arXiv subjects

Frank Steiner

Publications and source records attributed to Frank Steiner.

15 recordsLinked to original sources

arXiv2024

Betti Functionals as a Probe for Cosmic Topology

The question of the global topology of the Universe (cosmic topology) is still open. In the $Λ$CDM concordance model it is assumed that the space of the Universe possesses the trivial topology of $\mathbb{R}^3$ and thus that the Universe has an infinite volume. As an alternative, we study in this paper one of the simplest non-trivial topologies given by a cubic 3-torus describing a universe with a finite volume. To probe cosmic topology, we analyse certain structure properties in the cosmic microwave background (CMB) using Betti Functionals and the Euler Characteristic evaluated on excursions sets, which possess a simple geometrical interpretation. Since the CMB temperature fluctuations $δT$ are observed on the sphere $\mathbb{S}^2$ surrounding the observer, there are only three Betti functionals $β_k(ν)$, $k=1,2,3$. Here $ν=δT/σ_0$ denotes the temperature threshold normalized by the standard deviation $σ_0$ of $δT$. Analytic approximations of the Gaussian expectations for the Betti functionals and an exact formula for the Euler characteristic are given. It is shown that the amplitudes of $β_0(ν)$ and $β_1(ν)$ decrease with increasing volume $V=L^3$ of the cubic 3-torus universe. Since the computation of the $β_k$'s from observational sky maps is hindered due to the presence of masks, we suggest a method yielding lower and upper bounds for them and apply it to four Planck 2018 sky maps. It is found that the $β_k$'s of the Planck maps lie between those of the torus universes with side-lengths $L=2.0$ and $L=3.0$ in units of the Hubble length and above the infinite $Λ$CDM case. These results give a further hint that the Universe has a non-trivial topology.

astro-ph.CO↗

arXiv2021

The variance of the CMB temperature gradient: a new signature of a multiply connected Universe

In this work we investigate the standard deviation of the Cosmic Microwave Background (CMB) temperature gradient field as a signature for a multiply connected nature of the Universe. CMB simulations of a spatially infinite universe model within the paradigm of the standard cosmological model present non-zero two-point correlations at any angular scale. This is in contradiction with the extreme suppression of correlations at scales above $60^{\circ}$ in the observed CMB maps. Universe models with spatially multiply connected topology contain typically a discrete spectrum of the Laplacian with a specific wave-length cut-off and thus lead to a suppression of the correlations at large angular scales, as observed in the CMB (in general there can be also an additional continuous spectrum). Among the simplest examples are 3-dimensional tori which possess only a discrete spectrum. To date, the universe models with non-trivial topology such as the toroidal space are the only models that possess a two-point correlation function showing a similar behaviour as the one derived from the observed Planck CMB maps. In this work it is shown that the normalized standard deviation of the CMB temperature gradient field does hierarchically detect the change in size of the cubic 3-torus, if the volume of the Universe is smaller than $\simeq 2.5 \cdot 10^3$ Gpc$^3$. It is also shown that the variance of the temperature gradient of the Planck maps is consistent with the median value of simulations within the standard cosmological model. All flat tori are globally homogeneous, but are globally anisotropic. However, this study also presents a test showing a level of homogeneity and isotropy of all the CMB map ensembles for the different torus sizes considered that are nearly at the same weak level of anisotropy revealed by the CMB in the standard cosmological model.

astro-ph.CO↗

arXiv2017

Model-independent analyses of non-Gaussianity in Planck CMB maps using Minkowski Functionals

Despite the wealth of $Planck$ results, there are difficulties in disentangling the primordial non-Gaussianity of the Cosmic Microwave Background (CMB) from the secondary and the foreground non-Gaussianity (NG). For each of these forms of NG the lack of complete data introduces model-dependencies. Aiming at detecting the NGs of the CMB temperature anisotropy $δT$, while paying particular attention to a model-independent quantification of NGs, our analysis is based upon statistical and morphological univariate descriptors, respectively: the probability density function $P(δT)$, related to ${\mathrm v}_{0}$, the first Minkowski Functional (MF), and the two other MFs, ${\mathrm v}_{1}$ and ${\mathrm v}_{2}$. From their analytical Gaussian predictions we build the discrepancy functions $Δ_{k}$ ($k=P,0,1,2$) which are applied to an ensemble of $10^{5}$ CMB realization maps of the $Λ$CDM model and to the $Planck$ CMB maps. In our analysis we use general Hermite expansions of the $Δ_{k}$ up to the $12^{th}$ order, where the coefficients are explicitly given in terms of cumulants. Assuming hierarchical ordering of the cumulants, we obtain the perturbative expansions generalizing the $2^{nd}$ order expansions of Matsubara to arbitrary order in the standard deviation $σ_0$ for $P(δT)$ and ${\mathrm v}_0$, where the perturbative expansion coefficients are explicitly given in terms of complete Bell polynomials. The comparison of the Hermite expansions and the perturbative expansions is performed for the $Λ$CDM map sample and the $Planck$ data. We confirm the weak level of non-Gaussianity ($1$-$2$)$σ$ of the foreground corrected masked $Planck$ $2015$ maps.

astro-ph.CO↗

arXiv2016

Do Black Holes Exist in a Finite Universe Having the Topology of a Flat 3-Torus?

Based on perturbation theory, we present the exact first-order solution to the Einstein equations for the exterior static gravitational field of an isolated non-rotating star in a spatially finite universe having the topology of a flat 3-torus. Since the method of images leads to a divergent Poincare' series, one needs a regularization which we achieve by using the Appell respectively the Epstein zeta function. The solution depends on a new positive constant which is completely fixed by the mass of the star and the spatial volume of the universe. The physical interpretation is that a stable or metastable equilibrium requires a topological dark energy which fills the whole universe with positive energy density and negative pressure. The properties of the gravitational field are discussed in detail. In particular, its anisotropy is made explicit by deriving an exact multipole expansion which shows that in this case Birkhoff's theorem does not hold. While the monopole describes the Newtonian potential, there is no dipole but always a non-vanishing quadrupole which leads to a repulsive force experienced by a planet at rest. Finally, we put forward the conjecture that black holes exist in a toroidal universe and that their gravitational field is in the weak-field limit well approximated by the first-order field.

gr-qc↗

arXiv2014

Zero modes of quantum graph Laplacians and an index theorem

We study zero modes of Laplacians on compact and non-compact metric graphs with general self-adjoint vertex conditions. In the first part of the paper the number of zero modes is expressed in terms of the trace of a unitary matrix $\mathfrak{S}$ that encodes the vertex conditions imposed on functions in the domain of the Laplacian. In the second part a Dirac operator is defined whose square is related to the Laplacian. In order to accommodate Laplacians with negative eigenvalues it is necessary to define the Dirac operator on a suitable Kre\uın space. We demonstrate that an arbitrary, self-adjoint quantum graph Laplacian admits a factorisation into momentum-like operators in a Kre\uın-space setting. As a consequence, we establish an index theorem for the associated Dirac operator and prove that the zero-mode contribution in the trace formula for the Laplacian can be expressed in terms of the index of the Dirac operator.

math-ph↗

arXiv2011

The Berry-Keating operator on $L^2(\rz_>,\ud x)$ and on compact quantum graphs with general self-adjoint realizations

The Berry-Keating operator $H_{\mathrm{BK}}:= -\ui\hbar(x\frac{\ud\phantom{x}}{\ud x}+{1/2})$ [M. V. Berry and J. P. Keating, SIAM Rev. 41 (1999) 236] governing the Schrödinger dynamics is discussed in the Hilbert space $L^2(\rz_>,\ud x)$ and on compact quantum graphs. It is proved that the spectrum of $H_{\mathrm{BK}}$ defined on $L^2(\rz_>,\ud x)$ is purely continuous and thus this quantization of $H_{\mathrm{BK}}$ cannot yield the hypothetical Hilbert-Polya operator possessing as eigenvalues the nontrivial zeros of the Riemann zeta function. A complete classification of all self-adjoint extensions of $H_{\mathrm{BK}}$ acting on compact quantum graphs is given together with the corresponding secular equation in form of a determinant whose zeros determine the discrete spectrum of $H_{\mathrm{BK}}$. In addition, an exact trace formula and the Weyl asymptotics of the eigenvalue counting function are derived. Furthermore, we introduce the "squared" Berry-Keating operator $H_{\mathrm{BK}}^2:= -x^2\frac{\ud^2\phantom{x}}{\ud x^2}-2x\frac{\ud\phantom{x}}{\ud x}-{1/4}$ which is a special case of the Black-Scholes operator used in financial theory of option pricing. Again, all self-adjoint extensions, the corresponding secular equation, the trace formula and the Weyl asymptotics are derived for $H_{\mathrm{BK}}^2$ on compact quantum graphs. While the spectra of both $H_{\mathrm{BK}}$ and $H_{\mathrm{BK}}^2$ on any compact quantum graph are discrete, their Weyl asymptotics demonstrate that neither $H_{\mathrm{BK}}$ nor $H_{\mathrm{BK}}^2$ can yield as eigenvalues the nontrivial Riemann zeros. Some simple examples are worked out in detail.

math-ph↗

arXiv2011

A new proof of the Voronoï summation formula

We present a short alternative proof of the Voronoï summation formula which plays an important role in Dirichlet's divisor problem and has recently found an application in physics as a trace formula for a Schrödinger operator on a non-compact quantum graph \mathfrak{G} [S. Egger né Endres and F. Steiner, J. Phys. A: Math. Theor. 44 (2011) 185202 (44pp)]. As a byproduct we give a new proof of a non-trivial identity for a particular Lambert series which involves the divisor function d(n) and is identical with the trace of the Euclidean wave group of the Laplacian on the infinite graph \mathfrak{G}.

math-ph↗

arXiv2011

An exact trace formula and zeta functions for an infinite quantum graph with a non-standard Weyl asymptotics

We study a quantum Hamiltonian that is given by the (negative) Laplacian and an infinite chain of $δ$-like potentials with strength $κ>0$ on the half line $\rz_{\geq0}$ and which is equivalent to a one-parameter family of Laplacians on an infinite metric graph. This graph consists of an infinite chain of edges with the metric structure defined by assigning an interval $I_n=[0,l_n]$, $n\in\nz$, to each edge with length $l_n=\fracπ{n}$. We show that the one-parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each $κ>0$ and prove that the Dirichlet Laplacian is the limit of the one-parameter family in the strong resolvent sense. The spectrum of the resulting Dirichlet quantum graph is also purely discrete. The eigenvalues are given by $λ_n=n^2$, $n\in\nz$, with multiplicities $d(n)$, where $d(n)$ denotes the divisor function. We thus can relate the spectral problem of this infinite quantum graph to Dirichlet's famous divisor problem and infer the non-standard Weyl asymptotics $\mathcal{N}(λ)=\frac{\sqrtλ}{2}\lnλ+\Or(\sqrtλ)$ for the eigenvalue counting function. Based on an exact trace formula, the Voronoï summation formula, we derive explicit formulae for the trace of the wave group, the heat kernel, the resolvent and for various spectral zeta functions. These results enable us to establish a well-defined (renormalized) secular equation and a Selberg-like zeta function defined in terms of the classical periodic orbits of the graph, for which we derive an exact functional equation and prove that the analogue of the Riemann hypothesis is true.

math-ph↗

arXiv2010

Ellipticity of Structures in CMB Sky Maps

We study the ellipticity of contour lines in the sky maps of the cosmic microwave background (CMB) as well as other measures of elongation. The sensitivity of the elongation on the resolution of the CMB maps which depends on the pixelization and the beam profile of the detector, is investigated. It is shown that the current experimental accuracy does not allow to discriminate between cosmological models which differ in curvature by Delta Omega_tot=0.05. Analytical expressions are given for the case that the statistical properties of the CMB are those of two-dimensional Gaussian random fields.

astro-ph.CO↗

arXiv2010

Quantum Cosmology Close to the Classical Big Bang Singularity and in the Semiclassical Limit

We investigate a cosmological model whose energy content is described by a Chaplygin gas represented by a scalar field $ϕ$ with an associated potential producing a big bang singularity such that for vanishing scale factor, $a\to 0$, one has $|ϕ|\to \infty$. The classical version of the model is discussed in detail, however, our main interest lies in its quantization. Upon quantization of this model in the Schrödinger picture, we get the Wheeler--DeWitt equation which can be solved exactly in the two limits $a\to 0$ and $a\to\infty$, respectively. Employing the DeWitt criterium that the wave function should vanish at the classical singularity in order to avoid the big bang, we show that a solution to the Wheeler-DeWitt equation fulfilling this condition can indeed be found. In addition to DeWitt's initial condition at the big bang, we postulate an asymptotic condition to be imposed on the wave function which guarantees that the quantum wave function is strongly peaked at the classical field configurations in the semiclassical limit. We also investigate a universe filled with dust (describing baryonic and dark matter) and show that in this case there exists an exact solution to the Wheeler-DeWitt equation which describes the evolution of the universe during its whole history. Also for this model one can construct a wave packet which avoids the big bang singularity at small scale factors and is strongly peaked at the classical field in the far future of the universe. Finally, we discuss the significance of our treatment of the Wheeler-DeWitt equation and mention some open problems.

gr-qc↗

arXiv2008

Do we Live in a "Small Universe"?

We compute the effects of a compact flat universe on the angular correlation function, the angular power spectrum, the circles-in-the-sky signature, and the covariance matrix of the spherical harmonics coefficients of the cosmic microwave background radiation using the full Boltzmann physics. Our analysis shows that the Wilkinson Microwave Anisotropy Probe (WMAP) three-year data are well compatible with the possibility that we live in a flat 3-torus with volume ~5x10^3 Gpc^3.

astro-ph↗

arXiv2007

Study of Lorentz violation in INTEGRAL Gamma-Ray Bursts

We search for possible time lags caused by quantum gravitational (QG) effects using gamma-ray bursts (GRBs) detected by INTEGRAL. The advantage of this satellite is that we have at our disposal the energy and arrival time of every detected single photon, which enhances the precision of the time resolution. We present a new method for seeking time lags in unbinned data using a maximum likelihood method and support our conclusions with Monte Carlo simulations. The analysis of the data yields a mass scale well below the Planck mass whose value may however increase if better statistics of GRBs were available. Furthermore, we disagree with previous studies in which a non-monotonic function of the redshift was used to perform a linear fit.

gr-qc↗

arXiv2004

Hyperbolic Universes with a Horned Topology and the CMB Anisotropy

We analyse the anisotropy of the cosmic microwave background (CMB) in hyperbolic universes possessing a non-trivial topology with a fundamental cell having an infinitely long horn. The aim of this paper is twofold. On the one hand, we show that the horned topology does not lead to a flat spot in the CMB sky maps in the direction of the horn as stated in the literature. On the other hand, we demonstrate that a horned topology having a finite volume could explain the suppression of the lower multipoles in the CMB anisotropy as observed by COBE and WMAP.

astro-ph↗

arXiv1994

Quantum Chaos

A short historical overview is given on the development of our knowledge of complex dynamical systems with special emphasis on ergodicity and chaos, and on the semiclassical quantization of integrable and chaotic systems. The general trace formula is discussed as a sound mathematical basis for the semiclassical quantization of chaos. Two conjectures are presented on the basis of which it is argued that there are unique fluctuation properties in quantum mechanics which are universal and, in a well defined sense, maximally random if the corresponding classical system is strongly chaotic. These properties constitute the quantum mechanical analogue of the phenomenon of chaos in classical mechanics. Thus quantum chaos has been found.

chao-dyn↗

arXiv1993

Classification of Solvable Feynman Path Integrals

A systematic classification of Feynman path integrals in quantum mechanics is presented and a table of solvable path integrals is given which reflects the progress made during the last ten years or so, including, of course, the main contributions since the invention of the path integral by Feynman in 1942. An outline of the general theory is given. Explicit formulæ for the so-called basic path integrals are presented on which our general scheme to classify and calculate path integrals in quantum mechanics is based.

hep-th↗