arXiv subjects

Ashley Wilkins

Publications and source records attributed to Ashley Wilkins.

7 recordsLinked to original sources

arXiv2023

Spectators no more! How even unimportant fields can ruin your Primordial Black Hole model

In this work we terminate inflation during a phase of Constant Roll by means of a waterfall field coupled to the inflaton and a spectator field. The presence of a spectator field means that inflation does not end at a single point, $ϕ_e$, but instead has some uncertainty resulting in a stochastic end of inflation. We find that even modestly coupled spectator fields can drastically increase the abundance of Primordial Black Holes (PBHs) formed by many orders of magnitude. The power spectrum created by the inflaton can be as little as $10^{-4}$ during a phase of Ultra Slow-Roll and still form a cosmologically relevant number of PBHs. We conclude that the presence of spectator fields, which very generically will alter the end of inflation, is an effect that cannot be ignored in realistic models of PBH formation.

astro-ph.CO↗

arXiv2023

Stochastic Processes in Mesoscale Physics and the Early Universe

This thesis is dedicated to the study of stochastic processes; non-deterministic physical phenomena that can be well described by classical physics. The stochastic processes we are interested in are akin to Brownian Motion and can be described by an overdamped Langevin equation comprised of a deterministic drift term and a random noise term. In Part I we examine stochastic processes in the Mesoscale. For us this means that the Langevin equation is driven by thermal noise, with amplitude proportional to the temperature $T$ and there exists a genuine equilibrium thermal state. We apply a technique known as the Functional Renormalisation Group (FRG) which allows us to coarse-grain in temporal scales. We describe how to obtain effective equations of motion for the 1- and 2-point functions of a particle evolving in highly non-trivial potentials and verify their accuracy by comparison to direct numerical simulations. In this way we outline a novel procedure for describing the behaviour of stochastic processes without having to resort to time consuming numerical simulations. In Part II we turn to the Early Universe and in particular examine stochastic processes occurring during a period of accelerated expansion known as inflation. This inflationary period is driven by a scalar field called the inflaton which also obeys a Langevin equation in the Stochastic Inflation formalism. We use this to study the formation of Primordial Black Holes during a period of Ultra Slow-Roll. We finish this thesis by applying the techniques developed in Part I to a spectator field during inflation. FRG techniques can compute cosmologically relevant observables such as the power spectrum and spectral tilt. We also extend the FRG formalism to solve first-passage time problems and verify that it gives the correct prediction for the average time taken for a field (or particle) to overcome a barrier in the potential.

astro-ph.CO↗

arXiv2023

Computing First-Passage Times with the Functional Renormalisation Group

We use Functional Renormalisation Group (FRG) techniques to analyse the behaviour of a spectator field, $σ$, during inflation that obeys an overdamped Langevin equation. We briefly review how a derivative expansion of the FRG can be used to obtain Effective Equations of Motion (EEOM) for the one- and two-point function and derive the EEOM for the three-point function. We show how to compute quantities like the amplitude of the power spectrum and the spectral tilt from the FRG. We do this explicitly for a potential with multiple barriers and show that in general many different potentials will give identical predictions for the spectral tilt suggesting that observations are agnostic to localised features in the potential. Finally we use the EEOM to compute first-passage time (FPT) quantities for the spectator field. The EEOM for the one- and two-point function are enough to accurately predict the average time taken $\left\langle \mathcal{N}\right\rangle$ to travel between two field values with a barrier in between and the variation in that time $δ\mathcal{N}^2$. It can also accurately resolve the full PDF for time taken $ρ(\mathcal{N})$, predicting the correct exponential tail. This suggests that an extension of this analysis to the inflaton can correctly capture the exponential tail that is expected in models producing Primordial Black Holes.

astro-ph.CO↗

arXiv2022

Coarse-graining in time with the Functional Renormalisation Group: Relaxation in Brownian Motion

We apply the functional Renormalisation Group (fRG) to study relaxation in a stochastic process governed by an overdamped Langevin equation with one degree of freedom, exploiting the connection with supersymmetric quantum mechanics in imaginary time. After reviewing the functional integral formulation of the system and its underlying symmetries, including the resulting Ward-Takahashi identities for arbitrary initial conditions, we compute the effective action $Γ$ from the fRG, approximated in terms of the leading and subleading terms in the gradient expansion: the Local Potential Approximation and Wavefunction Renormalisation respectively. This is achieved by coarse-graining the thermal fluctuations in time resulting in e.g. an effective potential incorporating fluctuations at all timescales. We then use the resulting effective equations of motion to describe the decay of the covariance, and the relaxation of the average position and variance towards their equilibrium values at different temperatures. We use as examples a simple polynomial potential, an unequal Lennard-Jones type potential and a more complex potential with multiple trapping wells and barriers. We find that these are all handled well, with the accuracy of the approximations improving as the relaxation's spectral representation shifts to lower eigenvalues, in line with expectations about the validity of the gradient expansion. The spectral representation's range also correlates with temperature, leading to the conclusion that the gradient expansion works better for higher temperatures than lower ones. This work demonstrates the ability of the fRG to expedite the computation of statistical objects in otherwise long-timescale simulations, acting as a first step to more complicated systems.

cond-mat.stat-mech↗

arXiv2021

Inflation is always semi-classical: Diffusion domination overproduces Primordial Black Holes

We use the Hamilton-Jacobi (H-J) formulation of stochastic inflation to describe the evolution of the inflaton during a period of Ultra-Slow Roll (USR), taking into account the field's velocity and its gravitational backreaction. We demonstrate how this formalism allows one to modify existing slow-roll (SR) formulae to be fully valid outside of the SR regime. We then compute the mass fraction, $β$, of Primordial Black Holes (PBHs) formed by a plateau in the inflationary potential. By fully accounting for the inflaton velocity as it enters the plateau, we find that PBHs are generically overproduced before the inflaton's velocity reaches zero, ruling out a period of free diffusion or even stochastic noise domination on the inflaton dynamics. We also examine a local inflection point and similarly conclude that PBHs are overproduced before entering a quantum diffusion dominated regime. We therefore surmise that the evolution of the inflaton is always predominantly classical with diffusion effects always subdominant. Both the plateau and the inflection point are characterized by a very sharp transition between the under- and over-production regimes. This can be seen either as severe fine-tunning on the inflationary production of PBHs, or as a very strong link between the fraction $β$ and the shape of the potential and the plateau's extent.

astro-ph.CO↗

arXiv2021

Functional Renormalisation Group for Brownian Motion I: The Effective Equations of Motion

We use the functional Renormalisation Group (fRG) to describe the in and out of equilibrium dynamics of stochastic processes, governed by an overdamped Langevin equation. Exploiting the connection between Langevin dynamics and supersymmetric quantum mechanics in imaginary time, we write down renormalisation flow equations for the effective action, approximated in terms of the Local Potential Approximation and Wavefunction Renormalisation. We derive \textit{effective equations of motion} (EEOM) from the effective action (EA) $Γ$ for the average position $\left\langle x\right\rangle$, variance $\langle \left(x- \langle x \rangle\right)^2\rangle$ and covariance. The fRG flow equations outlined here provide a concrete way to compute the EA and thus solve the derived EEOM. The obtained effective potential should determine directly the exact equilibrium statistics, name the position, the variance, as well as all higher order cumulants of the equilibrium Boltzmann distribution. This first paper of a two part series is mostly concerned with setting up the necessary formalism while in part two we will numerically solve the equations derived her and assess their validity both in and out of equilibrium.

cond-mat.stat-mech↗

arXiv2015

Quantum estimation via parametric amplification in circuit QED arrays

We propose a scheme for quantum estimation by means of parametric amplification in circuit Quantum Electrodynamics. The modulation of a SQUID interrupting a superconducting waveguide transforms an initial thermal two-mode squeezed state in such a way that the new state is sensitive to the features of the parametric amplifier. We find the optimal initial parameters which maximize the Quantum Fisher Information. In order to achieve a large number of independent measurements we propose to use an array of non-interacting resonators. We show that the combination of both large QFI and large number of measurements enables -in principle- the use of this setup for Quantum Metrology applications.

quant-ph↗