Background information: A study on the sensitivity of astrophysical gravitational-wave background searches

Background information: A study on the sensitivity of astrophysical

gravitational-wave background searches

Arianna I. Renzini ,

1,2,3,4

Tom Callister ,

†

Katerina Chatziioannou ,

1,2

‡

and Will M. Farr

6,7

,§

LIGO Laboratory,

California Institute of Technology

, Pasadena, California 91125, USA

Department of Physics,

California Institute of Technology

, Pasadena, California 91125, USA

Dipartimento di Fisica

“

G. Occhialini,

”

Universit`

a degli Studi di Milano-Bicocca

Piazza della Scienza 3, 20126 Milano, Italy

INFN, Sezione di Milano-Bicocca

, Piazza della Scienza 3, 20126 Milano, Italy

Kavli Institute for Cosmological Physics,

The University of Chicago

, Chicago, Illinois 60637, USA

Department of Physics and Astronomy,

Stony Brook University

, Stony Brook, New York 11794, USA

Center for Computational Astrophysics,

Flatiron Institute

, New York, New York 10010, USA

(Received 21 March 2024; accepted 14 June 2024; published 9 July 2024)

The vast majority of gravitational-wave signals from stellar-mass compact binary mergers are too weak

to be individually detected with present-day instruments and instead contribute to a faint, persistent

background. This astrophysical background is targeted by searches that model the gravitational-wave

ensemble collectively with a small set of parameters. The traditional search models the background as a

stochastic field and estimates its amplitude by

cross-correlating

data from multiple interferometers. A

different search uses gravitational-wave

templates

to marginalize over all individual event parameters and

measure the duty cycle and population properties of binary mergers. Both searches ultimately estimate the

total merger rate of compact binaries and are expected to yield a detection in the coming years. Given the

conceptual and methodological differences between them, though, it is not well understood how their

results should be mutually interpreted. In particular, when a detection of an astrophysical compact binary

background is claimed by either approach, which portion of the population is in fact contributing to this

detection? In this paper, we use the Fisher information to study the implications of a background detection

in terms of which region of the Universe each approach probes. Specifically, we quantify how information

about the compact binary merger rate is accumulated by each search as a function of the event redshift. For

the LIGO design sensitivity and a uniform-in-comoving-volume distribution of equal-mass

⊙

binaries,

the traditional cross-correlation search obtains 99% of its information from binaries up to redshift 2.5

(average signal-to-noise ratio

), and the template-based search from binaries up to redshift 1.0 (average

signal-to-noise ratio

∼

). While we do not calculate the

total

information accumulated by each search, our

analysis emphasizes the need to pair any claimed detection of the stochastic background with an assessment

of which binaries contribute to said detection. In the process, we also clarify the astrophysical assumptions

imposed by each search.

DOI:

10.1103/PhysRevD.110.023014

I. INTRODUCTION

For every compact binary merger

[1]

observed by the

LIGO

[2]

and Virgo

[3]

observatories, there are many more

that are too distant and too weak to be directly detected.

Although these distant binaries may be individually indis-

tinguishable from instrumental noise, their population may

be collectively detectable via the slight coherence it imparts

across networks of widely separated gravitational-wave

detectors

–

. The resulting collection of weak signals is

colloquially referred to as the astrophysical gravitational-

wave background. Current constraints based on the indi-

vidually detectable tail of the total population suggest that

this astrophysical background is several orders of magni-

tude larger than any expected cosmological stochastic

background in the relevant frequency range

–

detected, the stochastic background will offer indirect

information about the properties of compact binaries

Contact author: arenzini@caltech.edu

†

Contact author: tcallister@uchicago.edu

‡

Contact author: kchatziioannou@caltech.edu

Contact author:will.farr@stonybr

ook.edu, wfarr@flatironinstitute

.org

For this reason, we will drop the

“

astrophysical

”

designation

in the rest of the paper, with the understanding that unless

explicitly noted otherwise we refer to the astrophysical stochastic

gravitational-wave background.

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beyond the horizons of present-day detectors

–

13]

;itis

therefore a prime target for a variety of gravitational-wave

searches

[6,14,15]

Traditional searches for the background (both astrophysi-

cal and cosmological) rely on the cross-correlation between

detector pairs. These searches model the background

gravitational-wave strain as a mean-zero stochastic field

that is continuous and Gaussian and measure its variance.

The steep low-frequency noise of ground-based detectors

[16]

suggests that any autocorrelations induced by the

stochastic background are subdominant and can therefore

be neglected; this is often referred to as the weak signal limit

[4,5,17]

. But given sufficiently long observation times, the

stochastic background will manifest as excess cross power

between instruments. Given current predictions for the

power spectrum of the stochastic background

[6,7]

detection is unlikely in the current observing run, but

may be feasible during the next observing run if the

Advanced LIGO sensitivity target

[18]

is reached.

An alternative search strategy is motivated by the fact

that the background is neither continuous nor Gaussian.

Given current merger rate estimates, we expect that a black

hole binary (neutron star binary) merges every 5

–

10 min

–

60 s) in the mass range relevant for ground-based

detectors

[7]

. As black hole binary coalescences last for

seconds

in the ground-based detector frequency band

[1,19,20]

, the black hole background is expected to be

composed of distinct nonoverlapping transient signals. For

neutron star binaries the duration is

minutes

[21]

these signals overlap. However, given a low frequency

cutoff of 10 or 20 Hz, it is still unlikely that the relevant

background lies in the confusion noise limit

[22

–

26]

though subject to large uncertainty on the binary neutron

star merger rate

[6]

. While the non-Gaussianity of the

compact binary background does not bias the cross-

correlation search in the low-signal limit, it does imply

that it is suboptimal

[27]

Given distinct

—

though individually undetectable

—

transient signals, the author of Ref.

[28]

proposed a search

that relies on the matched-filter technique that has success-

fully resulted in the detection of individually resolved

binaries, e.g.,

[29]

. This template-based search utilizes

phase-coherent gravitational waveform models to margin-

alize over the properties of individual events that may (or

may not) be present within every segment of data, regard-

less of whether the events rise above the threshold for direct

detection. An extended version also infers and/or margin-

alizes over the properties of the compact-binary population

[30]

. In general, template-based techniques are expected to

be more sensitive than cross-correlation ones as the former

include (even vanishingly small) information about the

waveform phase, while the latter rely solely on excess

power. Whereas cross-correlation searches might require

years of integration to claim a detection of a gravitational-

wave background, it is argued that the template-based

search might reach a detection given only

days

of data at

design sensitivity

[28]

. If such an improvement is realized,

the template-based search would represent a remarkable

leap in sensitivity and enable imminent detection.

The stark difference in sensitivity motivates the main

question of this study: when we detect a background with

either search, which region of the Universe have we

successfully probed and what astrophysical assumption

have we made about this region? A direct time-to-detection

estimate does not fully address this question as the two

searches make different astrophysical assumptions and are

not necessarily sensitive to the same population of binaries

at the same regions in the Universe. Instead, the authors of

Ref.

[30]

examined the template-based search

’

s ability to

infer a cutoff in the binary redshift distribution and

concluded that there is some information from binaries

at the edge of resolvability. Additionally, the cross-

correlation and template-based formalisms are constructed

in terms of different physical quantities and rely on

different detection statistics, specifically the gravitational

wave

energy density

and the

event rate

, respectively.

The reliance on different statistics is not merely a

technical inconvenience, but also reveals a conceptual

incompatibility between approaches. The energy density

and event rate can be mathematically related to each other

only after

assuming a specific merger rate distribution

for

sources across the Universe. Gravitational-wave detectors

measure spacetime strain, which can be trivially converted

to intensity and energy density without further assump-

tions. This suggests that the cross-correlation search can

infer the gravitational-wave energy density directly from

the data, with no assumption on source distribution. The

template-based analysis, on the other hand, is based on the

measurement of a nonzero rate of events. The conversion

from measured strain to an event rate (or duty cycle) relies

on a Bayes factor that compares the hypotheses that a data

segment contains signal or noise, which in turn depends on

assumptions about the prior or astrophysical distribution of

sources, including the mass, spin, and redshift distribution.

While Ref.

[30]

relaxes the mass and spin dependence, the

assumption of knowledge of the redshift distribution

remains. If an analysis intrinsically assumes perfect knowl-

edge of how sources are distributed in the Universe, this

begs the question: how much information is actually

coming from the unresolved sources compared to what

is extrapolated from the resolved ones?

The goal of our work is to study the sensitivity of the

cross-correlation and template-based searches and identify

the astrophysical assumptions each search makes. We use

the Fisher information matrix to quantify the information

accumulated by each search and study how this information

is accumulated by observing binaries at different redshifts.

In lieu of a technically challenging full implementation of

the template-based search, we consider a simplified pop-

ulation of compact binaries where all parameters are known

RENZINI, CALLISTER, CHATZIIOANNOU, and FARR

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other than the distance/redshift. This simplified setup does

not allow us to compare the

total

information accumulated

by each search, but it yields the

relative

contribution by

binaries at different redshifts. For

⊙

binaries, the cross-

correlation (template-based) search accumulates 99% of its

information from binaries up to redshift 2.5 (1.2). The same

redshift estimates for

⊙

binaries are 1.5 (1.6). With the

hopefully imminent detection of a stochastic background in

LIGO data, our study emphasizes the need to carefully

assess the sensitivity and assumptions of our search

methods.

The rest of the paper presents the details of our

calculation. We begin in Sec.

by reviewing the stochastic

background and its characterization. In Sec.

III

we lay out

the two search methodologies, highlighting the targeted

observables. In Sec.

we describe the theory and calculate

the functional forms of the Fisher information for each

search with respect to a common parameter, and in Sec.

we study the information functions in different scenarios.

Finally, we draw our conclusions in Sec.

II. THE COMPACT BINARY BACKGROUND

In this section, we introduce the basics of the gravitational-

wave stochastic background and establish notation. The

familiar reader can skip ahead to Sec.

III

where we introduce

the search methods.

The background of unresolved compact binaries is

generally characterized by its dimensionless energy-density

spectrum

[4,8]

Þ¼

;

where

is the energy density in gravitational waves

(GW) and

is the Universe

’

s closure energy density. The

present-day

in the frequency range

<f<

Hz is dominated by the integrated merger history of

compact binaries over all redshifts. For simplicity,

we consider a single class of compact binaries with

ensemble-averaged source-frame energy spectra

=df

(a subscript

denotes source-frame quantities), and define

to be the number of mergers per unit comoving

volume

per unit source-frame time

, i.e., the source-

frame merger rate density

Þ¼

Given gravitational-wave sources up to redshift

max

, the

present-day dimensionless energy-density spectrum is

[6]

Þ¼

max

dz;

where

≈

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

is the Hubble

parameter at redshift

and

is the Hubble constant.

The total detector-frame merger rate

max

Here

=dz

is the comoving volume per unit redshift. The

factor of

converts the source-frame rate into the

detector-frame rate as measured at Earth.

Both the gravitational-wave energy-density and the total

merger rate depend on the source-frame rate density,

. It is convenient to write the latter as the product

Þ¼

, where

is the local merger rate at

and

is the merger rate density function normalized to 1 at

. Then, in terms of the total rate

, the local merger rate

becomes

max

−

≡

max

−

;

where we have defined the integral

max

which quantifies

the ratio between total and local merger rate, given

max

.We

also define the normalized probability distribution for source

redshift,

Þ¼

max

≡

;

defining

to be the event rate per unit detector-

frame time.

So far, searches targeting the stochastic background via

have operated under the assumption that the signal

is continuous and Gaussian. However, the background is

only Gaussian in the limit of large numbers of sources that

saturate the detector time stream and satisfy the central limit

theorem. Given the current set of compact binary detections

[7]

, it is clear that this is not the case

[24]

. As merging black

holes are expected to strongly contribute to the GW energy

density

[31]

, this suggests the astrophysical stochastic

background is non-Gaussian and intermittent in the detec-

tor frequency band, with a cadence dictated by the

prevalence of black hole binary mergers in the Universe.

This formalism is largely inspired by the cosmological

background, but it is applied to the astrophysical background

as well.

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To quantify the non-Gaussianity and intermittence of the

signal and inspired by

[27]

, Ref.

[28]

split the data in

segments of duration

and denoted the probability that a

given segment contains a signal as

. This duty cycle

can

be related to the merger rate. Given the total detector-frame

merger rate

, the probability that

gravitational-wave

signals are present in a data segment with duration

Poisson distributed

Þ¼

−

Then the duty cycle

is simply the probability that

≥

signals are present in the data segment:

∞

−

max

In the limit where

≪

, this becomes simply

≈

Assuming a segment length

∼

the expected duty

cycle of the black hole background is

∼

−

The cross-correlation analysis (described in Sec.

III A

)is

typically expressed in terms of the

spectrum, as it

was conceived for Gaussian and continuous backgrounds.

The template-based search (described in Sec.

III B

), on the

other hand, was proposed with a non-Gaussian and

intermittent background in mind and is hence framed in

terms of

. In Sec.

we also work in terms of independent

parameters that the observables of both searches can be

mapped to. Specifically, we use Eqs.

(3)

(4)

, and

(8)

express the cross-correlation and template-based searches

in terms of the same quantities,

and

max

, for direct

comparison.

III. STOCHASTIC BACKGROUND

SEARCH METHODS

We consider two searches that target the subthreshold

population of binary mergers in ground-based detectors:

the cross-correlation search (relevant quantities are labeled

“

”

) and the template-based search (

“

”

A. Cross-correlation search

The most common search for the stochastic gravitational-

wave background relies on the cross-correlation spectrum

between the (frequency domain) data

̃

and

̃

measured by two gravitational-wave detectors to construct

an optimal (i.e., unbiased minimum-variance) statistic for

[4,5,17]

ˆ

Þ¼

R

̃

̃

ÞÞ

seg

;

where

seg

is the time segment duration over which data are

measured. Here and throughout this discussion we use

“

hatless

”

symbols to denote physical quantities and

“

hatted

”

symbols to denote their estimates based on data. The data

are Fourier transforms of strain data and hence have units

̃

−

. The function

is defined as

Þ¼

;

and converts the strain power spectrum to a dimensionless

energy-density spectrum. The factor

is the overlap

reduction function

[32,33]

, which quantifies the geometrical

sensitivity of the cross-correlated detector pair to an

isotropic background. The factor of 2 in Eq.

(9)

accounts

for the contribution of negative frequencies. The variance of

the statistic in Eq.

(9)

[6]

ˆ

Þ¼

ˆ

ˆ

;

where

ˆ

and

ˆ

are the one-sided strain power

spectra of the data in detectors 1 and 2, respectively,

defined as

ˆ

Þ¼

seg

̃

Þj

The energy density spectrum can be decomposed as

Þ¼

ref

f=f

ref

;

with an overall amplitude

ref

≡

ref

at reference

frequency

ref

. The spectral shape

f=f

ref

can be

assumed known or parametrized

f=f

ref

and inferred

[34]

. For example, for compact binary sources

f=f

ref

should be universal, as the inspiral frequency evolution is

independent of

. This is true up to a turnover frequency

that corresponds to the redshifted merger frequency of the

binaries. If we are not sensitive to the spectrum turnover, as

is the case with current ground-based interferometers, we

can treat the spectral shape of the signal as redshift-

independent and set

f=f

ref

∝

f=f

ref

In practice, cross-correlation spectra are estimated inde-

pendently for a large number of short time segments, each of

100

, and combined via a weighted average. Since a

large number (

–

)ofsuchtime segmentsarecombined,

the resulting cross-correlation measurements

ˆ

are

With current detector sensitivity, and in order to avoid double-

counting, it is reasonable to pick a segment duration such that

there is at most one event per data segment. The segment duration

is therefore chosen to be comparable to the time a signal spends in

the detector frequency band.

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well-described by Gaussian statistics even in the case of an

intermittent signal. As shown in

[17]

, this cross-correlation

statistic is a sufficient statistic for the stochastic signal in the

case where the signal is Gaussian and weak compared to

detector noise

and can be derived from the mean-zero

Gaussian likelihood

̃

n;I

Þ¼

Þj

−

̃

†

−

̃

Here,

̃

is thedatavector

̃

¼ð

̃

;

̃

,and

appears

in the data covariance as the intrinsic variance of the

gravitational-wave signal

[17]

Þ¼

seg

;

where

Þ¼

≡

ref

f=f

ref

For stationary noise within each analysis segment the noise

covariance is diagonal in frequency; hence, the total like-

lihood is a product of Gaussian likelihoods evaluated at each

frequency. The

parameters are the expected power

spectra in each detector and can be explicitly written as the

sum of the noise and gravitational-wave power spectra,

Þ¼

n;I

Þþ

This implies both diagonal and off-diagonal terms of the

covariance in Eq.

(15)

depend on the gravitational-wave

energy density, giving rise to autocorrelation and cross-

correlation terms, respectively. The noise power spectra

n;I

are additional parameters of the search and may in

principle be estimated alongside

In the weak signal limit, the autocorrelation terms in

Eq.

(14)

can be neglected, such that

≈

n;I

and

the likelihood reduces to

ˆ

∝

exp

−

ˆ

−

ÞÞ

ˆ

;

where hatted quantities are directly estimated from the data.

To simplify parameter estimation, we fix

f=f

ref

Þ¼

f=f

ref

such that the likelihood of Eq.

(18)

refers to

a single parameter,

ref

, and depends only on quantities

derived from the data as well as the assumed

.Itis

possible to remove this second dependence by reexpressing

the likelihood in the full spectrum

and constrain

each frequency bin independently; however,

ref

is typi-

cally preferred as this allows us to marginalize over the

spectrum and improve detection statistics.

B. Template-based search

The template-based search adopts a different approach

that is more similar to the traditional matched-filter

searches for individually detectable signals

[28,30,35]

.In

this search, the entire strain time series measured by a

gravitational-wave detector network is divided into

time

segments, each with duration

. For instance, the search for

a stochastic background from black-hole binaries divides

one year of data into approximately

segments of

s duration, given the reasonable expectation that

each segment contains

≪

signal on average

[28]

. Here the

segment length is selected by considering the expected

single-event duration at present sensitivity. This is in stark

contrast with the typical choices made for the cross-

correlation analysis, where the segment duration is typi-

cally on the timescale of a few minutes to access lower

frequency content, handle noise nonstationarity, and min-

imize the computational cost of the search

[36]

Within every time segment

, a template-based analysis

computes the marginalized likelihood (or

“

evidence

”

) that a

compact binary merger is (hypothesis

) or is not

(hypothesis

) present. The marginalized likelihood for

the signal hypothesis is obtained via marginalization over

all source parameters

of the binary,

Þ¼

;

where

is the data comprising segment

;

is the

likelihood of having obtained these data in the presence of a

source with parameters

(binary masses, spins, redshift,

etc.), and

is the prior on the source parameters

With the segment-by-segment signal and noise evidences

in hand, the template-based analysis then seeks to measure

the duty cycle

, i.e., the fraction of time segments

containing a gravitational-wave signal (see discussion in

Sec.

). The relevant likelihood across all

segments is

ðf

Þ¼

Þþ

Þþð

−

;

where, by definition,

Þ¼

is the probability that a

signal is present in segment

; correspondingly,

Þ¼

−

is the probability that a signal is absent. Factoring out

, we can rewrite the likelihood as

A weak signal is a signal that does not appreciably contribute

to the power measured by a single detector

[5]

, such that the

variance of the data can be equated to the variance of the noise.

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ðf

Þ¼

Þ½

þð

−

;

where

is the Bayes factor between signal and noise

hypotheses in segment

Within the framework of the template-based search, obser-

vation of the stochastic background amounts to con-

straining

away from zero. Equivalently, the Bayes

factor

between the

“

signal

”

hypothesis that allows

≤

and the

“

noise

”

hypotheses in which

ðf

;

can be used as a detection statistic given some prior on the

duty cycle

In contrast to Eq.

(18)

, which only depends on the data

and the inferred quantities, Eq.

(23)

also depends on a

collection of binary parameter priors. These include priors

on compact binary masses

, spins

, and redshift

(or,

equivalently, distance). These priors encode our belief

about the underlying population of compact binaries.

Choosing a particular prior

when evaluating

Eq.

(21)

amounts to assuming that the population distri-

bution of these parameters is perfectly known. An extended

version of the template-based analysis relaxes the

assumption of a known population distribution for masses

and spins

[30]

. Instead, the population prior

parametrized, and the resulting set of hyperparameters are

added to the search. Conceptually, this is equivalent to

performing a search for compact binary mergers while

simultaneously performing a population analysis to mea-

sure their ensemble properties

[7,37]

. However, a fixed

prior on the binary redshift remains: the current formulation

of the search assumes perfect knowledge of how compact

binaries are distributed with redshift. This assumption,

combined with the presence of resolved low-redshift

binaries, raises the question of whether a measurement

is informed by binaries at high redshift, or is

instead dominated primarily by foreground binaries.

This question motivates our study: below we revisit the

sensitivity of stochastic searches and quantify the impact of

resolved sources in the template-based case.

IV. INFORMATION CONTENT

The different assumptions, methodology, and formalism

between the cross-correlation and template-based searches

make a direct comparison in terms of a simple time-to-

detection difficult. We instead consider the

Fisher infor-

mation

for each search and examine how this information is

gathered as a function of source redshift and for different

astrophysical assumptions about the event distribution.

Given a likelihood

for data

conditioned on

parameters

¼f

, the Fisher information matrix is

defined as the expectation value over data realizations

Þ¼

−

∂

∂

∂

In the high signal-to-noise ratio limit where the likelihood

becomes approximately Gaussian, the inverse Fisher matrix

−

corresponds to the covariance matrix quantifying the

uncertainties on parameters

. The Fisher information,

meanwhile, is defined as the matrix determinant:

Þ¼

det

The strong signal-to-noise assumption of the Fisher for-

malism is not in tension with the previously defined low-

signal limit employed for the cross-correlation search. The

latter refers to the fact that the autocorrelation of strain data

is dominated by detector noise, rather than astrophysical

signals. The former, in contrast, refers to the detectability of

excess cross power due to the gravitational-wave back-

ground, after integrating over a sufficiently long period of

time. In other words, even though the individual signals are

subthreshold, the applicability of the Fisher formalism

refers to the stochastic signal as a whole.

The cross-correlation and template-based searches are

framed in terms of different observables; i.e.,

is different

in each case. The cross-correlation search reports a meas-

urement of the amplitude

ref

, i.e.,

ref

. The

template-based search, meanwhile, reports a measurement

of the duty cycle

, i.e.,

. The duty cycle of a

population and its total fractional energy density can be

related by combining Eqs.

(3)

(5)

, and

(8)

; however, this

requires prior knowledge of the source redshift distribution

. Equivalently, knowledge of

is required to convert

the fractional energy density emitted by a population to its

local merger rate density

, and therefore to

. Comparing

the two searches thus relies on prior knowledge of

which raises doubts as to what is actually being measured,

as opposed to extrapolated, to produce the individual search

results as well as in their comparison. To make progress,

our strategy here is to parametrize

and infer it along-

side each search parameter.

Extending each search to also probe the source redshift

distribution amounts to a parametrization of

≡

;

and additional parameters

. Here, we adopt the redshift

distribution parametrization of Eq.

(5)

, with

max

similar to

[30]

. For the template-based search this results

¼f

max

. In terms of the likelihood in Eq.

(21)

the duty cycle

is explicit, while

max

enters through the

distance/redshift prior used in the Bayes factor calculation,

max

. However, varying

max

while keeping

constant

would result in a change in the redshift distribution of the

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local, individually detectable sources. Since this is expected

to be well-constrained by the time of detection of the

stochastic background, we instead reparametrize the analy-

sis to explicitly separate the local merger rate

from the

source redshift distribution,

¼f

max

In the case of the cross-correlation search, the only

parameter considered in the likelihood of Eq.

(18)

ref

For direct comparison with the template-based search, we

also need to express this likelihood in terms of the same

and

max

parameters, linked to

ref

through Eqs.

(3)

and

(5)

We therefore also consider a

version of the cross-

correlation search with

¼f

max

, though both

parameters enter the likelihood only through their

ref

combination. In what follows, we discuss both the case

where the maximum cutoff

max

is assumed known and

fixed (

searches), and the case where it is a free parameter

(

searches). For the

comparison, we present results in

ref

, as this naturally quantifies the background amplitude;

while for the

case, we use

and

max

for the reasons

detailed above.

A back-of-the-envelope calculation clarifies the

ensuing detailed calculation in the

case. Here, the

cross-correlation search is sensitive to the GW energy

density while the template-based one is sensitive to the

local merger rate. In the cross-correlation case, the energy

density of

events contained in a volume

∝

scales as

∼

, as the single event energy is

∝

−

.This

means that the total energy density scales roughly linearly

with event distance. Meanwhile, for the template-based

search the likelihood is dominated by the

term,

which scales as

∝

SNR

∼

−

,where

is an event Bayes factor as in

(21)

, and we have assumed

that

∝

SNR

implying a loud signal. This indicates that

events at large

will contribute less to this search, as the

Bayes factor decreases rapidly with distance. In what

follows, to avoid taking this loud signal approximation for

events around/below the detection threshold, we evaluate

Bayes factors numerically.

A. Cross-correlation search

We calculate the Fisher information for the cross-

correlation search starting from the full likelihood of

Eq.

(14)

which may be expanded as

[17]

̃

ref

∼−

seg

Þjþ

seg

ˆ

Þþ

ˆ

−

ref

f=f

ˆ

Þj

;

where the determinant of the covariance is

Þj¼

seg

−

;

and

seg

is the total number of segments used in the

analysis. In what follows, we consider the noise terms

n;I

entering the data power spectra

in Eq.

(17)

to be known.

We proceed to calculate the Fisher matrix in log

ref

and

ref

log

ref

and

ref

, respectively. Here

log

ref

corre-

sponds to the fractional uncertainty on

ref

, and thus scales

with the sensitivity of the search. In contrast,

ref

scales

with both the sensitivity and the value of

ref

itself. In terms

of log

ref

we have

log

ref

−

∂

∂

log

ref

̃

ref

;

where the angle brackets imply calculating the expectation

value at maximum likelihood over many realizations;

hence, first derivatives of the likelihood are dropped.

Under the noise stationarity assumption, each frequency

contributes to the likelihood calculation independently;

hence, the total contribution is the sum over individual

frequencies:

log

ref

;

where

−

∂

∂

̃

;

computed at the individual frequency

, where

ref

. Taking the maximum likelihood limit

(which amounts to setting hatted quantities equal to their

unhatted counterparts, assuming these are unbiased estima-

tors), defining

Þ¼

−

, and dropping the explicit

frequency dependence from the functions

n;I

, and

for conciseness, we find

seg

−

Þþ

−

ÞÞ

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In accordance with the standard cross-correlation search,

we take the low-signal limit and expand

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

, which yields to first order,

Þþ

;

where

Þ¼

seg

;

Þ¼

−

seg

The term

log

ref

is then

log

ref

Þ½

Þþ

ref

Since we are restricting to a single parameter here, Eq.

(35)

gives the Fisher information gathered in a cross-correlation

search for the fractional uncertainty in

ref

and has two

limiting cases. In the limit where the signal

ref

→

, i.e., a

Universe with no compact binaries, the absolute informa-

tion per frequency bin is

→

. Hence,

quantifies the information inherent in a nondetection of

the stochastic background, and only depends on the search

sensitivity

∼

. The relative information, Eq.

(35)

,is

then zero as expected for vanishingly small signals, as in

this case; if the stochastic background is undetectable, then

we have infinite fractional uncertainty on its size. This

makes sense qualitatively, and though the Fisher formalism

is only strictly applicable in the strong signal limit, it

provides the Cramer-Rao lower bound on the variance of

the estimator. The term

, on the other hand, quantifies

the information contributed by a measurable stochastic

signal. The Fisher information

ref

itself scales intuitively

with the size of the stochastic background: given the

negative sign of

,as

grows

ref

decreases

However, the Fisher information

log

ref

can in principle

increase or decrease, depending on which term dominates

Eq.

(35)

. That is, as

is increased, absolute uncertainty

ref

grows, but fractional uncertainty

ref

decreases as long as we are in the weak-signal approxi-

mation.

Finally, information increases as observing time

grows. At fixed analysis segment length

seg

, the number of

segments grows linearly with time and

ref

∝

seg

. This is

expected as the variance of the measured stochastic field

power scales as

obs

[5]

Figure

shows

and

−

, for the two-detector

network of Advanced LIGO detectors, each with design

sensitivity

[38]

assuming one year of data. As seen in

Eqs.

(33)

and

(34)

, the functional forms of these spectra

depend on both the overlap reduction function and the

individual detector sensitivities. The 5 orders of magnitude

between the two curves imply that for typical values of the

background amplitude (

−

) and at this sensitivity

the

term dominates

, verifying the weak-

signal approximation. In the case in question, the majority

of the information is gathered at

∼

Hz, where

maximum for this specific configuration.

As sources at

cosmological distances are redshifted, this

“

most inform-

ative

”

frequency can be translated into a most informative

redshift, given the source mass. At the top of Fig.

we have

added axes with the

“

merger redshift

”

at which equal-

mass binaries of different masses merge, at the correspond-

ing frequency on the

axis. We select three component

mass values for this example,

⊙

, which

we reprise in Sec.

to calculate the Fisher information.

Binaries with larger mass merge at lower frequencies, and

conversely binaries with smaller mass contribute to lower

FIG. 1.

and

spectra from Eq.

(32)

as a function of

frequency. The top

axes mark the redshift

at which equal-

component-mass binaries merge emitting at the frequency in the

lower

axis for different binary masses. The frequency corre-

sponding to maximum stochastic sensitivity, 45 Hz, is marked

with a gray dashed line.

This may be seen qualitatively by observing that, in Eq.

(32)

the

term can compete with the

term when

is of

the order of the noise terms

n;i

As discussed, the spectral shapes of

and

will

necessarily vary for different detector pairs. In general, the larger

the separation between detectors, the lower we expect these

functions to peak in frequency, due to the form of the overlap

reduction function

[5]

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frequencies only when highly redshifted. For example, a

binary composed of two

⊙

black holes merges at

around

∼

Hz at

, and can contribute to lower

frequencies in the band (

<f<

Hz) when redshifted

. On the other hand, a

⊙

equal-mass binary

system can only contribute to a signal at frequencies

Hz if redshifted by

. This implies, in general,

that

at peak sensitivity

∼

Hz is larger for higher-

mass binary populations.

Finally, we calculate the Fisher matrix for the

extension to the cross-correlation search with parameters

and

max

≡

max

Each of the Fisher terms here can be related to the Fisher

matrix in

ref

using the chain rule. The first diagonal

term is

−

∂

∂

ref

≡

−

log

ref

;

using the fact that

∂

∂

≡

ref

;

as per Eqs.

(3)

and

(13)

The second diagonal term is similarly derived as

max

−

∂

∂

max

;

where

≡

∂

∂

max

is the integrand of Eq.

(3)

evaluated at

max

Finally, the off-diagonal terms are

max

−

∂

∂

∂

max

ref

B. Template-based search

The template-based search likelihood, Eq.

(21)

, depends

on the redshift distribution parameters

and

max

explic-

itly through the

max

parameter, Eq.

(8)

, and

implicitly through the Bayes factors

max

, Eq.

(A2)

which depend on prior distributions on source parameters

that in turn rely on

max

. The likelihood is then rewritten as

ðf

max

Þ½

max

Þþð

−

max

ÞÞ

;

and the Fisher matrix is

max

≡

max

Starting with the diagonal term in

and writing

≡

ðf

max

for conciseness, we find

−

∂

∂

−

∂

∂

∂

∂

∂

∂

∂

∂

The brackets signify ensemble averaging over data real-

izations, which we interpret in practice as utilizing all

available data and evaluating the derivatives at the maxi-

mum of the likelihood. Since the first derivative vanishes at

the maximum we get

−

∂

∂

∂

∂

Substituting Eq.

(42)

we obtain

−

ÞÞ

;

where

≡

∂

∂

max

−

max

−

;

where

max

is defined as in Eq.

(5)

. The Fisher matrix

diagonal term in

max

−

ÞÞ

−

þð

−

ÞÞ

;

where

≡

∂

∂

max

;

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∂

∂

max

Þð

−

Finally, the off-diagonal terms are

max

−

ÞÞ

V. SENSITIVITY REACH

We quantify the dependence of the Fisher information on

the merger rate redshift distribution with simulated popula-

tions. We adopt a constant normalized merger rate density

such that the merger rate

is uniform in comoving

volume, a local merger rate of

Gpc

−

[7]

, and

vary

max

such that

Þ¼

≤

max

;z>z

max

We consider three distinct populations, made up of equal-

mass, nonspinning black hole binaries with source-frame

masses of

⊙

, and

⊙

, respectively. We

calculate the Fisher information for each population at

varying

max

, over one year of data from a network of two

LIGO detectors at design sensitivity

[38]

A. Cross-correlation search

For the cross-correlation search, both the

and the

Fisher matrices in Eqs.

(28)

and

(36)

can be calculated

analytically. The spectrum

is calculated for each

population and at each

max

using Eq.

(3)

. The emitted

energy spectrum

=df

is a function of source-frame

chirp mass

[39]

;

where the function

may be found, e.g., in

[39]

. During

the inspiral phase, i.e., for frequencies lower than the merger

frequency,

≈

−

, implying

∝

, which

is generally a good approximation to the low-frequency

background spectral shape. To model the background spec-

trum at higher frequencies, we adopt an analytical inspiral-

merger-ringdown approximation for

[39,40]

We start by considering the single-parameter,

case.

We compute the Fisher matrix in log

ref

of Eq.

(35)

varying the cutoff

max

in the computation of

ref

and plot

log

ref

as a function of

max

in the left panel of Fig.

. This

plot illustrates how the Fisher matrix (equivalently, the

Fisher information as we are in

) increases with

max

further distant binaries contribute to the total

ref

. The

information plateaus at varying

max

, depending on the

mass, reach higher values for higher masses as expected.

The

max

at which

log

ref

plateaus is a combination of the

redshift at which the binaries emitting at the most sensitive

frequency merge (

in Fig.

) and the redshift at which

binaries no longer appreciably contribute to

ref

.For

example, for

max

, 99% of the background amplitude

is accumulated from binaries within

, independently

of mass.

To understand how binaries in each redshift bin con-

tribute to the total information, we consider the case of a

Universe with binaries up to

max

. We calculate the

contribution of a redshift shell

z; z

to the background

Þ¼

−

and plug this into the

FIG. 2. Left: Single-parameter Fisher matrix

log

ref

, Eq.

(35)

, calculated for varying

max

, for each equal-mass binary population

considered. Right: Cumulative sum of the information

log

ref

contributed from binaries in a redshift shell

z; z

as a function of

for a Universe with binaries up to

max

and for each mass. The dots indicate the redshift at which 99% of information is

accumulated. Both plots indicate that the information saturates at a mass-dependent redshift

∼

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