Stochastic diffusion within expanding space-time

The NASA mission WMAP and the ESA (European Space Agency) mission Planck have been instrumental in collecting highly accurate cosmological data, resulting in a precise map depicting the distribution of Cosmic Microwave Background Radiation (CMB) [4, 5]. It is expected that new experiments, in particular, within CMB-S4 Collaboration and ESA’s Euclid mission, will provide measurements of the CMB at unprecedented precision.

The CMB spectrum indicates that since the last scattering around 380,000 years after the big bang, the universe has been transparent to electromagnetic radiation. However the universe is not transparent to charged cosmic ray particles. They are deviated by magnetic fields as they pass close to galaxies. Recent estimates of the number of galaxies in the observable universe range from $2\times 10^{12}$ to $6\times 10^{12}$. In any angular aperture of observation, there will be part of at least one galaxy. Extragalactic cosmic ray particles reach the earth in a small number of showers each year. They are distinguished from local galactic cosmic rays by their high particle energy, greater than $5\times 10^{18}eV$. This corresponds to charged particles typically arriving at more than half the speed of light. They have likely been travelling for a very long time in cosmic terms, during which they would have been deviated by a number of galaxies. This results in a long-term diffusive redistribution of matter throughout the universe.

This dynamical process is very complicated. Effective diffusion occurs partly by scattering due to magnetic fields [22] and also by motion of the magnetic field lines themselves [25]. It has also been argued that the temperature gradients formed from galaxies can effectively repel particles, enough to avoid gravitational capture [37].

Due to the long distances between galaxy clusters, scattering is intermittent, reasonably described by a fractional $\alpha$-stable Lévy distribution with tail probabilities of order $|x|^{-\alpha}$ for exceeding a large displacement, and the distribution due to a local disturbance broadening asymptotically in proportion to $t^{1/\alpha}.$ Such a distribution can result from a fractional super-diffusion that is of order $\alpha<2$. Data from intra-galactic cosmic rays evidence values of $\alpha$ less than 0.5 [15].

The cosmological missions have yielded high resolution maps of CMB. The necessity of modelling and analysing them have recently attracted increasing attention to the theory of spherical random fields. From the mathematical point of view and for modelling purposes, a map depicting the distribution of CMB can be regarded as a single realization of a random field on a sphere of a large radius. The sphere plays a role of the underlying space and expands in time. For the Dark-energy-dominated era, the expansion factor has the exponential form [11]. This expansion impacts the stochastic diffusion and should be incorporated in a model for the evolution of CMB. Contrary to the other models in the literature, this paper takes into consideration the expansion of space-time, which leads to a stochastic diffusion equation with coefficients that are non-constant and evolve with the expansion factor.

We refer to the monograph [31] for the systematic exposition of the main results of the theory of spherical random fields. The paper [26] studied isotropic random fields on high-dimensional spheres $\mathbb{S}^{n}$ and established connections between the smoothness of the covariance kernel and the decay of their angular power spectrum, derived conditions for almost sure sample continuity and sample differentiability of spherical random fields, and obtained sufficient conditions for their $L_{2}$ continuity in terms of the decay of the angular power spectrum.

Another important direction of the modern theory of random fields is the exploration of extremes of random fields including fields given on manifolds, see the classical results in [38], [17], [35] and their modern generalisations. The monograph [14] examined extremes of sub-Gaussian fields while the expected Euler characteristic method developed by Adler and Taylor was demonstrated in [6]. The publication [16] provided the asymptotics of excursion probabilities for both smooth and non-smooth spherical Gaussian fields. Some inequalities for excursion probabilities for spherical sub-Gaussian random fields were obtained in [36]. Another approach utilized limit theorems for sojourn measures, see [28] and [30].

Stochastic partial differential equations (SPDE) are the main tool to model the evolution of spherical random fields over time. They have been extensively studied, see, for example, [19], [26], [34] and the references therein. Several models were recently presented in [7, 12, 13, 27] that employed stochastic hyperbolic diffusion equations and modelled various types of evolution of spherical random fields.

This article integrates the aforementioned approaches and extend them to the context of SPDEs within an expanding space-time. The equations studied in this paper differ from the mentioned SPDEs and are given as a hyperbolic diffusion on the expanding sphere. Motivated by cosmological applications, the exponential expansion is used in the considered model, which leads to a stochastic diffusion equation with non-constant coefficients.

The article also conducts the numerical analysis of the solution of the considered model. The numerical analysis section investigates the evolution of the solution of the studied SPDE over space-time, the structure of its space-time dependencies, and its extremes. The CMB intensity map from the mission Planck and its spectrum are used as initial conditions. The numerical analysis confirms and visualizes the obtained theoretical results.

The main novelties of the paper include:

• •

  The consideration of diffusion within an expanding space-time framework;

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  The examination of equations with non-constant coefficients dependent on the expansion factor values;

• •

  An exploration of both the local and asymptotic properties of the solutions and their respective approximations;

• •

  An analysis of excursion probabilities associated with the solutions and their approximations.

The paper is structured as follows: Section 2 provides the main definitions and notations. Section 3 presents the diffusion model within an expanding time-space universe. Then, this section investigates the initial-value problem for this diffusion equation and explores the properties of the derived non-random solutions. Section 3 is dedicated to the examination of the equation with random initial conditions. The solution to the equation and its associated covariance function are derived. Section 5 investigates properties of stochastic solutions and their corresponding approximations. Sections 6 studies excursion probabilities related to the solutions and their approximations. Finally, Section 7 presents simulation studies that illustrate the properties of the solutions and the obtained results.

This section reviews the main definitions and notations used in this paper and provides required background knowledge from the theory of random fields.

$C$ with subindices represents a generic finite positive constant. The values of constants are not necessarily the same in each appearance and may be changed depending on the expression. $||\cdot||$ is the Euclidean norm in $\mathbb{R}^{3},$ $Leb(\cdot)$ stands for the Lebesgue measure on the unit sphere $\mathbb{S}^{2}=\{\textbf{{x}}:\ ||\textbf{{x}}||=1,\ \textbf{{x}}\in\mathbb{R}^{3}\}.$

The notation $\textit{{x}}=(x_{1},x_{2},x_{3})\in\mathbb{S}^{2}$ is used to define Euclidean coordinates of points, while the notation $(\theta,\varphi),\ 0\leq\theta\leq\pi,\ 0\leq\varphi<2\pi,$ is used for the corresponding spherical coordinates. They are related by the following transformations

In what follows, $\Theta$ denotes the angular distance between two points with spherical coordinates $(\theta,\varphi)$ and $(\theta^{\prime},\varphi^{\prime})$ on the unit sphere $\mathbb{S}^{2}.$

By $L^{2}(\mathbb{S}^{2})$ we denote the Hilbert space of square integrable functions on $\mathbb{S}^{2}$ with the following canonical inner product [9, Page 8]

where $g^{*}(\cdot)$ denotes a complex conjugate of $g(\cdot),$ and the induced norm

The spherical harmonics $Y_{lm}(\theta,\varphi),\ l=0,1,...,m=-l,..l,$ are given as

where

$P_{l}^{m}(\cdot)$ are the associated Legendre polynomials with the indices $l$ and $m,$ and $P_{l}(\cdot)$ is the $l$th Legendre polynomial

The spherical harmonics $Y_{lm}(\theta,\varphi),\ l=0,1,...,m=-l,..l,$ form an orthogonal basis in the Hilbert space $L^{2}(\mathbb{S}^{2}),$ i.e.

where $\delta_{ll^{\prime}}$ is the Kronecker delta function.

The addition formula for the spherical harmonics states that

For any $f\in L^{2}(\mathbb{S}^{2})$ it holds

where $f_{lm}\in\mathbb{C},\ l=0,1,...,\ m=-l,...,l.$

A spherical random field $T(\theta,\varphi),$ is a collection of random variables given on a common complete probability space $\{\Omega,\mathfrak{F},P\}$ and indexed by parameters $\theta,\varphi.$ In this paper, we consider real-valued spherical random fields that are continuous and twice-differentiable with probability $1.$

By $L^{2}(\Omega\times\mathbb{S}^{2})$ we denote the Hilbert space of spherical random fields that have a finite norm

A spherical random field $T(\theta,\varphi)$ is called isotropic if its finite-dimensional distributions are invariant with respect to rotation transformations, i.e. if

for any $\textit{{x}}_{i}\in\mathbb{S}^{2},$ $a_{i}\in\mathbb{R},\ i=1,2,...k,\ k\in\mathbb{N}$ and any rotation matrix $A.$

An isotropic spherical random field $T(\theta,\varphi)\in L^{2}(\Omega\times\mathbb{S}^{2})$ allows a representation as the following Laplace series [26]

where the convergence is in the space $L^{2}(\Omega\times\mathbb{S}^{2}),$ i.e..

The random variables $a_{lm},\ l=0,1,...,\ m=-l,...,l,$ are given by the next stochastic integrals defined in the mean-square sense

Note that a spherical random field $T(\theta,\varphi)$ takes a constant random value, i.e. $T(\theta,\varphi)=\xi,\ \xi\in L^{2}(\Omega),\forall\theta,\varphi,$ with probability 1, if and only if $a_{lm}=0$ a.s. for $l=1,2,...$ Indeed, by the properties of spherical harmonics, for $T(\theta,\varphi)=\xi$ the integrals in (3) equal to $0$ a.s. for $l=1,2,...$ The inverse statement is trivial due to (2).

If $T(\theta,\varphi)\in L^{2}(\Omega\times\mathbb{S}^{2})$ is a centered real-valued Gaussian random field, then it allows the representation (2), where $a_{lm},\ l=0,1,...,\ m=-l,...,l,$ are Gaussian random variables such that

The sequence $\{C_{l},l=0,1,...\}$ is called the angular power spectrum of the isotropic random field $T(\theta,\varphi).$ The series (2) converges in the $L_{2}(\Omega\times\mathbb{S}^{2})$ sense if it holds true [26, Section 2] that

We assume that condition (4) remains valid throughout all subsequent sections of the paper.

The Bessel function of the first kind $J_{\nu}(x),\ x\geq 0,\ \nu\in\mathbb{R},$ is given by the following series [3, 9.1.10]

It has a finite value at the origin if $\nu\geq 0$ and a singularity if $\nu<0.$

For a noninteger $\nu,$ the Bessel function of the second kind $Y_{\nu}(x),\ x\geq 0,$ is given as [3, 9.1.2]

The Bessel function of the second kind $Y_{n}(x),\ x\geq 0,$ of a positive integer order $n\in\mathbb{N}$ is obtained as the limit $Y_{n}(x)=\lim\limits_{\nu\to n}Y_{\nu}(x),$ see [3, 9.1.11].

This section provides a justification for the model and obtain certain properties of solutions to the non-random version of the considered diffusion equations.

The universe is observed to have spatial cross sections with zero curvature. The Friedmann–Lemaître–
Robertson–Walker metric (FLRW) on the sphere of radius $r$ is

in the spherical space coordinates $(\theta,\varphi)$ and the cosmic time coordinate $t,$ where the term $a(\cdot)$ is the expansion factor. Note that for the Dark-energy-dominated era [23, 24], the expansion factor has the exponential form of the maximally symmetric de Sitter universe, $a(t)=e^{\sqrt{\frac{\Lambda}{3}}ct}$ [11, 42].

The spherical diffusion is given by the following equation

where $\nabla_{\mu}$ is the covariant derivative operator and $g^{\mu\nu}$ are the elements of the contravariant metric tensor where the indices take values from 0 to 2. We also impose the following initial conditions

The covariant derivatives do not commute when they act on vectors, see [10]. However, the Laplace-Beltrami operator on the sphere can be expanded unambiguously in terms of partial derivatives as

where $g$ is the covariant metric tensor. For the above defined FLRW metric, $g$ is the diagonal matrix with the elements $c^{2},-r^{2}a^{2}(t),-r^{2}a^{2}(t)sin^{2}\theta,$ and the contravariant metric tensor is the inverse of $g.$

Changing to a conformal time coordinate

the FLRW metric becomes

where $e(\eta):=\frac{\eta_{\infty}}{\eta_{\infty}-\eta}=a(t),\ \eta_{\infty}=\frac{1}{c}\sqrt{\frac{3}{\Lambda}},$ and the covariant metric tensor $g$ becomes the diagonal matrix with the elements $c^{2}e^{2}(\eta),-r^{2}e^{2}(\eta),-r^{2}e^{2}(\eta)\sin^{2}\theta.$

Using (7) and the above expression of the covariant metric tensor in terms of the conformal time, the equation (5) can be written in the following form

As for $t=0$ it holds that $\eta=0$ and $\frac{d\eta}{dt}\big{|}_{t=0}=1,$ the initial conditions (6) become

The equation (8) is different compared to the models studied in the literature, see [7, 12, 13, 27] and the references therein. Namely, for the underlying FLRW space-time metric the coefficient of the term $\frac{\partial\widetilde{u}}{\partial\eta}$ in (8) is not constant and depends on the evolution of the expansion factor $e(\eta).$

The following results derive some properties of the functions $F_{l}(\eta),$ which will be used later.