The Probability Distribution of the Astrophysical Gravitational-Wave - - PowerPoint PPT Presentation

The Probability Distribution of the Astrophysical Gravitational-Wave Background

YONADAV BARRY GINAT WITH: VINCENT DESJACQUES, ROBERT REISCHKE, AND HAGAI PERETS IPS CONFERENCE, 2020.

Gravitational Waves are Space-Time Perturbations

• Gravitational waves are tensor perturbations to the space-time metric, defined by

𝑕𝑏𝑐 = ҧ 𝑕𝑏𝑐 + ℎ𝑏𝑐.

• They satisfy a wave equation

𝐸𝑑𝐸𝑑ℎ𝑏𝑐 = − 16𝜌𝐻 𝑑4 𝑈𝑏𝑐 − 1 2 ҧ 𝑕𝑏𝑐𝑈

• They propagate along null geodesics of ҧ

𝑕, and their amplitude scales like 𝐸−1.

Source: [7]

What Is the Astrophysical SGWB?

• Gravitational waves from all over the universe constantly bathe our detectors, forming a background.
• The background is essentially stochastic, due to random nature of emission [5].
• Should be detectable directly with LISA, but current, ground based detectors can only find it with cross-

correlations [6].

Modelling The Background is Important

• Can teach us about cosmology [e.g. 1,6].
• Can be used to learn about binaries in the Universe, star formation history, phase transitions and even

inflation [e.g. 5,6].

• Previous studies focus on power-spectra and implicitly assume Gaussianity [2,4], even for the

astrophysical component.

Gravitational Waves Add Like a Random Walk

• Treat a wave with amplitude ℎ(𝑢) and phase 𝜚(𝑢) as z = ℎ𝑓i𝜚.

Gravitational Waves Add Like a Random Walk

• Treat a wave with amplitude ℎ(𝑢) and phase 𝜚(𝑢) as z = ℎ𝑓i𝜚.

𝑨1

Gravitational Waves Add Like a Random Walk

• Treat a wave with amplitude ℎ(𝑢) and phase 𝜚(𝑢) as z = ℎ𝑓i𝜚.

𝑨1 𝑨2

Gravitational Waves Add Like a Random Walk

• Treat a wave with amplitude ℎ(𝑢) and phase 𝜚(𝑢) as z = ℎ𝑓i𝜚.

𝑨1 𝑨2 𝑨3

Gravitational Waves Add Like a Random Walk

• Treat a wave with amplitude ℎ(𝑢) and phase 𝜚(𝑢) as z = ℎ𝑓i𝜚.

𝑨1 𝑨2 𝑨3 𝑨4

Gravitational Waves Add Like a Random Walk

• Treat a wave with amplitude ℎ(𝑢) and phase 𝜚(𝑢) as z = ℎ𝑓i𝜚.

𝑨1 𝑨2 𝑨3 𝑨4

Gravitational Waves Add Like a Random Walk

• Treat a wave with amplitude ℎ(𝑢) and phase 𝜚(𝑢) as z = ℎ𝑓i𝜚.
• Total strain from 𝑂(𝑢) sources is position of random walker after 𝑂(𝑢) steps.

𝑨1 𝑨2 𝑨3 𝑨4 𝑨𝑂

Sources Considered Here

• Sources are binary black holes or neutron starts.
• The wave-form is determined by source parameters 𝜊 which are randomly distributed.
• Assume sources are i.i.d, and that they are homogeneously distributed and Poisson-clustered with

mean number 𝑂0.

Fourier Transform Gives P(h)

• Assume initial phases are random.
• Then
• 𝐻(𝑡) is the single-source characteristic function, given by an expected value of exp i𝑡ℎ𝑙 (computed as

an average over both position and source parameters).

Irregularity of G Determines High-Strain Asymptotics

• The large ℎ limit is useful for determining

Gaussianity.

• The steepest descents method enables one to

compute it asymptotically.

• By the Paley-Wiener theorem, the irregularity of

𝐻(𝑡) is related to how fast 𝑄 ℎ declines.

• Mellin transform approximation of 𝐻(𝑡) at small

𝑡 gives From: [3]

P(h) Has Universal Power-Law Tail

• Mellin transform arguments again imply
• where 𝑐3

describes the mean single-source amplitude for a source at origin.

• Scaling is universal in-so-far-as it is independent of source parameter distribution details.
• Valid for any 𝑂0.

From: [3]

Interferences Are Important

• Random phases imply that small strains are

much more important.

• GW luminosity is square of sum of i.i.d.

variables, not sum of i.i.d.s. From: [3]

Conclusions and Outlook

• A method to calculate the full probability density of the SGWB was presented.
• The astrophysical SGWB has a universal scaling at large strains.
• It is non-Gaussian for any number of sources.
• The method can be extended to frequency space [3].
• In future work assumptions on homogeneity and isotropy of source distribution will be relaxed.
• Also in future work: explicit subtraction of bright (i.e. resolved) sources.

Works cited

1. Bertacca et al. (2019), arXiv:1909.11627. 2. Cusin et al. (2017), PRD. 3. Ginat et al. (2019), 1910.04587. 4. Jenkins et al. (2019), arXiv: 1907.06642. 5. Kosenko & Postnov (2000), A&A. 6. Maggiore, Gravitational Waves, vols. 1 (2008) and 2 (2018), OUP. 7. Pössel, Markus, in Einstein-Online https://www.einstein-online.info/en/spotlight/gw_waves/