Stochastic gravitational wave background methods and Implications

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Stochastic gravitational wave background: methods and Implications

Nick van Remortela,∗, Kamiel Janssensa,b, Kevin Turbangc,a

aUniversiteit Antwerpen, Prinsstraat 13, 2000 Antwerpen, Belgium

bArtemis, Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, F-06304 Nice, France

cVrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium

Abstract

Beyond individually resolvable gravitational wave events such as binary black hole and binary neutron star mergers, the

superposition of many more weak signals coming from a multitude of sources is expected to contribute to an overall

background, the so-called stochastic gravitational wave background. In this review, we give an overview of possible

detection methods in the search for this background and provide a detailed review of the data-analysis techniques,

focusing primarily on current Earth-based interferometric gravitational-wave detectors. In addition, various validation

techniques aimed at reinforcing the claim of a detection of such a background are discussed as well. We conclude this

review by listing some of the astrophysical and cosmological implications resulting from current upper limits on the

stochastic background of gravitational waves.

Keywords: general relativity, gravitational waves, stochastic background, cosmology, astrophysics, laser interferometers

∗Corresponding author

Email address: nick.vanremortel@uantwerpen.be (Nick van Remortel)

Preprint submitted to Progress in Particle and Nuclear Physics October 4, 2022

arXiv:2210.00761v1 [gr-qc] 3 Oct 2022

Contents

1 Introduction to gravitational waves 4

2 Stochastic background: deﬁnitions hypotheses and properties 5

3 Detection methods 7

3.1 Planetary bodies as resonant gravitational wave detectors ........................... 7

3.2 Pulsar Timing .................................................... 8

3.3 Laser Interferometry ................................................ 10

4 Analysis techniques 13

4.1 Cross-correlation analysis for the search for an isotropic SGWB ........................ 13

4.2 Anisotropic backgrounds .............................................. 15

4.2.1 Spherical harmonics decomposition .................................... 18

4.2.2 Radiometer - narrowband and broadband ................................ 19

4.2.3 All sky all frequency ............................................ 23

4.3 The Bayesian Search (TBS) ............................................ 24

5 Validation techniques 28

5.1 Typical noise sources at Earth-based gravitational wave detectors ....................... 28

5.1.1 Transient noise sources ........................................... 28

5.1.2 Narrowband noise sources ......................................... 29

5.1.3 Broadband noise sources .......................................... 30

5.2 Instrumental and environmental monitoring ................................... 30

5.3 Gravitational wave geodesy - a validation tool for the SGWB ......................... 32

5.4 Subtraction of noise sources ............................................ 33

5.4.1 Wiener ﬁlter ................................................ 34

5.4.2 Noise subtraction .............................................. 36

5.4.3 Case study of noise subtraction – Schumann resonances ........................ 36

5.5 Joint Bayesian modeling of noise sources and a SGWB ............................. 37

5.6 The null channel: a gravitational wave insensitive channel for a triangular conﬁguration of interferometers 38

6 Implications 41

6.1 Astrophysical Implications ............................................. 41

6.2 Cosmological Implications ............................................. 43

7 Outlook 49

List of abbreviations

ASAF: all-sky all-frequency

ASD: amplitude spectral density

BBH: binary black hole

BBN: Big Bang nucleosynthesis

BBR: broadband radiometer

BNS: binary neutron star

CBC: compact binary coalescence

CMB: cosmic microwave background

ET: Einstein Telescope

FOPT: ﬁrst order phase transition

GW: gravitational wave

LISA: Laser Interferometer Space Antenna

LVK: LIGO-Virgo-KAGRA

MSP: millisecond pulsar

NBR: narrowband radiometer

ORF: overlap reduction function

PBH: primordial black holes

PI: power-law integrated (sensitivity curve)

PSD: power spectral density

PTA: pulsar timing array

SGWB: stochastic gravitational wave background

SMBHB: supermassive binary black hole binaries

SNR: signal-to-noise ratio

TBS: the Bayesian search

1. Introduction to gravitational waves

The discovery of the ﬁrst gravitational wave (GW) signal due to the merger of two black holes [1] made by the LIGO

and Virgo collaborations in 2015 propelled the ﬁeld of GW astronomy into a new era. Since then, several tens of binary

coalescence signals have been detected, including the merger of two black holes with a wide and continuous range of

composite stellar size masses [2,3,4,5]. In addition, evidences for the binary merger of a black hole and a neutron star

[6] and two neutron stars [7,8] have been collected as well. The latter observation was followed up by several observations

in the electromagnetic spectrum [9,10], which kick-started the ﬁeld of gravitational-wave multi-messenger astronomy.

The signals of all these events, which have characteristic transient features, are of relative short duration (O(1-100 s)),

corresponding to the time during which the signal remains within the sensitive frequency band of current earth bound

interferometers, and have relatively large amplitudes that exceed the intrinsic noise levels of the detectors. Many other

astrophysical sources are predicted to yield detectable GW signals that are either continuous in nature and which typically

originate from asymmetrical rotating compact objects such as pulsars, or are burst-like, such as supernovae type objects.

None of these signals have been detected so far, but could well be in reach of operational and planned Earth-based and

satellite borne GW detectors [11]. Finally, there is also bound to be a stochastic gravitational wave background (SGWB)

in our Universe that can contain several components. The ﬁrst component is of astrophysical nature and consists of

the random superposition of individually unresolved signals from the entire population of astrophysical sources listed

above. In addition, signals from cosmological events or structures could be present as well [12]. The SGWB is persistent

but can have an intermittent nature, has no phase coherence, and is in several experimental conditions, such as for

unresolved binary coalescences and cosmological signals searched for by current earth based interferometers, buried under

the intrinsic noise level of a single detector. Such a signal has a small but non-negligible contribution to the total energy

content of our Universe. Its detection, and in particular a primordial or cosmological component, would be as signiﬁcant

as the discovery of the cosmic microwave background (CMB) [13,14]. Its spectral structure will yield information on the

dynamical properties of its contributors and on the cosmological evolution of our Universe, and up to its earliest time

scales, way before the decoupling of the CMB.

In the past decades, many review papers on the SGWB have been published [15,16,17,18,19,20,21], where the

philosophy and explanation of the mathematical framework in this paper has the most overlap with the earlier work in

[19]. In this paper we discuss the properties of the possible components of the SGWB, their theoretical and experimental

bounds, the state-of-the-art of detection techniques and an outlook for future observations. We will refrain from giving an

extensive review of the astrophysical, cosmological and particle physics inspired models for the generation of a SGWB, as

these have been presented in other recent reviews, see e.g. [22,23,20,12,24,25,26]. We try to complement previous work

by giving an update on the latest results and upper limits achieved by Earth-based interferometric gravitational-wave

detectors as well as pulsar timing arrays. The main focus of this paper will be on the analysis techniques that are used or

being investigated for data analysis of Earth-based interferometers. Since we are nearing the ﬁrst detection of a SGWB

with the continuously increasing sensitivity of the detectors, we will discuss several techniques that can be used to prove

the observed signal is due to GWs and not to a terrestrial or instrumental noise source.

2. Stochastic background: deﬁnitions hypotheses and properties

In addition to individually detectable GW sources with generally deterministic signal properties, the Universe is perme-

ated by a SGWB. Gradual understanding of the properties of this type of GW signal and of its detectability has been

accumulated since the 1980’s by the works of Michelson [27], Christensen [28,15,17,20] and Flanagan [16], and has

been extensively reviewed and expanded upon by Allen and Romano [18], Romano and Cornish [19], and more recently

in [21]. The discussion on the general properties of the SGWB is still ongoing, but in general terms it is the result of

the incoherent sum of a large amount of weak, unresolvable sources. If you identify these sources with the way galaxies

are distributed within the universe at its largest scales, they are distributed almost isotropically across the Universe.

Depending on the intrinsic sensitivity of current and planned observatories, and the accessible GW frequency range, the

isotropy of the detectable background is not completely obvious. For example, the LISA space mission [29] will observe

a galactic ’foreground’ of binary white dwarfs [29], and Earth bound observatories could detect signiﬁcant contributions

of binary pulsars in nearby galaxy clusters [30]. In addition, the direct detection of binary mergers with deterministic

signal characteristics should be subtracted from the weaker, unresolved background radiation. In that sense, ’weak’ is

a relative and evolving concept. By virtue of the central limit theorem, the amplitudes of the stochastic background

originating from an incoherent superposition of independent sources should be Gaussian, but only in the limit of large

numbers. Current estimates on the population and merger rates [5] of binary systems of black holes, neutron stars and

black hole - neutron star systems indicate that for current observatories, the binary black hole merger signals that are

detectable given the limited frequency band at which the detectors have maximal sensitivity, have an intermittent nature

whose time structure is more ‘popcorn’ like, rather than stationary [31,32,22,23,33]. This translates in a so-called small

duty cycle, which corresponds to the probability of occurrence of a GW signal from these sources in a data analysis time

window.

The SGWB signal is in general approximated as a weak signal. Its power is small compared to the power spec-

tral densities of individual detectors. One therefore relies in most cases on the cross-correlation of outputs from two

interferometers, deﬁning a baseline that is given by their locations, relative separation, and relative orientations of the

interferometer arms [18,19]. Using the cross-correlation technique, the property of independence and stationarity implies

that any correlation depends on time diﬀerences, i.e. if hA(t)and h0

A0(t0)are the strain output of two detectors with

respective polarizations Aand A0, then the statistical correlator hhA(t)h0

A0(t0)iwill only depend on t−t0, or in the

frequency domain

h˜

hA(f)˜

A0(f0)i ∝ δ(f−f0)δAA0(2.1)

If the amplitudes are Gaussian, then all N-point correlators should reduce to products of two-point correlators, or the

expectation values of a single strain amplitude. These approximations are currently followed in the vast majority of

SGWB data analyses.

In the absence of detection of a SGWB signal, one generally puts upper limits on the total energy density of the

SGWB which, as originally outlined in [34], relates to the cross-correlation of ﬁrst time derivative of the strain amplitudes

in frequency space of two observatories that point at a location in the sky, ˆ

n, at any given siderial time, t:

ρGW =c2

32πG h˙

h(ˆ

n, t)˙

h0(ˆ

n, t)i.(2.2)

The energy density, ρGW , is generally expressed as a dimensionless quantity by normalizing it to the critical density of

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Stochasticgravitationalwavebackground:methodsandImplicationsNickvanRemortela,,KamielJanssensa,b,KevinTurbangc,aaUniversiteitAntwerpen,Prinsstraat13,2000Antwerpen,BelgiumbArtemis,UniversitéCôted'Azur,ObservatoiredelaCôted'Azur,CNRS,F-06304Nice,FrancecVrijeUniversiteitBrussel,Pleinlaan2,1050Brussel,B...

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