Abstract
The paper starts by giving a motivation for this research and justifying the considered stochastic diffusion models for cosmic microwave background (CMB) radiation studies. Then it derives the exact solution in terms of a series expansion to a hyperbolic diffusion equation on the unit sphere. The Cauchy problem with random initial conditions is studied. All assumptions are stated in terms of the angular power spectrum of the initial conditions. An approximation to the solution is given and analysed by finitely truncating the series expansion. The upper bounds for the convergence rates of the approximation errors are derived. Smoothness properties of the solution and its approximation are investigated. It is demonstrated that the sample Hölder continuity of these spherical fields is related to the decay of the angular power spectrum. Numerical studies of approximations to the solution and applications to CMB data are presented to illustrate the theoretical results.
Similar content being viewed by others
References
Adam, R., et al.: Planck 2015 results. I. Overview of products and scientific results. Astron. Astrophys. 594, A16 (2016)
Ade, P.A.R., et al.: Planck 2015 results. XVI. Isotropy and statistics of the CMB, Planck Collaboration. Astron. Astrophys. 594, A16 (2016)
Ali, Y.M., Zhang, L.C.: Relativistic heat conduction. Int. J. Heat Mass Transf. 48(12), 2397–2406 (2005)
Anh, V.V., Leonenko, N.N.: Spectral analysis of fractional kinetic equations with random data. J. Stat. Phys. 104(5–6), 1349–1387 (2001)
Anh, V.V., Broadbridge, P., Olenko, A., Wang, Y.G.: On approximation for fractional stochastic partial differential equations on the sphere. Stoch. Environ. Res. Risk Assess. 32, 2585–2603 (2018)
Applegate, J.H., Hogan, C.J., Scherrer, R.J.: Cosmological baryon diffusion and nucleosynthesis. Phys. Rev. D 35(4), 1151–1159 (1987)
Barrow, J.D., Scherrer, R.J.: Constraining density fluctuations with Big-Bang nucleosynthesis in the era of precision cosmology. Phys. Rev. D 98(4), art. 043534 (2018)
Berg, E.J.: Heaviside’s Operational Calculus. McGraw-Hill, New York (1936)
Birrell, N.D., Davies, P.C.W.: Quantum Fields in Curved Space. Cambridge University Press, London (1984)
Bjørken, J.D., Drell, S.D.: Relativistic Quantum Mechanics. McGraw-Hill, New York (1964)
Broadbridge, P., Zulkowski, P.: Dark energy states from quantization of boson fields in a universe with unstable modes. Rep. Math. Phys. 7(1), 27–40 (2006)
Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids. Oxford University Press, London (1959)
Cattaneo, C.R.: Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée. Comptes Rendus 247(4), 431–433 (1958)
Dodelson, S.: Modern Cosmology. Academic, New York (2003)
Dunkel, J., Hänggi, P.: Relativistic Brownian motion. Phys. Rep. 471, 1–73 (2009)
Fryer, D., Olenko, A., Li, M.: rcosmo: R Package for Analysis of Spherical, HEALPix and Cosmological Data (2019). arxiv:1907.05648 (2019). Accessed 27 Sept 2019
Fryer, D., Olenko, A., Li, M., Wang, Yu.: rcosmo: Cosmic Microwave Background Data Analysis. R package version 1.0.0. https://CRAN.R-project.org/package=rcosmo (2018). Accessed 27 Sept 2019
Gorski, K.M., Hivon, E., Banday, A.J., Wandelt, B.D., Hansen, F.K., Reinecke, M., Bartelmann, M.: HEALPix: a framework for high-resolution discretization and fast analysis of data distributed on the sphere. Astrophys. J. 622, 759–771 (2005)
Higgs, P.: My life as a boson: the story of \(\text{``}\)The Higgs\(\text{'' }\). Int. J. Mod. Phys. A 17(Suppl. 01), 86–88 (2002)
Hirata, C.: The Standard Model—Cosmology. Caltech lecture notes. www.tapir.caltech.edu/~chirata/ph217 (2017). Accessed 27 Sept 2019
Iocco, F., Mangano, G., Miele, G., Pisanti, O., Serpico, P.D.: Primordial nucleosynthesis: from precision cosmology to fundamental physics. Phys. Rep. 472(1–6), 1–76 (2009)
Ivanov, A.V., Leonenko, N.N.: Statistical Analysis of Random Fields. Kluwer Academic Publishers, Dordrecht (1989)
Kolesnik, A.D., Ratanov, N.: Telegraph Processes and Option Pricing. Springer, Heidelberg (2013)
Kozachenko, YuV, Kozachenko, L.F.: Modeling Gaussian isotropic random fields on a sphere. J. Math. Sci. 107, 3751–3757 (2001)
Kurki-Suonio, H., Jedamzik, K., Matthews, G.J.: Stochastic isocurvature baryon fluctuations, baryon diffusion, and primordial nucleosynthesis. Astrophys. J. 479, 31–39 (1997)
Lan, X., Xiao, Y.: Regularity properties of the solution to a stochastic heat equation driven by a fractional Gaussian noise on \({\mathbb{S}}^2\). J. Math. Anal. Appl. 476(1), 27–52 (2019)
Lang, A., Schwab, C.: Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochastic partial differential equations. Ann. Appl. Probab. 25, 3047–3094 (2015)
Leonenko, N.N.: Limit Theorems for Random Fields with Singular Spectrum. Kluwer Academic Publishers, Dordrecht (1999)
Marinucci, D., Peccati, G.: Random Fields on the Sphere. Representation, Limit Theorems and Cosmological Applications. Cambridge University Press, Cambridge (2011)
Marinucci, D., Peccati, G.: Mean-square continuity on homogeneous spaces of compact groups. Electron. Commun. Probab. 18, 1–10 (2013)
NASA/WMAP Science Team: What is the Universe Made of? https://wmap.gsfc.nasa.gov/universe/uni_matter.html. Accessed 27 Sept 2019
Schweber, S.: An Introduction to Relativistic Quantum Field Theory. Dover, New York (2005)
Thompson, P.A.: Compressible-Fluid Dynamics. McGraw-Hill, New York (1971)
Terasawa, N., Sato, K.: Neutron diffusion and nucleosynthesis in the inhomogeneous universe. Prog. Theor. Phys. 81(2), 254–259 (1989)
Weinberg, S.: Cosmology. Oxford University Press, Oxford (2008)
Wigner, E.P.: On unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40(1), 149–204 (1939)
Yadrenko, M.I.: Spectral Theory of Random Fields. Optimization Software, Inc., New York (1983)
Acknowledgements
This research was supported under the Australian Research Council’s Discovery Project DP160101366. N. Leonenko was supported in part by Cardiff Incoming Visiting Fellowship Scheme, International Collaboration Seedcorn Fund, Data Innovation URI Seedcorn Fund. We are also grateful for the use of data of the Planck/ESA Mission from the Planck Legacy Archive. The authors are also grateful to the referees for their careful reading of the paper and suggestions that helped to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Irene Giardina.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A: Diffusion Length of a Local Disturbance
Consider a density disturbance u of total mass Q originating at the origin. The well-known point source solution to linear diffusion in three space dimensions is given by
The density level set at some low significance value u is at
so
The level set reaches its maximum extent when \(\frac{dr}{dt}=0,\) implying \(t=\frac{(Q/u)^{2/3}}{8e\pi D},\) so the diffusion length is
For example, for a mass disturbance the size of a solar mass, and a neutron diffusion length of 0.3 light-year at the temperature of neutrino dissociation from weak nuclear interactions, estimated from [6] and [25], the marginal disturbance density u is around 1 solar mass per cubic light year. This is meant to have occurred at a time when the cosmological expansion factor a(t) was less than \(10^{-3},\) so after expansion to the current level, the equivalent marginal density would be less than one nucleon mass per cubic metre, around the current mean density of the universe.
Appendix B: Proofs
Proof of Theorem 1
By substituting (19) into Eq. (15) and using (18), we obtain
To find particular solutions of (31), we need to solve the ordinary differential equation
The initial conditions for this equation can be determined from (20) and (16) and they are
The characteristic equation of (32) is \(\frac{1}{c^{2}}z^{2}+\frac{1}{D}z+l(l+1)k^{2}=0, \) with the roots \(z_{1,2}=-{c^{2}}/(2D)\pm K_l. \) Therefore, the general solution of Eq. (32) is given by the formula:
where \(M_{1},M_{2}\) are some constants. From the initial conditions in (33) we obtain
Thus, the solution of the Cauchy problem (32)–(33) is given by
Returning now to (19), we obtain the solution of the Cauchy problem (15)–(16) in the form
Note that the multiplier of \(Q_{l}({\mathbf {x}})\) on the right-hand side of (34) equals
By substituting this expression into (34), we get
Finally, using \(K_l'\) and rewriting the Green function we obtain the statement of the theorem. \(\square \)
Proof of Theorem 2
The solution of the initial value problem (22)–(24) can be written as a spherical convolution of the Green function \(p(\theta ,\varphi ,t)\) from Sect. 4 and the random field \(T(\theta ,\varphi ),\) if the corresponding Laplace series converges in the Hilbert space \(L_{2}(\varOmega \times {\mathbb {S}}^2,\sin \theta d\theta d\varphi ).\)
Let the two functions \(f_1(\cdot )\) and \(f_2(\cdot )\) on the sphere \({\mathbb {S}}^2\) belong to the space \(L_{2}({\mathbb {S}}^2,\sin \theta d\theta d\varphi )\) and have the Fourier–Laplace coefficients
Recall (see, i.e., [15]) that their non-commutative spherical convolution is defined as the Laplace series
with the Fourier–Laplace coefficients given by
provided that the series (35) converges in the corresponding Hilbert space.
Thus, the random solution \(u(\theta ,\varphi ,t)\) of Eq. (22) with the initial values determined by (23) and (24) can be written as a spherical random field with the following Laplace series representation
provided that this series is convergent in the Hilbert space \(L_{2}(\varOmega \times {\mathbb {S}}^2,\sin \theta d\theta d\varphi ),\) where \(p_{t}=p(\theta ,\varphi ,t)\) is given by Theorem 1 and T is given by (23). The complex Gaussian random variables \(a_{lm}^{(t)}\) are given by
where \(a_{l0}^{(p_{t})}=Y_{l0}^{*}({\mathbf {0}})d_{l}(\theta ,\varphi ,t)\) and
It gives the first statement of the theorem.
By the addition formula for spherical harmonics (see, i.e., [30, p.66])
where \(P_l(\cdot )\) is the lth Legendre polynomial [see (8)], and \(\cos \varTheta \) is the angular distance between the points \((\theta ,\varphi )\) and \( (\theta ^{\prime },\varphi ^{\prime })\) on \({\mathbb {S}}^2.\)
Using (10) we obtain that the random field \(u(\theta ,\varphi ,t)\) is isotropic if and only if the covariance structure of the solution (25) can be written in the form
which gives the result in (29) provided the series (29) converges for every fixed t and \(t^{\prime },\) that is
Noting that \(\left| P_l(\cos \varTheta )\right| \le 1,\) only a finite number of terms \(A_{l}\) is non-zero, and there is a constant C such that \(\sup _{t\ge 0} |B(t)|<C,\) we obtain that condition (38) follows from (13). This condition on the angular spectrum \(C_{l}, l\ge 0,\) guarantees the convergence of the series (36) in the Hilbert space \(L_{2}(\varOmega \times {{\mathbb {S}}}^{2},\sin \theta d\theta d\varphi ).\)\(\square \)
Proof of Theorem 3
The approximation \(u_L(\theta ,\varphi ,t)\) is a centered Gaussian random field, i.e. \( {\mathbf {E}}u_L(\theta ,\varphi ,t)=0\) for all \(L\in {\mathbb {N}}, \theta \in [0,\pi ), \varphi \in [0,2\pi ),\) and \(t>0.\) Therefore,
Hence, for all \(L\in {\mathbb {N}}\) it holds
For \(l>\frac{\sqrt{D^{2}k^{2}+c^{2}}-Dk}{2Dk}\) it follows from (27) that \(A_{l}(t)\equiv 0.\) Therefore, by (39) and (40) we obtain
\(\square \)
Proof of Corollary 2
The statement (i) immediately follows from (30) and the estimate
Then, applying Chebyshev’s inequality, we get the upper bound in (ii).
Finally, (iii) follows from statement (ii) and the Borel–Cantelli lemma as
\(\square \)
Proof of Theorem 4
Let h belong to a bounded neighbourhood of the origin. It follows from (14), (26), (27), (28) and (37) that
We start by showing how to estimate the first summand in (41). By (27), for the case \(l=0\) we obtain
For \(l>0\) we will use the upper bound
By properties of \(\cosh (\cdot )\) and \(\sinh (\cdot )\) we get
Then, applying (27) and noting that only a finite number of \(A_l\) is non-vanished (namely, only if \(l\in \left[ 0, \frac{\sqrt{D^{2}k^{2}+c^{2}}-Dk}{2Dk}\right] \)) we obtain the following estimates
Now we estimate the second summand in (41) as
Using (40) and applying the inequalities \(|\cos (x)-\cos (y)|\le 2\left| \sin \left( \frac{x-y}{2}\right) \right| \le |x-y|\) and \(|\sin (x)-\sin (y)|\le |x-y|\) we obtain
Note that for all \(l\ge 0\) it holds
Applying the above estimates to (41) we obtain
which completes the proof. \(\square \)
Proof of Corollary 4
Note that \(u(\theta ,\varphi ,t)\) is a centered Gaussian random field and for any centered Gaussian random variable X it holds
Applying this result to the statement of Theorem 4 we obtain
\(\square \)
Proof of Corollary 5
By (29) it holds
Applying the next property of Legendre polynomials (see, for example, [27, p.16]) \(|1 - P_l (x)| \le 2|1 - x|^\gamma (l(l + 1))^\gamma , \gamma \in [0,1],\) and the upper bounds (40), we obtain that uniformly in \(t\ge 0\)
\(\square \)
Appendix C: Sensitivity to Parameters
To further understand the impact of time and the model parameters on the difference of the mean \({L_2(\varOmega \times {\mathbb S}^{2})}\)-errors and their upper bound (30) we produced 3D-plots showing the difference as a function of the truncation degree L and each parameter provided that other parameters are fixed. These plots are displayed in Figs. 12, 13, 14, and 15.
In all cases the difference between the error and its upper bound asymptotically vanish when L increases. Figure 12 demonstrates that the difference is a decreasing function of time \(t',\) which is expected as the series representation (25) of the solutions \(u(\theta ,\varphi ,t')\) has the multiplication factor \(\exp (-t')=\exp (-{c^{2}t}/(2D))\) exponentially decaying in time. The differences are extreme at the origin and decrease when time or the parameter c increases, see Fig. 13. For the parameter D the situation depicted in Fig. 14 is opposite and the difference is increasing in D which is expected as the multiplication factor is exponentially decaying in \(D^{-1}.\) Finally, Fig. 15 suggests that the parameter k seems have no substantial impact on the difference.
Rights and permissions
About this article
Cite this article
Broadbridge, P., Kolesnik, A.D., Leonenko, N. et al. Random Spherical Hyperbolic Diffusion. J Stat Phys 177, 889–916 (2019). https://doi.org/10.1007/s10955-019-02395-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-019-02395-0
Keywords
- Cosmic microwave background
- Stochastic partial differential equations
- Hyperbolic diffusion equation
- Spherical random field
- Hölder continuity
- Approximation errors