Higgs Boson equation in de Sitter spacetime: Numerical investigation of bubbles using GPUs

The Higgs field, along with its corresponding boson, represent a milestone for modern day particle physics. In this work we consider the Higgs boson equation in de Sitter spacetime. Previous work by K. Yagdjian [23] has formulated sufficient conditions for the existence of the zeros of global solutions in the interior of their supports. In searching for such solutions, we turn to heterogeneous parallel computing, which allows for faster computation through graphical processing units (GPUs). Armed with general-purpose computation on graphics hardware (GPGPU) techniques and explicit numerical schemes, we approximate solutions of the equation for the Higgs boson along with the creation, growth, and interaction of the zeros, or bubbles

The art of simulating the early Universe -- Part I

We present a comprehensive discussion on lattice techniques for the simulation of scalar and gauge field dynamics in an expanding universe. After reviewing the continuum formulation of scalar and gauge field interactions in Minkowski and FLRW backgrounds, we introduce basic tools for the discretization of field theories, including lattice gauge invariant techniques. Following, we discuss and classify numerical algorithms, ranging from methods of $O(dt^2)$ accuracy like $staggered~leapfrog$ and $Verlet$ integration, to $Runge-Kutta$ methods up to $O(dt^4)$ accuracy, and the $Yoshida$ and $Gauss-Legendre$ higher-order integrators, accurate up to $O(dt^{10})$. We adapt these methods for their use in classical lattice simulations of the non-linear dynamics of scalar and gauge fields in an expanding grid in $3+1$ dimensions, including the case of `self-consistent' expansion sourced by the volume average of the fields' energy and pressure densities. We present lattice formulations of canonical cases of: $i)$ Interacting scalar fields, $ii)$ Abelian $U(1)$ gauge theories, and $iii)$ Non-Abelian $SU(2)$ gauge theories. In all three cases we provide symplectic integrators, with accuracy ranging from $O(dt^2)$ up to $O(dt^{10})$. For each algorithm we provide the form of relevant observables, such as energy density components, field spectra and the Hubble constraint. Remarkably, all our algorithms for gauge theories respect the Gauss constraint to machine precision, including when `self-consistent' expansion is considered. As a numerical example we analyze the post-inflationary dynamics of an oscillating inflaton charged under $SU(2)\times U(1)$. The present manuscript is meant as part of the theoretical basis for $CosmoLattice$, a modern C++ MPI-based package for simulating the non-linear dynamics of scalar-gauge field theories in an expanding universe, publicly available at www.cosmolattice.netComment: Minor corrections to match published version, and one more algorithm added. Still 79 pages, 8 figures, 1 appendix, and many algorithm

Non-Thermal Fixed Point in a Holographic Superfluid

We study the far-from-equilibrium dynamics of a (2+1)-dimensional superfluid at finite temperature and chemical potential using its holographic description in terms of a gravitational system in 3+1 dimensions. Starting from various initial conditions corresponding to ensembles of vortex defects we numerically evolve the system to long times. At intermediate times the system exhibits Kolmogorov scaling the emergence of which depends on the choice of initial conditions. We further observe a universal late-time regime in which the occupation spectrum and different length scales of the superfluid exhibit scaling behaviour. We study these scaling laws in view of superfluid turbulence and interpret the universal late-time regime as a non-thermal fixed point of the dynamical evolution. In the holographic superfluid the non-thermal fixed point can be understood as a stationary point of the classical equations of motion of the dual gravitational description.Comment: 37 pages, 10 figures; v2: discussion and figures added, matches published version; movies and additional material at http://www.thphys.uni-heidelberg.de/holographic-superflui

Scalar Fields in Numerical General Relativity

Einstein's field equation of General Relativity (GR) has been known for over 100 years, yet it remains challenging to solve analytically in strongly relativistic regimes, particularly where there is a lack of a priori symmetry. Numerical Relativity (NR) - the evolution of the Einstein Equations using a computer - is now a relatively mature tool which enables such cases to be explored. In this thesis, a description is given of the development and application of a new Numerical Relativity code, GRChombo. GRChombo uses the standard BSSN formalism, incorporating full adaptive mesh refinement (AMR) and massive parallelism via the Message Passing Interface (MPI). The AMR capability permits the study of physics which has previously been computationally infeasible in a full 3+1 setting. The functionality of the code is described, its performance characteristics are demonstrated, and it is shown that it can stably and accurately evolve standard spacetimes such as black hole mergers. We use GRChombo to study the effects of inhomogeneous initial conditions on the robustness of small and large field inflationary models. and investigate the critical behaviour which occurs in the collapse of both spherically symmetric and asymmetric scalar field bubbles.Comment: PhD Thesis 2017, 232 page

Numerical relativity, compact objects, and fundamental fields

Las recientes detecciones de ondas gravitacionales están abriendo una nueva ventana al Universo. La naturaleza de los agujeros negros y las estrellas de neutrones ahora puede ser desvelada, pero la radiación gravitacional también puede conducir a descubrimientos emocionantes de nuevos y exóticos objetos compactos, ajenos a las ondas electromagnéticas. En esta tesis, he investigado tres temas principales que involucran campos bosónicos escalares y vectoriales fundamentales acoplados a la gravedad dentro de la Relatividad General y en simetría esférica: (i) configuraciones cuasiestacionarias de campos escalares reales alrededor de agujeros negros de Schwarzschild como modelos de materia oscura como campo escalar, (ii) la inestabilidad superradiante y la formación de agujeros negros cargados con pelo, y (iii) estrellas bosónicas. Estos sistemas podrían tener una relevancia astrofísica importante, si existen campos bosónicos ultraligeros en la Naturaleza. En 2012, se descubrió la primera partícula de bosón no gauge, el bosón de Higgs, en el Gran Colisionador de Hadrones (LHC). El trabajo principal en esta tesis consiste en evoluciones de relatividad numéricas de campos bosónicos en el régimen de campo intenso de la gravedad. Recientemente, se han estudiado configuraciones de campo escalar alrededor de agujeros negros en el régimen linealizado, tomando el espacio-tiempo fijo. Se descubrió que se pueden formar estados cuasiligados de campo escalar de muy larga vida alrededor del agujero negro. Para investigar las evoluciones temporales en escenarios altamente dinámicos, se requiere realizar simulaciones numéricas de los sistemas acoplados no lineales Einstein-Klein-Gordon o Einstein-Proca. Con este objetivo he extendido códigos de relatividad numérica en 1D y 3D que usan coordenadas esféricas y que resuelven las ecuaciones de hidrodinámica relativista acopladas a las ecuaciones de Einstein, implementando las ecuaciones fundamentales que describen los campos bosónicos. En primer lugar, he llevado a cabo evoluciones numéricas de los campos escalares alrededor de los agujeros negros, teniendo en cuenta la reacción del campo escalar en la dinámica del campo gravitacional. Por lo tanto, el espacio-tiempo podía cambiar dinámicamente: aumento de la masa del agujero negro debido a la absorción de parte del campo escalar autogravitante o debido a la acreción adiabática de alguna otra forma de materia, o formación de agujeros negros del colapso gravitacional de una estrella politrópica descrita con un ecuación de estado (EOS) similar a la de una estrella supermasiva. En todos los casos, mis simulaciones no lineales han revelado una frecuencia de oscilación bien determinada que apunta a la presencia de los estados cuasiligados descritos en la literatura en el régimen linealizado. Otro fenómeno interesante que involucra campos bosónicos y agujeros negros es la inestabilidad súperradiante, que puede ser desencadenada por la dispersión del campo por el agujero negro. Un campo bosónico puede extraer energía del agujero negro y, si se introduce un mecanismo de captura, puede crecer exponencialmente. Sobre la base de trabajos numéricos previos en el régimen linealizado, he estudiado la inestabilidad de un campo escalar cargado alrededor de un agujero negro de Reissner-Nordström (cargado) en una cavidad, mostrando que el punto final es una solución en la que el agujero negro y el campo bosónico están en equilibrio, es decir, un agujero negro con pelo. Además, descubrí que el colapso de los solitones de campo escalar cargados en una cavidad también puede formar estas soluciones. Finalmente, he considerado los modelos de equilibrio de las estrellas bosónicas autogravitantes, condensados Bose-Einstein no singulares y sin horizontes, de campos masivos, en concreto, estrellas de bosones con y sin autointeracción y estrellas de Proca. Estos objetos compactos son considerados imitadores de agujeros negros ya que solo interactúan con la gravedad. Observamos que las estrellas Proca se parecen en muchos aspectos a sus primos escalares, incluso con un término de interacción propia. Una separación entre configuraciones estables e inestables que ocurre en la solución con la máxima masa ADM ha sido obtenido por estudios previos de la teoría de perturbaciones lineal. Mis simulaciones no solo confirman este resultado sino que además muestran que los diferentes resultados de los modelos inestables, es decir, la migración a la rama estable, la dispersión total del campo escalar o el colapso a un agujero negro de Schwarzschild, están presentes en ambos campos. En este último caso, un remanente del campo permanece fuera del horizonte, formando un estado cuasiligado. Se puede establecer un paralelismo adicional con las estrellas de neutrones, para las cuales también se ha encontrado numéricamente el colapso y la migración, pero no la dispersión. Con respecto al marco de la relatividad numérica, en esta tesis he modificado la formulación CCZ4, que es una descomposición conforme y sin traza de las ecuaciones de Einstein, para hacer que sus ecuaciones de evolución sean adecuadas para coordenadas curvilíneas. Descubrí que las violaciones de restricción Hamiltoniana podían reducirse de uno a tres órdenes de magnitud para los espacios espaciales al vacío con respecto a la formulación BSSN estándar. Para los agujeros negros de Schwarzschild, sin embargo, los resultados no fueron significativamente mejores. Esta tesis también contiene algunos trabajos de investigación sobre dos temas adicionales, a saber, estrellas (fermiónicas) de rotación lenta y mi contribución a la Colaboración de Virgo. Esta última ha consistido en producir patrones de onda gravitacionales a partir de simulaciones numéricas de estrellas que colapsan descritas por una ecuación de estado no convexa (EOS). Para el primero, he estudiado numéricamente el modelo de Hartle modificado recientemente de estrellas que giran lentamente dentro de la teoría de perturbaciones, que correctamente toma en cuenta las discontinuidades de densidad en la superficie de la estrella para la corrección de la masa, dM, para diferentes EOS . He ayudado a desarrollar un código numérico que proporciona modelos iniciales de estrellas en rotación para una cantidad de EOS, más allá de estrellas polítrópicas y la idealización con densidad constante. Pudimos determinar e incluir la universalidad de dM en las llamadas relaciones I-Love-Q.The recent detections of gravitational waves are opening a new window to the Universe. The nature of black holes and neutron stars may now be unveiled, but gravitational radiation may also lead to exciting discoveries of new exotic compact objects, oblivious to electromagnetic waves. In this thesis, I have investigated three main topics involving fundamental scalar and vector bosonic fields coupled to gravity within General Relativity and under the assumption of spherical symmetry: (i) quasistationary configurations of real scalar fields around Schwarzschild black holes as scalar field dark matter models, (ii) the superradiant instability and the formation of charged hairy black holes, and (iii) bosonic stars. These systems could have important astrophysical relevance, if ultralight bosonic fields exist in Nature. In 2012, the first non-gauge boson particle, the Higgs boson, was discovered in the Large Hadron Collider (LHC). The main work in this thesis deals with numerical-relativity evolutions of bosonic fields in the strong-field regime of gravity. Recently, scalar field configurations around black holes have been studied in the linearized regime, taking the spacetime as a background. It was found that very long-lived scalar field quasibound states may form around the black hole. To investigate time evolutions in highly dynamical scenarios, it is required to perform numerical simulations of the fully non-linear Einstein-Klein-Gordon or Einstein-Proca coupled systems. To this aim I have extended numerical-relativity codes in 1D and 3D using spherical coordinates that solve the relativistic hydrodynamics equations coupled to the Einstein equations, implementing the fundamental equations describing the bosonic fields. Firstly, I have carried out numerical evolutions of scalar fields around black holes, taking into account the back-reaction of the field onto the gravitational field dynamics. Therefore, the spacetime could dynamically change: mass growth due to the absorption of part of a self-gravitating scalar field or from the adiabatic accretion of some other form of matter, or black hole formation from the gravitational collapse of a polytropic star described with an equation of state (EOS) similar to that of a supermassive star. In all cases, my non-linear simulations have revealed a well-determined oscillating frequency that pointed out to the presence of the quasibound states described in the linearized literature. Another interesting phenomena involving bosonic fields and black holes is the superradiant instability, that can be triggered by the scattering of the field off the black hole. A bosonic field may then extract energy from the black hole and, if a trapping mechanism is introduced, it can grow exponentially. Building on previous numerical works in the linearized regime, I have studied the instability for a charged scalar field around a Reissner-Nordström (charged) black hole in a cavity, showing that the endpoint is a solution in which the black hole and the bosonic field are in equilibrium, i.e. a hairy black hole. Moreover, I found that the collapse of charged scalar field solitons in a cavity may also form these solutions. Finally, I have considered equilibrium models of bosonic stars as self-gravitating, everywhere non-singular, horizonless Bose-Einstein condensates of massive fields, namely boson stars with and without self interaction and Proca stars. These compact objects are regarded as black hole mimickers as they only interact through gravity. By performing accurate numerical simulations, I have observed that Proca stars resemble in many ways its scalar cousins, even with a self-interaction term. A separation between stable and unstable configurations occuring at the solution with maximal ADM mass has been indicated by previous results from linear perturbation theory. My simulations not only confirm this result but furthermore they show that the different outcomes of unstable models, namely migration to the stable branch, total dispersion of the scalar field, or collapse to a Schwarzschild black hole, are present for both fields. In the latter case, a field remnant lingers outside the horizon, forming a quasibound state. A further parallelism can be established with neutron stars, for which the collapse and the migration, but not the dispersion, has also been found numerically. Regarding the numerical-relativity framework, in this thesis I have modified the CCZ4 formulation, which is a conformal and traceless decomposition of the Einstein equations, to make its evolution equations suitable for curvilinear coordinates. I have found that the Hamiltonian constraint violations could be reduced by one to three orders of magnitude for non-vacuum spacetimes with respect to the standard BSSN formulation. For Schwarzschild black holes, however, the results were not significantly better. This thesis also contains some miscellaneous research work on two topics, namely slowly-rotating (fermionic) stars and my contribution to the Virgo Collaboration. The latter has consisted in producing gravitational waveforms from numerical simulations of collapsing stars described by a non-convex equation of state (EOS). For the former I have studied numerically the recently amended Hartle's model of slowly-rotating stars within perturbation theory, which correctly takes into account density discontinuities in the surface of the star for the correction of the mass, dM, for different EOS. I have helped to develop a numerical code providing initial models of rotating stars for a number of EOS, beyond polytropes and the constant-density idealization. We were able to determine and include the universality of dM in the so-called I-Love-Q relations

Application of Black-Hole Physics to Vortex Dynamics in Superfluids

This thesis is concerned with the investigation of non-equilibrium dynamics and turbulence in two- and three-dimensional superfluids using an intrinsically non-perturbative holographic description in terms of field theories in higher-dimensional black-hole-anti-de Sitter spacetimes. We perform numerical real-time simulations of these systems on large numerical domains. In a first part, we study the kinematics of vortex dipoles in two dimensions. To this end, we introduce a high-precision track- ing routine to locate their cores. By matching to the vortex trajectories solutions of the dissipative Gross–Pitaevskii equation and of equations for the motion of point vortices, we quantify the strong dissipation of the superfluid, which in holography is related to the absorption of modes by the black hole. We conjecture holography to be applicable to vortex dynamics in films of superfluid helium and in oblate cold quantum gases. In a second part, we study for the first time vortex lines and rings in the three-dimensional holographic superfluid. We investigate their dynamics, and interactions, including the famous leapfrogging motion and scattering events of rings, as well as Kelvin-wave excitations of their cores. Further, we study the evolution of the superfluid starting from far-from-equilibrium initial conditions characterised by dense vortex tangles. We analyse the dynamics in terms of scaling behaviour in correlation functions and observe signatures of universal turbulent behaviour during different regimes of the evolution. This work constitutes the first ab initio study of the mentioned phenomena in a strongly dissipative three-dimensional superfluid