## Tesi etd-09182018-183813 |

Thesis type

Tesi di laurea magistrale

Author

BRESCHI, MATTEO

URN

etd-09182018-183813

Title

Hierarchical Bayesian Analysis of Post-Newtonian Gravitational Waveforms

Struttura

FISICA

Corso di studi

FISICA

Supervisors

**relatore**Dott. Del Pozzo, Walter

Parole chiave

- gravitational waves
- data analysis
- Bayesian statistic
- General Relativity
- black holes
- binary coalescences

Data inizio appello

17/10/2018;

Consultabilità

Secretata d'ufficio

Riassunto analitico

In this thesis we prove that by means of a hierarchical Bayesian analysis, we are able to measure the values of the post-Newtonian (PN) coefficients of a gravitational waveform from simulations.

This analysis is motivated by several facts: first of all, we have a several models to describe the inspiral of a compact binary coalescence (such as the PN series, EOB theory and phenomeno- logical models). All of these are incomplete and must be adjusted by introducing and fitting extra-parameters on Numerical Relativity. Our purpose is to obtain a posterior distribution for the waveform’s model, in order to have a band which describe the strain of the wave. Using such a model, we could be able to include all the previous models in a single one and if this is true, we can conclude that all the templates are different realizations of the same posterior distribution. This approach should also avoid systematic errors or bias due to theoretical approximations. Moreover, another important aspect is that we cannot compute the values of certain parameters, such as the higher PN terms, and so we have to estimate them from simulations either from observations. We can learn from the observations the values we need to perform GR tests or we can extract informations from numerical simulations and use these values for the parameter estimation.

We employ a hierarchical Bayesian model to recover the PN coefficients as functions of the symmetric mass ratio η = m1m2/M2 (where M = m1 + m2) using the frequency-domain wave- form TaylorF2. We use this waveform because we can keep under control every terms since it is totally analytically determined. The method starts with the measurement of the n-th PN coefficient for different values of the physical variable η using a parallelized nested sampling and assuming to know the waveform up to the (n − 1)-th PN order and the frequency power. We get a set of points that relates the PN coefficients with the physical variable η. Then we use the assumption that the PN coefficient is a polynomial in η and we fit on these data different polynomial relations and compute the evidence (also called marginalized likelihood). Finally, either define the Bayes’ factor of keep using the evidences, and we select the most predictive model.

This method is completely general and can be used on simulations to learn the necessary values for the parameters estimation or it can be employed directly on observations to measure (and compare with) the theoretical expected values. Moreover we obtain posterior probability distributions for those parameters and we reconstruct a waveform with the associated error bars, which include the uncertainty of the model. The approach is predictive and gives the expected results; so, we can extend it to other physical effects such as spins and tidal contributions. It can be also used in the analysis of merger and of ring-down.

This analysis is motivated by several facts: first of all, we have a several models to describe the inspiral of a compact binary coalescence (such as the PN series, EOB theory and phenomeno- logical models). All of these are incomplete and must be adjusted by introducing and fitting extra-parameters on Numerical Relativity. Our purpose is to obtain a posterior distribution for the waveform’s model, in order to have a band which describe the strain of the wave. Using such a model, we could be able to include all the previous models in a single one and if this is true, we can conclude that all the templates are different realizations of the same posterior distribution. This approach should also avoid systematic errors or bias due to theoretical approximations. Moreover, another important aspect is that we cannot compute the values of certain parameters, such as the higher PN terms, and so we have to estimate them from simulations either from observations. We can learn from the observations the values we need to perform GR tests or we can extract informations from numerical simulations and use these values for the parameter estimation.

We employ a hierarchical Bayesian model to recover the PN coefficients as functions of the symmetric mass ratio η = m1m2/M2 (where M = m1 + m2) using the frequency-domain wave- form TaylorF2. We use this waveform because we can keep under control every terms since it is totally analytically determined. The method starts with the measurement of the n-th PN coefficient for different values of the physical variable η using a parallelized nested sampling and assuming to know the waveform up to the (n − 1)-th PN order and the frequency power. We get a set of points that relates the PN coefficients with the physical variable η. Then we use the assumption that the PN coefficient is a polynomial in η and we fit on these data different polynomial relations and compute the evidence (also called marginalized likelihood). Finally, either define the Bayes’ factor of keep using the evidences, and we select the most predictive model.

This method is completely general and can be used on simulations to learn the necessary values for the parameters estimation or it can be employed directly on observations to measure (and compare with) the theoretical expected values. Moreover we obtain posterior probability distributions for those parameters and we reconstruct a waveform with the associated error bars, which include the uncertainty of the model. The approach is predictive and gives the expected results; so, we can extend it to other physical effects such as spins and tidal contributions. It can be also used in the analysis of merger and of ring-down.

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