Supplement To: Search for Tensor, Vector, and Scalar Polarizations in the Stochastic
Gravitational-Wave Background
B. P. Abbott
et al.
(LIGO Scientific Collaboration & Virgo Collaboration)
This documents contains additional information and results supplemental to the material presented in Ref. [1].
In particular, we illustrate the contributions of different frequency bands to Advanced LIGO’s stochastic search
sensitivity, further describe the hypothesis testing procedure used in Ref. [1], and discuss our treatment of
calibration uncertainties. Finally, we show complete parameter estimation results under a variety of hypotheses
for the polarization content of the stochastic gravitational-wave background.
Sensitive Frequency Bands
Although the stochastic search for non-standard polariza-
tions 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 contri-
bution from each frequency bin to a background’s optimal
signal-to-noise ratio (SNR), given by [2]
SNR
2
=
3
H
2
0
10
π
2
2
T
∫
∞
0
[
∑
A
γ
A
(
f
)Ω
A
(
f
)
]
2
f
6
P
1
(
f
)
P
2
(
f
)
df.
(A1)
Up to additive constants, SNR and
O
SIG
N
are related by
ln
O
SIG
N
∼
SNR
2
/
2
.
Using the measured O1 search sensitivity, Fig. 1 illustrates
the cumulative fraction of the squared-SNR of several repre-
sentative hypothetical backgrounds, obtained by integrating
Eq. (A1) 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
α
=
−
8
,
0
, and
8
.
As seen in Fig. 1, the most sensitive frequency band for a
given background is highly dependent on the background’s
spectral index. For steeply negatively-sloped backgrounds
(
α
=
−
8
), the majority of the measured SNR is obtained at
very low frequencies between
∼
20
−
30
Hz. Meanwhile, the
∼
20
−
100
Hz band is most sensitive to flat backgrounds, and
high frequencies above
∼
700
Hz are most sensitive to steeply
positively-sloped backgrounds. Although trends are generally
independent of polarization, Fig. 1 does show somewhat dif-
ferent behaviors for tensor, vector, and scalar modes. These
differences are due to the different overlap reduction functions
for each polarization sector.
Model Construction
Here, we briefly summarize the construction of our Sig-
nal, Gaussian noise, Non-standard polarization, and Tensor-
polarization hypotheses; see Ref. [3] for further details.
Gaussian noise
: We assume that no signal is present and
the observed cross-power
ˆ
C
(
f
)
is Gaussian distributed about
zero with variance given by Eqs. (9) and (12) of Ref. [1].
Although Advanced LIGO instrumental noise is neither sta-
tionary nor Gaussian, searches for the stochastic background
10
2
10
3
f
(Hz)
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
Cumulative
SNR
2
α
=
−
8
α
= 0
α
= 8
Tensor
Vector
Scalar
FIG. 1. Cumulative squared signal-to-noise ratios as a function of
frequency for hypothetical backgrounds of tensor (blue), vector (red),
scalar (green) polarizations with spectral indices
α
=
−
8
,
0
, and
8
(solid, dashed, and dot-dashed, respectively). The three
α
=
−
8
curves lie nearly on top of one another, as do the three
α
= 8
curves.
The Advanced LIGO network is most sensitive to negatively-sloped
backgrounds at low frequencies, while high frequencies contribute
the most sensitively to positively-sloped backgrounds.
are nonetheless well-described by Gaussian statistics due to
the large number of time-segments combined to form the final
cross-power spectrum
ˆ
C
(
f
)
[4].
Signal
: The Signal hypothesis is the union of seven sub-
hypotheses, which together allow for each unique combina-
tion of tensor, vector, and scalar polarizations. The “TVS”
sub-hypothesis, for example, assumes the simultaneous pres-
ence of all polarization modes, with a canonical energy-
density spectrum of the form:
Ω
TVS
(
f
) = Ω
T
0
(
f
f
0
)
α
T
+ Ω
V
0
(
f
f
0
)
α
V
+ Ω
S
0
(
f
f
0
)
α
S
.
(A2)
The “TS” sub-hypothesis, meanwhile, assumes only the exis-
tence of tensor and scalar modes:
Ω
TS
(
f
) = Ω
T
0
(
f
f
0
)
α
T
+ Ω
S
0
(
f
f
0
)
α
S
.
(A3)
In this fashion, we can construct seven unique sub-
hypotheses:
{
T,V,S,TV,TS,VS,TVS
}
. The union of these
2
seven possibilities is the Signal hypothesis.
Non-standard polarization (NGR)
– Analogous to the Sig-
nal 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 back-
ground is present and purely-tensor polarized, with the
energy-density spectrum
Ω
GR
= Ω
T
0
(
f
f
0
)
α
T
;
(A4)
this hypothesis is identical the “T” signal sub-hypothesis
above.
Odds Ratios
Here, we review the procedure for constructing odds
O
SIG
N
between Signal and Gaussian noise hypotheses, and odds
O
NGR
GR
between the NGR and GR hypotheses. As above, see
Ref. [3] for further details.
The odds between two hypotheses
M
and
N
is the ratio of
posterior probabilities for each hypothesis, given data
d
:
O
M
N
=
p
(
M|
d
)
p
(
N|
d
)
=
B
M
N
π
(
M
)
π
(
N
)
,
(A5)
where
B
M
N
is the Bayes factor between the two hypotheses
and
π
(
M
)
and
π
(
N
)
are the prior probabilities on
M
and
N
, respectively. The ratio
π
(
M
)
/π
(
N
)
is known as the prior
odds.
To obtain
O
SIG
N
, we first compute the Bayes factor
B
A
N
be-
tween each Signal sub-hypothesis
A ∈ {
T
,
V
,
S
,...
}
and the
Noise hypothesis. Because each sub-hypothesis is indepen-
dent,
O
SIG
N
is then just the sum
O
SIG
N
=
∑
A
∈
SIG
O
A
N
=
∑
A
∈
SIG
B
A
N
π
(
A|
SIG
)
π
(
SIG
)
π
(
N
)
,
(A6)
where we have expanded
π
(
A
) =
π
(
A|
SIG
)
π
(
SIG
)
. We
choose equal prior probabilities on the Signal and Noise hy-
potheses, such that
π
(
SIG
)
/π
(
N
) = 1
. Within the Signal hy-
pothesis, we assign equal probabilities to each sub-hypothesis,
giving
π
(
A|
SIG
) = 1
/
7
.
The odds
O
NGR
GR
is analogously given by
O
NGR
GR
=
∑
A
∈
NGR
B
A
GR
π
(
A|
NGR
)
π
(
NGR
)
π
(
GR
)
.
(A7)
We set
π
(
NGR
)
/π
(
GR
) = 1
and again choose equal prior
probabilities for each sub-hypotheses within NGR, such that
π
(
A|
NGR
) = 1
/
6
.
TABLE I. Bayes factors between each signal sub-hypothesis and the
Gaussian noise hypothesis, as computed by
MultiNest
. These
Bayes factors are combined following Eqs. (A6) and (A7) to ob-
tain odds
O
SIG
N
between Signal and Gaussian noise hypotheses, and
odds
O
NGR
GR
between NGR and GR hypotheses.
Hypothesis
ln
B
A
N
T
−
0
.
33
V
−
0
.
33
S
−
0
.
31
TV
−
0
.
66
TS
−
0
.
65
VS
−
0
.
65
TVS
−
0
.
99
Our chosen prior odds between hypotheses are necessar-
ily somewhat arbitrary, and different choices will yield differ-
ent values of
O
SIG
N
and
O
NGR
GR
. For completeness, Table I pro-
vides the Bayes factors between each signal sub-hypothesis
and Gaussian noise. These Bayes factors allow readers to re-
compute odds
O
SIG
N
and
O
NGR
GR
using different choices of prior
odds.
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
ˆ
C
(
f
)
are not estimators of
∑
A
γ
A
(
f
)Ω
A
(
f
)
, but rather of
λ
∑
A
γ
A
(
f
)Ω
A
(
f
)
, where
λ
is some multiplicative factor [5]. Perfect calibration would
yield
λ
= 1
, but in general
λ
is unknown. We include the cal-
ibration factor
λ
as an additional parameter in
MultiNest
,
so that the likelihood function becomes
L
(
ˆ
C
(
f
)
|
Ω
A
0
,α
A
,λ
)
∝
∏
f
exp
−
(
ˆ
C
(
f
)
−
λ
∑
A
γ
A
(
f
)Ω
A
0
(
f/f
0
)
α
A
)
2
2
σ
2
(
f
)
,
(A8)
with
ˆ
C
(
f
)
and
σ
2
(
f
)
given by Eqs. (11) and (12) of Ref. [1],
respectively.
We place a Gaussian prior on
λ
, centered at
λ
= 1
:
π
(
λ
)
∝
exp
(
−
(
λ
−
1)
2
2
2
)
.
(A9)
The standard deviation
encapsulates the amplitude calibra-
tion uncertainty. Within the 20-1726 Hz frequency band,
LIGO-Hanford and LIGO-Livingston have maximum esti-
mated amplitude uncertainties of
4
.
8%
and
5
.
4%
, respectively
[6]. These uncertainty estimates have been improved relative
3
to the the uncertainties previously adopted in Refs. [7–9]. For
our analysis, we take
= 0
.
072
, the quadrature sum of the
Hanford and Livingston uncertainties [7]. All results are given
after marginalization over
λ
.
In the above prescription we have made two simplifying
assumptions. First, we have neglected phase calibration un-
certainty, which is expected to be a sub-dominant source of
uncertainty in the stochastic analysis [5, 6]. Secondly, al-
though calibration uncertainties are frequency-dependent, for
simplicity we adopt uniform amplitude uncertainties across all
frequencies. Our quoted amplitude uncertainties are conserva-
tive, encompassing the largest calibration uncertainties in the
stochastic sensitivity band [6, 7].
Detailed Parameter Estimation Results
In Ref. [1], we presented marginalized posteriors for the
tensor, vector, and scalar background amplitudes under the
“TVS” hypothesis. In Fig. 2 we show the full six-dimensional
parameter estimation results obtained when choosing log-
uniform amplitude priors. Diagonal subplots show marginal-
ized posteriors on the amplitudes and slopes of each polariza-
tion, while the interior subplots show joint posteriors between
each pair of parameters. The spectral index posteriors (not
shown in Ref. [1]) 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 [3]. The
non-detection of a stochastic background therefore constrains
larger amplitudes to have small and/or negative spectral in-
dices; see, for instance, the joint
log Ω
T
0
–
α
T
posterior in Fig.
2.
Figure 3, meanwhile, shows full parameter estimation re-
sults when alternatively assuming uniform amplitude priors.
Here, the posterior preference towards small or negative spec-
tral indices is far more pronounced. The joint 2D posteriors
(e.g.
Ω
T
0
–
α
T
) again illustrate that large, positive slopes are
preferentially ruled out in case of large background ampli-
tudes.
As stated in Ref. [1], upper limits obtained under one hy-
pothesis are not, in general, equal to those obtained under
some different hypothesis. While we presented upper lim-
its only for the TVS hypothesis, results from other hypothe-
ses may be desired as well (the TS results, for instance, are
best suited for comparison to predictions from scalar-tensor
theories). In Tables II and III we have therefore listed the
95% credible upper limits corresponding to each signal sub-
hypothesis, for both log-uniform and uniform amplitude pri-
ors. We have also listed 95% credible bounds on spectral in-
dices for each choice of amplitude prior.
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