# Posts Tagged perturbation

## Recent Postings from perturbation

### Practical application of KAM theory to galactic dynamics: I. Motivation and methodology

Our understanding of the mechanisms governing the structure and secular evolution galaxies assume nearly integrable Hamiltonians with regular orbits; our perturbation theories are founded on the averaging theorem for isolated resonances. On the other hand, it is well-known that dynamical systems with many degrees of freedom are irregular in all but special cases. The best developed framework for studying the breakdown of regularity and the onset is the Kolmogorov-Arnold-Moser (KAM) theory. Here, we use a numerical version of the KAM procedure to construct regular orbits (tori) and locate irregular orbits (broken tori). Irregular orbits are most often classified in astronomical dynamics by their exponential divergence using Lyapunov exponents. Although their computation is numerically challenging, the procedure is straightforward and they are often used to estimate the measure of regularity. The numerical KAM approach has several advantages: 1) it provides the morphology of perturbed orbits; 2) its constructive nature allows the tori to be used as basis for studying secular evolution; 3) for broken tori, clues to the cause of the irregularity may be found by studying the largest, diverging Fourier terms; and 4) it is more likely to detect weak chaos and orbits close to bifurcation. Conversely, it is not a general technique and works most cleanly for small perturbations. We develop a perturbation theory that includes chaos by retaining an arbitrary number of interacting terms rather than eliminating all but one using the averaging theorem. The companion papers show that models with significant stochasticity seem to be the rule, not the exception.

### Scalar field cosmology modified by the Generalized Uncertainty Principle [Cross-Listing]

We consider quintessence scalar field cosmology in which the Lagrangian of the scalar field is modified by the Generalized Uncertainty Principle. We show that the perturbation terms which arise from the deformed algebra are equivalent with the existence of a second scalar field, where the two fields interact in the kinetic part. Moreover, we consider a spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker spacetime (FLRW), and we derive the gravitational field equations. We show that the modified equation of state parameter $w_{GUP}$ can cross the phantom divide line; that is $w_{GUP}<-1$. Furthermore, we derive the field equations in the dimensionless parameters, the dynamical system which arises is a singular perturbation system in which we study the existence of the fixed points in the slow manifold. Finally, we perform numerical simulations for some well known models and we show that for these models with the specific initial conditions, the parameter $w_{GUP}$ crosses the phantom barrier.

### Scalar field cosmology modified by the Generalized Uncertainty Principle [Cross-Listing]

We consider quintessence scalar field cosmology in which the Lagrangian of the scalar field is modified by the Generalized Uncertainty Principle. We show that the perturbation terms which arise from the deformed algebra are equivalent with the existence of a second scalar field, where the two fields interact in the kinetic part. Moreover, we consider a spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker spacetime (FLRW), and we derive the gravitational field equations. We show that the modified equation of state parameter $w_{GUP}$ can cross the phantom divide line; that is $w_{GUP}<-1$. Furthermore, we derive the field equations in the dimensionless parameters, the dynamical system which arises is a singular perturbation system in which we study the existence of the fixed points in the slow manifold. Finally, we perform numerical simulations for some well known models and we show that for these models with the specific initial conditions, the parameter $w_{GUP}$ crosses the phantom barrier.

### Nonminimal derivative coupling scalar-tensor theories: odd-parity perturbations and black hole stability [Cross-Listing]

We derive the odd parity perturbation equation in scalar-tensor theories with a non minimal kinetic coupling sector of the general Horndeski theory, where the kinetic term is coupled to the metric and the Einstein tensor. We derive the potential of the perturbation, by identifying a master function and switching to tortoise coordinates. We then prove the mode stability under linear odd-parity perturbations of hairy black holes in this sector of Horndeski theory. Finally, we comment on the existence of slowly rotating black hole solutions in this setup and discuss their implications on the physics of compact objects configurations, such as neutron stars.

### Nonminimal derivative coupling scalar-tensor theories: odd-parity perturbations and black hole stability [Cross-Listing]

We derive the odd parity perturbation equation in scalar-tensor theories with a non minimal kinetic coupling sector of the general Horndeski theory, where the kinetic term is coupled to the metric and the Einstein tensor. We derive the potential of the perturbation, by identifying a master function and switching to tortoise coordinates. We then prove the mode stability under linear odd-parity perturbations of hairy black holes in this sector of Horndeski theory. Finally, we comment on the existence of slowly rotating black hole solutions in this setup and discuss their implications on the physics of compact objects configurations, such as neutron stars.

### Reflection and transmission of conformal perturbation defects

We consider reflection and transmission of interfaces which implement renormalisation group flows between conformal fixed points in two dimensions. Such an RG interface is constructed from the identity defect in the ultraviolet CFT by perturbing the theory on one side of the defect line. We compute reflection and transmission coefficients in perturbation theory to third order in the coupling constant and check our calculations against exact constructions of RG interfaces between coset models.

### Perturbations in some models of tachyonic inflation

In the present work an inflationary tachyon field model of the early universe in the braneworld scenario is considered. Several cosmological effects produced by a particular potential in this tachyonic era are studied, under the approximation of slow-roll inflation. In particular, the evolution of the spectral index $n_s$ with time is obtained. The equations for the cosmological scalar perturbations are analytically solved in order to show that the power spectrum for small $k$ values is $P_{\zeta}\sim 1/k^{\frac{1}{2}+\nu_2}$, where $\nu_2$ depends on the barotropic index $\gamma_0$. For large $k$ values we find that the power spectrum is well approximated by the standard inflation model. Additionally, the three-point correlation function is calculated in order to get the primordial non-Gaussianity of the perturbation. The result is that $f_{NG} \simeq 0$ so the non-gaussianities generated by this tachyon field are negligible.

### Perturbations in some models of tachyonic inflation [Cross-Listing]

In the present work an inflationary tachyon field model of the early universe in the braneworld scenario is considered. Several cosmological effects produced by a particular potential in this tachyonic era are studied, under the approximation of slow-roll inflation. In particular, the evolution of the spectral index $n_s$ with time is obtained. The equations for the cosmological scalar perturbations are analytically solved in order to show that the power spectrum for small $k$ values is $P_{\zeta}\sim 1/k^{\frac{1}{2}+\nu_2}$, where $\nu_2$ depends on the barotropic index $\gamma_0$. For large $k$ values we find that the power spectrum is well approximated by the standard inflation model. Additionally, the three-point correlation function is calculated in order to get the primordial non-Gaussianity of the perturbation. The result is that $f_{NG} \simeq 0$ so the non-gaussianities generated by this tachyon field are negligible.

### Measurement of the Nodal Precession of WASP-33 b via Doppler Tomography

We have analyzed new and archival time series spectra taken six years apart during transits of the hot Jupiter WASP-33 b, and spectroscopically resolved the line profile perturbation caused by the Rossiter-McLaughlin effect. The motion of this line profile perturbation is determined by the path of the planet across the stellar disk, which we show to have changed between the two epochs due to nodal precession of the planetary orbit. We measured rates of change of the impact parameter and the sky-projected spin-orbit misalignment of $db/dt=-0.0228_{-0.0018}^{+0.0050}$ yr$^{-1}$ and $d\lambda/dt=-0.487_{-0.076}^{+0.089}$ $^{\circ}$ yr$^{-1}$, respectively, corresponding to a rate of nodal precession of $d\Omega/dt=0.117_{-0.029}^{+0.012}$ $^{\circ}$ yr$^{-1}$. This is only the second measurement of nodal precession for a confirmed exoplanet transiting a single star. Finally, we used the rate of precession to set limits on the stellar gravitational quadrupole moment of $0.0017\leq J_2\leq0.011$.

### Measurement of the Nodal Precession of WASP-33 b via Doppler Tomography [Replacement]

We have analyzed new and archival time series spectra taken six years apart during transits of the hot Jupiter WASP-33 b, and spectroscopically resolved the line profile perturbation caused by the Rossiter-McLaughlin effect. The motion of this line profile perturbation is determined by the path of the planet across the stellar disk, which we show to have changed between the two epochs due to nodal precession of the planetary orbit. We measured rates of change of the impact parameter and the sky-projected spin-orbit misalignment of $db/dt=-0.0228_{-0.0018}^{+0.0050}$ yr$^{-1}$ and $d\lambda/dt=-0.487_{-0.076}^{+0.089}$~$^{\circ}$ yr$^{-1}$, respectively, corresponding to a rate of nodal precession of $d\Omega/dt=0.373_{-0.083}^{+0.031}$~$^{\circ}$ yr$^{-1}$. This is only the second measurement of nodal precession for a confirmed exoplanet transiting a single star. Finally, we used the rate of precession to set limits on the stellar gravitational quadrupole moment of $0.0054\leq J_2\leq0.035$.

### Emerging lattice approach to the K-Unitarity Triangle [Cross-Listing]

It has been clear for past many years that in low energy observables new physics can only appear as a perturbation. Therefore precise theoretical predictions and precise experimental measurements have become mandatory. Here we draw attention to the significant advances that have been made on the lattice in recent years in $K\to \pi \pi$, $\Delta M_K$, the long-distance part of $\varepsilon$ and rare K-decays. Thus, in conjunction with experiments, the construction of a unitarity triangle purely from Kaon physics should soon become feasible. Along with the B-unitarity triangle, this should allow for more stringent tests of the Standard Model and tighter constraints on new physics.

### Emerging lattice approach to the K-Unitarity Triangle [Cross-Listing]

It has been clear for past many years that in low energy observables new physics can only appear as a perturbation. Therefore precise theoretical predictions and precise experimental measurements have become mandatory. Here we draw attention to the significant advances that have been made on the lattice in recent years in $K\to \pi \pi$, $\Delta M_K$, the long-distance part of $\varepsilon$ and rare K-decays. Thus, in conjunction with experiments, the construction of a unitarity triangle purely from Kaon physics should soon become feasible. Along with the B-unitarity triangle, this should allow for more stringent tests of the Standard Model and tighter constraints on new physics.

### Emerging lattice approach to the K-Unitarity Triangle

It has been clear for past many years that in low energy observables new physics can only appear as a perturbation. Therefore precise theoretical predictions and precise experimental measurements have become mandatory. Here we draw attention to the significant advances that have been made on the lattice in recent years in $K\to \pi \pi$, $\Delta M_K$, the long-distance part of $\varepsilon$ and rare K-decays. Thus, in conjunction with experiments, the construction of a unitarity triangle purely from Kaon physics should soon become feasible. Along with the B-unitarity triangle, this should allow for more stringent tests of the Standard Model and tighter constraints on new physics.

### Gravity Dual of Quantum Information Metric

We study a quantum information metric (or fidelity susceptibility) in conformal field theories with respect to a small perturbation by a primary operator. We argue that its gravity dual is approximately given by a volume of maximal time slice in an AdS spacetime when the perturbation is exactly marginal. We confirm our claim in several examples.

### Gravity Dual of Quantum Information Metric [Replacement]

We study a quantum information metric (or fidelity susceptibility) in conformal field theories with respect to a small perturbation by a primary operator. We argue that its gravity dual is approximately given by a volume of maximal time slice in an AdS spacetime when the perturbation is exactly marginal. We confirm our claim in several examples.

### Matter Mixing in Core-collapse Supernova Ejecta: Large Density Perturbations in the Progenitor Star?

Matter mixing is one important topic in the study of core-collapse supernova (CCSN) explosions. In this paper, we perform two-dimensional hydrodynamic simulations to reproduce the high velocity $^{56}$Ni clumps observed in SN 1987A. This is the first time that large density perturbation is proposed in the CCSN progenitor to generate Rayleigh-Taylor (RT) instability and make the effective matter mixing. In the case of a spherical explosion, RT instability is efficient at both C+O/He and He/H interfaces of the SN progenitor. Radial coherent structures shown in perturbation patterns are important for obtaining high velocity $^{56}$Ni clumps. We can also obtain matter mixing features and high velocity $^{56}$Ni clumps in some cases of aspherical explosion. We find that one of the most favorable models in our work has a combination of bipolar and equatorially asymmetric explosions in which at least 25\% of density perturbation is introduced at different composition interfaces of the CCSN progenitor. These simulation results are comparable to the observational findings of SN 1987A.

### Path-integral Evidence

Here we present a Bayesian formalism for the goodness-of-fit that is the evidence for a fixed functional form over the evidence for all functions that are a general perturbation about this form. This is done under the assumption that the statistical properties of the data can be modelled by a multivariate Gaussian distribution. We use this to show how one can optimise an experiment to find evidence for a fixed function over perturbations about this function. We apply this formalism to an illustrative problem of measuring perturbations in the dark energy equation of state about a cosmological constant.

### WIMP isocurvature perturbation and small scale structure

The adiabatic component of perturbations is damped during the kinetic decoupling due to the collision with relativistic component on sub-horizon scales. However the isocurvature part is free from the damping and could be large enough to make a substantial contribution to the formation of small scale structure. We explicitly study the weakly interacting massive particles as dark matter with an early matter dominated period before radiation domination and show that the isocurvature perturbation is generated during the phase transition and leaves imprint in the observable signatures for the small scale structure.

### WIMP isocurvature perturbation and small scale structure [Cross-Listing]

The adiabatic component of perturbations is damped during the kinetic decoupling due to the collision with relativistic component on sub-horizon scales. However the isocurvature part is free from the damping and could be large enough to make a substantial contribution to the formation of small scale structure. We explicitly study the weakly interacting massive particles as dark matter with an early matter dominated period before radiation domination and show that the isocurvature perturbation is generated during the phase transition and leaves imprint in the observable signatures for the small scale structure.

### Scale Invariant Resummed Perturbation at Finite Temperatures

We use the scalar model with quartic interaction to illustrate how a nonperturbative variational technique combined with renormalization group (RG) properties efficiently resums perturbative expansions in thermal field theories. The resulting convergence and scale dependence of optimized thermodynamical quantities, here illustrated up to two-loop order, are drastically improved as compared to standard perturbative expansions, as well as to other related methods such as the screened perturbation or (resummed) hard-thermal-loop perturbation, that miss RG invariance as we explain. Being very general and easy to implement, our method is a potential analytical alternative to deal with the phase transitions of field theories such as thermal QCD.

### Baryo-Leptogenesis induced by modified gravities in the primordial Universe [Cross-Listing]

The long-standing problem of the asymmetry between matter and antimatter in the Universe is, in this paper, analysed in the context of the modified theories of gravity. In particular we study two models of $f(R)$ theories of gravitation that, with the opportune choice of the free parameters, introduce little perturbation to the scale factor of the Universe in the radiation dominated (RD) phase predicted by general relativity (GR), i.e., $a(t)\sim t^{1/2}$. This little perturbation generates a Ricci scalar different by zero, i.e., $R\neq 0$ that reproduces the correct magnitude for the asymmetry factor $\eta$ computed in the frame of the theories of the gravitational baryogenesis and gravitational leptogenesis. The opportune choice of the free parameters is discussed in order to obtain results coherent with experimental data. Furthermore, the form of the potential $V$, for the scalar-tensor theory conformally equivalent to the $f(R)$ theory which reproduces the right asymmetry factor, is here obtained.

### Baryo-Leptogenesis induced by modified gravities in the primordial Universe

The long-standing problem of the asymmetry between matter and antimatter in the Universe is, in this paper, analysed in the context of the modified theories of gravity. In particular we study two models of $f(R)$ theories of gravitation that, with the opportune choice of the free parameters, introduce little perturbation to the scale factor of the Universe in the radiation dominated (RD) phase predicted by general relativity (GR), i.e., $a(t)\sim t^{1/2}$. This little perturbation generates a Ricci scalar different by zero, i.e., $R\neq 0$ that reproduces the correct magnitude for the asymmetry factor $\eta$ computed in the frame of the theories of the gravitational baryogenesis and gravitational leptogenesis. The opportune choice of the free parameters is discussed in order to obtain results coherent with experimental data. Furthermore, the form of the potential $V$, for the scalar-tensor theory conformally equivalent to the $f(R)$ theory which reproduces the right asymmetry factor, is here obtained.

### Baryo-Leptogenesis induced by modified gravities in the primordial Universe [Cross-Listing]

The long-standing problem of the asymmetry between matter and antimatter in the Universe is, in this paper, analysed in the context of the modified theories of gravity. In particular we study two models of $f(R)$ theories of gravitation that, with the opportune choice of the free parameters, introduce little perturbation to the scale factor of the Universe in the radiation dominated (RD) phase predicted by general relativity (GR), i.e., $a(t)\sim t^{1/2}$. This little perturbation generates a Ricci scalar different by zero, i.e., $R\neq 0$ that reproduces the correct magnitude for the asymmetry factor $\eta$ computed in the frame of the theories of the gravitational baryogenesis and gravitational leptogenesis. The opportune choice of the free parameters is discussed in order to obtain results coherent with experimental data. Furthermore, the form of the potential $V$, for the scalar-tensor theory conformally equivalent to the $f(R)$ theory which reproduces the right asymmetry factor, is here obtained.

### The chiral condensate from renormalization group optimized perturbation [Cross-Listing]

Our recently developed variant of variationnally optimized perturbation (OPT), in particular consistently incorporating renormalization group properties (RGOPT), is adapted to the calculation of the QCD spectral density of the Dirac operator and the related chiral quark condensate $\langle \bar q q \rangle$ in the chiral limit, for $n_f=2$ and $n_f=3$ massless quarks. The results of successive sequences of approximations at two-, three-, and four-loop orders of this modified perturbation, exhibit a remarkable stability. We obtain $\langle \bar q q\rangle^{1/3}_{n_f=2}(2\, {\rm GeV}) = -(0.833-0.845) \bar\Lambda_2$, and $\langle\bar q q\rangle^{1/3}_{n_f=3}(2\, {\rm GeV}) = -(0.814-0.838) \bar\Lambda_3$ where the range spanned by the first and second numbers (respectively four- and three-loop order results) defines our theoretical error, and $\bar\Lambda_{n_f}$ is the basic QCD scale in the $\overline{MS}$-scheme. We obtain a moderate suppression of the chiral condensate when going from $n_f=2$ to $n_f=3$. We compare these results with some other recent determinations from other nonperturbative methods (mainly lattice and spectral sum rules).

### The chiral condensate from renormalization group optimized perturbation

Our recently developed variant of variationnally optimized perturbation (OPT), in particular consistently incorporating renormalization group properties (RGOPT), is adapted to the calculation of the QCD spectral density of the Dirac operator and the related chiral quark condensate $\langle \bar q q \rangle$ in the chiral limit, for $n_f=2$ and $n_f=3$ massless quarks. The results of successive sequences of approximations at two-, three-, and four-loop orders of this modified perturbation, exhibit a remarkable stability. We obtain $\langle \bar q q\rangle^{1/3}_{n_f=2}(2\, {\rm GeV}) = -(0.833-0.845) \bar\Lambda_2$, and $\langle\bar q q\rangle^{1/3}_{n_f=3}(2\, {\rm GeV}) = -(0.814-0.838) \bar\Lambda_3$ where the range spanned by the first and second numbers (respectively four- and three-loop order results) defines our theoretical error, and $\bar\Lambda_{n_f}$ is the basic QCD scale in the $\overline{MS}$-scheme. We obtain a moderate suppression of the chiral condensate when going from $n_f=2$ to $n_f=3$. We compare these results with some other recent determinations from other nonperturbative methods (mainly lattice and spectral sum rules).

### Neutrino Mixing with Non-Zero $\theta_{13}$ and CP Violation in the 3-3-1 Model Based on $S_4$ Flavor Symmetry

The 3-3-1 model proposed in 2011 based on discrete symmetry $S_4$ responsible for the neutrino and quark masses is updated, in which the non-zero $\theta_{13}$ is focused. Neutrino masses and mixings are consistent with the most recent data on neutrino oscillations without perturbation. The new feature is adding a new $SU(3)_L$ anti-sextet lying in doublet under $S_4$ which can result the non-zero $\theta_{13}$ without perturbation, and consequently, the number of Higgs multiplets required is less than those of other models based on non-Abelian discrete symmetries and the 3-3-1 models. The exact tribimaximal form obtained with the breaking $S_4 \rightarrow Z_3$ in charged lepton sector and $S_4 \rightarrow \mathcal{K}$ in neutrino sector. If both breakings $S_4\rightarrow \mathcal{K}$ and $\mathcal{K} \rightarrow Z_2$ are taken place in neutrino sector, the realistic neutrino spectrum is obtained without perturbation. The upper bound on neutrino mass and the effective mass governing neutrinoless double beta decay at the tree level are presented. The model predicts the Dirac CP violation phase $\delta=292.45^\circ$ in the normal spectrum (with $\theta_{23}\neq \frac{\pi}{4}$) and $\delta=303.14^\circ$ in the inverted spectrum.

### Metric perturbations produced by eccentric equatorial orbits around a Kerr black hole

We present the first numerical calculation of the (local) metric perturbation produced by a small compact object moving on an eccentric equatorial geodesic around a Kerr black hole, accurate to first order in the mass ratio. The procedure starts by first solving the Teukolsky equation to obtain the Weyl scalar $\psi_4$ using semi-analytical methods. The metric perturbation is then reconstructed from $\psi_4$ in an (outgoing) radiation gauge, adding the appropriate non-radiative contributions arising from the shifts in mass and angular momentum of the spacetime. As a demonstration we calculate the generalized redshift $U$ as a function of the orbital frequencies $\Omega_r$ and $\Omega_\phi$ to linear order in the mass ratio, a gauge invariant measure of the conservative corrections to the orbit due to self-interactions. In Schwarzschild, the results surpass the existing result in the literature in accuracy, and we find new estimates for some of the unknown 4PN and 5PN terms in the post-Newtonian expansion of $U$. In Kerr, we provide completely novel values of $U$ for eccentric equatorial orbits. Calculation of the full self-force will appear in a forthcoming paper.

### Metric perturbations produced by eccentric equatorial orbits around a Kerr black hole [Replacement]

We present the first numerical calculation of the (local) metric perturbation produced by a small compact object moving on an eccentric equatorial geodesic around a Kerr black hole, accurate to first order in the mass ratio. The procedure starts by first solving the Teukolsky equation to obtain the Weyl scalar $\psi_4$ using semi-analytical methods. The metric perturbation is then reconstructed from $\psi_4$ in an (outgoing) radiation gauge, adding the appropriate non-radiative contributions arising from the shifts in mass and angular momentum of the spacetime. As a demonstration we calculate the generalized redshift $U$ as a function of the orbital frequencies $\Omega_r$ and $\Omega_\phi$ to linear order in the mass ratio, a gauge invariant measure of the conservative corrections to the orbit due to self-interactions. In Schwarzschild, the results surpass the existing result in the literature in accuracy, and we find new estimates for some of the unknown 4PN and 5PN terms in the post-Newtonian expansion of $U$. In Kerr, we provide completely novel values of $U$ for eccentric equatorial orbits. Calculation of the full self-force will appear in a forthcoming paper.

### Fast spectral source integration in black hole perturbation calculations

This paper presents a new technique for achieving spectral accuracy and fast computational performance in a class of black hole perturbation and gravitational self-force calculations involving extreme mass ratios and generic orbits. Called \emph{spectral source integration} (SSI), this method should see widespread future use in problems that entail (i) point-particle description of the small compact object, (ii) frequency domain decomposition, and (iii) use of the background eccentric geodesic motion. Frequency domain approaches are widely used in both perturbation theory flux-balance calculations and in local gravitational self-force calculations. Recent self-force calculations in Lorenz gauge, using the frequency domain and method of extended homogeneous solutions, have been able to accurately reach eccentricities as high as $e \simeq 0.7$. We show here SSI successfully applied to Lorenz gauge. In a double precision Lorenz gauge code, SSI enhances the accuracy of results and makes a factor of three improvement in the overall speed. The primary initial application of SSI–for us its \emph{raison d’\^{e}tre}–is in an arbitrary precision \emph{Mathematica} code that computes perturbations of eccentric orbits in the Regge-Wheeler gauge to extraordinarily high accuracy (e.g., 200 decimal places). These high accuracy eccentric orbit calculations would not be possible without the exponential convergence of SSI. We believe the method will extend to work for inspirals on Kerr, and will be the subject of a later publication. SSI borrows concepts from discrete-time signal processing and is used to calculate the mode normalization coefficients in perturbation theory via sums over modest numbers of points around an orbit. A variant of the idea is used to obtain spectral accuracy in solution of the geodesic orbital motion.

### Stability of the Early Universe in Bigravity Theory [Cross-Listing]

We study the stability of a spherically symmetric perturbation around the flat Friedmann-Lema$\hat{\i}$tre-Robertson-Walker spacetime in the ghost-free bigravity theory, retaining nonlinearities of the helicity-$0$ mode of the massive graviton. It has been known that, when the graviton mass is smaller than the Hubble parameter, homogeneous and isotropic spacetimes suffer from the Higuchi-type ghost or the gradient instability against the linear perturbation in the bigravity. Hence, the bigravity theory has no healthy massless limit for cosmological solutions at linear level. In this paper we show that the instabilities can be resolved by taking into account nonlinear effects of the scalar graviton mode for an appropriate parameter space of coupling constants. The growth history in the bigravity can be restored to the result in general relativity in the early stage of the Universe, in which the St\"uckelberg fields are nonlinear and there is neither ghost nor gradient instability. Therefore, the bigravity theory has the healthy massless limit, and cosmology based on it is viable even when the graviton mass is smaller than the Hubble parameter.

### Stability of the Early Universe in Bigravity Theory [Cross-Listing]

We study the stability of a spherically symmetric perturbation around the flat Friedmann-Lema$\hat{\i}$tre-Robertson-Walker spacetime in the ghost-free bigravity theory, retaining nonlinearities of the helicity-$0$ mode of the massive graviton. It has been known that, when the graviton mass is smaller than the Hubble parameter, homogeneous and isotropic spacetimes suffer from the Higuchi-type ghost or the gradient instability against the linear perturbation in the bigravity. Hence, the bigravity theory has no healthy massless limit for cosmological solutions at linear level. In this paper we show that the instabilities can be resolved by taking into account nonlinear effects of the scalar graviton mode for an appropriate parameter space of coupling constants. The growth history in the bigravity can be restored to the result in general relativity in the early stage of the Universe, in which the St\"uckelberg fields are nonlinear and there is neither ghost nor gradient instability. Therefore, the bigravity theory has the healthy massless limit, and cosmology based on it is viable even when the graviton mass is smaller than the Hubble parameter.

### Stability of the Early Universe in Bigravity Theory

We study the stability of a spherically symmetric perturbation around the flat Friedmann-Lema$\hat{\i}$tre-Robertson-Walker spacetime in the ghost-free bigravity theory, retaining nonlinearities of the helicity-$0$ mode of the massive graviton. It has been known that, when the graviton mass is smaller than the Hubble parameter, homogeneous and isotropic spacetimes suffer from the Higuchi-type ghost or the gradient instability against the linear perturbation in the bigravity. Hence, the bigravity theory has no healthy massless limit for cosmological solutions at linear level. In this paper we show that the instabilities can be resolved by taking into account nonlinear effects of the scalar graviton mode for an appropriate parameter space of coupling constants. The growth history in the bigravity can be restored to the result in general relativity in the early stage of the Universe, in which the St\"uckelberg fields are nonlinear and there is neither ghost nor gradient instability. Therefore, the bigravity theory has the healthy massless limit, and cosmology based on it is viable even when the graviton mass is smaller than the Hubble parameter.

### Stability of the Early Universe in Bigravity Theory [Replacement]

We study the stability of a spherically symmetric perturbation around the flat Friedmann-Lema$\hat{\i}$tre-Robertson-Walker spacetime in the ghost-free bigravity theory, retaining nonlinearities of the helicity-$0$ mode of the massive graviton. It has been known that, when the graviton mass is smaller than the Hubble parameter, homogeneous and isotropic spacetimes suffer from the Higuchi-type ghost or the gradient instability against the linear perturbation in the bigravity. Hence, the bigravity theory has no healthy massless limit for cosmological solutions at linear level. In this paper we show that the instabilities can be resolved by taking into account nonlinear effects of the scalar graviton mode for an appropriate parameter space of coupling constants. The growth history in the bigravity can be restored to the result in general relativity in the early stage of the Universe, in which the St\"uckelberg fields are nonlinear and there is neither ghost nor gradient instability. Therefore, the bigravity theory has the healthy massless limit, and cosmology based on it is viable even when the graviton mass is smaller than the Hubble parameter.

### Stability of the Early Universe in Bigravity Theory [Replacement]

We study the stability of a spherically symmetric perturbation around the flat Friedmann-Lema$\hat{\i}$tre-Robertson-Walker spacetime in the ghost-free bigravity theory, retaining nonlinearities of the helicity-$0$ mode of the massive graviton. It has been known that, when the graviton mass is smaller than the Hubble parameter, homogeneous and isotropic spacetimes suffer from the Higuchi-type ghost or the gradient instability against the linear perturbation in the bigravity. Hence, the bigravity theory has no healthy massless limit for cosmological solutions at linear level. In this paper we show that the instabilities can be resolved by taking into account nonlinear effects of the scalar graviton mode for an appropriate parameter space of coupling constants. The growth history in the bigravity can be restored to the result in general relativity in the early stage of the Universe, in which the St\"uckelberg fields are nonlinear and there is neither ghost nor gradient instability. Therefore, the bigravity theory has the healthy massless limit, and cosmology based on it is viable even when the graviton mass is smaller than the Hubble parameter.

### Stability of the Early Universe in Bigravity Theory [Replacement]

We study the stability of a spherically symmetric perturbation around the flat Friedmann-Lema$\hat{\i}$tre-Robertson-Walker spacetime in the ghost-free bigravity theory, retaining nonlinearities of the helicity-$0$ mode of the massive graviton. It has been known that, when the graviton mass is smaller than the Hubble parameter, homogeneous and isotropic spacetimes suffer from the Higuchi-type ghost or the gradient instability against the linear perturbation in the bigravity. Hence, the bigravity theory has no healthy massless limit for cosmological solutions at linear level. In this paper we show that the instabilities can be resolved by taking into account nonlinear effects of the scalar graviton mode for an appropriate parameter space of coupling constants. The growth history in the bigravity can be restored to the result in general relativity in the early stage of the Universe, in which the St\"uckelberg fields are nonlinear and there is neither ghost nor gradient instability. Therefore, the bigravity theory has the healthy massless limit, and cosmology based on it is viable even when the graviton mass is smaller than the Hubble parameter.

### Phenomenology of dark energy: general features of large-scale perturbations [Cross-Listing]

We present a systematic exploration of dark energy and modified gravity models containing a single scalar field non-minimally coupled to the metric. Even though the parameter space is large, by exploiting an effective field theory (EFT) formulation and by imposing simple physical constraints such as stability conditions and (sub-)luminal propagation of perturbations, we arrive at a number of generic predictions. (1) The linear growth rate of matter density fluctuations is generally suppressed compared to $\Lambda$CDM at intermediate redshifts ($0.5 \lesssim z \lesssim 1$), despite the introduction of an attractive long-range scalar force. This is due to the fact that, in self-accelerating models, the background gravitational coupling weakens at intermediate redshifts, over-compensating the effect of the attractive scalar force. (2) At higher redshifts, the opposite happens; we identify a period of super-growth when the linear growth rate is larger than that predicted by $\Lambda$CDM. (3) The gravitational slip parameter $\eta$ – the ratio of the space part of the metric perturbation to the time part – is bounded from above. For Brans-Dicke-type theories $\eta$ is at most unity. For more general theories, $\eta$ can exceed unity at intermediate redshifts, but not more than about $1.5$ if, at the same time, the linear growth rate is to be compatible with current observational constraints. We caution against phenomenological parametrization of data that do not correspond to predictions from viable physical theories. We advocate the EFT approach as a way to constrain new physics from future large-scale-structure data.

### Phenomenology of dark energy: general features of large-scale perturbations [Replacement]

We present a systematic exploration of dark energy and modified gravity models containing a single scalar field non-minimally coupled to the metric. Even though the parameter space is large, by exploiting an effective field theory (EFT) formulation and by imposing simple physical constraints such as stability conditions and (sub-)luminal propagation of perturbations, we arrive at a number of generic predictions. (1) The linear growth rate of matter density fluctuations is generally suppressed compared to $\Lambda$CDM at intermediate redshifts ($0.5 \lesssim z \lesssim 1$), despite the introduction of an attractive long-range scalar force. This is due to the fact that, in self-accelerating models, the background gravitational coupling weakens at intermediate redshifts, over-compensating the effect of the attractive scalar force. (2) At higher redshifts, the opposite happens; we identify a period of super-growth when the linear growth rate is larger than that predicted by $\Lambda$CDM. (3) The gravitational slip parameter $\eta$ – the ratio of the space part of the metric perturbation to the time part – is bounded from above. For Brans-Dicke-type theories $\eta$ is at most unity. For more general theories, $\eta$ can exceed unity at intermediate redshifts, but not more than about $1.5$ if, at the same time, the linear growth rate is to be compatible with current observational constraints. We caution against phenomenological parametrization of data that do not correspond to predictions from viable physical theories. We advocate the EFT approach as a way to constrain new physics from future large-scale-structure data.

### Phenomenology of dark energy: general features of large-scale perturbations [Replacement]

We present a systematic exploration of dark energy and modified gravity models containing a single scalar field non-minimally coupled to the metric. Even though the parameter space is large, by exploiting an effective field theory (EFT) formulation and by imposing simple physical constraints such as stability conditions and (sub-)luminal propagation of perturbations, we arrive at a number of generic predictions. (1) The linear growth rate of matter density fluctuations is generally suppressed compared to $\Lambda$CDM at intermediate redshifts ($0.5 \lesssim z \lesssim 1$), despite the introduction of an attractive long-range scalar force. This is due to the fact that, in self-accelerating models, the background gravitational coupling weakens at intermediate redshifts, over-compensating the effect of the attractive scalar force. (2) At higher redshifts, the opposite happens; we identify a period of super-growth when the linear growth rate is larger than that predicted by $\Lambda$CDM. (3) The gravitational slip parameter $\eta$ – the ratio of the space part of the metric perturbation to the time part – is bounded from above. For Brans-Dicke-type theories $\eta$ is at most unity. For more general theories, $\eta$ can exceed unity at intermediate redshifts, but not more than about $1.5$ if, at the same time, the linear growth rate is to be compatible with current observational constraints. We caution against phenomenological parametrization of data that do not correspond to predictions from viable physical theories. We advocate the EFT approach as a way to constrain new physics from future large-scale-structure data.

### Phenomenology of dark energy: general features of large-scale perturbations

We present a systematic exploration of dark energy and modified gravity models containing a single scalar field non-minimally coupled to the metric. Even though the parameter space is large, by exploiting an effective field theory (EFT) formulation and by imposing simple physical constraints such as stability conditions and (sub-)luminal propagation of perturbations, we arrive at a number of generic predictions. (1) The linear growth rate of matter density fluctuations is generally suppressed compared to $\Lambda$CDM at intermediate redshifts ($0.5 \lesssim z \lesssim 1$), despite the introduction of an attractive long-range scalar force. This is due to the fact that, in self-accelerating models, the background gravitational coupling weakens at intermediate redshifts, over-compensating the effect of the attractive scalar force. (2) At higher redshifts, the opposite happens; we identify a period of super-growth when the linear growth rate is larger than that predicted by $\Lambda$CDM. (3) The gravitational slip parameter $\eta$ – the ratio of the space part of the metric perturbation to the time part – is bounded from above. For Brans-Dicke-type theories $\eta$ is at most unity. For more general theories, $\eta$ can exceed unity at intermediate redshifts, but not more than about $1.5$ if, at the same time, the linear growth rate is to be compatible with current observational constraints. We caution against phenomenological parametrization of data that do not correspond to predictions from viable physical theories. We advocate the EFT approach as a way to constrain new physics from future large-scale-structure data.

### Phenomenology of dark energy: general features of large-scale perturbations [Cross-Listing]

We present a systematic exploration of dark energy and modified gravity models containing a single scalar field non-minimally coupled to the metric. Even though the parameter space is large, by exploiting an effective field theory (EFT) formulation and by imposing simple physical constraints such as stability conditions and (sub-)luminal propagation of perturbations, we arrive at a number of generic predictions. (1) The linear growth rate of matter density fluctuations is generally suppressed compared to $\Lambda$CDM at intermediate redshifts ($0.5 \lesssim z \lesssim 1$), despite the introduction of an attractive long-range scalar force. This is due to the fact that, in self-accelerating models, the background gravitational coupling weakens at intermediate redshifts, over-compensating the effect of the attractive scalar force. (2) At higher redshifts, the opposite happens; we identify a period of super-growth when the linear growth rate is larger than that predicted by $\Lambda$CDM. (3) The gravitational slip parameter $\eta$ – the ratio of the space part of the metric perturbation to the time part – is bounded from above. For Brans-Dicke-type theories $\eta$ is at most unity. For more general theories, $\eta$ can exceed unity at intermediate redshifts, but not more than about $1.5$ if, at the same time, the linear growth rate is to be compatible with current observational constraints. We caution against phenomenological parametrization of data that do not correspond to predictions from viable physical theories. We advocate the EFT approach as a way to constrain new physics from future large-scale-structure data.

### Phenomenology of dark energy: general features of large-scale perturbations [Replacement]

We present a systematic exploration of dark energy and modified gravity models containing a single scalar field non-minimally coupled to the metric. Even though the parameter space is large, by exploiting an effective field theory (EFT) formulation and by imposing simple physical constraints such as stability conditions and (sub-)luminal propagation of perturbations, we arrive at a number of generic predictions. (1) The linear growth rate of matter density fluctuations is generally suppressed compared to $\Lambda$CDM at intermediate redshifts ($0.5 \lesssim z \lesssim 1$), despite the introduction of an attractive long-range scalar force. This is due to the fact that, in self-accelerating models, the background gravitational coupling weakens at intermediate redshifts, over-compensating the effect of the attractive scalar force. (2) At higher redshifts, the opposite happens; we identify a period of super-growth when the linear growth rate is larger than that predicted by $\Lambda$CDM. (3) The gravitational slip parameter $\eta$ – the ratio of the space part of the metric perturbation to the time part – is bounded from above. For Brans-Dicke-type theories $\eta$ is at most unity. For more general theories, $\eta$ can exceed unity at intermediate redshifts, but not more than about $1.5$ if, at the same time, the linear growth rate is to be compatible with current observational constraints. We caution against phenomenological parametrization of data that do not correspond to predictions from viable physical theories. We advocate the EFT approach as a way to constrain new physics from future large-scale-structure data.

### Phenomenology of dark energy: general features of large-scale perturbations [Cross-Listing]

We present a systematic exploration of dark energy and modified gravity models containing a single scalar field non-minimally coupled to the metric. Even though the parameter space is large, by exploiting an effective field theory (EFT) formulation and by imposing simple physical constraints such as stability conditions and (sub-)luminal propagation of perturbations, we arrive at a number of generic predictions. (1) The linear growth rate of matter density fluctuations is generally suppressed compared to $\Lambda$CDM at intermediate redshifts ($0.5 \lesssim z \lesssim 1$), despite the introduction of an attractive long-range scalar force. This is due to the fact that, in self-accelerating models, the background gravitational coupling weakens at intermediate redshifts, over-compensating the effect of the attractive scalar force. (2) At higher redshifts, the opposite happens; we identify a period of super-growth when the linear growth rate is larger than that predicted by $\Lambda$CDM. (3) The gravitational slip parameter $\eta$ – the ratio of the space part of the metric perturbation to the time part – is bounded from above. For Brans-Dicke-type theories $\eta$ is at most unity. For more general theories, $\eta$ can exceed unity at intermediate redshifts, but not more than about $1.5$ if, at the same time, the linear growth rate is to be compatible with current observational constraints. We caution against phenomenological parametrization of data that do not correspond to predictions from viable physical theories. We advocate the EFT approach as a way to constrain new physics from future large-scale-structure data.

### Phenomenology of dark energy: general features of large-scale perturbations [Replacement]

We present a systematic exploration of dark energy and modified gravity models containing a single scalar field non-minimally coupled to the metric. Even though the parameter space is large, by exploiting an effective field theory (EFT) formulation and by imposing simple physical constraints such as stability conditions and (sub-)luminal propagation of perturbations, we arrive at a number of generic predictions. (1) The linear growth rate of matter density fluctuations is generally suppressed compared to $\Lambda$CDM at intermediate redshifts ($0.5 \lesssim z \lesssim 1$), despite the introduction of an attractive long-range scalar force. This is due to the fact that, in self-accelerating models, the background gravitational coupling weakens at intermediate redshifts, over-compensating the effect of the attractive scalar force. (2) At higher redshifts, the opposite happens; we identify a period of super-growth when the linear growth rate is larger than that predicted by $\Lambda$CDM. (3) The gravitational slip parameter $\eta$ – the ratio of the space part of the metric perturbation to the time part – is bounded from above. For Brans-Dicke-type theories $\eta$ is at most unity. For more general theories, $\eta$ can exceed unity at intermediate redshifts, but not more than about $1.5$ if, at the same time, the linear growth rate is to be compatible with current observational constraints. We caution against phenomenological parametrization of data that do not correspond to predictions from viable physical theories. We advocate the EFT approach as a way to constrain new physics from future large-scale-structure data.

### Primordial Power Spectra of EiBI Inflation in Strong Gravity Limit [Cross-Listing]

We investigate the scalar and the tensor perturbations of the $\varphi^2$ inflation model in the strong-gravity limit of Eddington-inspired Born-Infeld (EiBI) theory. In order to consider the strong EiBI-gravity effect, we take the value of $\kappa$ large, where $\kappa$ is the EiBI theory parameter. The energy density of the Universe at the early stage is very high, and the Universe is in a strong-gravity regime. Therefore, the perturbation feature is not altered from what was investigated earlier. At the attractor inflationary stage, however, the feature is changed in the strong EiBI-gravity limit. The correction to the scalar perturbation in this limit comes mainly via the background matter field, while that to the tensor perturbation comes directly from the gravity ($\kappa$) effect. The change in the value of the scalar spectrum is little compared with that in the weak EiBI-gravity limit, or in GR. The form of the tensor spectrum is the same with that in the weak limit, but the value of the spectrum can be suppressed down to zero in the strong limit. Therefore, the resulting tensor-to-scalar ratio can also be suppressed in the same way, which makes $\varphi^2$ model in EiBI theory viable.

### Primordial Power Spectra of EiBI Inflation in Strong Gravity Limit

We investigate the scalar and the tensor perturbations of the $\varphi^2$ inflation model in the strong-gravity limit of Eddington-inspired Born-Infeld (EiBI) theory. In order to consider the strong EiBI-gravity effect, we take the value of $\kappa$ large, where $\kappa$ is the EiBI theory parameter. The energy density of the Universe at the early stage is very high, and the Universe is in a strong-gravity regime. Therefore, the perturbation feature is not altered from what was investigated earlier. At the attractor inflationary stage, however, the feature is changed in the strong EiBI-gravity limit. The correction to the scalar perturbation in this limit comes mainly via the background matter field, while that to the tensor perturbation comes directly from the gravity ($\kappa$) effect. The change in the value of the scalar spectrum is little compared with that in the weak EiBI-gravity limit, or in GR. The form of the tensor spectrum is the same with that in the weak limit, but the value of the spectrum can be suppressed down to zero in the strong limit. Therefore, the resulting tensor-to-scalar ratio can also be suppressed in the same way, which makes $\varphi^2$ model in EiBI theory viable.

### Primordial Power Spectra of EiBI Inflation in Strong Gravity Limit [Cross-Listing]

We investigate the scalar and the tensor perturbations of the $\varphi^2$ inflation model in the strong-gravity limit of Eddington-inspired Born-Infeld (EiBI) theory. In order to consider the strong EiBI-gravity effect, we take the value of $\kappa$ large, where $\kappa$ is the EiBI theory parameter. The energy density of the Universe at the early stage is very high, and the Universe is in a strong-gravity regime. Therefore, the perturbation feature is not altered from what was investigated earlier. At the attractor inflationary stage, however, the feature is changed in the strong EiBI-gravity limit. The correction to the scalar perturbation in this limit comes mainly via the background matter field, while that to the tensor perturbation comes directly from the gravity ($\kappa$) effect. The change in the value of the scalar spectrum is little compared with that in the weak EiBI-gravity limit, or in GR. The form of the tensor spectrum is the same with that in the weak limit, but the value of the spectrum can be suppressed down to zero in the strong limit. Therefore, the resulting tensor-to-scalar ratio can also be suppressed in the same way, which makes $\varphi^2$ model in EiBI theory viable.

### A relativistic signature in large-scale structure: Scale-dependent bias from single-field inflation

In General Relativity, the constraint equation relating metric and density perturbations is inherently nonlinear, leading to an effective non-Gaussianity in the density field on large scales — even if the primordial metric perturbation is Gaussian. This imprints a relativistic signature in large-scale structure which is potentially observable, for example via a scale-dependent galaxy bias. The effect has been derived and then confirmed by independent calculations, using second-order perturbation theory. Recently, the physical reality of this relativistic effect has been disputed. The counter-argument is based on the claim that a very long wavelength curvature perturbation can be removed by a coordinate transformation. We argue that while this is true locally, the large-scale curvature cannot be removed by local coordinate transformations. The transformation itself contains the long-wavelength modes and thus includes the correlation. We show how the separate universe approach can be used to understand this correlation, confirming the results of perturbation theory.

### A relativistic signature in large-scale structure: Scale-dependent bias from single-field inflation [Cross-Listing]

In General Relativity, the constraint equation relating metric and density perturbations is inherently nonlinear, leading to an effective non-Gaussianity in the density field on large scales — even if the primordial metric perturbation is Gaussian. This imprints a relativistic signature in large-scale structure which is potentially observable, for example via a scale-dependent galaxy bias. The effect has been derived and then confirmed by independent calculations, using second-order perturbation theory. Recently, the physical reality of this relativistic effect has been disputed. The counter-argument is based on the claim that a very long wavelength curvature perturbation can be removed by a coordinate transformation. We argue that while this is true locally, the large-scale curvature cannot be removed by local coordinate transformations. The transformation itself contains the long-wavelength modes and thus includes the correlation. We show how the separate universe approach can be used to understand this correlation, confirming the results of perturbation theory.

### Inclusion of isospin breaking effects in lattice simulations [Cross-Listing]

Isospin symmetry is explicitly broken in the Standard Model by the mass and electric charge of the up and down quarks. These effects represent a perturbation of hadronic amplitudes at the percent level. Although these contributions are small, they play a crucial role in hadronic and nuclear physics. Moreover, as lattice computations are becoming increasingly precise, it is becoming more and more important to include these effects in numerical simulations. We summarize here how to properly define QCD and QED on a finite and discrete space-time so that isospin corrections to hadronic observables can be computed ab-initio and we review the main results on the isospin corrections to the hadron spectrum. We mainly focus on the recent work going beyond the electro-quenched approximation.

### On the bounds of RMS radii of heavy flavored mesons in a new approach to potential model [Replacement]

We report the results for bounds on r.m.s. radii of heavy flavored mesons in a Potential model with the Cornell potential $V(r)=-4\alpha_s/3r + br + c$. As the potential is not analytically solvable, we use the Dalgarno’s method of perturbation to obtain the total wave function with the Coulombic part as parent and linear part as perturbation for a particular short distance scale and then with linear part as parent and coulombic part as perturbation to contribute the long distance effect.