• binary
• Gravitational waves
• magnetic field
• neutron stars
• post newtonian

General relativity serves as the cornerstone of our contemporary understanding of
gravity, providing a comprehensive framework for analyzing the movements and interactions
of celestial bodies, as well as for describing the intricate structure of spacetime and
its curvature. Gravitational waves represent a directly measurable phenomenon that is
unique to general relativity; their detection and analysis can significantly enhance our
understanding and refinement of the current gravitational framework - more importantly
their study is fundamental in studying the properties of astrophysical objects such as
neutron stars.
Neutron stars can have very large magnetic fields, as in the case of magnetars, the
objective of this thesis is to present and study numerically an effect to probe the value of
the magnetic field in neutron star binaries: a post Newtonian correction to gravitational
waves caused by the magnetic deformation of neutron stars.
Gravitational waves are generated by astrophysical systems such as binary neutron
stars and black holes. To effectively describe these complex systems, it is necessary
to develop a formalism that extends beyond the simplistic Newtonian limit. However,
establishing a new formalism presents challenges due to the inherent non-linearity of Einstein’s
field equations. Directly solving these equations for gravitational radiation is not
feasible; thus, a perturbative approach has become the state of the art in gravitational
wave research, with the Post-Newtonian (PN) expansion serving as a powerful technique
to describe progressively more relativistic effects in the gravitational wave signals produced
by such systems. While Newtonian gravity provides a baseline for gravitational
radiation, the PN formalism enables increasingly precise descriptions of these effects.
The corrections to the Newtonian solution are ordered in powers of a parameter describing
the departure from Newtonian gravity: from a study of the equation of motion
one would calculate correction to the metric, curvature and stress tensor, which would
then be used to calculate the corrected radiative solution to Einstein’s equations.
Finally the calculated gravitational wave, including its corrections, can be taken to
the frequency domain by using the stationary phase approximation. The final result is
the signal and is ultimately what is measured using gravitational waves detectors.
Among the diverse properties of stellar objects and binary systems that influence
gravitational wave production, this thesis investigates the role of magnetic deformation
in neutron stars binaries, specifically focusing on its contribution to the gravitational
wave signal through a magnetic quadrupole correction in the Post-Newtonian expansion.
Compact binaries emit enormous quantities of energy throughout their evolutionary
processes, and their inspiral can be observed and tracked over approximately 104
revolutions. Notably, some neutron stars, magnetars, have been found to possess extraordinarily
strong magnetic fields - so strong, in fact, that they may induce significant
deformations in the star itself, thereby altering its quadrupole moment. Consequently,
it becomes possible to derive a magnetic quadrupole moment correction for gravitational
waves; this correction would be of the 2PN order.
The deformation to one of the binary’s companion can be expressed by its quadrupole
moment tensor, the deformed mass distribution reflects as a perturbation to the gravitational
field, thus modifying the motion of the other star. This leads to a correction
2PN to the gravitational radiation of the binary, with the parameter σqm in the post
Newtonian expansions containing the quadrupole moment.
Notably a similar correction is already know: rapidly rotating neutron stars can experience
deformation to their mass distribution, leading to a similar 2PN correction to
gravitation wave in the σqm; only the source of the mass deformation would be different
between the two correction - with the quadrupole moment inside σqm carrying the
information on what of the two effect was studied.This allows for a useful comparison
between the spin quadrupole correction and the magnetic quadrupole correction.
A different 2PN correction to the gravitational wave signal caused by the magnetic
field is already present in literature, this correction is due to the interaction between the
magnetic dipoles of the companions and the electromagnetic flux. A direct comparison
between the two magnetic correction is carried out.
The majority of neutron stars are expected to have lower magnetic field than the one
possesed by magnetars but, as the bounds for the magnetic field in neutron star binaries
remain uncertain, this gravitational wave correction could be useful in their study.
The upcoming third generation gravitational wave detector - the Einstein telescope
- will provide a much greater sensitivity in measuring gravitational radiation. This
increase in sensitivity could prove crucial in enabling the measuribility of higher order
PN corrections.
An introductory explanation of gravitational wave detection is also presented, highlighting
the sensitivity functions of interferometers like LIGO and the anticipated Einstein
Telescope. The matched-filtering technique and statistical measures such as the
Faithfulness are explained to provide a framework in which to estimate the measurability
of the magnetic quadrupole correction.
Finally, the detectability of this magnetic quadrupole correction is evaluated using
current detectors and anticipated third-generation detectors. It will be shown that, while
this correction is undetectable by current interferometers, it is potentially observable
with the upcoming Einstein Telescope.
The correction’s magnitude is quantified and compared to existing 2PN spin and
magnetic corrections in the literature.
The spin correction will be shown to be of similar importance to the magnetic quadrupole
correction, with the two correction having similar mismatch for values in the B ∼ 10^16G,
χ ∼ 0.2 region.
The analysis shows that the magnetic quadrupole correction is more significant than
the magnetic dipole and flux correction - with this second correction only becoming
significant for unrealistically high magnetic magnitude.