Curvature Expansion Research Articles

The AdS Veneziano amplitude at small curvature

We compute the AdS Veneziano amplitude for type IIB gluon scattering in AdS5 × S3 to all orders in α′ in a small curvature expansion. This is achieved by combining a dispersion relation in the dual 4dN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFT with an ansatz for the amplitude as a worldsheet integral in terms of multiple polylogarithms. The first curvature correction is fully fixed in this way and satisfies consistency checks in the high energy limit, the low energy expansion as previously fixed using supersymmetric localisation, and for the energy of massive string operators, which we independently compute using a semiclassical expansion. We also combine localisation with this first curvature correction to fix the unprotected D4F4 correction to the amplitude at finite curvature.

Local space time constant mean curvature and constant expansion foliations

Inspired by the small sphere-limit for quasi-local energy we study local foliations of surfaces with prescribed mean curvature. Following the strategy used by Ye in [23] to study local constant mean curvature foliations we use a Lyapunov Schmidt reduction in an n+1 dimensional manifold equipped with a symmetric 2-tensor to construct the foliations around a point, prove their uniqueness and show their nonexistence conditions. To be specific, we study two foliation conditions. First we consider constant space-time mean curvature surfaces. They are used to characterizing the center of mass in general relativity [4]. Second, we study local foliations of constant expansion surfaces [18].

Testing spatial curvature and anisotropic expansion on top of the ΛCDM model

We explore the possible advantages of extending the $\Lambda$CDM model by more realistic backgrounds compared to its spatially flat RW spacetime assumption, while preserving the underpinning physics; in particular, by simultaneously allowing non-zero spatial curvature and anisotropic expansion on top of it, viz., the An-$o\Lambda$CDM model. This is to test whether the latest data support spatial flatness and/or isotropic expansion, and, if not, to explore the roles of spatial curvature and expansion anisotropy (due to its stiff fluid-like behavior) in addressing some of the cosmological tensions. We first present the theoretical background and explicit mathematical construction of An-$o\Lambda$CDM; combining the simplest anisotropic generalizations of the RW spacetime, viz., the Bianchi type I, V, and IX spacetimes. Then we constrain this model and its particular cases, viz., An-$\Lambda$CDM, $o\Lambda$CDM, and $\Lambda$CDM, by using the data sets from different probes, viz., Planck CMB(+Lens), BAO, SnIa Pantheon, and CC data, and discuss the results. Ultimately, we conclude that, within the setup under consideration, (i) the data confirm the spatial flatness and isotropic expansion, though a very small amount of present-day expansion anisotropy cannot be excluded, e.g., $\Omega_{\sigma0}\lesssim10^{-18}$ (95\% C.L.) for An-$\Lambda$CDM from CMB+Lens, (ii) the introduction of spatial curvature or anisotropic expansion, or both, on top $\Lambda$CDM does not offer a possible relaxation to the $H_0$ tension, and (iii) the introduction of anisotropic expansion neither affects the closed space prediction from CMB(+Lens) nor does it improve the drastically reduced value of $H_0$ led by the closed space. We discuss why it is important and indispensable to maintain the geometric generalization work program, especially in models that offer solutions to cosmological tensions. [abridged]

Feedback linking cell envelope stiffness, curvature, and synthesis enables robust rod-shaped bacterial growth

Bacterial growth is remarkably robust to environmental fluctuations, yet the mechanisms of growth-rate homeostasis are poorly understood. Here, we combine theory and experiment to infer mechanisms by which Escherichia coli adapts its growth rate in response to changes in osmolarity, a fundamental physicochemical property of the environment. The central tenet of our theoretical model is that cell-envelope expansion is only sensitive to local information, such as enzyme concentrations, cell-envelope curvature, and mechanical strain in the envelope. We constrained this model with quantitative measurements of the dynamics of E. coli elongation rate and cell width after hyperosmotic shock. Our analysis demonstrated that adaptive cell-envelope softening is a key process underlying growth-rate homeostasis. Furthermore, our model correctly predicted that softening does not occur above a critical hyperosmotic shock magnitude and precisely recapitulated the elongation-rate dynamics in response to shocks with magnitude larger than this threshold. Finally, we found that, to coordinately achieve growth-rate and cell-width homeostasis, cells employ direct feedback between cell-envelope curvature and envelope expansion. In sum, our analysis points to cellular mechanisms of bacterial growth-rate homeostasis and provides a practical theoretical framework for understanding this process.

Sparse expanders have negative curvature

We prove that bounded-degree expanders with non-negative Ollivier–Ricci curvature do not exist, thereby solving a long-standing open problem suggested by A. Naor and E. Milman and publicized by Y. Ollivier (2010). In fact, this remains true even if we allow for a vanishing proportion of large degrees, large eigenvalues, and negatively-curved edges. Moreover, the same conclusion applies to the Bakry–Émery curvature condition CD $$(0,\infty )$$ , thereby settling a recent conjecture of D. Cushing, S. Liu and N. Peyerimhoff (2019). To establish those results, we work directly at the level of Benjamini–Schramm limits, and exploit the entropic characterization of the Liouville property on stationary random graphs to show that non-negative curvature and uniform expansion are incompatible “at infinity”. We then transfer this conclusion to finite graphs via local weak convergence. Our approach also shows that the class of finite graphs with non-negative curvature and degrees at most d is hyperfinite, for any fixed $$d\in {\mathbb {N}}$$ .

A viscous active shell theory of the cell cortex

The cell cortex is a thin layer beneath the plasma membrane that gives animal cells mechanical resistance and drives most of their shape changes, from migration, division to multicellular morphogenesis. It is mainly composed of actin filaments, actin binding proteins, and myosin molecular motors. Constantly stirred by myosin motors and under fast renewal, this material may be well described by viscous and contractile active-gel theories. Here, we assume that the cortex is a thin viscous shell with non-negligible curvature and use asymptotic expansions to find the leading-order equations describing its shape dynamics, starting from constitutive equations for an incompressible viscous active gel. Reducing the three-dimensional equations leads to a Koiter-like shell theory, where both resistance to stretching and bending rates are present. Constitutive equations are completed by a kinematical equation describing the evolution of the cortex thickness with turnover. We show that tension and moment resultants depend not only on the shell deformation rate and motor activity but also on the active turnover of the material, which may also exert either contractile or extensile stress. Using the finite-element method, we implement our theory numerically to study two biological examples of drastic cell shape changes: osmotic shocks and cell division. Our work provides a numerical implementation of thin active viscous layers and a generic theoretical framework to develop shell theories for slender active biological structures.

Convective Velocity Perturbations and Excess Gain in Flame Response as a Result of Flame-Flow Feedback

Convective velocity perturbations (CVPs) are known to play an important role in the response of flames to acoustic perturbations and in thermoacoustic combustion instabilities. In order to elucidate the flow-physical origin of CVPs, the present study models the response of laminar premixed slit flames to low amplitude perturbations of the upstream flow velocity with a reduced order flow decomposition approach: A linearized G-equation represents the shape and heat release rate of the perturbed flame, while the velocity perturbation field is decomposed into irrotational and solenoidal contributions. The former are determined with a conformal mapping from geometry and boundary conditions, whereas the latter are governed by flame front curvature and flow expansion across the flame, which generates baroclinic vorticity. High-resolution CFD analysis provides values of model parameters and confirms the plausibility of model results. This flow decomposition approach makes it possible to explicitly evaluate and analyze the respective contributions of irrotational and solenoidal flows to the flame response, and conversely the effect of flame perturbations on the flow. The use of the popular ad hoc hypothesis of convected velocity perturbation is avoided. It is found that convected velocity perturbations do not result from immediate acoustic-to-hydrodynamic mode conversion, but are generated by flame-flow feedback. In this sense, models for flame dynamics that rely on ad-hoc models for CVPs do not respect causality. Furthermore, analysis of the flame impulse response reveals that for the configuration investigated, flame-flow feedback is also responsible for “excess gain” of the flame response, that is, the magnitude of the flame frequency response above unity.

Formulation of Novelty and Complexity in Shape and its Application to Design

Design aesthetics is one of the most important factors in the attractiveness of industrial products. In this study, we focused on the theory of aesthetics, which states that moderate novelty and complexity maximize pleasant feelings, and formulated the novelty and complexity perceived from a closed curve contour shape using curvature and Fourier expansion coefficients. Using this formulation, we developed "Hybrid-GAN," a shape generation system that iterates shape generation, feature computation, and shape optimization. The system is capable of generating a variety of shapes of arbitrary novelty and complexity. In a subsequent experiment, subjective evaluations of novelty and complexity, as well as beauty and interest, were obtained for the shape samples generated by the system. The results showed that the novelty and complexity formulations had significant positive correlations with the subjective evaluation, and the sum of novelty and complexity had a strong positive correlation with interestingness. The results of this study are expected to be used to support the design of attractive and long-loved shapes by providing feedback to designers on the degree of novelty and complexity that users find most pleasant.

Monte Carlo simulations and mean-field modeling of electric double layers at weakly and moderately charged spherical macroions.

Monte Carlo simulations are employed to determine the differential capacitance of an electric double layer formed by small size-symmetric anions and cations in the vicinity of weakly to moderately charged macroions. The influence of interfacial curvature is deduced by investigating spherical macroions, ranging from flat to moderately curved. We also calculate the differential capacitance using a previously developed mean-field model where, in addition to electrostatic interactions, the excluded volumes of the ions are taken into account using either the lattice-gas or the Carnahan-Starling equation of state. For both equations of state, we compare the mean-field model for arbitrary curvature with a recently developed second-order curvature expansion. Our Monte Carlo simulations predict an increase in the differential capacitance with growing macroion curvature if the surface charge density is small, whereas for moderately charged macroions the differential capacitance passes through a local minimum. Both mean-field models tend to somewhat overestimate the differential capacitance as compared with Monte Carlo simulations. At the same time, they do reproduce the curvature dependence of the differential capacitance, especially for small surface charge density. Our study suggests that the quality of mean-field modeling does not worsen when weakly or moderately charged macroions exhibit spherical curvature.

Non-perturbative quantum Galileon in the exact renormalization group

We investigate the non-perturbative renormalization group flow of the scalar Galileon model in flat space. We discuss different expansion schemes of the Galileon truncation, including a heat-kernel based derivative expansion, a vertex expansion in momentum space and a curvature expansion in terms of a covariant geometric formulation. We find that the Galileon symmetry prevents a quantum induced renormalization group running of the Galileon couplings. Consequently, the Galileon truncation only features a trivial Gaussian fixed point.

Graviton-mediated scattering amplitudes from the quantum effective action

We employ the curvature expansion of the quantum effective action for gravity-matter systems to construct graviton-mediated scattering amplitudes for non-minimally coupled scalar fields in a Minkowski background. By design, the formalism parameterises all quantum corrections to these processes and is manifestly gauge-invariant. The conditions resulting from UV-finiteness, unitarity, and causality are analysed in detail and it is shown by explicit construction that the quantum effective action provides sufficient room to meet these structural requirements without introducing non-localities or higher-spin degrees of freedom. Our framework provides a bottom-up approach to all quantum gravity programs seeking for the quantisation of gravity within the framework of quantum field theory. Its scope is illustrated by specific examples, including effective field theory, Stelle gravity, infinite derivative gravity, and Asymptotic Safety.

Lagrangian density and local symmetries of inhomogeneous hyperconical universes

Hyperconical universes can be represented by means of an inhomogeneous metric with positive curvature and linear expansion, that is isomorphic to flat universes with acceleration thanks to an appropriate transformation. Various symmetry properties of this metric are analysed, primarily at the local scale. In particular, the Lagrangian formalism and the Arnowitt–Deser–Misner (ADM) equations are applied. To this extent, a modified gravity Lagrangian density is derived, from which the comoving paths as solutions of the Euler–Lagrange equations leading to a stationary linear expansion are deduced. It is shown that the evolution of this alternate metric is compatible with the ADM formalism when applied to the modified Lagrangian density, thanks to a redefinition of the energy density baseline (according to the global curvature). Finally, results on symmetry properties imply that only the angular momenta are global symmetries. The radial inhomogeneity of the metric is interpreted as an apparent radial acceleration, which breaks all the non-rotational local symmetries at large distances.

Singularities in quantum corrected space-times

In this paper we consider a static and regular fluid generating a locally spherically symmetric and time-independent space-time and calculate the leading quantum corrections to the metric to first order in curvature. Starting from a singularity free classical solution of general relativity, we show that singularities can be introduced in the curvature invariants by quantum gravitational corrections calculated using an effective field theory approach to quantum gravity. We identify non-trivial conditions that ensure that curvature invariants remain singularity free to leading order in the curvature expansion of the effective action.

Black hole one-loop determinants in the large dimension limit

We calculate the contributions to the one-loop determinant for transverse traceless gravitons in an n + 3-dimensional Schwarzschild black hole background in the large dimension limit, due to the SO(n + 2)-type tensor and vector fluctuations, using the quasinormal mode method. Accordingly we find the quasinormal modes for these fluctuations as a function of a fiducial mass parameter ∆. We show that the behavior of the one-loop determinant at large ∆ accords with a heat kernel curvature expansion in one lower dimension, lending further evidence towards a membrane picture for black holes in the large dimension limit.

Numerical metrics, curvature expansions and Calabi-Yau manifolds

We discuss the extent to which numerical techniques for computing approximations to Ricci-flat metrics can be used to investigate hierarchies of curvature scales on Calabi-Yau manifolds. Control of such hierarchies is integral to the validity of curvature expansions in string effective theories. Nevertheless, for seemingly generic points in moduli space it can be difficult to analytically determine if there might be a highly curved region localized somewhere on the Calabi-Yau manifold. We show that numerical techniques are rather efficient at deciding this issue.

Tolman lengths and rigidity constants from free-energy functionals-General expressions and comparison of theories.

The leading order terms in a curvature expansion of surface tension, the Tolman length (first order), and rigidities (second order) have been shown to play an important role in the description of nucleation processes. This work presents general and rigorous expressions to compute these quantities for any nonlocal density functional theory (DFT). The expressions hold for pure fluids and mixtures and reduce to the known expressions from density gradient theory (DGT). The framework is applied to a Helmholtz energy functional based on the perturbed chain polar statistical associating fluid theory (PCP-SAFT) and is used in an extensive investigation of curvature corrections for pure fluids and mixtures. Predictions from the full DFT are compared to two simpler theories: predictive DGT, which has a density and temperature dependent influence matrix derived from DFT, and DGT, where the influence parameter reproduces the surface tension predicted from DFT. All models are based on the same equation of state and predict similar Tolman lengths and spherical rigidities for small molecules, but the deviations between DFT and DGT increase with chain length for alkanes. For all components except water, we find that DGT underpredicts the value of the Tolman length but overpredicts the value of the spherical rigidity. An important basis for the calculation is an accurate prediction of the planar surface tension. Therefore, further work is required to accurately extract Tolman lengths and rigidities of alkanols because DFT with PCP-SAFT does not accurately predict surface tensions of these fluids.

Free energy dependence on spatial geometry for (2 + 1)-dimensional QFTs

We consider (2 + 1)-QFT at finite temperature on a product of time with a static spatial geometry. The suitably defined difference of thermal vacuum free energy for the QFT on a deformation of flat space from its value on flat space is a UV finite quantity, and for reasonable fall-off conditions on the deformation is IR finite too. For perturbations of flat space we show this free energy difference goes quadratically with perturbation amplitude and may be computed from the linear response of the stress tensor. As an illustration we compute it for a holographic CFT finding that at any temperature, and for any perturbation, the free energy decreases. Similar behaviour was previously found for free scalars and fermions, and for unitary CFTs at zero temperature, suggesting (2 + 1)-QFT may generally energetically favour a crumpled spatial geometry. We also treat the deformation in a hydrostatic small curvature expansion relative to the thermal scale. Then the free energy variation is determined by a curvature correction to the stress tensor and for these theories is negative for small curvature deformations of flat space.

Elasticity in curved topographies: Exact theories and linear approximations

Almost all available results in elasticity on curved topographies are obtained within either a small curvature expansion or an empirical covariant generalization that accounts for screening between Gaussian curvature and disclinations. In this paper, we present a formulation of elasticity theory in curved geometries that unifies its underlying geometric and topological content with the theory of defects. The two different linear approximations widely used in the literature are shown to arise as systematic expansions in reference and actual space. Taking the concrete example of a two-dimensional crystal, with and without a central disclination, constrained on a spherical cap, we compare the exact results with different approximations and evaluate their range of validity. We conclude with some general discussion about the universality of nonlinear elasticity.

Curvature instability of membranes near rigid inclusions.

In multicomponent membranes, internal scalar fields may couple to membrane curvature, thus renormalizing the membrane elastic constants and destabilizing the flat membranes. Here, a general elasticity theory of membranes is considered that employs a quartic curvature expansion. The shape of the membrane and its deformation energy near a long rod-like inclusion are studied analytically. In the limit where one can neglect the end effects, the nonlinear response of the membrane to such inclusions is found in exact form. Notably, exact shape solutions are found when the membrane is curvature unstable, manifested by a negative rigidity. Near the instability point (i.e., at vanishing rigidity), the membrane is stabilized by the quartic term, giving rise to a different length scale and scale exponents for the shape and its energy profile than those found for stable membranes. The contact angle induced by an applied force at the inclusion provides a method to experimentally determine the quartic curvature modulus.

Tensor perturbations in anisotropically curved cosmologies

Besides expanding anisotropically, the universe can also be anisotropic at the level of its (spatial) curvature. In particular, models with anisotropic curvature and isotropic expansion leads both to a ΛCDM-like phenomenology and to an isotropic and homogeneous CMB at the background level. Thus, they offer an interesting and viable example where the cosmological principle does not follow from the isotropy of observational data. In this paper we extract the linear dynamics of tensor perturbations in two classes of cosmologies with anisotropic spatial curvature. Two difficulties arise in comparison to the same computation in isotropic cosmologies. First, the two tensor polarizations do not behave as a spin-2 field, but rather as the spin-0 and spin-1 irreducible components of a symmetric, traceless and transverse tensor field, each with its own dynamics. Second, because metric perturbations are algebraically coupled, one cannot ignore scalar and vector modes and focus just on tensors—even if one is only interested in the latter—under the penalty of obtaining the wrong equations of motion. We illustrate our results by finding analytical solutions and evaluating the power-spectra of tensor polarizations in a radiation dominated universe. We conclude with some comments on how these models could be constrained with future experiments on CMB polarization.