Fast analysis

of long gravitational wave signals

Jacopo Tissino

GSSI

2026-07-20

Outline

  • GW250114: what we learned from the best\(^*\) black hole coalescence ever;
  • as seen by the Lunar Gravitational Wave Antenna:
    • the arrival time problem;
  • as seen by the Einstein Telescope:
    • multimodalities and configuration comparison.

GW250114: the clearest signal yet

11 years ago: GW150914

(LIGO Scientific Collaboration and Virgo Collaboration et al. 2016)

A decade of improvement

LIGO, J. Tissino, R. Hurt (Caltech-IPAC).

What the detectors saw

Adapted from Abac et al. (2025)

How we modelled it

(Abac et al. 2025)

How we interpreted it

Two black holes colliding, with masses \(m_1 = 33.6_{-0.8}^{+1.2} M_{\odot}\) and \(m_2 = 32.2_{-1.3}^{+0.8} M_{\odot}\).

Small spins \(|\chi_{1, 2}| \leq 0.26\), negligible eccentricity \(e_{13 \text{Hz}} \leq 0.03\).

(Abac et al. 2025)

How we analyzed it

Bayesian inference, accounting for our uncertainty on:

  • source masses \(m_i\) and spins \(\vec{\chi}_i\);
  • source location \((\alpha, \delta)\) and distance \(d_L\);
  • source inclination \(\theta_{\text{JN}}\) and polarization \(\psi\);
  • coalescence phase \(\phi_c\) and merger time \(t_c\);
  • eccentricity \(e\) and eccentric anomaly \(\zeta\);
  • detector calibration.

The observed ringdown

Modelling the ringdown

A superposition of damped sinusoids: quasinormal modes.

\[ \begin{aligned} h_{+}(t)-ih_{\times}(t) &\approx \sum_{\ell=2}^{\infty} \sum_{m=-\ell}^{\ell} \sum_{n=0}^{\infty} h_{\ell m n} S_{\ell mn} \\ h_{\ell m n} &= A_{\ell mn} \exp\left[i(\omega_{\ell mn}(t-t_\text{ref})+\phi^{x}_{\ell mn})\right] \\ \omega_{\ell mn} &= 2 \pi f_{\ell mn} + i / \tau _{\ell mn} \end{aligned} \]

For earlier signals, \(\ell mn = 220\) was sufficient.

Testing the Kerr hypothesis

For GW250114, the overtone \(\ell mn = 221\) was also required to explain the data.

Under the Kerr assumption, perturbation theory predicts the spectrum \(f_{\ell mn} (M_f, \chi_f)\).

We parameterize deviations as

\[ f_{221} = e^{\delta f_{221}} f_{221}^{\text{Kerr}} \]

Testing the Kerr hypothesis

(Abac et al. 2025)

We’ll come back to GW250114.

To get long gravitational wave signals, we need lower frequencies, which require a quiet place.

Lunar Gravitational Wave Antenna

The lunar north pole

Detector configuration

Measuring seismic motion at 4 locations inside a lunar crater, with cryogenics and seismic noise cancellation.

(Joris van Heijningen et al. 2023)

Watt’s linkage

Pendulum + inverted pendulum.

We measure horizontal displacement between the ground and this inertial mass, along two axes.

(Bertolini et al. 2006)

The Moon as a sensor

The displacement measured at the detector is

\[ s(t _{\text{det}}; f) = h_{ij}(t; f) D_{ij}(t _{\text{det}}) L(f) + \text{noise} \]

with

\[ D = \text{vertical} \otimes \text{horizontal} \]

\[ L(f) = \frac{\text{ground displacement}}{\text{GW strain}} \]

Sensitivity curve

(Ajith et al. 2025)

The deci-Hertz band

Adapted from Cozzumbo et al. (2024)

GW250114 with the LGWA

(Tissino et al. 2026)

The signal’s arrival time

When did the signal arrive?

Detector motion causes a nonlinear relation between source-frame and detector-frame time,

\[ t _{\text{det}} = t + t_0 - \frac{\hat{n} \cdot \vec{r}(t _{\text{det}})}{c}\,, \]

  • \(\vec{r}\) connects the origin and the detector’s position,
  • \(\hat{n}\) points to the GW source.

The timing parameter is:

\[ t _{\text{ref}} = \text{time when a reference event reaches the origin}\,. \]

What origin?

Standard approaches:

  • space-based detectors: the Solar System Barycenter (SSB),
  • ground-based detectors: the center of the Earth,
    • only valid assuming a short observation, such that \(\vec{r}_{\text{Earth}} \approx \vec{r}_0 + \vec{v} t\).

New approach:

  • SSB with a constant shift of origin.

Why does it matter? An experiment

  • correlations between \(t_{\text{ref}}\) and sky position lead to:
    • increased uncertainty on \(t_{\text{ref}}\) (requires a larger prior)
    • a more complex likelihood surface (difficult sampling)
Likelihood evaluations Sampling time [min] Timing uncertainty [s] Prior width [s] Origin
\(0.8\times 10^6\) 17 0.075 0.5 shifted
\(1.5\times 10^6\) 33 0.075 27 shifted
\(9.1\times 10^6\) 199 4.733 27 SSB

What reference event?

Impact of a different reference event

Recap: combining the two

Uncertainty on \(t_{\text{ref}}\) is a proxy for sampling efficiency.

It depends on origin position \(\vec{r}_0\) and reference frequency \(f\).

As a function of \(f\), we can compute

\[ \vec{r}_{\text{opt}}(f) = \text{argmin}_{\vec{r}_0} \text{var} [t_{\text{ref}}(\vec{r}_0, f)] \]

and reframe this as \(\vec{r}_{\text{opt}}(t)\) using the signal’s \(t(f)\).

Multi-band with the Einstein Telescope

The multi-band concept

(Ajith et al. 2025)

The multi-band horizon

(Ajith et al. 2025)

Einstein Telescope configurations

  • ET-\(\Delta\): equilateral triangle with 10km arms;
    • location: EMR region.
  • ET-2L: two L-shaped detectors with 15km arms;
    • locations: Sardinia and Saxony;
    • misaligned arms by \(45^\circ\).

GW250114, now and in the future

We compare:

  • The LVK detection of GW250114 (SNR 77–80);
  • a zero-noise injection with the LGWA (SNR 35);
  • a zero-noise injection with ET-\(\Delta\) (SNR 682);
  • a zero-noise injection with ET-2L (SNR 835).

Intrinsic parameters

  • LGWA complementary with ET on mass;
  • LGWA more precise than LVK on effective spin.
  • ET extremely precise on mass ratio and spin.

(Tissino et al. 2026)

Extrinsic parameters

  • LGWA localization tighter and “rounder” than LVK;
  • inclination not well constrained by LGWA;
  • ET-2L: tight, near Gaussian.
  • ET-\(\Delta\): tight, bimodal!

(Tissino et al. 2026)

Multimodalities with Einstein Telescope

GW250114 with ET-\(\Delta\)

Adapted from Tissino et al. (2026)

Symmetry in the antenna patterns

The GW signal is obtained as

\[ h(t) = F_+ h_+(t) + F_{\times} h_{\times}(t) \]

The antenna patterns \(F_{+ / \times}^{\text{ET-}\Delta}\), a function of the local sky coordinates, altitude \(\beta\) and azimuth \(\lambda\), are symmetric1 under:

  • \(\beta \to - \beta\),
  • \(\lambda \to \lambda + \pi / 2\).

Massive BBH: sky posterior

(Santoliquido et al. 2025)

Massive BBH: multimodalities

Sky localization modes for high mass

(\(150 M_{\odot} \lesssim \mathcal{M} \leq 1100 M_{\odot}\))

and distant (\(D_L \gtrsim 6 \text{Gpc}\)) BBH.

(Santoliquido et al. 2026)

How to resolve these multimodalities?

  • imprint of the rotation of the Earth;
  • imprint of frequency-dependent antenna patterns;
  • short-baseline triangulation;
  • other detectors.

GW250114: arrival times at the vertices

Summary

  • The Kerr metric is still compatible with observations;
  • the LGWA could have measured GW250114 and provided strong constraints:
    • standard analysis assumptions need to be revised for deci-Hertz observations;
  • the Einstein Telescope could have exquisitely measured GW250114:
    • but in its \(\Delta\) configuration it would have exhibited a multimodality.

Extra information

Ask me about

  • Hawking’s area theorem constrained with GW250114;
  • caveats to LGWA analyses;
  • the LGWA’s localization capabilities;
  • the variation of timing uncertainty with position.

The thesis is available as a website at https://jacopok.github.io/thesis.

Testing Hawking’s area theorem

The area of a black hole is:

\[ \mathcal{A}(M, \chi) = 8 \pi \left( \frac{GM}{c^{2}} \right)^{2} \left( 1+\sqrt{ 1-\chi^{2} } \right) \]

Hawking’s area theorem implies:

\[ \mathcal{A_i} = \mathcal{A}_1 + \mathcal{A}_2 < \mathcal{A_f} \]

We tested this ignoring the strong-field portion of the signal.

Testing Hawking’s area theorem

  • Data before \(t_<\) leads to \((1+z)^2 \mathcal{A_i}\).
  • Data after \(t_>\) leads to \((1+z)^2 \mathcal{A_f}\).

A violation of the theorem would be

\[ \frac{\mathcal{A_f} - \mathcal{A_i}}{\mathcal{A_i}} < 0. \]

(Abac et al. 2025)

Main caveats for LGWA

  • Assuming a known lunar response \(L(f)\);
  • assuming no data gaps due to lunar events;
  • assuming stationary noise;
  • assuming a simple geometry for seismic wave propagation;
  • working directly in the frequency domain with a heterodyned likelihood;
  • only considering the dominant, \(\ell |m| = 22\) mode of GW emission with a Post-Newtonian model;
  • performing zero-noise injections.

Moonquakes

(Ajith et al. 2025)

On the zero-noise assumption

On the zero-noise assumption

Analytical timing uncertainty

Length scale of the timing uncertainty variation:

\[ r \sim c \frac{\sigma_t}{\sigma_\theta} \]

  • \(r\) is the displacement to get a doubling of the timing variance;
  • \(\sigma_t\) is the minimal timing uncertainty;
  • \(\sigma_\theta\) is the angular parameter uncertainty along a given axis.

Timing uncertainty ellipses

(Tissino et al. 2026)

Analytical angular resolution

\[ \Delta \Omega \approx \frac{c^{2}}{(f_0 \rho_{T})^{2} A_s(T)} \]

\[ A_s(T)= 2\pi \sqrt{\mathrm{Var}(r_\theta)\mathrm{Var}(r_\phi)-\mathrm{Cov}^2(r_\theta,r_\phi)} \]

  • \(f_0\) is the monochromatic signal’s frequency
  • \(\rho_T\) is the signal-to-noise ratio

(Wen & Chen 2010), (Tissino et al. 2026)

Validating the expression

Trajectories and areas

(Tissino et al. 2026)

References

Abac AG, Abouelfettouh I, Acernese F, Ackley K, Adamcewicz C, et al. 2025. Phys. Rev. Lett. 135(11):111403
Ajith P, Seoane PA, Sedda MA, Arcodia R, Badaracco F, et al. 2025. J. Cosmol. Astropart. Phys. 2025(01):108
Bertolini A, DeSalvo R, Fidecaro F, Francesconi M, Marka S, et al. 2006. Nucl. Instrum. Methods Phys. Res. Sect. Accel. Spectrometers Detect. Assoc. Equip. 556(2):616–23
Cozzumbo A, Mestichelli B, Mirabile M, Paiella L, Tissino J, Harms J. 2024. Philos. Trans. R. Soc. Math. Phys. Eng. Sci. 382(2271):20230066
Joris van Heijningen, Marcel ter Brake, Oliver Gerberding, Shreevathsa Chalathadka Subrahmanya, Harms J, et al. 2023. The payload of the Lunar Gravitational-wave Antenna. http://arxiv.org/abs/2301.13685
LIGO Scientific Collaboration and Virgo Collaboration, Abbott BP, Abbott R, Abbott TD, Abernathy MR, et al. 2016. Phys. Rev. Lett. 116(6):061102
Santoliquido F, Tissino J, Dupletsa U, Branchesi M, Harms J, et al. 2025. Phys. Rev. D. 112(10):103015
Santoliquido F, Tissino J, Dupletsa U, Branchesi M, Harms J. 2026. Astron. Astrophys. 708:A175
Tissino J, Santoliquido F, Iacovelli F, Dupletsa U, Canton TD, et al. 2026. The geometry of lunar gravitational wave detection. http://arxiv.org/abs/2606.04918
Wen L, Chen Y. 2010. Phys. Rev. D. 81(8):082001