Supplement–Directional limits on persistent gravitational waves from Advanced
LIGO’s first observing run
In this supplement we describe how we use the narrow-
band, directed radiometer search[
?
] to make a state-
ment on the gravitational wave (GW) strain amplitude
h
0
of a persistent source given some power described by
our cross correlation statistic. We take into account the
expected modulation of the quasi-monochromatic source
frequency over the duration of the observation. We do
this by combining individual search frequency bins into
combined bins
that cover the extent of the possible mod-
ulation.
Source Model.
—We can relate GW frequency emitted
in the source frame
f
s
to the observed frequency in the
detector frame
f
d
using the relation
f
d
= [1
−
A
(
t
)
−
B
(
t
)
−
C
(
t
)]
f
s
(1)
where
A
(
t
) takes into account the modulation of the sig-
nal due to Earth’s motion with respect to the source,
B
(
t
)
takes into account the orbital modulation for a source in
a binary orbit, and
C
(
t
) takes into account any other
modulation due to intrinsic properties of the source (for
example any spin-down terms for isolated neutron stars).
The Earth modulation term is given by
A
(
t
) =
~v
E
(
t
)
·
ˆ
k
c
(2)
where
~v
E
is the velocity of the Earth. In equatorial co-
ordinates:
~v
E
(
t
) =
ωR
[sin
θ
(
t
)ˆ
u
−
cos
θ
(
t
) cos
φ
ˆ
v
−
cos
θ
(
t
) sin
φ
ˆ
w
]
,
(3)
in which
R
is the mean distance between Earth and the
Sun,
ω
is the angular velocity of the Earth around the
Sun and
φ
= 23
◦
, 26 min, 21.406 sec is the obliquity of
the ecliptic. The time dependent phase angle
θ
(
t
) is given
by
θ
(
t
) = 2
π
(
t
−
T
VE
)
/T
year
, where
T
year
is the number
of seconds in a year and
T
VE
is the time at the Vernal
equinox. The unit vector
ˆ
k
pointing from the source to
the earth is given by
ˆ
k
=
−
cos
δ
cos
α
ˆ
u
−
cos
δ
sin
α
ˆ
v
−
sin
δ
ˆ
w
, where
δ
is the declination and
α
is the right as-
cension of the source on the sky.
In the case of a source in a binary system, the binary
term (for a circular orbit) is given by
B
(
t
) =
2
π
P
orb
a
sin
i
×
cos
(
2
π
t
−
T
asc
P
orb
)
(4)
where
a
sin
i
is the projection of the semi-major axis (in
units of light seconds) of the binary orbit on the line of
sight,
T
asc
is the time of the orbital ascending node and
P
orb
is the binary orbital period.
In the case of an isolated source we set
B
(
t
) = 0, while
C
(
t
) can take into account any spin modulation expected
to occur during an observation time. In the absence of
a model for this behaviour, a statement can be made
on the maximum allowable spin modulation that can be
tolerated by our search.
Search.
—The narrowband radiometer search is run
with 192 s segments and 1/32 Hz frequency bins. For
each 1/32 Hz frequency bin we combine the number of
bins required to account for the extent of any signal fre-
quency modulation The source frequency
f
s
is taken as
the center of a frequency bin. We calculate the minimum
and maximum detector frequency
f
d
over the time of the
analysis corresponding to the respective
edges
of the bin
in order to define our combined bins.
We combine the detection statistic
Y
i
and variance
σ
Y,i
into a new combined statistic
Y
c
for each representative
frequency bin via
Y
c
=
a
∑
i
=
−
b
Y
i
and
σ
2
Y,c
=
a
∑
i
=
−
b
σ
2
Y,i
,
(5)
where
i
represents the index for each of the frequency bins
we want to combine. If we assign
i
= 0 to the bin where
the source frequency falls, then
a
and
b
are the number of
frequency bins we want to combine above and below the
source frequency bin, respectively. The overlapping bins,
which ensure we do not lose signal due to edge effects,
create correlations between our
combined
bins.
Significance.
—To establish significance, we assume
that the strain power in each frequency bin is consis-
tent with Gaussian noise and simulate
>
1000 noise re-
alizations. For each realization, we generate values of
Y
i
in each frequency bin
i
by drawing from a Gaus-
sian distribution with
σ
=
σ
Y,i
. We then combine these
bins into combined bins as we do in the actual analysis
and calculate the maximum of the signal to noise ratio,
SNR =
Y
c
/σ
Y,c
, across all of the combined bins. We
use the distribution of maximum SNR to establish the
significance of our results.
Upper limits.
—In the absence of a significant detection
statistic, we set upper limits on the tensor strain ampli-
tude
h
0
of a gravitational wave source with frequency
f
s
.
To take into account the unknown parameters of the sys-
tem, such as the polarization
ψ
and inclination angle
ι
,
and consider reduced sensitivity to signals that are not
circularly polarized, we calculate a direction-dependent
and time-averaged value
μ
ι,ψ
. This value represents a
scaling between the true value of the amplitude
h
0
and
what we would measure with our search, and is given by
μ
ι,ψ
=
∑
M
j
=1
[
(
A
+
/h
0
)
2
F
+
d
j
+ (
A
×
/h
0
)
2
F
×
d
j
](
F
+
d
j
+
F
×
d
j
)
∑
M
j
=1
(
F
+
d
j
+
F
×
d
j
)
2
(6)
2
for each time segment
j
. Here
A
+
=
1
2
h
0
(1 + cos
2
ι
) and
A
×
=
h
0
cos
ι,
(7)
and
ψ
dependence is implicit in
F
A
d
j
=
F
A
1
j
F
A
2
j
, where
A
indicates (+ or
×
) polarization and the response func-
tions
F
A
1
j
and
F
A
2
j
for the LIGO detectors are defined in
[
?
] (see also [
?
]). We calculate
μ
ι,ψ
many times for a
uniform distribution of cos
ι
and
ψ
, and then marginalize
over it. We also marginalize over calibration uncertainty,
where we assume (as in the past) that calibration uncer-
tainty is manifest in a multiplicative factor (
l
+ 1)
>
0
where
l
is normally distributed around 0 with uncertainty
given by the calibration uncertainty,
σ
l
= 0
.
18. The full
expression for our posterior distribution given a measure-
ment,
Y
and its uncertainty
σ
Y
in a single combined bin
is given by
p
(
h
0
|
Y,σ
Y
) =
∫
1
−
1
d
(cos
ι
)
∫
π/
4
−
π/
4
dψ
∫
∞
−
1
dle
L
(
l
)
,
(8)
where
L
(
l
) =
−
1
2
{
(
l
σ
l
)
2
+
[
Y
(
l
+ 1)
−
μ
ι,ψ
h
2
0
σ
Y
(
l
+ 1)
]
2
}
.
(9)
We set upper limits on
h
0
at a 90% credible level for
frequencies within each combined bin that correspond to
each source frequency
f
s
. Denoted
h
UL
0
, upper limits are
calculated via 0
.
9 =
∫
h
UL
0
0
dh
0
p
(
h
0
|
Y,σ
Y
)
.
Frequency Notches.
—For frequency bins flagged to be
removed from the analysis due to instrumental artifacts,
we set our statistic to zero, so they will not contribute
to the combined statistics described in Eq
??
. We re-
quire that more than half of the frequency bins are still
available when generating a combined bin. When setting
upper limits, the noise
σ
Y,c
in any combined bin that
contained notched frequency bins is rescaled to account
for the missing bins to provide a more accurate represen-
tation of the true sensitivity in that combined bin.