A Search for Tensor, Vector, and Scalar Polarizations in the Stochastic Gravitational-Wave Background

TL;DR

This paper searches for different polarization modes in the stochastic gravitational-wave background using LIGO data, setting the first direct bounds on vector and scalar contributions, and finds no evidence for non-tensor polarizations.

Contribution

It provides the first direct constraints on vector and scalar polarizations in the stochastic gravitational-wave background, expanding tests of gravitational theories beyond general relativity.

Findings

No evidence for non-tensor polarizations was found.

Upper bounds were placed on the energy densities of vector and scalar modes.

Constraints improve understanding of gravitational wave polarization properties.

Abstract

The detection of gravitational waves with Advanced LIGO and Advanced Virgo has enabled novel tests of general relativity, including direct study of the polarization of gravitational waves. While general relativity allows for only two tensor gravitational-wave polarizations, general metric theories can additionally predict two vector and two scalar polarizations. The polarization of gravitational waves is encoded in the spectral shape of the stochastic gravitational-wave background, formed by the superposition of cosmological and individually-unresolved astrophysical sources. Using data recorded by Advanced LIGO during its first observing run, we search for a stochastic background of generically-polarized gravitational waves. We find no evidence for a background of any polarization, and place the first direct bounds on the contributions of vector and scalar polarizations to the stochastic background. Under log-uniform priors for the energy in each polarization, we limit the energy-densities of tensor, vector, and scalar modes at 95% credibility to $\Omega^T_0 < 5.6 \times 10^{-8}$, $\Omega^V_0 < 6.4\times 10^{-8}$, and $\Omega^S_0 < 1.1\times 10^{-7}$ at a reference frequency $f_0 = 25$ Hz.

Tables

Table 1. Table 1: 95% credible upper limits on the log amplitudes of tensor, vector, and scalar modes in the stochastic background at reference frequency f 0 = 25 subscript 𝑓 0 25 f_{0}=25 Hz. We assume an energy-density spectrum in which all three modes are present, and present limits following marginalization over the spectral index of each component [see Eq. ( 13 )]. We show results for two different amplitude priors: a log-uniform prior ( d ​ p / d ​ log ⁡ Ω 0 ∝ 1 proportional-to 𝑑 𝑝 𝑑 subscript Ω 0 1 dp/d\log\Omega_{0}\propto 1 ; top row) and a uniform prior ( d ​ p / d ​ Ω 0 ∝ 1 proportional-to 𝑑 𝑝 𝑑 subscript Ω 0 1 dp/d\Omega_{0}\propto 1 ; bottom row). Additional parameter estimation results are shown in the Supplemental Material.

Prior	$\log {\Omega }_0^T$	$\log {\Omega }_0^V$	$\log {\Omega }_0^S$	${\Omega }_0^T$	${\Omega }_0^V$	${\Omega }_0^S$
Log uniform	$-7.25$	$-7.20$	$-6.96$	$5.58\text{×}{10}^{-08}$	$6.35\text{×}{10}^{-08}$	$1.08\text{×}{10}^{-07}$
Uniform	$-6.70$	$-6.59$	$-6.07$	$2.02\text{×}{10}^{-07}$	$2.54\text{×}{10}^{-07}$	$8.44\text{×}{10}^{-07}$

Table 2. Table 2: Bayes factors between each signal sub-hypothesis and the Gaussian noise hypothesis, as computed by MultiNest . These Bayes factors are combined following Eqs. ( 19 ) and ( 20 ) to obtain odds 𝒪 n sig subscript superscript 𝒪 sig n \mathcal{O}^{\textsc{sig}}_{\textsc{n}} between Signal and Gaussian noise hypotheses, and odds 𝒪 gr ngr subscript superscript 𝒪 ngr gr \mathcal{O}^{\textsc{ngr}}_{\textsc{gr}} between NGR and GR hypotheses.

Hypothesis	$\ln {ℬ}_{\text{n}}^{𝒜}$
T	$-0.33$
V	$-0.33$
S	$-0.31$
TV	$-0.66$
TS	$-0.65$
VS	$-0.65$
TVS	$-0.99$

Table 3. Table 3: Parameter estimation results for each signal sub-hypothesis, obtained with log-uniform priors on the amplitude of each polarization component. Columns 2-4 give 95% credible upper limits on log ⁡ Ω 0 A subscript superscript Ω 𝐴 0 \log\Omega^{A}_{0} , and columns 5-7 show the corresponding limits on Ω 0 A subscript superscript Ω 𝐴 0 \Omega^{A}_{0} for convenience. Columns 8-10, meanwhile, show the central 95% credible bounds on spectral indices α A subscript 𝛼 𝐴 \alpha_{A} . The parameter estimation results for the TVS sub-hypothesis (final row) correspond to those given in Table 1 of the main text.

Hypothesis	$\log {\Omega }_0^{T,95\%}$	$\log {\Omega }_0^{V,95\%}$	$\log {\Omega }_0^{S,95\%}$	${\Omega }_0^{T,95\%}$	${\Omega }_0^{V,95\%}$	${\Omega }_0^{S,95\%}$	${\alpha }_T$	${\alpha }_V$	${\alpha }_S$
T	$-7.21$	-	-	$6.23\text{×}{10}^{-08}$	-	-	$-{0.4}_{-5.9}^{+5.5}$	-	-
V	-	$-7.16$	-	-	$6.96\text{×}{10}^{-08}$	-	-	$-{0.4}_{-5.8}^{+4.9}$	-
S	-	-	$-6.86$	-	-	$1.38\text{×}{10}^{-07}$	-	-	$-{0.5}_{-5.8}^{+5.4}$
TV	$-7.25$	$-7.16$	-	$5.56\text{×}{10}^{-08}$	$6.95\text{×}{10}^{-08}$	-	$-{0.4}_{-5.9}^{+5.6}$	$-{0.4}_{-5.9}^{+4.8}$	-
TS	$-7.25$	-	$-6.93$	$5.68\text{×}{10}^{-08}$	-	$1.18\text{×}{10}^{-07}$	$-{0.4}_{-5.9}^{+5.5}$	-	$-{0.5}_{-5.8}^{+5.4}$
VS	-	$-7.17$	$-6.90$	-	$6.74\text{×}{10}^{-08}$	$1.25\text{×}{10}^{-07}$	-	$-{0.4}_{-5.9}^{+5.0}$	$-{0.5}_{-5.8}^{+5.4}$
TVS	$-7.25$	$-7.20$	$-6.96$	$5.58\text{×}{10}^{-08}$	$6.35\text{×}{10}^{-08}$	$1.08\text{×}{10}^{-07}$	$-{0.4}_{-5.9}^{+5.5}$	$-{0.4}_{-5.9}^{+4.8}$	$-{0.6}_{-5.8}^{+5.4}$

Table 4. Table 4: Parameter estimation results for each signal sub-hypothesis, obtained with uniform priors on the background amplitudes. Columns are defined as in Table 3 above. The parameter estimation results for the TVS sub-hypothesis (final row) correspond to those given in Table 1 of the main text.

Hypothesis	$\log {\Omega }_0^{T,95\%}$	$\log {\Omega }_0^{V,95\%}$	$\log {\Omega }_0^{S,95\%}$	${\Omega }_0^{T,95\%}$	${\Omega }_0^{V,95\%}$	${\Omega }_0^{S,95\%}$	${\alpha }_T$	${\alpha }_V$	${\alpha }_S$
T	$-6.60$	-	-	$2.53\text{×}{10}^{-07}$	-	-	$-{0.4}_{-4.7}^{+3.8}$	-	-
V	-	$-6.48$	-	-	$3.29\text{×}{10}^{-07}$	-	-	$-{1.5}_{-4.9}^{+3.3}$	-
S	-	-	$-5.97$	-	-	$1.08\text{×}{10}^{-06}$	-	-	$-{2.6}_{-4.0}^{+4.1}$
TV	$-6.66$	$-6.55$	-	$2.18\text{×}{10}^{-07}$	$2.85\text{×}{10}^{-07}$	-	$-{1.8}_{-4.7}^{+3.9}$	$-{1.7}_{-4.9}^{+3.6}$	-
TS	$-6.65$	-	$-6.03$	$2.23\text{×}{10}^{-07}$	-	$9.28\text{×}{10}^{-07}$	$-{1.7}_{-4.7}^{+3.8}$	-	$-{2.6}_{-4.1}^{+4.0}$
VS	-	$-6.53$	$-6.04$	-	$2.96\text{×}{10}^{-07}$	$9.21\text{×}{10}^{-07}$	-	$-{1.7}_{-4.7}^{+3.6}$	$-{2.8}_{-4.0}^{+4.1}$
TVS	$-6.70$	$-6.59$	$-6.07$	$2.02\text{×}{10}^{-07}$	$2.54\text{×}{10}^{-07}$	$8.44\text{×}{10}^{-07}$	$-{1.7}_{-4.8}^{+3.8}$	$-{1.7}_{-4.7}^{+3.7}$	$-{2.6}_{-4.1}^{+4.0}$

Equations

$${\hat{e}}_{+}=\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 0\end{array}\right){\hat{e}}_{\times }=\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right)$$

$${\hat{e}}_x$$

$${\hat{e}}_b$$

$$⟨{\tilde{s}}_1(f){\tilde{s}}_2^{\ast }(f^{\prime })⟩=\frac{1}{2}\delta (f-f^{\prime })\sum _A{\Gamma }_A(f)S_h^A(f).$$

$${\Gamma }_A(f)=\frac{1}{8\pi }\sum _{a\in A}\int d\hat{n}{F}_1^a(\hat{n})F_2^a(\hat{n})e^{2\pi if\hat{n}\cdot \Delta x/c}.$$

$$\Omega (f)=\frac{1}{{\rho }_c}\frac{d{\rho }_{\text{textsc}gw}}{d\ln f},$$

$$\Omega (f)=\frac{2{\pi }^2}{3H_0^2}{f}^3{S}_h(f).$$

$${\hat{C}}_i(f)=\frac{1}{\Delta T}\frac{20{\pi }^2}{3H_0^2}{f}^3{\tilde{s}}_{1,i}^{\ast }(f){\tilde{s}}_{2,i}(f),$$

$$⟨{\hat{C}}_i(f)⟩=\sum _A{\gamma }_A(f){\Omega }^A(f)$$

$${\sigma }_i^2(f)=\frac{1}{2\Delta Tdf}{\left(\frac{10{\pi }^2}{3H_0^2}\right)}^2{f}^6{P}_{1,i}(f)P_{2,i}(f),$$

$$⟨{\tilde{s}}_{I,i}^{\ast }(f){\tilde{s}}_{I,i}(f^{\prime })⟩=\frac{1}{2}\delta (f-f^{\prime })P_{I,i}(f).$$

$$\hat{C}(f)=\frac{{\sum }_i{\hat{C}}_i(f){\sigma }_i^{-2}(f)}{{\sum }_i{\sigma }_i^{-2}(f)},$$

$${\sigma }^{-2}(f)=\sum _i{\sigma }_i^{-2}(f).$$

$$\Omega (f)={\Omega }_0^T{\left(\frac{f}{f_0}\right)}^{{\alpha }_T}+{\Omega }_0^V{\left(\frac{f}{f_0}\right)}^{{\alpha }_V}+{\Omega }_0^S{\left(\frac{f}{f_0}\right)}^{{\alpha }_S}.$$

$${\text{SNR}}^2=\frac{3H_0^2}{10{\pi }^2}2T{\int }_0^{\infty }\frac{{\left[{\sum }_A{\gamma }_A(f){\Omega }^A(f)\right]}^2}{f^6{P}_1(f)P_2(f)}df.$$

$${\Omega }_{\text{textsc}tvs}(f)={\Omega }_0^T{\left(\frac{f}{f_0}\right)}^{{\alpha }_T}+{\Omega }_0^V{\left(\frac{f}{f_0}\right)}^{{\alpha }_V}+{\Omega }_0^S{\left(\frac{f}{f_0}\right)}^{{\alpha }_S}.$$

$${\Omega }_{\text{textsc}ts}(f)={\Omega }_0^T{\left(\frac{f}{f_0}\right)}^{{\alpha }_T}+{\Omega }_0^S{\left(\frac{f}{f_0}\right)}^{{\alpha }_S}.$$

$${\Omega }_{\text{textsc}gr}={\Omega }_0^T{\left(\frac{f}{f_0}\right)}^{{\alpha }_T};$$

$$\displaystyle\mathcal{O}^{\scalebox{0.6}{$\mathcal{M}$}}_{\scalebox{0.6}{$\mathcal{N}$}}$$

$$\displaystyle=\mathcal{B}^{\scalebox{0.6}{$\mathcal{M}$}}_{\scalebox{0.6}{$\mathcal{N}$}}\frac{\pi(\mathcal{M})}{\pi(\mathcal{N})},$$

$${\mathcal{O}}_{\text{textsc}n}^{\text{textsc}sig}$$

$$\displaystyle=\sum_{{\scalebox{0.6}{$\mathcal{A}$}}\,\in\,\textsc{sig}}\mathcal{B}^{\scalebox{0.6}{$\mathcal{A}$}}_{\textsc{n}}\frac{\pi(\mathcal{A}|\text{SIG})\pi(\text{SIG})}{\pi(\text{N})},$$

$$\mathcal{O}^{\textsc{ngr}}_{\textsc{gr}}=\sum_{{\scalebox{0.6}{$\mathcal{A}$}}\,\in\,\textsc{ngr}}\mathcal{B}^{\scalebox{0.6}{$\mathcal{A}$}}_{\textsc{gr}}\frac{\pi(\mathcal{A}|\text{NGR})\pi(\text{NGR})}{\pi(\text{GR})}.$$

$$\mathcal{L}($$

$$\propto \prod _f\exp \left[-\frac{{\left(\hat{C}(f)-\lambda {\sum }_A{\gamma }_A(f){\Omega }_0^A(f/f_0{)}^{{\alpha }_A}\right)}^2}{2{\sigma }^2(f)}\right],$$

$$\pi (\lambda )\propto \exp \left(-\frac{{\left(\lambda -1\right)}^2}{2{ϵ}^2}\right).$$

Full text

Introduction. – The direct detection of gravitational waves offers novel opportunities to test general relativity in previously unexplored regimes. Already, the compact binary mergers Abbott et al. (2016a, b, 2017a, 2017b, 2017c) observed by Advanced LIGO (the Laser Interferometer Gravitational Wave Observatory) Aasi et al. (2015); Abbott et al. (2016c) and Advanced Virgo Acernese et al. (2015) have enabled improved limits on the graviton mass, experimental measurements of post-Newtonian parameters, and inference of the speed of gravitational waves, among other tests Abbott et al. (2016d, e, 2017a, 2017d).

Another central prediction of general relativity is the existence of only two gravitational-wave polarizations: the tensor plus and cross modes, with spatial strain tensors

[TABLE]

(assuming waves propagating in the $+\hat{z}$ direction). Generic metric theories of gravity, however, can allow for up to four additional polarizations: the $x$ and $y$ vector modes and the breathing and longitudinal scalar modes, with basis strain tensors Eardley et al. (1973a, b); Will (2014)

[TABLE]

The observation of vector or scalar modes would be in direct conflict with general relativity, and so the direct measurement of gravitational-wave polarizations offers a promising avenue by which to test theories of gravity Will (2014).

Recently, the Advanced LIGO-Virgo network has succeeded in making the first direct statement about the polarization of gravitational waves. The gravitational-wave signal GW170814, observed by both the Advanced LIGO and Virgo detectors, significantly favored a model assuming pure tensor polarization over models with pure vector or scalar polarizations Abbott et al. (2017b); Isi and Weinstein (2017). In general, however, the ability of the Advanced LIGO-Virgo network to study the polarization of gravitational-wave transients is limited by several factors. First, the LIGO-Hanford and LIGO-Livingston detectors are nearly co-oriented, preventing Advanced LIGO from sensitively measuring more than a single polarization mode Abbott et al. (2016d, e, 2017b); Isi and Weinstein (2017). Second, at least five detectors are needed to fully characterize the five polarization degrees of freedom accessible to quadrupole detectors. Quadrupole detectors (those measuring differential arm motion) have degenerate responses to breathing and longitudinal modes, and can therefore measure only a single linear combination of scalar breathing and longitudinal polarizations Chatziioannou et al. (2012); Will (2014); Berti et al. (2015); Isi and Weinstein (2017).

Beyond compact binary mergers, another target for Advanced LIGO and Virgo is the stochastic gravitational-wave background. An astrophysical stochastic background is expected to arise from the population of distant compact binary mergers Rosado (2011); Zhu et al. (2011); Wu et al. (2012); Zhu et al. (2013); Abbott et al. (2016f, 2018a), core-collapse supernovae Buonanno et al. (2005); Zhu et al. (2010); Crocker et al. (2015), and rapidly rotating neutron stars Rosado (2012); Wu et al. (2013); Talukder et al. (2014). In particular, the astrophysical background from compact binary mergers is likely to be detected by LIGO and Virgo at their design sensitivities Abbott et al. (2018a). A background of cosmological origin may also be present, due to cosmic strings Ölmez et al. (2010); Aasi et al. (2014a), inflation Maggiore (2000); Giblin and Thrane (2014); Easther et al. (2007); Cook and Sorbo (2012), and phase transitions in the early Universe Maggiore (2000); Giblin and Thrane (2014); Caprini et al. (2008, 2009); Lopez and Freese (2015).

Long duration gravitational-wave sources, like the stochastic background Nishizawa et al. (2009, 2010); Nishizawa and Hayama (2013); Callister et al. (2017) or persistent signals from rotating neutron stars Isi et al. (2015, 2017); Abbott et al. (2018b), offer a viable means of searching for nonstandard gravitational-wave polarizations. Unlike gravitational-wave transients, which sample only a single point on the LIGO/Virgo antenna response patterns, long-duration signals contain information about many points on the antenna patterns. Long-duration signals, therefore, enable the direct measurement of gravitational-wave polarizations using the current generation of gravitational-wave detectors, without the need for additional detectors or an independent electromagnetic counterpart. The stochastic background is thus a valuable laboratory for polarization-based tests of general relativity Callister et al. (2017).

In this Letter, we present the first direct search for vector and scalar polarizations in the stochastic gravitational-wave background. We analyze data recorded during Advanced LIGO’s first observing run (O1), which has previously been searched for both isotropic and anisotropic backgrounds of standard tensor polarizations Abbott et al. (2017e, f). First, we describe the O1 data set and its initial processing. We then discuss the stochastic analysis, including the construction of Bayesian odds that indicate the nondetection of a generically polarized stochastic background in our data. Finally, we present upper limits on the joint contributions of tensor, vector, and scalar polarizations to the stochastic gravitational-wave background. Additional details and results are presented in the Supplemental Material, available online.

Data. – We search Advanced LIGO data for evidence of a stochastic background, analyzing data recorded between September 18, 2015 15:00 UTC and January 12, 2016 16:00 UTC during LIGO’s O1 observing run. We do not include several days of O1 data recorded prior to September 18, but this has negligible impact on our results. We exclude times containing the binary black hole signals GW150914 and GW151226, as well as the signal candidate LVT151012.

The initial data processing proceeds as in previous analyses Aasi et al. (2014b); Abbott et al. (2017e). Time-domain strain measurements from the LIGO-Hanford and LIGO-Livingston detectors are down-sampled from 16384 Hz to 4096 Hz and divided into half-overlapping 192 s segments. Each time segment is then Hann-windowed, Fourier transformed, and high-pass filtered using a 16th order Butterworth filter with a knee frequency of 11 Hz. Finally, the strain data are coarse-grained to a frequency resolution of $0.03125$ Hz and restricted to a frequency band from 20–1726 Hz. Within each segment, we compute the LIGO-Hanford and LIGO-Livingston strain auto-power spectral densities using Welch’s method Welch (1967).

Standard data quality cuts are performed in both the time and frequency domains to mitigate the effects of non-Gaussian instrumental and environmental noise Abbott et al. (2017e, g, f). In the time domain, 35% of data is discarded due to nonstationary detector noise, leaving 29.85 days of coincident observing time. In the frequency domain, an additional 21% of data is discarded to remove correlated narrow-band features between LIGO-Hanford and LIGO-Livingston Abbott et al. (2017e, g, f). These narrow-band correlations are due to a variety of sources, including injected calibration signals, power mains, and GPS timing systems. To estimate possible contamination due to terrestrial Schumann resonances Thrane et al. (2013, 2014); Himemoto and Taruya (2017), we additionally monitored coherences between magnetometers installed at both detectors. Schumann resonances were found to contribute negligibly to the stochastic measurement Abbott et al. (2017e, g).

We assume conservative 4.8% and 5.4% calibration uncertainties on the strain amplitude measured by LIGO-Hanford and LIGO-Livingston, respectively Cahillane et al. (2017). Phase calibration is a much smaller source of uncertainty and is therefore neglected Abbott et al. (2017e); Whelan et al. (2014). All results below are obtained after marginalization over amplitude uncertainties; see the Supplemental Material for details.

Method. – To search for a generically polarized stochastic background, we will apply the methodology presented in Ref. Callister et al. (2017). This method is summarized below, and additional details are discussed in the Supplemental Material.

The stochastic background may be detected in the form of a correlated signal between pairs of gravitational-wave detectors. We will assume that the stochastic background is stationary, isotropic, and Gaussian. For simplicity, we also assume that the background is uncorrelated between polarization modes. Finally, we assume that the tensor and vector contributions to the background are individually unpolarized (with equal contributions, for instance, from the tensor plus and cross modes). Certain theories may violate one or more of these assumptions. For example, the stochastic background is unlikely to remain strictly unpolarized in the presence of gravitational-wave birefringence, as in Chern-Simons gravity Jackiw and Pi (2003); Alexander et al. (2008); Alexander and Yunes (2009), while theories violating Lorentz invariance may yield a departure from isotropy Kostelecký (2004); Kostelecký and Mewes (2016). The violation of one or more of our assumptions would likely reduce our search’s sensitivity to the stochastic background.

Given the above assumptions, the expected cross-correlation between two detectors in the presence of a stochastic background is of the form Allen and Romano (1999); Nishizawa et al. (2009, 2010); Nishizawa and Hayama (2013)

[TABLE]

Here, $S_{h}^{A}(f)$ is the one-sided gravitational-wave strain power spectral density of the net tensor ($A=T$), vector ($V$), and scalar ($S$) contributions to the stochastic background. The detectors’ geometry is encoded in the overlap reduction functions $\Gamma_{A}(f)$, defined Nishizawa et al. (2009); Allen and Romano (1999); Christensen (1992); Callister et al. (2017)

[TABLE]

$F^{a}_{I}(\hat{n})$ is the antenna response function of detector $I$ to signals of polarization $a$, $\Delta x$ is the separation vector between detectors, and $c$ is the speed of light. The integral is taken over all sky directions $\hat{n}$.

We will work not directly with $\Gamma_{A}(f)$, but rather with the normalized overlap reduction functions $\gamma_{A}(f)\propto\Gamma_{A}(f)/\Gamma_{0}$, where the constant $\Gamma_{0}$ is chosen such that $\gamma_{T}(f)=1$ for co-located and co-oriented detectors. For Advanced LIGO, $\Gamma_{0}=1/5$, but in general its value will vary for other experiments like LISA and pulsar timing arrays Romano and Cornish (2017). The normalized overlap reduction functions for LIGO’s Hanford-Livingston baseline are shown in Fig. 1. Because tensor, vector, and scalar modes each have distinct overlap reduction functions, the shape of a measured cross-correlation spectrum [Eq. (3)] will reflect the polarization content of the stochastic background Nishizawa et al. (2009); Callister et al. (2017). Of the three curves in Fig. 1, the scalar overlap reduction function is smallest in magnitude. This reflects the fact that the Advanced LIGO detectors have weaker geometrical responses to scalar-polarized gravitational waves than to tensor- and vector-polarized signals.

Conventionally, gravitational-wave backgrounds are parameterized by their energy-density spectra Allen and Romano (1999); Romano and Cornish (2017)

[TABLE]

where $d\rho_{\textsc{gw}}$ is the energy density in gravitational waves per logarithmic frequency interval $d\ln f$. We normalize Eq. (5) by $\rho_{c}=3H_{0}^{2}c^{2}/8\pi G$, the closure energy density of the Universe. Here, $G$ is Newton’s constant and $H_{0}$ is the Hubble constant; we take $H_{0}=68\,\text{km}\,\text{s}^{-1}\,\text{Mpc}^{-1}$ Ade et al. (2016). The precise relationship between $\Omega(f)$ and $S_{h}(f)$ is theory dependent. Under any theory obeying Isaacson’s formula for the stress-energy of gravitational waves Isaacson (1968), the energy-density spectrum is related to $S_{h}(f)$ by Allen and Romano (1999); Callister et al. (2017); Isi and Stein (2018)

[TABLE]

Equation (6) does not hold in general, however Isi and Stein (2018). For ease of comparison with previous studies, we will instead take Eq. (6) as the definition of the canonical energy-density spectra $\Omega^{A}(f)$. The canonical energy-density spectra can be directly identified with true energy densities under any theory obeying Isaacson’s formula. For other theories, $\Omega^{A}(f)$ can instead be understood simply as a function of the detector-frame observable $S_{h}^{A}(f)$.

Within each 192 s time segment (indexed by $i$), we form an estimator of the visible cross power between LIGO-Hanford and LIGO-Livingston:

[TABLE]

normalized such that the estimator’s mean and variance are Callister et al. (2017)

[TABLE]

and

[TABLE]

respectively. Within Eqs. (7) and (9), $\Delta T$ is the segment duration, $df$ the frequency bin width, and $P_{I,i}(f)$ is the one-sided auto-power spectral density of detector $I$ in time segment $i$, defined by

[TABLE]

The normalization of $\hat{C}(f)$ is chosen such that the contribution from each polarization appears symmetrically in Eq. (8); this choice differs by a factor of $\gamma_{T}(f)$ from the point estimate $\hat{Y}(f)$ typically used in stochastic analyses Aasi et al. (2014b); Abbott et al. (2017e); Callister et al. (2017). Finally, the cross-power estimators from each segment are optimally combined via a weighted sum to form a single cross-power spectrum for the O1 observing run,

[TABLE]

with the corresponding variance

[TABLE]

Note that, unlike transient gravitational-wave searches, searches for the stochastic background are well described by Gaussian statistics due to the large number of time segments contributing to the final cross-power spectrum Meacher et al. (2015).

Given the measured cross-power spectrum $\hat{C}(f)$, we compute Bayesian evidence for various hypotheses describing the presence and polarization of a possible stochastic signal within our data. Evidences are computed using PyMultiNest Buchner et al. (2014), a Python interface to the nested sampling code MultiNest Feroz and Hobson (2008); Feroz et al. (2009, 2013); Skilling (2004, 2006). We consider several different hypotheses:

• $\bullet$

Gaussian noise (N): No stochastic signal is present in our data, and the measured cross power is due entirely to Gaussian noise.

• $\bullet$

Signal (SIG): A stochastic background of any polarization(s) is present.

• $\bullet$

Tensor-polarized (GR): The data contains a purely tensor-polarized stochastic signal, consistent with general relativity.

• $\bullet$

Nonstandard polarizations (NGR): The data contains a stochastic signal with vector and/or scalar contributions.

These evidences are combined to form two Bayesian odds Callister et al. (2017): (1) Odds $\mathcal{O}^{\textsc{sig}}_{\textsc{n}}$ for the presence of a stochastic signal relative to pure noise, and (2) odds $\mathcal{O}^{\textsc{ngr}}_{\textsc{gr}}$ for the presence of nonstandard polarizations versus ordinary tensor modes. $\mathcal{O}^{\textsc{sig}}_{\textsc{n}}$ quantifies evidence for the detection of a generically polarized stochastic background, and generally depends only on a background’s total power, not its polarization content. $\mathcal{O}^{\textsc{ngr}}_{\textsc{gr}}$ indicates if the background’s polarization is inconsistent with general relativity. In particular, the sensitivity of $\mathcal{O}^{\textsc{ngr}}_{\textsc{gr}}$ to nonstandard polarizations is not significantly affected by the strength of any tensor polarization which may also be present Callister et al. (2017). See the Supplemental Material for further details about our hypotheses and odds ratio construction, including the priors placed on these hypotheses and their parameters.

Results. – Using the cross power measured between LIGO-Hanford and LIGO-Livingston during Advanced LIGO’s O1 observing run, we obtain odds $\ln\mathcal{O}^{\textsc{sig}}_{\textsc{n}}=-0.53$ between Signal and Gaussian noise hypotheses, indicating a nondetection of the stochastic gravitational-wave background. Additionally, we find $\ln\mathcal{O}^{\textsc{ngr}}_{\textsc{gr}}=-0.25$, consistent with values expected in the presence of Gaussian noise Callister et al. (2017). (We will use $\ln$ and $\log$ to denote base-$e$ and base-$10$ logarithms, respectively.)

Given our nondetection, we place upper limits on the presence of tensor, vector, and scalar contributions to the stochastic background. To simultaneously constrain the properties of each polarization, we will restrict our analysis to a model assuming the presence of tensor, vector, and scalar-polarized signals (this is the TVS hypothesis in the notation of the Supplemental Material). Under this hypothesis, we model the total canonical energy density of the stochastic background as a sum of power laws:

[TABLE]

Here, $\Omega^{A}_{0}$ is the amplitude of polarization $A$ at a reference frequency $f_{0}$, and $\alpha_{A}$ is the corresponding spectral index. We take $f_{0}=25$ Hz Abbott et al. (2017e). Standard tensor-polarized stochastic backgrounds are predicted to be well described by power laws in the Advanced LIGO band. The expected astrophysical background from compact binary mergers, for instance, is well modeled by a power law with $\alpha_{T}=2/3$ Rosado (2011); Zhu et al. (2011); Wu et al. (2012); Callister et al. (2016).

We will consider two different prior distributions for the background amplitudes: a log-uniform prior between $10^{-13}\leq\Omega^{A}_{0}\leq 10^{-5}$ and a uniform prior between $0\leq\Omega^{A}_{0}\leq 10^{-5}$. The former (log-uniform) corresponds to the prior adopted in Ref. Callister et al. (2017). The latter (uniform) implicitly reproduces the maximum likelihood analysis used in previous studies, and is included to allow direct comparison to previous stochastic results Aasi et al. (2014b); Abbott et al. (2017e). The upper amplitude bound ($10^{-5}$) is consistent with limits placed by Initial LIGO and Virgo Aasi et al. (2014b). In order to be normalizable, the log-uniform prior requires a nonzero lower bound; although parameter estimation results will depend on the specific choice of lower bound, in general this dependence is weak Isi et al. (2017). Our lower bound ($10^{-13}$) is chosen to encompass small energy densities well below the reach of LIGO and Virgo at design sensitivity Abbott et al. (2017e, 2018a).

Following Ref. Callister et al. (2017), we take our spectral index priors to be $p(\alpha_{A})\propto 1-|\alpha_{A}|/\alpha_{\textsc{max}}$ for $|\alpha_{A}|\leq\alpha_{\textsc{max}}$ and $p(\alpha_{A})=0$ elsewhere. This prior preferentially weights flat energy-density spectra, penalizing spectra which are more steeply positively or negatively sloped in the Advanced LIGO band. We conservatively choose $\alpha_{\textsc{max}}=8$, allowing for energy-density spectra significantly steeper than backgrounds predicted from known astrophysical sources (like compact binary mergers).

We perform parameter estimation using posterior samples obtained by PyMultiNest. Figure 2 shows posteriors on the tensor, vector, and scalar background amplitudes, under each choice of amplitude prior. The dashed and dot-dashed curves are proportional to the log-uniform and uniform amplitude priors, respectively; each prior curve has been renormalized by a constant factor to illustrate consistency between our priors and posteriors at small $\Omega^{A}_{0}$. We can now place upper limits on the amplitude of each component at $f_{0}=25$ Hz. The 95% credible upper limits on the amplitude of each polarization are listed in Table 1 for each choice of prior (for convenience, we list limits in terms of both $\log\Omega^{A}_{0}$ and $\Omega^{A}_{0}$). As no signal was detected, our posteriors on the spectral indices $\alpha_{A}$ are dominated by our prior. Full parameter estimation results, including posteriors on $\alpha_{A}$, are given in the Supplemental Material.

Care should be taken when comparing these upper limits to those obtained in previous analyses (e.g., Table I of Ref. Abbott et al. (2017e)). Three important distinctions should be kept in mind. First, the amplitude posteriors in Fig. 2 (and hence the limits in Table 1) are obtained after marginalization over spectral index. Previous analysis, on the other hand, typically assume specific fixed spectral indices or present exclusion curves in the $\Omega^{T}_{0}-\alpha_{T}$ plane Abbott et al. (2017e). Second, Bayesian upper limits may be strongly influenced by one’s adopted prior. Uniform amplitude priors, for instance, preferentially weight larger signals and hence yield larger upper limits, while log-uniform priors support smaller signal amplitudes, giving tighter limits. Finally, our results are obtained under a specific signal hypothesis allowing simultaneously for tensor, vector, and scalar polarizations. These limits are not generically identical to those that would be obtained if we allowed for tensor modes alone. In the Supplemental Material, we have tabulated upper limits under a variety of signal hypotheses allowing for each unique combination of gravitational-wave polarizations (our results, though, do not vary considerably between hypotheses). We have additionally verified that, under the GR (tensor-only) hypothesis with delta-function priors on the background’s spectral index, we recover upper limits identical to results previously published in Ref. Abbott et al. (2017e).

Conclusion. – The direct measurement of gravitational-wave polarizations may open the door to powerful new tests of gravity. Such measurements largely depend only on the geometry of a gravitational wave’s strain and its direction of propagation, not on the details of any specific theory of gravity. Recently, the Advanced LIGO-Virgo observation of the binary black hole merger GW170814 has enabled the first direct study of gravitational-wave polarizations Abbott et al. (2017b); Isi and Weinstein (2017). While LIGO and Virgo are limited in their ability to discern the polarization of gravitational-wave transients, the future construction of additional detectors, like KAGRA Aso et al. (2013); Akutsu et al. (2018) and LIGO-India Iyer et al. (2011), will help to break existing degeneracies and allow for increasingly precise polarization measurements.

Long-duration signals offer further opportunities to study gravitational-wave polarizations. Detections of continuous sources like rotating neutron stars Isi et al. (2017); Abbott et al. (2018b) and the stochastic background Callister et al. (2017) will offer the ability to directly measure and/or constrain gravitational-wave polarizations, even in the absence of additional detectors. In this Letter, we have conducted a search for a generically polarized stochastic background of gravitational waves using data from Advanced LIGO’s O1 observing run. Although we find no evidence for the presence of a background (of any polarization), we have succeeded in placing the first direct upper limits (listed in Table 1) on the contributions of vector and scalar modes to the stochastic background.

The authors gratefully acknowledge the support of the United States National Science Foundation (NSF) for the construction and operation of the LIGO Laboratory and Advanced LIGO as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. The authors gratefully acknowledge the Italian Istituto Nazionale di Fisica Nucleare (INFN), the French Centre National de la Recherche Scientifique (CNRS) and the Foundation for Fundamental Research on Matter supported by the Netherlands Organisation for Scientific Research, for the construction and operation of the Virgo detector and the creation and support of the EGO consortium. The authors also gratefully acknowledge research support from these agencies as well as by the Council of Scientific and Industrial Research of India, the Department of Science and Technology, India, the Science & Engineering Research Board (SERB), India, the Ministry of Human Resource Development, India, the Spanish Agencia Estatal de Investigación, the Vicepresidència i Conselleria d’Innovació, Recerca i Turisme and the Conselleria d’Educació i Universitat del Govern de les Illes Balears, the Conselleria d’Educació, Investigació, Cultura i Esport de la Generalitat Valenciana, the National Science Centre of Poland, the Swiss National Science Foundation (SNSF), the Russian Foundation for Basic Research, the Russian Science Foundation, the European Commission, the European Regional Development Funds (ERDF), the Royal Society, the Scottish Funding Council, the Scottish Universities Physics Alliance, the Hungarian Scientific Research Fund (OTKA), the Lyon Institute of Origins (LIO), the Paris Île-de-France Region, the National Research, Development and Innovation Office Hungary (NKFI), the National Research Foundation of Korea, Industry Canada and the Province of Ontario through the Ministry of Economic Development and Innovation, the Natural Science and Engineering Research Council Canada, the Canadian Institute for Advanced Research, the Brazilian Ministry of Science, Technology, Innovations, and Communications, the International Center for Theoretical Physics South American Institute for Fundamental Research (ICTP-SAIFR), the Research Grants Council of Hong Kong, the National Natural Science Foundation of China (NSFC), the Leverhulme Trust, the Research Corporation, the Ministry of Science and Technology (MOST), Taiwan and the Kavli Foundation. The authors gratefully acknowledge the support of the NSF, STFC, MPS, INFN, CNRS and the State of Niedersachsen/Germany for provision of computational resources.

I Sensitive Frequency Bands

Although this search utilizes the full 20-1726 Hz frequency band, different frequency sub-bands contribute variously to our overall search sensitivity. To illustrate this, we can investigate the contribution from each frequency bin to a background’s optimal signal-to-noise ratio (SNR), given by Allen and Romano (1999)

[TABLE]

Up to additive constants, SNR and $\mathcal{O}^{\textsc{sig}}_{\textsc{n}}$ are related by $\ln\mathcal{O}^{\textsc{sig}}_{\textsc{n}}\sim\text{SNR}^{2}/2$.

Using the measured O1 search sensitivity, Fig. 3 illustrates the cumulative fraction of the squared-SNR of several representative hypothetical backgrounds, obtained by integrating Eq. (14) from 20 Hz up to a cutoff frequency $f$. Results are shown for purely tensor- (blue), vector- (red), and scalar-polarized (green) backgrounds, with spectral indices $\alpha=-8$, [math], and $8$.

As seen in Fig. 3, the most sensitive frequency band for a given background is highly dependent on the background’s spectral index. For steeply negatively-sloped backgrounds ($\alpha=-8$), the majority of the measured SNR is obtained at very low frequencies between $\sim 20-30$ Hz. Meanwhile, the $\sim 20-100$ Hz band is most sensitive to flat backgrounds, and high frequencies above $\sim 700$ Hz are most sensitive to steeply positively-sloped backgrounds. Although trends are generally independent of polarization, Fig. 3 does show somewhat different behaviors for tensor, vector, and scalar modes. These differences are due to the different overlap reduction functions for each polarization sector.

II Model Construction

Here, we briefly summarize the construction of our Signal, Gaussian noise, Non-standard polarization, and Tensor-polarization hypotheses; see Ref. Callister et al. (2017) for further details.

Gaussian noise: We assume that no signal is present and the observed cross-power $\hat{C}(f)$ is Gaussian distributed about zero with variance given by Eqs. (9) and (12). Although Advanced LIGO instrumental noise is neither stationary nor Gaussian, searches for the stochastic background are nonetheless well-described by Gaussian statistics due to the large number of time-segments combined to form the final cross-power spectrum $\hat{C}(f)$ Meacher et al. (2015).

Signal: The Signal hypothesis is the union of seven sub-hypotheses, which together allow for each unique combination of tensor, vector, and scalar polarizations. The “TVS” sub-hypothesis, for example, assumes the simultaneous presence of all polarization modes, with a canonical energy-density spectrum of the form:

[TABLE]

The “TS” sub-hypothesis, meanwhile, assumes only the existence of tensor and scalar modes:

[TABLE]

In this fashion, we can construct seven unique sub-hypotheses: {T,V,S,TV,TS,VS,TVS}. The union of these seven possibilities is the Signal hypothesis.

Non-standard polarization (NGR) – Analogous to the Signal hypothesis above, this is the union of the six sub-hypotheses {V,S,TV,TS,VS,TVS} containing non-standard polarizations.

Tensor-polarization (GR) – We assume the stochastic background is present and purely-tensor polarized, with the energy-density spectrum

[TABLE]

this hypothesis is identical the “T” signal sub-hypothesis above.

III Odds Ratios

Here, we review the procedure for constructing odds $\mathcal{O}^{\textsc{sig}}_{\textsc{n}}$ between Signal and Gaussian noise hypotheses, and odds $\mathcal{O}^{\textsc{ngr}}_{\textsc{gr}}$ between the NGR and GR hypotheses. As above, see Ref. Callister et al. (2017) for further details.

The odds between two hypotheses $\mathcal{M}$ and $\mathcal{N}$ is the ratio of posterior probabilities for each hypothesis, given data $d$:

[TABLE]

where \mathcal{B}^{\scalebox{0.6}{\mathcal{M}}}_{\scalebox{0.6}{\mathcal{N}}} is the Bayes factor between the two hypotheses and $\pi(\mathcal{M})$ and $\pi(\mathcal{N})$ are the prior probabilities on $\mathcal{M}$ and $\mathcal{N}$, respectively. The ratio $\pi(\mathcal{M})/\pi(\mathcal{N})$ is known as the prior odds.

To obtain $\mathcal{O}^{\textsc{sig}}_{\textsc{n}}$, we first compute the Bayes factor \mathcal{B}^{\scalebox{0.6}{\mathcal{A}}}_{\textsc{n}} between each Signal sub-hypothesis $\mathcal{A}\in\{\text{T},\text{V},\text{S},...\}$ and the Noise hypothesis. Because each sub-hypothesis is independent, $\mathcal{O}^{\textsc{sig}}_{\textsc{n}}$ is then just the sum

[TABLE]

where we have expanded $\pi(\mathcal{A})=\pi(\mathcal{A}|\text{SIG})\pi(\text{SIG})$. We choose equal prior probabilities on the Signal and Noise hypotheses, such that $\pi(\text{SIG})/\pi(\text{N})=1$. Within the Signal hypothesis, we assign equal probabilities to each sub-hypothesis, giving $\pi(\mathcal{A}|\text{SIG})=1/7$.

The odds $\mathcal{O}^{\textsc{ngr}}_{\textsc{gr}}$ is analogously given by

[TABLE]

We set $\pi(\text{NGR})/\pi(\text{GR})=1$ and again choose equal prior probabilities for each sub-hypotheses within NGR, such that $\pi(\mathcal{A}|\text{NGR})=1/6$.

Our chosen prior odds between hypotheses are necessarily somewhat arbitrary, and different choices will yield different values of $\mathcal{O}^{\textsc{sig}}_{\textsc{n}}$ and $\mathcal{O}^{\textsc{ngr}}_{\textsc{gr}}$. For completeness, Table 2 provides the Bayes factors between each signal sub-hypothesis and Gaussian noise. These Bayes factors allow readers to recompute odds $\mathcal{O}^{\textsc{sig}}_{\textsc{n}}$ and $\mathcal{O}^{\textsc{ngr}}_{\textsc{gr}}$ using different choices of prior odds.

IV Calibration Uncertainty

The strain measured by LIGO-Hanford and LIGO-Livingston is not known perfectly, but is subject to non-zero calibration uncertainty. For imperfectly calibrated data, the cross-power measurements $\hat{C}(f)$ are not estimators of $\sum_{A}\gamma_{A}(f)\Omega^{A}(f)$, but rather of $\lambda\sum_{A}\gamma_{A}(f)\Omega^{A}(f)$, where $\lambda$ is some multiplicative factor Whelan et al. (2014). Perfect calibration would yield $\lambda=1$, but in general $\lambda$ is unknown. We include the calibration factor $\lambda$ as an additional parameter in MultiNest, so that the likelihood function becomes

[TABLE]

with $\hat{C}(f)$ and $\sigma^{2}(f)$ given by Eqs. (11) and (12), respectively.

We place a Gaussian prior on $\lambda$, centered at $\lambda=1$:

[TABLE]

The standard deviation $\epsilon$ encapsulates the amplitude calibration uncertainty. Within the 20-1726 Hz frequency band, LIGO-Hanford and LIGO-Livingston have maximum estimated amplitude uncertainties of $4.8\%$ and $5.4\%$, respectively Cahillane et al. (2017). These uncertainty estimates have been improved relative to the the uncertainties previously adopted in Refs. Abbott et al. (2017e, g, f). For our analysis, we take $\epsilon=0.072$, the quadrature sum of the Hanford and Livingston uncertainties Abbott et al. (2017e). All results are given after marginalization over $\lambda$.

In the above prescription we have made two simplifying assumptions. First, we have neglected phase calibration uncertainty, which is expected to be a sub-dominant source of uncertainty in the stochastic analysis Cahillane et al. (2017); Whelan et al. (2014). Secondly, although calibration uncertainties are frequency-dependent, for simplicity we adopt uniform amplitude uncertainties across all frequencies. Our quoted amplitude uncertainties are conservative, encompassing the largest calibration uncertainties in the stochastic sensitivity band Abbott et al. (2017e); Cahillane et al. (2017).

V Detailed Parameter Estimation Results

In our Letter, we presented marginalized posteriors for the tensor, vector, and scalar background amplitudes under the “TVS” hypothesis. In Fig. 4 we show the full six-dimensional parameter estimation results obtained when choosing log-uniform amplitude priors. Diagonal subplots show marginalized posteriors on the amplitudes and slopes of each polarization, while the interior subplots show joint posteriors between each pair of parameters. The spectral index posteriors (not shown in the main Letter) are largely consistent with our choice of prior, but indicate a slight bias against large positive spectral indices. This reflects the fact that Advanced LIGO is most sensitive to backgrounds of large, positive slopes Callister et al. (2017). The non-detection of a stochastic background therefore constrains larger amplitudes to have small and/or negative spectral indices; see, for instance, the joint $\log\Omega^{T}_{0}\text{-- }\alpha_{T}$ posterior in Fig. 4.

Figure 5, meanwhile, shows full parameter estimation results when alternatively assuming uniform amplitude priors. Here, the posterior preference towards small or negative spectral indices is far more pronounced. The joint 2D posteriors (e.g. $\Omega^{T}_{0}\text{-- }\alpha_{T}$) again illustrate that large, positive slopes are preferentially ruled out in case of large background amplitudes.

As stated in the Letter, upper limits obtained under one hypothesis are not, in general, equal to those obtained under some different hypothesis. While we presented upper limits only for the TVS hypothesis, results from other hypotheses may be desired as well (the TS results, for instance, are best suited for comparison to predictions from scalar-tensor theories). In Tables 3 and 4 we have therefore listed the 95% credible upper limits corresponding to each signal sub-hypothesis, for both log-uniform and uniform amplitude priors. We have also listed 95% credible bounds on spectral indices for each choice of amplitude prior.