A practical theorem on gravitational-wave background statistics
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918460048539648 |
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| author | Ali-Haïmoud, Yacine |
| author_facet | Ali-Haïmoud, Yacine |
| contents | Inspiralling supermassive black-hole binaries (SMBHBs) are expected to be the main source of the nanohertz gravitational-wave background (GWB) targeted by pulsar timing arrays (PTAs). We provide a simple and general analytic expression for the probability distribution function (PDF) of the GWB characteristic strain squared $h_c^2$ in the limit of a large but finite effective number of sources, $N$, relevant for the lowest-frequency bands where PTAs are most sensitive. Explicitly, we show that for $N \gg 1$, the PDF of the rescaled variable $y \equiv h_c^2/\overline{h_c^2}$ takes the universal self-similar form $P(y) \simeq N^{1/3} \mathcal{P}(N^{1/3} (y -1))$, where $\mathcal{P}$ is the reflected map-Airy distribution. The effective number of in-band sources $N$ is fully specified by the mean $\overline{h_c^2}$ and the cubic shot-noise strain scale $\overline{h_0^3}$, a new summary statistic of the GWB that depends only on the local properties of the SMBHB population. This result is universal: it applies to any population of SMBHBs, regardless of whether they are circular or eccentric, and of the mechanism dominating orbital hardening. We explicitly quantify the accuracy of the large-source-count PDF for a simple but physically realistic SMBHB model, and outline its practical application to PTA data analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19701 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A practical theorem on gravitational-wave background statistics Ali-Haïmoud, Yacine Cosmology and Nongalactic Astrophysics General Relativity and Quantum Cosmology Inspiralling supermassive black-hole binaries (SMBHBs) are expected to be the main source of the nanohertz gravitational-wave background (GWB) targeted by pulsar timing arrays (PTAs). We provide a simple and general analytic expression for the probability distribution function (PDF) of the GWB characteristic strain squared $h_c^2$ in the limit of a large but finite effective number of sources, $N$, relevant for the lowest-frequency bands where PTAs are most sensitive. Explicitly, we show that for $N \gg 1$, the PDF of the rescaled variable $y \equiv h_c^2/\overline{h_c^2}$ takes the universal self-similar form $P(y) \simeq N^{1/3} \mathcal{P}(N^{1/3} (y -1))$, where $\mathcal{P}$ is the reflected map-Airy distribution. The effective number of in-band sources $N$ is fully specified by the mean $\overline{h_c^2}$ and the cubic shot-noise strain scale $\overline{h_0^3}$, a new summary statistic of the GWB that depends only on the local properties of the SMBHB population. This result is universal: it applies to any population of SMBHBs, regardless of whether they are circular or eccentric, and of the mechanism dominating orbital hardening. We explicitly quantify the accuracy of the large-source-count PDF for a simple but physically realistic SMBHB model, and outline its practical application to PTA data analysis. |
| title | A practical theorem on gravitational-wave background statistics |
| topic | Cosmology and Nongalactic Astrophysics General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2604.19701 |