# Posts Tagged perturbation

## Recent Postings from perturbation

### Tensor perturbations of $f(T)$-branes

We explore the tensor perturbation of the $f(T)$ brane embedded in an AdS$_5$ spacetime. With the transverse-traceless condition, we get the tensor perturbation equation of the $f(T)$ brane and show that the stability of this brane system can be ensured. In addition, we take $f(T)=T+\alpha T^2$ as an example to analyse the localization problem of the graviton zero mode. It is shown that the graviton zero mode can be localized on the brane.

### Instability of de Sitter Reissner-Nordstrom black hole in the 1/D expansion

We study large D effective theory for D dimensional charged (Anti) de Sitter black holes. Then we show that de Sitter Reissner-Nordstrom black hole becomes unstable against gravitational perturbations at larger charge than certain critical value in higher dimension. Furthermore we find that there is a non-trivial zero-mode static perturbation at the critical charge. The existence of static perturbations suggests the appearance of non-spherical symmetric solution branches of static charged de Sitter black hole. This expectation is confirmed by constructing the non-spherical symmetric static solutions of large D effective equations.

### Mode coupling in solar spicule oscillations

In a real medium which has oscillations, the perturbations can cause the energy transfer between different modes. The perturbation interpreted as an interaction between the modes is inferred as mode coupling. Mode coupling process in an inhomogeneous medium such as solar spicules may lead to the coupling of kink waves to local Alfven waves. This coupling occurs practically in any conditions when there is smooth variation in density in the radial direction. This process is seen as the decay of transverse kink waves in the medium. To study the damping of kink waves due to mode coupling, a 2.5-dimensional numerical simulation of the initial wave is considered in spicules. The initial perturbation is assumed to be in a plane perpendicular to the spicule axis. The considered kink wave is a standing wave which shows an exponential damping in the inhomogeneous layer after occurrence of the mode coupling.

### Effects of nonlinear inhomogeneity on the cosmic expansion with numerical relativity [Cross-Listing]

We construct a three-dimensional, fully relativistic numerical model of a universe filled with an inhomogeneous pressureless fluid, starting from initial data that represent a perturbation of the Einstein-de~Sitter model. We then measure the departure of the average expansion rate with respect to this Friedmann-Lema\^itre-Robertson-Walker reference model, comparing local quantities to the predictions of linear perturbation theory and of the averaging formalism. We find local deviations from the homogeneous expansion that can be as high as $15\%$ for an initial density contrast of $10^{-2}$. We also study, for the first time, the non-perturbative behavior of the backreaction term ${\cal Q}_{\cal D}$, measuring its sign and scaling during the evolution. We find that this term scales as the second-order perturbative prediction for small values of the initial perturbations, and that it becomes negative with a linearly-growing absolute value for larger perturbation amplitudes. Its magnitude, however, remains very small even for relatively large perturbations.

### Search for Compensated Isocurvature Perturbations with Planck Power Spectra

In the standard inflationary scenario, primordial perturbations are adiabatic. The amplitudes of most types of isocurvature perturbations are generally constrained by current data to be small. If, however, there is a baryon-density perturbation that is compensated by a dark-matter perturbation in such a way that the total matter density is unperturbed, then this compensated isocurvature perturbation (CIP) has no observable consequence in the cosmic microwave background (CMB) at linear order in the CIP amplitude. Here we search for the effects of CIPs on CMB power spectra to quadratic order in the CIP amplitude. An analysis of the Planck temperature data leads to an upper bound $\Delta_{\rm rms}^2 \leq 7.1\times 10^{-3}$, at the 68\% confidence level, to the variance $\Delta_{\rm rms}^2$ of the CIP amplitude. This is then strengthened to $\Delta_{\rm rms}^2\leq 5.0\times 10^{-3}$ if Planck small-angle polarization data are included. A cosmic-variance-limited CMB experiment could improve the $1\sigma$ sensitivity to CIPs to $\Delta^2_{\rm rms} \lesssim 9\times 10^{-4}$. It is also found that adding CIPs to the standard $\Lambda$CDM model can improve the fit of the observed smoothing of CMB acoustic peaks just as much as adding a non-standard lensing amplitude.

### Spatially covariant theories of gravity: disformal transformation, cosmological perturbations and the Einstein frame [Cross-Listing]

We investigate the cosmological background evolution and perturbations in a general class of spatially covariant theories of gravity, which propagates two tensor modes and one scalar mode. We show that the structure of the theory is preserved under the disformal transformation. We also evaluate the primordial spectra for both the gravitational waves and the curvature perturbation, which are invariant under the disformal transformation. Due to the existence of higher spatial derivatives, the quadratic Lagrangian for the tensor modes itself cannot be transformed to the form in the Einstein frame. Nevertheless, there exists a one-parameter family of frames in which the spectrum of the gravitational waves takes the standard form in the Einstein frame.

### Spatially covariant theories of gravity: disformal transformation, cosmological perturbations and the Einstein frame [Cross-Listing]

We investigate the cosmological background evolution and perturbations in a general class of spatially covariant theories of gravity, which propagates two tensor modes and one scalar mode. We show that the structure of the theory is preserved under the disformal transformation. We also evaluate the primordial spectra for both the gravitational waves and the curvature perturbation, which are invariant under the disformal transformation. Due to the existence of higher spatial derivatives, the quadratic Lagrangian for the tensor modes itself cannot be transformed to the form in the Einstein frame. Nevertheless, there exists a one-parameter family of frames in which the spectrum of the gravitational waves takes the standard form in the Einstein frame.

### OPE Coefficients of the 3D Ising model with a trapping potential

Recently the OPE coefficients of the 3D Ising model universality class have been calculated by studying the two-point functions perturbed from the critical point with a relevant field. We show that this method can be applied also when the perturbation is performed with a relevant field coupled to a non uniform potential acting as a trap. This setting is described by the trap size scaling ansatz, that can be combined with the general framework of the conformal perturbation in order to write down the correlators $<\sigma (\mathbf {r})\sigma(0)>$, $<\sigma (\mathbf{r})\epsilon(0)>$ and $<\epsilon (\mathbf {r})\epsilon(0)>$, from which the OPE coefficients can be estimated. We find $C^{\sigma}_{\sigma\epsilon}= 1.051(3)$ , in agreement with the results already known in the literature, and $C^{\epsilon}_{\epsilon\epsilon}= 1.32 (15)$ , confirming and improving the previous estimate obtained in the uniform perturbation case.

### OPE Coefficients of the 3D Ising model with a trapping potential [Cross-Listing]

Recently the OPE coefficients of the 3D Ising model universality class have been calculated by studying the two-point functions perturbed from the critical point with a relevant field. We show that this method can be applied also when the perturbation is performed with a relevant field coupled to a non uniform potential acting as a trap. This setting is described by the trap size scaling ansatz, that can be combined with the general framework of the conformal perturbation in order to write down the correlators $<\sigma (\mathbf {r})\sigma(0)>$, $<\sigma (\mathbf{r})\epsilon(0)>$ and $<\epsilon (\mathbf {r})\epsilon(0)>$, from which the OPE coefficients can be estimated. We find $C^{\sigma}_{\sigma\epsilon}= 1.051(3)$ , in agreement with the results already known in the literature, and $C^{\epsilon}_{\epsilon\epsilon}= 1.32 (15)$ , confirming and improving the previous estimate obtained in the uniform perturbation case.

### Baryon Transition in Holographic QCD

We propose a mechanism of holographic baryon transition in the Sakai-Sugimoto (SS) model: baryons in this model can jump to different states under the mediated effect of gravitons (or glueballs by holography). We consider a time-dependent gravitational perturbation from M5-brane solution of D=11 supergravity and by employing the relations between 11D M-theory and IIA string theory, we get its 10 dimensional counterpart in the SS model. Such a perturbation is received by the D4-branes wrapped on the $S^{4}$ part of the 10D background, namely the baryon vertex. Technically, baryons in the SS model are described by BPST instanton ansatz and their dynamics can be analyzed using the quantum mechanical system in the instanton's moduli space. In this way, different baryonic states are marked by quantum numbers of moduli space quantum mechanics. By holographic spirit, the gravitational perturbation enters the Hamiltonian as a time-dependent perturbation and it is this time-dependent perturbative Hamiltonian produces the transition between different baryonic states. We calculate the transition probability and get the selection rule and also compute the condition for baryon transition and give the possible transition processes in the limit $\omega\gg\left|\vec{k}\right|^{2}$. Since in 10D language, the fluctuation from 11D metric are the perturbation of 10D metric and dilaton which are the modes carried by close strings, thus from the string theory point of view, our proposition can be accounted as the baryonic D4 brane jumps to different states by emitting or absorbing close strings coming from the bulk. In the viewpoints of QCD, it could be interpreted as that baryons transform to different states by interacting with glueballs as a low energy effective theory.

### Baryon Transition in Holographic QCD [Cross-Listing]

We propose a mechanism of holographic baryon transition in the Sakai-Sugimoto (SS) model: baryons in this model can jump to different states under the mediated effect of gravitons (or glueballs by holography). We consider a time-dependent gravitational perturbation from M5-brane solution of D=11 supergravity and by employing the relations between 11D M-theory and IIA string theory, we get its 10 dimensional counterpart in the SS model. Such a perturbation is received by the D4-branes wrapped on the $S^{4}$ part of the 10D background, namely the baryon vertex. Technically, baryons in the SS model are described by BPST instanton ansatz and their dynamics can be analyzed using the quantum mechanical system in the instanton's moduli space. In this way, different baryonic states are marked by quantum numbers of moduli space quantum mechanics. By holographic spirit, the gravitational perturbation enters the Hamiltonian as a time-dependent perturbation and it is this time-dependent perturbative Hamiltonian produces the transition between different baryonic states. We calculate the transition probability and get the selection rule and also compute the condition for baryon transition and give the possible transition processes in the limit $\omega\gg\left|\vec{k}\right|^{2}$. Since in 10D language, the fluctuation from 11D metric are the perturbation of 10D metric and dilaton which are the modes carried by close strings, thus from the string theory point of view, our proposition can be accounted as the baryonic D4 brane jumps to different states by emitting or absorbing close strings coming from the bulk. In the viewpoints of QCD, it could be interpreted as that baryons transform to different states by interacting with glueballs as a low energy effective theory.

### Baryon Transition in Holographic QCD [Replacement]

We propose a mechanism of holographic baryon transition in the Sakai-Sugimoto (SS) model: baryons in this model can jump to different states under the mediated effect of gravitons (or glueballs by holography). We consider a time-dependent gravitational perturbation from M5-brane solution of D=11 supergravity and by employing the relations between 11D M-theory and IIA string theory, we get its 10 dimensional counterpart in the SS model. Such a perturbation is received by the D4-branes wrapped on the $S^{4}$ part of the 10D background, namely the baryon vertex. Technically, baryons in the SS model are described by BPST instanton ansatz and their dynamics can be analyzed using the quantum mechanical system in the instanton's moduli space. In this way, different baryonic states are marked by quantum numbers of moduli space quantum mechanics. By holographic spirit, the gravitational perturbation enters the Hamiltonian as a time-dependent perturbation and it is this time-dependent perturbative Hamiltonian produces the transition between different baryonic states. We calculate the transition probability and get the selection rule and also compute the condition for baryon transition and give the possible transition processes in the limit $\omega\gg\left|\vec{k}\right|^{2}$. Since in 10D language, the fluctuation from 11D metric are the perturbation of 10D metric and dilaton which are the modes carried by close strings, thus from the string theory point of view, our proposition can be accounted as the baryonic D4 brane jumps to different states by emitting or absorbing close strings coming from the bulk. In the viewpoints of QCD, it could be interpreted as that baryons transform to different states by interacting with glueballs as a low energy effective theory.

### Baryon Transition in Holographic QCD [Replacement]

We propose a mechanism of holographic baryon transition in the Sakai-Sugimoto (SS) model: baryons in this model can jump to different states under the mediated effect of gravitons (or glueballs by holography). We consider a time-dependent gravitational perturbation from M5-brane solution of D=11 supergravity and by employing the relations between 11D M-theory and IIA string theory, we get its 10 dimensional counterpart in the SS model. Such a perturbation is received by the D4-branes wrapped on the $S^{4}$ part of the 10D background, namely the baryon vertex. Technically, baryons in the SS model are described by BPST instanton ansatz and their dynamics can be analyzed using the quantum mechanical system in the instanton's moduli space. In this way, different baryonic states are marked by quantum numbers of moduli space quantum mechanics. By holographic spirit, the gravitational perturbation enters the Hamiltonian as a time-dependent perturbation and it is this time-dependent perturbative Hamiltonian produces the transition between different baryonic states. We calculate the transition probability and get the selection rule and also compute the condition for baryon transition and give the possible transition processes in the limit $\omega\gg\left|\vec{k}\right|^{2}$. Since in 10D language, the fluctuation from 11D metric are the perturbation of 10D metric and dilaton which are the modes carried by close strings, thus from the string theory point of view, our proposition can be accounted as the baryonic D4 brane jumps to different states by emitting or absorbing close strings coming from the bulk. In the viewpoints of QCD, it could be interpreted as that baryons transform to different states by interacting with glueballs as a low energy effective theory.

### Baryon Transition in Holographic QCD [Replacement]

We propose a mechanism of holographic baryon transition in the Sakai-Sugimoto (SS) model: baryons in this model can jump to different states under the mediated effect of gravitons (or glueballs by holography). We consider a time-dependent gravitational perturbation from M5-brane solution of D=11 supergravity and by employing the relations between 11D M-theory and IIA string theory, we get its 10 dimensional counterpart in the SS model. Such a perturbation is received by the D4-branes wrapped on the $S^{4}$ part of the 10D background, namely the baryon vertex. Technically, baryons in the SS model are described by BPST instanton ansatz and their dynamics can be analyzed using the quantum mechanical system in the instanton's moduli space. In this way, different baryonic states are marked by quantum numbers of moduli space quantum mechanics. By holographic spirit, the gravitational perturbation enters the Hamiltonian as a time-dependent perturbation and it is this time-dependent perturbative Hamiltonian produces the transition between different baryonic states. We calculate the transition probability and get the selection rule and also compute the condition for baryon transition and give the possible transition processes in the limit $\omega\gg\left|\vec{k}\right|^{2}$. Since in 10D language, the fluctuation from 11D metric are the perturbation of 10D metric and dilaton which are the modes carried by close strings, thus from the string theory point of view, our proposition can be accounted as the baryonic D4 brane jumps to different states by emitting or absorbing close strings coming from the bulk. In the viewpoints of QCD, it could be interpreted as that baryons transform to different states by interacting with glueballs as a low energy effective theory.

### Growth index of matter perturbations in running vacuum models [Replacement]

We derive for the first time the growth index of matter perturbations of the FLRW flat cosmological models in which the vacuum energy depends on redshift. A particularly well motivated model of this type is the so-called quantum field vacuum, in which apart from a leading constant term $\Lambda_0$ there is also a $H^{2}$-dependence in the functional form of vacuum, namely $\Lambda(H)=\Lambda_{0}+3\nu (H^{2}-H^{2}_{0})$. Since $|\nu|\ll1$ this form endows the vacuum energy of a mild dynamics which affects the evolution of the main cosmological observables at the background and perturbation levels. Specifically, at the perturbation level we find that the growth index of the running vacuum cosmological model is $\gamma_{\Lambda_{H}} \approx \frac{6+3\nu}{11-12\nu}$ and thus it nicely extends analytically the result of the $\Lambda$CDM model, $\gamma_{\Lambda}\approx 6/11$.

### Growth index of matter perturbations in running vacuum models [Replacement]

We derive for the first time the growth index of matter perturbations of the FLRW flat cosmological models in which the vacuum energy depends on redshift. A particularly well motivated model of this type is the so-called quantum field vacuum, in which apart from a leading constant term $\Lambda_0$ there is also a $H^{2}$-dependence in the functional form of vacuum, namely $\Lambda(H)=\Lambda_{0}+3\nu (H^{2}-H^{2}_{0})$. Since $|\nu|\ll1$ this form endows the vacuum energy of a mild dynamics which affects the evolution of the main cosmological observables at the background and perturbation levels. Specifically, at the perturbation level we find that the growth index of the running vacuum cosmological model is $\gamma_{\Lambda_{H}} \approx \frac{6+3\nu}{11-12\nu}$ and thus it nicely extends analytically the result of the $\Lambda$CDM model, $\gamma_{\Lambda}\approx 6/11$.

### The growth index of matter perturbations in running vacuum models [Replacement]

We derive for the first time the growth index of matter perturbations of the FLRW flat cosmological models in which the vacuum energy depends on redshift. A particularly well motivated model of this type is the so-called quantum field vacuum, in which apart from a leading constant term $\Lambda_0$ there is also a $H^{2}$-dependence in the functional form of vacuum, namely $\Lambda(H)=\Lambda_{0}+3\nu (H^{2}-H^{2}_{0})$. Since $|\nu|\ll1$ this form endows the vacuum energy of a mild dynamics which affects the evolution of the main cosmological observables at the background and perturbation levels. Specifically, at the perturbation level we find that the growth index of the running vacuum cosmological model is $\gamma_{\Lambda_{H}} \approx \frac{6+3\nu}{11-12\nu}$ and thus it nicely extends analytically the result of the $\Lambda$CDM model, $\gamma_{\Lambda}\approx 6/11$.

### Growth index of matter perturbations in running vacuum models [Replacement]

We derive for the first time the growth index of matter perturbations of the FLRW flat cosmological models in which the vacuum energy depends on redshift. A particularly well motivated model of this type is the so-called quantum field vacuum, in which apart from a leading constant term $\Lambda_0$ there is also a $H^{2}$-dependence in the functional form of vacuum, namely $\Lambda(H)=\Lambda_{0}+3\nu (H^{2}-H^{2}_{0})$. Since $|\nu|\ll1$ this form endows the vacuum energy of a mild dynamics which affects the evolution of the main cosmological observables at the background and perturbation levels. Specifically, at the perturbation level we find that the growth index of the running vacuum cosmological model is $\gamma_{\Lambda_{H}} \approx \frac{6+3\nu}{11-12\nu}$ and thus it nicely extends analytically the result of the $\Lambda$CDM model, $\gamma_{\Lambda}\approx 6/11$.

### The growth index of matter perturbations in running vacuum models [Replacement]

We derive for the first time the growth index of matter perturbations of the FLRW flat cosmological models in which the vacuum energy depends on redshift. A particularly well motivated model of this type is the so-called quantum field vacuum, in which apart from a leading constant term $\Lambda_0$ there is also a $H^{2}$-dependence in the functional form of vacuum, namely $\Lambda(H)=\Lambda_{0}+3\nu (H^{2}-H^{2}_{0})$. Since $|\nu|\ll1$ this form endows the vacuum energy of a mild dynamics which affects the evolution of the main cosmological observables at the background and perturbation levels. Specifically, at the perturbation level we find that the growth index of the running vacuum cosmological model is $\gamma_{\Lambda_{H}} \approx \frac{6+3\nu}{11-12\nu}$ and thus it nicely extends analytically the result of the $\Lambda$CDM model, $\gamma_{\Lambda}\approx 6/11$.

### The growth index of matter perturbations in running vacuum models [Replacement]

We derive for the first time the growth index of matter perturbations of the FLRW flat cosmological models in which the vacuum energy depends on redshift. A particularly well motivated model of this type is the so-called quantum field vacuum, in which apart from a leading constant term $\Lambda_0$ there is also a $H^{2}$-dependence in the functional form of vacuum, namely $\Lambda(H)=\Lambda_{0}+3\nu (H^{2}-H^{2}_{0})$. Since $|\nu|\ll1$ this form endows the vacuum energy of a mild dynamics which affects the evolution of the main cosmological observables at the background and perturbation levels. Specifically, at the perturbation level we find that the growth index of the running vacuum cosmological model is $\gamma_{\Lambda_{H}} \approx \frac{6+3\nu}{11-12\nu}$ and thus it nicely extends analytically the result of the $\Lambda$CDM model, $\gamma_{\Lambda}\approx 6/11$.

### Entanglement Temperature and Perturbed AdS$_3$ Geometry

In analogy to the first law of thermodynamics, the increase in entanglement entropy $\delta S$ of a conformal field theory (CFT) is proportional to the increase in energy, $\delta E$, of the subsystem divided by an effective entanglement temperature, $T_E$. Extending this analogy, we study entanglement entropy when the subsystem is perturbed by applying an external field, expressed as a coupling to a local marginal operator in the CFT. We show that the resulting entropy change is associated with a change in the entanglement temperature itself, leading to an equation analogous to the Clausius relation. Using AdS/CFT duality we develop a relationship between a perturbation in the local entanglement temperature, $\delta T_E(x)$ of the CFT and the perturbation of the bulk AdS metric. Using the AdS$_3$ minimal surface as a probe, we can construct bulk metric perturbations from an exact numerical computation of the entanglement temperature in a two dimensional $c=1$ boundary theory deformed by a marginal perturbation.

### Entanglement Temperature and Perturbed AdS$_3$ Geometry [Replacement]

In analogy to the first law of thermodynamics, the increase in entanglement entropy $\delta S$ of a conformal field theory (CFT) is proportional to the increase in energy, $\delta E$, of the subsystem divided by an effective entanglement temperature, $T_E$. Extending this analogy, we study entanglement entropy when the subsystem is perturbed by applying an external field, expressed as a coupling to a local marginal operator in the CFT. We show that the resulting entropy change is associated with a change in the entanglement temperature itself, leading to an equation analogous to the Clausius relation. Using AdS/CFT duality we develop a relationship between a perturbation in the local entanglement temperature, $\delta T_E(x)$ of the CFT and the perturbation of the bulk AdS metric. Using the AdS$_3$ minimal surface as a probe, we can construct bulk metric perturbations from an exact numerical computation of the entanglement temperature in a two dimensional $c=1$ boundary theory deformed by a marginal perturbation.

### Axion as a cold dark matter candidate: Analysis to third order perturbation for classical axion [Cross-Listing]

We investigate aspects of axion as a coherently oscillating massive classical scalar field by analyzing third order perturbations in Einstein's gravity in the axion-comoving gauge. The axion fluid has its characteristic pressure term leading to an axion Jeans scale which is cosmologically negligible for a canonical axion mass. Our classically derived axion pressure term in Einstein's gravity is identical to the one derived in the non-relativistic quantum mechanical context in the literature. We show that except for the axion pressure term, the axion fluid equations are exactly the same as the general relativistic continuity and Euler equations of a zero-pressure fluid up to third order perturbation. The general relativistic density and velocity perturbations of the CDM in the CDM-comoving gauge are exactly the same as the Newtonian perturbations to the second order (in all scales), and the pure general relativistic corrections appearing from the third order are numerically negligible (in all scales as well) in the current paradigm of concordance cosmology. Therefore, here we prove that, in the super-Jeans scale, the classical axion can be handled as the Newtonian CDM fluid up to third order perturbation. We also show that the axion fluid supports the vector-type (rotational) perturbation from the third order. Our analysis includes the cosmological constant.

### 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.

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

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, when a cosmological constant term in the action is included. 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

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.

### 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

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 [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).

### 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.