eutron stars are remarkably compact objects, with diameters ranging between 20 to 30 km, and masses as large as twice that of the sun. Their mean mass density is about two to three times larger than the nuclear saturation density (the mass density of nucleons in heavy atomic nuclei). The equation of state (EoS)2 describing their interior composition is still unknown, and perhaps the main mystery of neutron stars. An unfamiliar reader might be tempted to guess from their name that these stars are composed solely by neutrons, which would indeed fulfill charge neutrality, but that is not the lowest energy state of dense neutral matter. Protons and electrons (and corresponding anti-particles) will also be present, in a fraction of about ∼ 10%. At the high densities of a neutron star, the appearance of other non-nucleonic particles is also expected as their formation becomes energetically favored, such as hyperons (these are baryons with strangeness), condensed states of mesons, in particular pions and kaons, and even a phase transition to deconfined quark matter is a possibility. The composition of matter is considerably diﬀerent at the star’s birth, when neutrinos are trapped in the neutrinosphere, and the star has a lepton fraction3 of order ∼ 0.4, and an entropy per baryon of order ∼ 2 (in units of Boltzmann constant), both values significantly larger than what is found in old neutron stars. We show, in figure 1.3, a sketch of the composition of an old (cold) neutron star.
How can we observe neutron stars? Which observable quantities can be used to constraint our theoretical models? As mentioned before, neutron stars will emit a large quantity of neutrinos when born. In 1987, in the advent of a galactic type II supernova, SN 1987A, the Japanese neutrino observatory Kamiokande-II detected a burst of 11 neutrinos originated an SN 1987A. Together with two other observatories located at the USA and the former USSR, a total of 24 events were detected. The total number of events was consistent with the theoretical predictions for the neutrino luminosity of this supernova, strengthening the current theoretical picture of a core-collapse. Supernova neutrinos are an important mean to observe proto-neutron stars.
2 This acronym will be used throughout this thesis, both for singular and plural forms.
Figure 1.3: A sketch of the interior of a cold neutron star. Adapted from 
Furthermore, a core-collapse supernova will also produce gravitational radiation. So far, no direct observation of gravitational waves related to an event involving neutron stars has been published, but such observations should be possible in an earth based observatory, such as LIGO or VIRGO, for a galactic core-collapse supernova. What else can we observe?
Pulsars are highly magnetized rotating neutron stars, which convert their rotational energy into electromagnetic radiation (as charged particles are accelerated along the spinning mag-netic field lines), as shown in the sketch of figure 1.4, pulsating pretty much like a cosmic lighthouse. When they were first observed in 1967, by Jocelyn Bell Burnell and Antony Hewish, due to the fact that the pulsation period was so short that it would invalidate most of the known astrophysical sources of radiation, the idea came to their mind that this could be a signal from an alien civilization. They therefore named the signal as LGM-1 (for “little green man”). When one month later the same group found a second pulsar in a diﬀerent location of the sky, this hypothesis was abandoned for good. That first observation is now known as PSR J1921+2153. There were several speculative ideas proposed back then, but the most natural theoretical explanation for pulsars was to identify them with rotating neutron stars. Simple arguments allow astronomers to find a lower limit for the mean density of star matter as a function of its rotation period (see e.g. ); for pulsars, the obtained minimum mean densities were about the density of nuclear matter. Neutron stars had however never been observed by then, despite their strong theoretical motivation. It was one year later, with the observation of PSR B0531+21, also known as Crab Pulsar (for being located inside the Crab Nebula, the remnant of the earlier mentioned supernova SN 1054), that this theoretical model was confirmed, as well as the existence of neutron stars themselves.
In 1974, the first binary pulsar system, PSR B1913+16, was discovered by Russell Hulse and Joseph Taylor. This discovery was of major importance, as it consisted in the first indirect observation of gravitational radiation : the observed orbital decay has a precise agreement (as illustrated in figure 1.5) with the prediction from general relativity, that orbital energy would be lost to gravitational waves emission. We now have access to direct observations of these waves from LIGO  (from collisions of black holes), further confirming the nature of this observation.
Albeit the large majority of observed pulsars are isolated pulsars, observations of pulsars in binary systems are of major importance as a tool to constrain theoretical models of neutron stars: to measure the mass of a pulsar, all current methods rely on tracking its orbital motion. A review of neutron star mass measurements and the used methods can be found in . We show, in figure 1.6, an up to date list of measured neutron star masses. Double pulsars systems will also allow astronomers to measure the moment of inertia, which might impose strong constraints on the internal structure of neutron stars.
Radii measurements, unfortunately, represent a rather diﬃcult task. The currently avail-able methods rely on observations of thermal emission from the stellar surface.
One approach is to perform spectroscopic measurements, in order to measure its apparent angular size, much like what is done to measure the radii of normal stars. This, of course, imply measuring the neutron star distance, which is already by itself a diﬃcult task, in general yielding results with large uncertainties, which dominate the uncertainties on the radius measurement. Because of their compactness, neutron stars gravitational field will cause gravitational lensing eﬀects on their own surface emission, introducing therefore mass and spin dependent corrections on the observed angular sizes. Also, the pulsar magnetic field might be strong enough to lead to a non-uniform temperature at the stellar surface; and emissions due to a magnetosphere might contaminate the surface thermal emission.
Another approach is to study the amplitude and spectra of periodic brightness oscillations, originated from temperature anisotropies on the pulsar surface. Such quantities depend not only on the temperature profile at the stellar surface, but also on the gravitational field of the neutron star (therefore its model), and a number of other parameters, leaving room for large uncertainties.
In short, pulsar observations allow us (among other things) to do very precise measure-ments of rotation rates, precise measurements of their masses, and (unfortunately less reliable as of today) measurements of their radii. A large discussion of how currently available neu-tron star observation techniques (excluding gravitational wave emission, which only recently became available) could be used to understand their interiors can be found in .
Neutron Star Mergers
As we saw in figure 1.5, the orbit of a neutron star binary system will decay due to gravita-tional waves emission. What happens when the neutron stars finally collide? A new, hot and massive neutron star (HMNS) is born. Assuming that the total mass of the binary system is greater than the maximum mass limit, the resulting HMNS will always collapse to form a stellar-mass black hole in a relatively short timescale ; at birth, the HMNS will support itself against gravity due to its entropy pressure, which will dissipate as the neutron star cools down by radiating its neutrinos. Neutron star mergers are thought to be responsible for the production of short gamma-ray bursts (SGRB) : a short-lived (lifetime of about & 2 s) spectacularly powerful burst of γ-rays, the most luminous electromagnetic event in the universe. The mechanism for production of SGRBs is still not understood, and a hot research topic in relativistic astrophysics. They are also thought to be a preferred site for r-process (rapid neutron capture by heavy seed nuclei) nucleosynthesis (which will also occur in core-collapse supernovae), at the origin of some of the heaviest (and rarest) elements of the universe, such as gold and platinum .
The first studies of relativistic stars date back to the 30’s of the twentieth century, when the equations for a static, stationary spacetime of a spherically symmetric perfect fluid were independently derived by Tolman , and by Oppehneimer and Volkoﬀ . These equations are still widely used in the literature of relativistic stars, for studies for which the star rotation is not relevant. It was, however, not before the late 60’s that the first studies of rotating relativistic stars appeared. The first approach was proposed by Hartle and Thorne [18, 19], who developed an approximation for the spacetime of a slowly rigidly rotating, stationary and axisymmetric body, based on a perturbation of the spherically symmetric body up to second order. The first numerical solutions appeared shortly after [46, 47]. Since then several codes have been developed to solve the structure of stationary rotating relativistic stars, and much interest has been devoted to the study of these solutions, e.g. among other things, to study oscillation modes and stability (of relevance to the analysis of gravitational waves emitted by compact stars), the construction of initial data for general relativistic dynamical simulations, and more recently, to the approximately EoS independent relations between compact stars multipole moments, recently discovered by Yagi and Yunes  (see  for a comprehensive review on the literature of rotating compact stars, and  for a review of the approximate no-hair relations of compact stars).
In this thesis, we are interested in studying stars for which thermal eﬀects are not neg-ligible. Finite temperature eﬀects play an important role in astrophysical extreme events, such as core-collapse supernovae, and compact binary mergers, namely neutron star merg-ers and black hole – neutron star mergers. As mentioned before, in these events, matter can reach temperatures as high as 100 MeV. It therefore has an important impact in its composition, as it favors the production of non-nucleonic degrees of freedom, such as hy-perons (baryons with at least one strange quark), nuclear resonances, or mesons. Even a transition to the quark-gluon plasma could take place, which could facilitate the supernova explosion, as well as explain some gamma-ray bursts, or – within the scenario of “quark-novae” – some unusual supernova lightcurves . The impact of such additional particles on the evolution of proto-neutron stars has received great attention since long time (see e.g. [58, 59, 60, 62, 64, 65, 66, 67, 68, 63]). Several models for proto-neutron stars employ an EoS taking into account only homogeneous matter, neglecting nuclear clustering in the outer layers and the formation of a crust. The reason might be that the inhomogeneities in the EoS have only a minor impact on global PNS properties. In addition, until recently only a few EoS were available [101, 112, 77], treating the full temperature, baryon density and electron fraction dependence needed for the description of these hot objects, including nuclear clustering. In particular, those models neglected any possibility of non-nucleonic degrees of freedom at high density and temperature, probably more important for PNS than nuclear cluster. The situation changed in recent years, since, triggered by the study of black hole formation, a number of new EoS models has been developed, including as well nuclear clustering as hyperons [70, 73, 74, 75] or quark degrees of freedom [71, 72].
Finite temperature EoS have been used since long time in the studies of core-collapse dynamical simulations; more recently, they have also started to be used for neutron star mergers simulations (see e.g.[109, 110]). A complete hydrodynamical relativistic simulation of a proto-neutron star, is a rather complicated task, which was performed by Fischer et al , including a Boltzmann neutrino transport code and consistently taking into account finite temperature eﬀects, but excluding rotation (the simulation was performed in one dimension). As for stationary models of compact stars, the inclusion of finite temperature eﬀects has, for most of the literature, been taken into account neglecting the star’s rotation. The first evolutionary relativistic studies of non-rotating proto-neutron stars date back to the mid 80’s, by Burrows and Lattimer . Improvements to this study came with Pons et al. , including non-nucleonic degrees of freedom such as kaon condensates , and quarks . Ferrari et al.  used the same models to study the quasi-normal modes evolution of proto-neutron stars. The literature of rotating compact stars at finite temperature is however smaller. In fact, is is diﬃcult to find self-consistent solutions for the equilibrium configuration of relativistic stars with non-trivial entropy gradients.
The earliest stationary models of generally rotating1 proto-neutron stars have been carried by Goussard et al [1, 2]. The authors restricted their analysis to the case of barotropic fluids, by either considering an isentropic fluid with constant entropy per baryon, or by considering an isothermal fluid with constant redshifted temperature, in which both cases it is possible to solve an analytical first integral of the equilibrium equations. Similar approaches have been used by other authors, e.g. to construct evolutionary sequences of rotating proto-neutron stars [118, 119, 120], and more recently to study the influence of strong magnetic fields in proto-neutron stars with constant lepton fractions . An alternative approach, less con-straining with respect to the thermodynamical profiles, is to build an eﬀectively barotropic EoS, by parameterizing temperature (and eventually lepton fraction) as functions of baryon number density, implying assumptions specific to the chosen parameterization. Such an ap-proach was employed, for instance, by Villain et al. , to build evolutionary sequences of generally rotating proto-neutron stars by extrapolating results from spherically symmetric simulations , and by Kaplan et al. , to study quasi-equilibrium configurations of hot hypermassive neutron stars, born in the aftermath of a neutron star merger. The Hartle-Thorne slow rotation expansion has also been considered in the literature of axisymmetric proto-neutron stars, with the earlier study by Romero et al. , considering isothermal fluids. Albeit only valid in the regime of slow rotation velocities as compared to the Kepler frequency, the Hartle-Thorne approximation has the advantage that the equilibrium equa-tions, being a system of ordinary diﬀerential equations, can be integrated for a general EoS, without requiring eﬀective barotropicity, as is the case for finding analytical first integrals of generally rotating relativistic stars. In recent studies, the Hartle-Thorne approximation was employed using non-barotropic EoS, by Martinon et al.  and Camelio et al. , to build evolutionary models of proto-neutron stars (the latter including Boltzmann neutrino transport).
In this thesis, we will introduce a new solution for generally rotating relativistic stars, employing a numerical scheme to find solutions of the equilibrium equations of axisymmet-ric stationary perfect fluid bodies, which does not require the EoS to be barotropic. We will test the code with an analytical temperature dependent EoS, and with a realistic finite temperature EoS including hyperonic degrees of freedom. Finally, we will discuss the rele-vance of this solution for further evolutionary studies of proto-neutron stars, deepening our understanding of rapid and diﬀerential rotation in their cooling, the quasi-normal modes of radiated gravitational waves, their spin-evolution, proto-neutron star winds, etc.
Table of contents :
1.1 The onion-like structure of a massive star at the end of it’s life (shells are not to scale), with the burning timescale of each nuclear species. Adapted from 
1.2 A sketch of the ongoing processes prior to the re-ignition of a supernova shock. ˙M stands for the accreting mass, PNS stands for proto-neutron star, Rns is the PNS radius, Rν is the neutrinosphere radius, Rs is the shock position, and Rg (the gain radius) is the location where the temperature is low enough to allow the absorption of neutrinos and antineutrinos to start exceeding the neutrino emission. Adapted from 
1.3 A sketch of the interior of a cold neutron star. Adapted from 
1.4 A sketch of a pulsar and its magnetic field. Adapted from 
1.5 Orbital decay of binary pulsar system. Figure from 
1.6 Measured neutron star masses with 1 − σ errors. Figure adapted from [16, 17]. 28
3.1 Foliation of spacetime by slices of space at constant instants of time
3.2 Geometrical features of the 3+1 decomposition
4.1 Profiles for the monopolar term of sb
4.2 Isocontour enthalpy lines for the s0 radial entropy profile (frot = 382.84Hz)
4.3 Isocontour lines of log-enthalpy and entropy per baryon for a rapidly rotating star with the s1 entropy per baryon monopolar profile (frot = 721.85Hz)
4.4 Isocontour lines of log-enthalpy and entropy per baryon for the fastest rotation up to which the code can provide a solution without approximations, using the s1 entropy radial profile (frot = 650Hz)
4.5 Isocontour lines of log-enthalpy and entropy per baryon for a rapidly rotating star with the s2 entropy per baryon monopolar profile (frot = 490Hz)
4.6 Isocontour lines of enthalpy and entropy for the fastest rotation up to which the code can provide a solution without approximations, using the s2 entropy radial profile (frot = 395Hz)
4.7 Mass-radius profiles for the ideal gas EoS
4.8 Isocontour log-enthalpy lines for the s0 entropy per baryon profile, a=2
4.9 Isocontour log-enthalpy lines for the s0 entropy per baryon profile, a=0.95
4.10 Isocontour log-enthalpy lines for the s0 entropy per baryon profile,a=0.75
4.11 Isocontour lines of log-enthalpy and entropy per baryon for a differentially rotating star with the s1 monopolar profile, a=1
4.12 Isocontour lines of log-enthalpy and entropy per baryon for a differentially rotating star with the s1 monopolar profile,a=0.6
4.13 Isocontour lines of log-enthalpy and entropy per baryon for a differentially rotating star with the s2 monopolar profile, a=2.85
4.14 Isocontour lines of log-enthalpy and entropy per baryon for a differentially rotating star with the s2 monopolar profile, a=1
5.1 Values of J and L in different nuclear interaction models. The two grey rectangles correspond to the range for J and L derived in Ref.  (light grey) and Ref.  (dark grey) from nuclear physics experiments
5.2 Pressure (left panel) and energy per baryon (right panel of pure neutron matter as function of baryon number density within different nuclear interaction models compared with the ab initio calculations of Ref. , indicated by the green band. 83
5.3 Gravitational mass versus circumferential radius for a cold spherically symmetric neutron stars within different EoS models. The two horizontal bars indicate the two recent precise NS mass determinations, PSRJ1614-2230  (hatched blue) and PSR J0348+0432  (yellow)
5.4 The lines delimit the regions in temperature and baryon number density for which the overall hyperon fraction exceeds 10−4, which are situated above the lines. The dark purple line corresponds to the BHB model and light red line to the DD2Y(I) model. Different charge fractions are shown as indicated within the panels
6.1 Isocontour enthalpy lines for the HS(DD2) EoS, using s0 profile (frot = 908Hz) 88
6.2 Isocontour enthalpy lines for the BHBφ EoS, using s0 profile (frot = 913.63Hz) 89
6.3 Isocontour enthalpy lines for the DD2Y(I) EoS, using s0 profile (frot = 916.3Hz) 90
6.4 Isocontour lines of log-enthalpy and entropy per baryon for a rapidly rotating star with the s2 profile, for the HS(DD2) EoS (frot = 933Hz)
6.5 Isocontour lines of log-enthalpy and entropy per baryon for a rapidly rotating star with the s2 profile, for the BHBφ EoS (frot = 945.5Hz)
6.6 Isocontour lines of log-enthalpy and entropy per baryon for a rapidly rotating star with the s2 profile, for the DD2Y(I) EoS (frot = 936Hz)
6.7 Isocontour lines of enthalpy and entropy for the fastest rotation up to which the code can provide a solution without approximations, using the s2 profile, with the HS(DD2) EoS (frot = 620Hz)
6.8 Isocontour lines of enthalpy and entropy for the fastest rotation up to which the code can provide a solution without approximations, using the s2 profile, with the BHBφ EoS (frot = 650Hz)
6.9 Isocontour lines of enthalpy and entropy for the fastest rotation up to which the code can provide a solution without approximations, using the s2 profile, with the DD2Y(I) EoS (frot = 310Hz)
6.10 Mass-radius profiles for the HS(DD2) EoS
6.11 Mass-radius profiles for the BHBφ EoS
6.12 Mass-radius profiles for the DD2Y(I) EoS
6.13 Mass-radius profiles for all EoS, rotating at 100 Hz
6.14 Mass-radius profiles for all EoS, rotating at 600 Hz
6.15 Mass-radius profiles for all EoS, rotating at 900 Hz
6.16 Isocontour enthalpy lines for the HS(DD2) EoS, using s0 profile
6.17 Isocontour enthalpy lines for the BHBφ EoS, using s0 profile
6.18 Isocontour enthalpy lines for the DD2Y(I) EoS, using s0 profile
6.19 Isocontour lines of log-enthalpy and entropy per baryon for a differentially rotating star with the s2 profile and a = 0.4, for the HS(DD2) EoS
6.20 Isocontour lines of log-enthalpy and entropy per baryon for a differentially rotating star with the s2 profile and a = 0.4, for the BHBφ EoS
6.21 Isocontour lines of enthalpy and entropy for a differentially rotating star with the s2 profile and a = 0.1, with the HS(DD2) EoS
6.22 Isocontour lines of enthalpy and entropy for a differentially rotating star with the s2 profile and a = 0.1, with the BHBφ EoS