Weyl Term Research Articles

Parity violation in primordial tensor non-Gaussianities from matter bounce cosmology

It has been shown that primordial tensor non-Gaussianities from a cubic Weyl action with anon-dynamical coupling are suppressed by the so-called slow-roll parameter in a conventional framework of slow-roll inflation. In this paper, we consider matter bounce cosmology in which the background spacetime is no longer quasi-de Sitter, and hence one might expect that the matter bounce models could predict non-suppressed non-Gaussianities. Nevertheless, we first show that the corresponding non-Gaussian amplitudes from the cubic Weyl term with a non-dynamical coupling are much smaller than those from the conventional slow-roll inflation, in spite of the fact that there is no slow-roll suppression. We then introduce a dynamical coupling that can boost the magnitude of graviton cubic interactions and clarify that there is a parameter region where the tensor non-Gaussianities can be enhanced and can potentially be tested by cosmic microwave background experiments.

Two instabilities of Schwarzschild-AdS black holes in Einstein–Weyl-scalar theory

Stability of Schwarzschild-AdS (SAdS) black hole is investigated in Einstein–Weyl-scalar (EWS) theory with a negative cosmological constant. Here, we introduce a quadratic scalar coupling to the Weyl term, instead of the Gauss–Bonnet term. The linearized EWS theory admits the Lichnerowicz equation for Einstein tensor as well as scalar equation. The linearized Einstein-tensor carries with a regular mass term (M2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {M}}}^2$$\\end{document}), whereas the linearized scalar has a tachyonic mass term (-3r02/m2r6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-3r_0^2/m^2r^6$$\\end{document}). Two instabilities of SAdS black hole in EWS theory are found as Gregory–Laflamme and tachyonic instabilities. It shows that the correlated stability conjecture holds for small SAdS black holes obtained from EWS theory by establishing a close relation between Gregory–Laflamme and thermodynamic instabilities. On the other hand, tachyonic instability of SAdS black hole can be used for making five branches of scalarized black holes when considering proper thermodynamic quantities of EWS theory (M2>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {M}}}^2>0$$\\end{document}).

Quasinormal modes in higher-derivative gravity: Testing the black hole parametrization and sensitivity of overtones

The fundamental quasinormal modes of black holes in higher-derivative gravity given by the Einstein-Weyl action are known to be moderately corrected by the Weyl term. Here we will show that the first several overtones are highly sensitive to even a relatively small Weyl correction, which might be important when representing the earlier stage of the black hole ringdown. In addition, we have solved the problem related to the analytical parametrized approximation of the numerical black hole solution in the Einstein-Weyl theory: In some range of parameters the approximation for the metric developed up to the third order leads to the unusual highly nonmonotonic behavior of the frequencies. We have shown that this problem can be solved via the extension of the parametrization of the metric to higher orders until reaching the regime when the frequencies do not change with further increasing of the order.

Spectral Properties of Laplacians on Snowflake Domains and Filled Julia Sets

We present eigenvalue data and pictures of eigenfunctions of the classic and quadratic snowflake fractal and of quadratic filled Julia sets. Furthermore, we approximate the area and box-counting dimension of selected Julia sets to compare the eigenvalue counting function with the Weyl term.

Flat galactic rotation curves from geometry in Weyl gravity

We searched for a resolution of the flat galactic rotation curve problem from geometry instead of assuming the existence of dark matter. We observed that the scale independence of the rotational velocity in the outer region of galaxies could point out to a possible existence of local scale symmetry and therefore the gravitational phenomena inside such regions should be described by the unique local scale symmetric theory, namely Weyl's theory of gravity. We solved field equations of Weyl gravity and determined the special geometry in the outer region of galaxies. In order to understand the effective description of gravitational phenomena, we compared individual terms of so called Einstein-Weyl theory and concluded that while the outer region of galaxies are described by the Weyl term, the inner region of galaxies are described by the Einstein-Hilbert term.

THE STEKLOV SPECTRUM OF CUBOIDS

The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension d greater or equal to 3. Apart from the standard Weyl term, we calculate explicitly the second term in the asymptotics, capturing the contribution of the (d-2)-dimensional facets of a cuboid. Our approach is based on lattice counting techniques. While this strategy is similar to the one used for the Dirichlet Laplacian, the Steklov case carries additional complications. In particular, it is not clear how to establish directly the completeness of the system of Steklov eigenfunctions admitting separation of variables. We prove this result using a family of auxiliary Robin boundary value problems. Moreover, the correspondence between the Steklov eigenvalues and lattice points is not exact, and hence more delicate analysis is required to obtain spectral asymptotics. Some other related results are presented, such as an isoperimetric inequality for the first Steklov eigenvalue, a concentration property of high frequency Steklov eigenfunctions and applications to spectral determination of cuboids.

Cosmology with a light ghost

We study the creation and evolution of cosmological perturbations in renormalizable quadratic gravity with a Weyl term. We adopt a prescription that implies the stability of the vacuum at the price of introducing a massive spin-two ghost state, leading to the loss of unitarity. The theory may still be predictive regardless the interpretation of non-unitary processes provided that their rate is negligible compared to the Universe expansion rate. This implies that the ghost is effectively stable. In such a setup, there are two scalar degrees of freedom excited during inflation. The first one is the usual curvature perturbation whose power spectrum appears to coincide with that of single-field inflation. The second one is a scalar component of the ghost encoded in the shift vector of the metric in the uniform inflaton gauge. The amplitudes of primordial tensor and vector perturbations are strongly suppressed. After inflation the ghost field starts to oscillate and its energy density shortly becomes dominant in the Universe. For all ghost masses allowed by laboratory constraints ghosts should have ``overclosed'' the Universe at temperatures higher than that of primordial nucleosynthesis. Thus, the model with the light Weyl ghost is ruled out.

Holographic metal-insulator transition in higher derivative gravity

We introduce a Weyl term into the Einstein-Maxwell-Axion theory in four dimensional spacetime. Up to the first order of the Weyl coupling parameter $\gamma$, we construct charged black brane solutions without translational invariance in a perturbative manner. Among all the holographic frameworks involving higher derivative gravity, we are the first to obtain metal-insulator transitions (MIT) when varying the system parameters at zero temperature. Furthermore, we study the holographic entanglement entropy (HEE) of strip geometry in this model and find that the second order derivative of HEE with respect to the axion parameter exhibits maximization behavior near quantum critical points (QCPs) of MIT. It testifies the conjecture in 1502.03661 and 1604.04857 that HEE itself or its derivatives can be used to diagnose quantum phase transition (QPT).

Inflation with light Weyl ghost

Inflationary perturbations are considered in a renormalizable but non-unitary theory of gravity with the additional Weyl term. We obtained that ghost degrees of freedom do not spoil the inflation and the scalar perturbation amplitude at the linear level even in a case of the ghost with mass smaller than Hubble parameter at inflation. The ghost impact to the observables is also estimated to be negligible for the range of masses allowed by the experiment. The non-linear level of the theory and its possible application are also discussed.

Refined semiclassical asymptotics for fractional powers of the Laplace operator

Abstract We consider the fractional Laplacian on a domain and investigate the asymptotic behavior of its eigenvalues. Extending methods from semi-classical analysis we are able to prove a two-term formula for the sum of eigenvalues with the leading (Weyl) term given by the volume and the subleading term by the surface area. Our result is valid under very weak assumptions on the regularity of the boundary.

Constraints on noncommutative spectral action from Gravity Probe B and torsion balance experiments

Noncommutative spectral geometry offers a purely geometric explanation for the standard model of strong and electroweak interactions, including a geometric explanation for the origin of the Higgs field. Within this framework, the gravitational, the electroweak and the strong forces are all described as purely gravitational forces on a unified noncommutative space-time. In this study, we infer a constraint on one of the three free parameters of the model, namely the one characterising the coupling constants at unification, by linearising the field equations in the limit of weak gravitational fields generated by a rotating gravitational source, and by making use of recent experimental data. In particular, using data obtained by Gravity Probe B, we set a lower bound on the Weyl term appearing in the noncommutative spectral action, namely β≳10−6m−1. This constraint becomes stronger once we use results from torsion balance experiments, leading to β≳104m−1. The latter is much stronger than any constraint imposed so far to curvature squared terms.

Lorentz-violating vs. ghost gravitons: the example of Weyl gravity

AbstractWe show that the ghost degrees of freedom of Einstein gravity with a Weyl term can be tamed by a simple mechanism that invokes local Lorentz symmetry breaking. We demonstrate how the mechanism works in a cosmological setting. The presence of the Weyl term forces a redefinition of the quantum vacuum state of the tensor perturbations. As a consequence the amplitude of their spectrum blows up when the Lorentz-violating scale becomes comparable to the Hubble radius. Such a behaviour is in sharp contrast to what happens in standard Weyl gravity where the gravitational ghosts smoothly damp out the spectrum of primordial gravitational waves.

Weyl law for open systems with sharply divided mixed phase space

A generalization of the Weyl law to systems with a sharply divided mixed phase space is proposed. The ansatz is composed of the usual Weyl term which counts the number of states in regular islands and a term associated with sticky regions in phase space. For a piecewise linear map, we numerically check the validity of our hypothesis, and find good agreement not only for the case with a sharply divided phase space but also for the case where tiny island chains surround the main regular island. For the latter case, a nontrivial power law exponent appears in the survival probability of classical escaping orbits, which may provide a clue to develop the Weyl law for more generic mixed systems.

Second Order Semiclassics with Self-Generated Magnetic Fields

We consider the semiclassical asymptotics of the sum of negative eigenvalues of the three-dimensional Pauli operator with an external potential and a self-generated magnetic field $B$. We also add the field energy $\beta \int B^2$ and we minimize over all magnetic fields. The parameter $\beta$ effectively determines the strength of the field. We consider the weak field regime with $\beta h^{2}\ge {const}>0$, where $h$ is the semiclassical parameter. For smooth potentials we prove that the semiclassical asymptotics of the total energy is given by the non-magnetic Weyl term to leading order with an error bound that is smaller by a factor $h^{1+\e}$, i.e. the subleading term vanishes. However, for potentials with a Coulomb singularity the subleading term does not vanish due to the non-semiclassical effect of the singularity. Combined with a multiscale technique, this refined estimate is used in the companion paper \cite{EFS3} to prove the second order Scott correction to the ground state energy of large atoms and molecules.

Inflation with a Weyl term, or ghosts at work

In order to assess the role of ghosts in cosmology, we study the evolution of linear cosmological perturbations during inflation when a Weyl term is added to the action. Our main results are that vector perturbations can no longer be ignored and that scalar modes diverge in the newtonian gauge but remain bounded in the comoving slicing.

Gravitational waves in the spectral action of noncommutative geometry

The spectral triple approach to noncommutative geometry allows one to develop the entire standard model (and supersymmetric extensions) of particle physics from a purely geometry stand point and thus treats both gravity and particle physics on the same footing. The bosonic sector of the theory contains a modification to Einstein-Hilbert gravity, involving a nonconformal coupling of curvature to the Higgs field and conformal Weyl term (in addition to a nondynamical topological term). In this paper we derive the weak field limit of this gravitational theory and show that the production and dynamics of gravitational waves are significantly altered. In particular, we show that the graviton contains a massive mode that alters the energy lost to gravitational radiation, in systems with evolving quadrupole moment. We explicitly calculate the general solution and apply it to systems with periodically varying quadrupole moments, focusing in particular on the the well know energy loss formula for circular binaries.

Counting function for a sphere of anisotropic quartz

We calculate the leading Weyl term of the counting function for a monocrystalline quartz sphere. In contrast to other studies of counting functions, the anisotropy of quartz is a crucial element in our investigation. Hence we do not obtain a simple analytical form, but we carry out a numerical evaluation. To this end we employ the Radon transform representation of the Green's function. We compare our result to a previously measured unique data set of several tens of thousands of resonances.

Braneworld cosmological models with anisotropy

For a cosmological Randall-Sundrum braneworld with anisotropy, i.e., of Bianchi type, the modified Einstein equations on the brane include components of the five-dimensional Weyl tensor for which there are no evolution equations on the brane. If the bulk field equations are not solved, this Weyl term remains unknown, and many previous studies have simply prescribed it as ad hoc. We construct a family of Bianchi braneworlds with anisotropy by solving the five-dimensional field equations in the bulk. We analyze the cosmological dynamics on the brane, including the Weyl term, and shed light on the relation between anisotropy on the brane and the Weyl curvature in the bulk. In these models, it is not possible to achieve geometric anisotropy for a perfect fluid or scalar field---the junction conditions require anisotropic stress on the brane. But the solutions can isotropize and approach a Friedmann brane in an anti--de Sitter bulk.

Density of states for vibrations of fractal drums.

Vibrations of membranes with fractal boundaries (fractal drums) are investigated. Numerical results are presented for Koch drums of fractal dimension D(f)=3/2 at prefractal generations 1-3, and for Koch snowflake drums (D(f)=ln 4/ln 3) at generations 3 and 4. The results show that the low-frequency integrated densities of states (IDOS's) of the drums are well approximated by a two-term asymptotic of the form given by the modified Weyl-Berry (MWB) conjecture, which predicts a correction of DeltaN(Omega) proportional, variant Omega(D(f)) to the leading-order Weyl term. In the high-frequency regime, where the half wavelength is smaller than the smallest features of the prefractal perimeter, the two-term Weyl asymptotic is applicable, with DeltaN(Omega) approximately Omega. The results also indicate that oscillations in DeltaN(Omega) arise due to localization of the wave amplitude near the prefractal perimeter. It is argued that for a self-similar fractal boundary, the amplitude of the oscillations is asymptotically proportional to Omega(D(f)), which implies an O(Omega(D(f))), rather than the conjectured o(Omega(D(f))), error term for the asymptotic IDOS given by the MWB conjecture.

Gravitation in the Codimension Two Brane World

We derive the gravitational field equations on the three-brane in six dimensions. In a covariant manner, we show how Einstein gravity is recovered in the low energy limit. The Einstein equations are modified in the same way as in the five-dimensional model, adding two terms: the quadratic stress-energy term and the Weyl term. We show that the Weyl term is negligible in llinearized gravity.