Covariant Lagrangians Research Articles

Mochi_class: Modelling Optimisation to Compute Horndeski In class

We introduce mochi_class, an extension of the Einstein-Boltzmann solver hi_class, designed to unlock the full phenomenological potential of Horndeski gravity. This extension allows for general input functions of time without the need for hard-coded parametrisations or covariant Lagrangians. By replacing the traditional α-parametrisation with a set of stable basis functions, mochi_class ensures that the resulting effective theories are inherently free from gradient and ghost instabilities. Additionally, mochi_class features a quasi-static approximation implemented at the level of modified metric potentials, enhancing prediction accuracy, especially for models transitioning between a super- and sub-Compton regime. mochi_class can robustly handle a wide range of models without fine-tuning, and introduces a new approximation scheme that activates modifications to the standard cosmology deep in the matter-dominated era. Furthermore, it incorporates viability conditions on the equation of motion for the scalar field fluctuations, aiding in the identification of numerical instabilities. Through comprehensive validation against other Einstein-Boltzmann solvers, mochi_class demonstrates excellent performance and accuracy, broadening the scope of hi_class by facilitating the study of specific modified gravity models and enabling exploration of previously inaccessible regions of the Horndeski landscape. The code is publicly available at this URL (https://github.com/mcataneo/mochi_class_public)

Manifestly covariant polynomial M5-brane lagrangians

We present polynomial and manifestly covariant M5-brane Lagrangians along with their analyses involving their dynamics, gauge symmetries and their nonlinear self-duality condition. Such Lagrangians can be particularly useful for developments that are otherwise hindered by a non-polynomial structure and singularity of the Lagrangian such as its quantisation. Although on integrating out some of the auxiliary fields these polynomial Lagrangians reduce to the M5-brane Lagrangian given by the Pasti-Sorokin-Tonin (PST) formalism, in the analysis of the polynomial Lagrangians the only remnant of the non-polynomial structure of the PST type Lagrangian appears in the gauge transformation corresponding to an infinitesimal shift of a Stückelberg field. This transformation does not affect the dynamics or the on-shell self-duality condition of the polynomial M5-brane Lagrangians.

Relativistic constraints on 3N contact interactions

In this paper we analyze the relativistic corrections to the leading order three-nucleon (3N) contact interactions. These boost corrections are derived first from the nonrelativistic reduction of covariant Lagrangians and later from the Poincaré algebra constraints on nonrelativistic theories. We show that in order to describe the 3N potential in reference frames other than the center-of-mass frame, the inclusion of five additional terms with fixed coefficients is required. These terms will be relevant in systems with mass number A>3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A>3$$\\end{document}. How they will affect EFT calculations of binding energies and scattering observables in these systems should be investigated.

Brane dynamics from the first law of entanglement

In this note, we study the first law of entanglement in a boundary conformal field theory (BCFT) dual to warped AdS cut off by a brane. Exploiting the symmetry of boundary-centered half-balls in the BCFT, and using Wald’s covariant phase space formalism in the presence of boundaries, we derive constraints from the first law for a broad range of covariant bulk Lagrangians. We explicitly evaluate these constraints for Einstein gravity, and find a local equation on the brane which is precisely the Neumann condition of Takayanagi [6] at linear order in metric perturbations. This is analogous to the derivation of Einstein’s equations from the first law of entanglement entropy. This machinery should generalize to give local linearized equations of motion for higher-derivative bulk gravity with additional fields.

Hi_class background evolution, initial conditions and approximation schemes

Cosmological datasets have great potential to elucidate the nature of dark energy and test gravity on the largest scales available to observation. Theoretical predictions can be computed with hi_class (www.hiclass-code.net), an accurate, fast and flexible code for linear cosmology, incorporating a wide range of dark energy theories and modifications to general relativity. We introduce three new functionalities into hi_class: (1) Support for models based on covariant Lagrangians, including a constraint-preserving integration scheme for the background evolution and a series of worked-out examples: Galileon, nKGB, quintessence (monomial, tracker) and Brans-Dicke. (2) Consistent initial conditions for the scalar-field perturbations in the deep radiation era, identifying the conditions under which modified-gravity isocurvature perturbations may grow faster than adiabatic modes leading to a loss of predictivity. (3) An automated quasi-static approximation scheme allowing order-of-magnitude improvement in computing performance without sacrificing accuracy for wide classes of models. These enhancements bring the treatment of dark energy and modified gravity models to the level of detail comparable to software tools restricted to standard ΛCDM cosmologies. The hi_class code is publicly available (https://github.com/miguelzuma/hi_class_public), ready to explore current data and prepare for next-generation experiments.

Generalized Galileon cosmology

We study the cosmology of a generalized Galileon field $\phi$ with five covariant Lagrangians in which $\phi$ is replaced by general scalar functions $f_{i}(\phi)$ (i=1,...,5). For these theories, the equations of motion remain at second-order in time derivatives. We restrict the functional forms of $f_{i}(\phi)$ from the demand to obtain de Sitter solutions responsible for dark energy. There are two possible choices for power-law functions $f_{i}(\phi)$, depending on whether the coupling $F(\phi)$ with the Ricci scalar $R$ is independent of $\phi$ or depends on $\phi$. The former corresponds to the covariant Galileon theory that respects the Galilean symmetry in the Minkowski space-time. For generalized Galileon theories we derive the conditions for the avoidance of ghosts and Laplacian instabilities associated with scalar and tensor perturbations as well as the condition for the stability of de Sitter solutions. We also carry out detailed analytic and numerical study for the cosmological dynamics in those theories.

Matter perturbations in Galileon cosmology

We study the evolution of matter density perturbations in Galileon cosmology where the late-time cosmic acceleration can be realized by a field kinetic energy. We obtain full perturbation equations at linear order in the presence of five covariant Lagrangians ${cal L}_i$ ($i=1,...,5$) satisfying a Galilean symmetry in the flat space-time. The equations for a matter perturbation as well as an effective gravitational potential are derived under a quasi-static approximation on sub-horizon scales. This approximation can reproduce full numerical solutions with high accuracy for the wavelengths relevant to large-scale structures. For the model parameters constrained by the background expansion history of the Universe the growth rate of matter perturbations is larger than that in the LCDM model, with the growth index $gamma$ today typically smaller than 0.4. We also find that, even on very large scales associated with the Integrated-Sachs-Wolfe (ISW) effect in Cosmic Microwave Background (CMB) temperature anisotropies, the effective gravitational potential exhibits a temporal growth during the transition from the matter era to the epoch of cosmic acceleration. These properties are useful to distinguish the Galileon model from the LCDM in future high-precision observations.

Observational constraints on Galileon cosmology

We study the cosmology of a covariant Galileon field with five covariant Lagrangians and confront this theory with the most recent cosmological probes: the type Ia supernovae data (Constitution and Union2 sets), cosmic microwave background (WMAP7) and the baryon acoustic oscillations (SDSS7). In the Galileon cosmology with a late-time de Sitter attractor, there is a tracker that attracts solutions with different initial conditions to a common trajectory. Including the cosmic curvature K, we place observational constraints on two distinct cases: (i) the tracker, and (ii) the generic solutions to the equations of motion. We find that the tracker solution can be consistent with the individual observational data, but it is disfavored by the combined data analysis. The generic solutions fare quite well when a non-zero curvature parameter is taken into account, but the Akaike and Bayesian information criteria show that they are not particularly favored over the LCDM model.

Theoretical study of elastic electron scattering off stable and exotic nuclei

Results for elastic electron scattering by nuclei, calculated with charge densities of Skyrme forces and covariant effective Lagrangians that accurately describe nuclear ground states, are compared against experiment in stable isotopes. Dirac partial-wave calculations are performed with an adapted version of the ELSEPA package. Motivated by the fact that studies of electron scattering off exotic nuclei are intended in future facilities in the commissioned GSI and RIKEN upgrades, we survey the theoretical predictions from neutron-deficient to neutron-rich isotopes in the tin and calcium isotopic chains. The charge densities of a covariant interaction that describes the low-energy electromagnetic structure of the nucleon within the Lagrangian of the theory are used to this end. The study is restricted to medium- and heavy-mass nuclei because the charge densities are computed in mean-field approach. Because the experimental analysis of scattering data commonly involves parameterized charge densities, as a surrogate exercise for the yet unexplored exotic nuclei, we fit our calculated mean-field densities with Helm model distributions. This procedure turns out to be helpful to study the neutron-number variation of the scattering observables and allows us to identify correlations of potential interest among some of these observables within the isotopic chains.

Transverse symmetry and spin-3/2 fields

We study the possible covariant Lagrangians that describe the propagation of pure spin-3/2 particles. We show that, apart from the well-known Rarita–Schwinger Lagrangian, there is another possibility where the field is described by a γ-traceless combination and that both Lagrangians yield the same physical predictions for the interaction of conserved sources. We also prove that for the case when the spin-2 field is described by a traceless field, there is no possible spin-3/2 action that makes the system supersymmetric. Nevertheless, the interaction between this field and the spin-3/2 field may be possible.

Galilei-invariant gauge symmetries in Fokker–Planck dynamics with logarithmic diffusion and drift terms

A tensorial approach to Galilean invariance is utilized, together with Lie symmetries of differential equations, in order to derive equations of Fokker–Planck type containing a logarithmic diffusion tensor and drift term. The formalism is based on the projection from an extended (by one space-like dimension) Minkowski manifold to the usual Newtonian spacetime, so that non-relativistic models are described by manifestly covariant Lagrangians. In this paper, we obtain the Fokker–Planck equations from the Euler–Lagrange equations with the extended manifold by using a specific choice of the gauge condition. We work in (1 + 1) spacetime and carry out the analysis for both Abelian and non-Abelian symmetries.

The bosonic string and superstring models in 26+2 and 10+2 dimensional space-time, and the generalized Chern-Simons action

We have covariantized the Lagrangians of the U(1)_V * U(1)_A models, which have U(1)_V * U(1)_A gauge symmetry in two dimensions, and studied their symmetric structures. The special property of the U(1)_V * U(1)_A models is the fact that all these models have an extra time coordinate in the target space-time. The U(1)_V * U(1)_A models coupled to two-dimensional gravity are string models in 26+2 dimensional target space-time for bosonic string and in 10+2 dimensional target space-time for superstring. Both string models have two time coordinates. In order to construct the covariant Lagrangians of the U(1)_V * U(1)_A models the generalized Chern-Simons term plays an important role. The supersymmetric generalized Chern-Simons action is also proposed. The Green-Schwarz type of U(1)_V * U(1)_A superstring model has another fermionic local symmetry as well as \kappa-symmetry. The supersymmetry of target space-time is different from the standard one.

On complexes related with calculus of variations

We consider the variational complex on infinite jet space and the complex of variational derivatives for Lagrangians of multidimensional paths and study relations between them. The discussion of the variational (bi)complex is set up in terms of a flat connection in the jet bundle. We extend it to supercase using a particular new class of forms. We establish relation of the complex of variational derivatives and the variational complex. Certain calculus of Lagrangians of multidimensional paths is developed. It is shown how covariant Lagrangians of higher order can be used to represent characteristic classes.

Covariant actions for chiral supersymmetric bosons

We review a recently developed covariant Lagrangian formulation for p–forms with (anti)self–dual field–strengths and present its extension to the supersymmetric case. As explicit examples we construct covariant Lagrangians for six–dimensional models with N = 1 rigid and curved supersymmetry.

Covariant first-order Lagrangians, energy-density and superpotentials in general relativity

A class of covariant first-order Lagrangians for General Relativity interacting with external matter fields is considered. These Lagrangians depend on the choice of a background connection which has no dynamics. The Poincare-Cartan form, energy density flows and energy-momentum tensors are derived for this class of Lagrangians by applying standard methods, and the results are compared with those appearing in current literature. We obtain new results concerning superpotentials and covariant conservation laws. As an example, these are applied to calculate the mass and angular momentum for the Schwarzschild and Kerr black-holes.

Quantum structure of the unified N = 1 supergravity and N = 1 super-Yang-Mills theory

We consider the unified n = 1 supergravity and n = 1 super-Yang-Mills theory in 10 dimensions recently introduced by Chapline and Manton. The Faddeev-Popov ansatz is inadequate for quantizing this theory. We therefore give an alternative prescription for quantizing it. We first derive the full set of BRS and anti-BRS equations associated with Chapline-Manton gauge symmetry. Then the complete nilpotency of BRS and anti-BRS symmetry generators allows a straightforward derivation of the quantum extension of the theory. Two covariant quantum lagrangians are displayed. The first one is BRS and anti-BRS invariant. The second one is only BRS invariant but, being simpler, it is better suited for computations. We give also the dual form of the Chapline-Manton lagrangian. Its gauge invariance is simpler, but it involves higher-rank abelian anti-symmetric tensors.

Classical scalar solitons in three space dimensions

A systematic study of a class of covariant scalar field Lagrangians with derivative self-coupling is made. Methods are found which allow for the selection of classes of Lagrangian models with three-space-dimensional soliton solutions. These methods allow, in particular, for the elaboration of models with solutions which are finite-energy confining potentials.

Reflection groups and internal-symmetry algebras

It is shown that to discrete Abelian reflection groups in pseudo-Euclidean space there correspond, in spinor index space, non-Abelian groups whose anticommuting elements build up Lie algebras; in particular, two reflections, with respect to orthogonal planes, and their product build up asu2,c algebra. Conditions are given for these «reflection algebras» to beinternal-symmetry algebras (commuting with the Poincare algebras). When covariance with respect to the «extended» conformal group (conformal transformations and reflections) is postulated, the full use of 8-component conformal spinors is necessary. The conformal spinor is either a doublet of canonical Dirac spinors (Dirac basis) or a doublet of conformal Cartan semi-spinors (eigenstates ofΓ7) (semi-spinor basis). The two bases correspond toequivalent representations in index space, but they give rise tophysically nonequivalent spinor equations in Minkowski space: Dirac spinors of the doublet may give rise separately to free particles; semi-spinors do not: they obey coupled spinor equations and they transform into each other for any reversal (including space reversal). The conformal reflection group may generate «internal symmetry algebras»,u2,c for massive spinors,u2,c,L⊕u2,c,R for massless ones. Possible conformal covariant Lagrangians with «internal-symmetry algebras» are discussed. It appears that the simplest conformally covariant interaction Lagrangians may naturally represent weak interactions. Next, the groupO4,4 (orO5,3 orO6,2) containing conformal and Poincare groups as subgroups is studied. Reflection groups inV9 give rise tou4 «internal symmetry algebra», which becomesu4,L⊕u4,R for massless spinors. Theu4 internal symmetry algebra acts onV8 spinors, which may either be quadruplets of Dirac spinors (Dirac basis) or three independent quadruplets of conformal semi-spinors (semi-spinor basis); on these acts au3 algebra orthogonal to theu4 above. It is shown that conformal semi-spinors may not exist as free fields individually in Minkowski space, since they obey coupled equations, but only in systems and precisely an even number (≥2) for bosons and odd (≥3) for fermions. Furthermore, these systems must be singlets of theu3 algebra. The analogy ofV8 spinor properties with leptons, when in the Dirac basis, and with coloured quarks, when in the Cartan semi-spinor basis, is emphasized and discussed.

Corrigendum to ’’Covariant inverse problem of the calculus of variations’’

Our recent work showing how to obtain covariant Lagrangians for certain classes of field equations is extended to cases where the field appears to order λm (m⩽−2) and also to cases where metric variations are taken.