11/25/25

Current results and future Current results and future scenarios for gravitational scenarios for gravitational

wave’s stochastic backgroundwave’s stochastic background

G. Cella – INFN sez. PisaG. Cella – INFN sez. Pisa

22/25/25

SummarySummary

IntroductionIntroductionData analysis strategiesData analysis strategiesCurrent collaborations & projectsCurrent collaborations & projectsCurrent upper limitsCurrent upper limitsSome future perspectivesSome future perspectives

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Stochastic backgroundStochastic backgroundh is a sum of statistically independent events

We can describe h using the correlation functions:

Using the theory of point processes we can show that this scales with the rate as:

As expected, at large rates h becomes a Gaussian stochastic field (central limit theorem)

Rate

Burst

Coalescence

Pulsar

Cosmological background

Astrophysical background

Stochastic Gaussian

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The gaussian caseThe gaussian caseEvery gravitational wave strain field can formally be developed in modes:

Polarizations

FrequenciesDirections

• We do not know the amplitudes , so we can only give a probabilistic description.

• In the gaussian case only the second order correlations are relevant. They fully describe the stochastic field.

• Unpolarized

• Stationary

• Isotropic With these additional simplification the background is fully described by its power spectrum.

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Gaussian case: space correlationsGaussian case: space correlationsSpace correlations can be evaluated analytically. They can be written as a sum over all the modes:

When the phase factor oscillates strongly on the full solid angle. Correlations go to zero.

These properties are detector independent. Now we must describe the real measurement.

Always in phase

Always out of phase

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Coupling to the detectorCoupling to the detectorA detector give us one or more projections of the strain hij :

Signal

Detector tensor

Strain at the detector position

Example: for a bar aligned to the direction:

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Detector tensor: interferometerDetector tensor: interferometer

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For the differential mode (arms along ):

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Common mode:

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Detector tensor: sphereDetector tensor: sphere

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Overlap reduction functionOverlap reduction functionThe signal is a linear combination of the elements of the strain tensor.

• Gaussian

• Stationary, at least if Dij is time independent

This can be written as

where the overlap reduction function AB is defined by

• Depends on distance (same features of strain correlations)

• Depends on orientation (via the overlap of detector tensors)

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ORF: examplesORF: examples

Trigo: =0.375 (frequency independent)

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DetectionDetectionBasic idea: two detectors.

1. noise not correlated with the signal

2. noise not correlated between the detectors (?)

Determine the optimal correlation, which is of the form:

Maximizing SNR…..

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Optimal correlationOptimal correlation

Can be seen as a matched filter for cross correlations.

It depends on:

• theoretical PSD of stochastic background

• overlap reduction function

• noise PSD of the two detectors

The optimal statistic is given by:

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Or, converting to theoretician’s units

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Many detectorsMany detectorsGaussian signals: optimal detector is a combination of optimal correlations from each pair.

• Not a large improvement factor, but

• Very important for ruling out spurious effects

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3 groups involved (Virgo, Auriga, ROG)3 groups involved (Virgo, Auriga, ROG)

Methodological studyMethodological study

Software injection of simulated signalSoftware injection of simulated signal

Use of real data (not syncronized) Use of real data (not syncronized)

Validation of standard detection algorithmValidation of standard detection algorithm

Noise power spectrum for the

detectors involved :

Overlap reduction

function :

(the coherence

between detectors)

Detection study in progressDetection study in progress Impact of non stationarityImpact of non stationarity Study of vetoesStudy of vetoes Comparison with simulated Comparison with simulated

gaussian noisegaussian noise Application to C7 dataApplication to C7 data

Perspectives: upper limit

when at design sensitivity.

Upper limit on the GW stochastic background

energy density

as a function of the upper

frequency of the band:

Virgo/Auriga/ROG collaborative data analysis: Stochastic background

1616/25/25

Virgo/LSC collaboration: the project Virgo/LSC collaboration: the project conceptconcept

Joint analysis between Virgo, LHO, LLO Joint analysis between Virgo, LHO, LLO GEO600GEO600

Background above 200 HzBackground above 200 Hz Standard correlation (isotropic model)Standard correlation (isotropic model) Targeted search (map of sky gw luminosity)Targeted search (map of sky gw luminosity)

Real data when comparable sensitivity will be Real data when comparable sensitivity will be reachedreached

Preparatory phase: work by projectPreparatory phase: work by project Phase 1: simulated or technical data, pipeline testPhase 1: simulated or technical data, pipeline test Phase 2: A4/C7 data Phase 2: A4/C7 data

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Virgo/LSC: standard correlation analysisVirgo/LSC: standard correlation analysis

Design sensitivities, 4 months of data, 5% FA & FD:

• LHO/LLO and *LO/Virgo: same sensitivity above 200 Hz

• GEO/Virgo good above 200 Hz

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SensitivitiesSensitivities

false alarm rate, detection rate

(constant gw)

@ 900 Hz feasible with final sensitivity of Virgo & italian bars

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AnisotropiesAnisotropiesGeneralized statistics of mode amplitudes:

Strategy:

• Expand in spherical harmonics

• Standard cross correlation analysis

• Signature: modulation generated by the earth rotation

Detection of higher order harmonics requires larger SNR

2020/25/25

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Targeted searchTargeted searchThe idea: reconstruct a map of the gravitational wave luminosity in the sky.

Recipe:

1. Decompose luminosity and detector response in spherical harmonics, in the sky and in the detector frames

2. Solve the inverse problem

• Detailed study needed (work in progress inside LSC)

Virgo+WA 200 Hz

• At least 3 detectors needed to close the inverse

problem.

• Angular resolution limited by

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Non gaussian signalsNon gaussian signals

• Several astrophysical stochastic background are far from the gaussian limit

• They are interesting in itself, but also an obstacle for the detection of cosmological backgrounds

Questions:

• Can they be optimally detected?

• Can they be disentangled from other backgrounds?

Some proposals, but work in progress. Theoretical input is important.

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Detector sensitivities:Detector sensitivities:

1 Virgo-like

2 Virgo-like

Virgo+bar

2 AdvancedLISA

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Upper limitsUpper limits

1 Virgo-like

2 Virgo-like

Virgo+bar

2 AdvancedLISA

• COBE

• Double pulsars

• ms pulsars

• Nucleosynthesis

• LIGO S3

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ConclusionsConclusions

Other relevant points, not discussed:

• Detection of scalar fields (dilatons, moduli, …)

• How to deal with non gaussian and non stationary noise

• How to deal with noise correlations between detectors

• Template banks, simulation issues, pipeline implementation

• LISA

• Probably only upper limits with the current generation of detectors

• Good perspectives for advanced detectors

Thank you!