral indices. This relects the act that Advanced LIGO is most sensitive to backgrounds o large, positive slopes []. The nondetection o a stochastic background thereore constrains larger amplitudes to have small and/or negative spectral indices; see, or instance, the joint log Ω T α T posterior in Fig.. Figure, meanwhile, shows ull parameter estimation results when alternatively assuming uniorm amplitude priors. Here, the posterior preerence towards small or negative spectral indices is ar more pronounced. The joint D posteriors (e.g. Ω T α T ) again illustrate that large, positive slopes are preerentially ruled out in case o large background amplitudes. As stated in Re. [], upper limits obtained under one hypothesis are not, in general, equal to those obtained under some dierent hypothesis. While we presented upper limits only or the TVS hypothesis, results rom other hypotheses may be desired as well (the TS results, or instance, are best suited or comparison to predictions rom scalartensor theories). In Tables II and III we have thereore listed the % credible upper limits corresponding to each signal subhypothesis, or both loguniorm and uniorm amplitude priors. We have also listed % credible bounds on spectral indices or each choice o amplitude prior. [] B. P. Abbott et al., Phys. Rev. Lett., (). [] B. Allen and J. D. Romano, Phys. Rev. D 9, (999). [] T. Callister et al., Phys. Rev. X 7, (7). [] D. Meacher et al., Phys. Rev. D 9, (). [] J. T. Whelan, E. L. Robinson, J. D. Romano, and E. H. Thrane, J. Phys. Con. Ser., 7 (). [] C. Cahillane et al., Phys. Rev. D 9, (7). [7] B. P. Abbott et al., Phys. Rev. Lett., (7). [] B. P. Abbott et al., Phys. Rev. Lett., (7), Online Supplement. [9] B. P. Abbott et al., Phys. Rev. Lett., (7).

4 log ΩV log ΩT log ΩS 9 log ΩV 9 log ΩS FIG.. Posterior probability distributions or the powerlaw amplitudes and slopes o tensor, vector, and scalar contributions to the stochastic background, assuming the TVS hypothesis and loguniorm amplitude priors. The subplots along the diagonal show marginalized posteriors or each parameter; dashed curves show the corresponding prior. The marginalized amplitude posteriors yield the % credible upper limits given in Table I. The remaining subplots, meanwhile, show the twodimensional posteriors between each pair o parameters, as well as contours containing the central % and % posterior probability. TABLE II. Parameter estimation results or each signal subhypothesis, obtained with loguniorm priors on the amplitude o each polarization A component. Columns give % credible upper limits on log ΩA, and columns 7 show the corresponding limits on Ω or convenience. Columns, meanwhile, show the central % credible bounds on spectral indices αa. The parameter estimation results or the TVS subhypothesis (inal row) correspond to those given in Table I o the main text. T V S TV TS VS TVS log ΩT,%. log ΩV,%. log ΩS,% ΩT,% ΩV,% ΩS,%

5 ΩV.. ΩS ΩT.. ΩV..9 ΩS... FIG.. As in Fig., but assuming a uniorm prior on background amplitudes. The marginalized amplitude posteriors yield the % credible upper limits given in Table I. TABLE III. Parameter estimation results or each signal subhypothesis, obtained with uniorm priors on the background amplitudes. Columns are deined as in Table II above. The parameter estimation results or the TVS subhypothesis (inal row) correspond to those given in Table I o the main text. log ΩT,% log ΩV,% log ΩS,% ΩT,% T. V. S. ΩV,% ΩS,% TV.. TS VS TVS..