# Posts Tagged perturbation

## Recent Postings from perturbation

### The B-mode polarization of CMB and Cosmic Neutrino Background [Cross-Listing]

It is known that in contrast with the E-mode polarization, the B-mode polarization of the Cosmic Microwave Background cannot be generated by the Compton scattering in the case of scalar mode of metric perturbation. However it is possible to generate the B-mode by the Compton scattering in the case of tensor mode of metric perturbation. For this reason, the ratio of tensor to scalar modes of metric perturbation ($r\sim C_{Bl}/C_{El}$) is estimated by comparing the B-mode power spectrum with the E-mode at least for small $l$. We study the CMB polarization specially B-mode due to the weak interaction of Cosmic Neutrino Background (CNB) and CMB, in addition to the Compton scattering in both cases of scalar and tensor metric perturbations. It is shown that the power spectrum $C_{Bl}$ of the B-mode polarization receives some contributions from scalar and tensor modes, which have effects on the value of $r$-parameter. We also show that the B-mode polarization power spectrum can be used as an indirect probe into the CNB.

### The B-mode polarization of CMB and Cosmic Neutrino Background [Cross-Listing]

It is known that in contrast with the E-mode polarization, the B-mode polarization of the Cosmic Microwave Background cannot be generated by the Compton scattering in the case of scalar mode of metric perturbation. However it is possible to generate the B-mode by the Compton scattering in the case of tensor mode of metric perturbation. For this reason, the ratio of tensor to scalar modes of metric perturbation ($r\sim C_{Bl}/C_{El}$) is estimated by comparing the B-mode power spectrum with the E-mode at least for small $l$. We study the CMB polarization specially B-mode due to the weak interaction of Cosmic Neutrino Background (CNB) and CMB, in addition to the Compton scattering in both cases of scalar and tensor metric perturbations. It is shown that the power spectrum $C_{Bl}$ of the B-mode polarization receives some contributions from scalar and tensor modes, which have effects on the value of $r$-parameter. We also show that the B-mode polarization power spectrum can be used as an indirect probe into the CNB.

### The B-mode polarization of CMB and Cosmic Neutrino Background

It is known that in contrast with the E-mode polarization, the B-mode polarization of the Cosmic Microwave Background cannot be generated by the Compton scattering in the case of scalar mode of metric perturbation. However it is possible to generate the B-mode by the Compton scattering in the case of tensor mode of metric perturbation. For this reason, the ratio of tensor to scalar modes of metric perturbation ($r\sim C_{Bl}/C_{El}$) is estimated by comparing the B-mode power spectrum with the E-mode at least for small $l$. We study the CMB polarization specially B-mode due to the weak interaction of Cosmic Neutrino Background (CNB) and CMB, in addition to the Compton scattering in both cases of scalar and tensor metric perturbations. It is shown that the power spectrum $C_{Bl}$ of the B-mode polarization receives some contributions from scalar and tensor modes, which have effects on the value of $r$-parameter. We also show that the B-mode polarization power spectrum can be used as an indirect probe into the CNB.

### On the Origin of Flux Ratio Anomaly in Quadruple Lens Systems [Cross-Listing]

We explore the origin of flux ratio anomaly in quadruple lens systems. Using a semi-analytic method based on $N$-body simulations, we estimate the effect of possible magnification perturbation caused by subhaloes with a mass scale of <~ $10^9\,h^{-1} \textrm{M}_\odot$ in lensing galaxy haloes. Taking into account astrometric shifts by perturbers, we find that the expected change to the flux ratios per a multiply lensed image is just a few percent and the mean of the expected convergence perturbation at the effective Einstein radius of the lensing galaxy halo is $\langle \delta \kappa_{\textrm{sub}} \rangle = 0.003$, corresponding to the mean of the ratio of a projected dark matter mass fraction in subhaloes $\langle f_{\textrm{sub}} \rangle = 0.006$ for observed 11 quadruple lens systems. In contrast, the expected change to the flux ratio caused by line-of-sight structures in intergalactic spaces is typically ~10 percent and the mean of the convergence perturbation is $\langle |\delta \kappa_{\textrm{los}}| \rangle = 0.008$, corresponding to $\langle f_{\textrm{los}} \rangle = 0.017$. The contribution of magnification perturbation caused by subhaloes is $\sim 40$ percent of the total at a source redshift $z_\textrm{S}= 0.7$ and decreases monotonically in $z_\textrm{S}$ to $\sim 20$ percent at $z_\textrm{S}= 3.6$. Assuming statistical isotropy, the convergence perturbation estimated from the 11 systems has a positive correlation with the source redshift $z_\textrm{S}$, which is much stronger than that with the lens redshift $z_{\textrm{L}}$. This feature also supports the idea that the flux ratio anomaly is caused mainly by line-of-sight structures rather than subhaloes. We also discuss about a possible imprint of line-of-sight structures in demagnification of minimum images due to locally underdense structures in the line of sight.

### On the Origin of Flux Ratio Anomaly in Quadruple Lens Systems

We explore the origin of flux ratio anomaly in quadruple lens systems. Using a semi-analytic method based on $N$-body simulations, we estimate the effect of possible magnification perturbation caused by subhaloes with a mass scale of <~ $10^9\,h^{-1} \textrm{M}_\odot$ in lensing galaxy haloes. Taking into account astrometric shifts by perturbers, we find that the expected change to the flux ratios per a multiply lensed image is just a few percent and the mean of the expected convergence perturbation at the effective Einstein radius of the lensing galaxy halo is $\langle \delta \kappa_{\textrm{sub}} \rangle = 0.003$, corresponding to the mean of the ratio of a projected dark matter mass fraction in subhaloes $\langle f_{\textrm{sub}} \rangle = 0.006$ for observed 11 quadruple lens systems. In contrast, the expected change to the flux ratio caused by line-of-sight structures in intergalactic spaces is typically ~10 percent and the mean of the convergence perturbation is $\langle |\delta \kappa_{\textrm{los}}| \rangle = 0.008$, corresponding to $\langle f_{\textrm{los}} \rangle = 0.017$. The contribution of magnification perturbation caused by subhaloes is $\sim 40$ percent of the total at a source redshift $z_\textrm{S}= 0.7$ and decreases monotonically in $z_\textrm{S}$ to $\sim 20$ percent at $z_\textrm{S}= 3.6$. Assuming statistical isotropy, the convergence perturbation estimated from the 11 systems has a positive correlation with the source redshift $z_\textrm{S}$, which is much stronger than that with the lens redshift $z_{\textrm{L}}$. This feature also supports the idea that the flux ratio anomaly is caused mainly by line-of-sight structures rather than subhaloes. We also discuss about a possible imprint of line-of-sight structures in demagnification of minimum images due to locally underdense structures in the line of sight.

### Attractor non-equilibrium stationary states in perturbed long-range interacting systems [Cross-Listing]

Isolated long-range interacting particle systems appear generically to relax to non-equilibrium states ("quasi-stationary states" or QSS) which are stationary in the thermodynamic limit. A fundamental open question concerns the "robustness" of these states when the system is not isolated. In this paper we explore, using both analytical and numerical approaches to a paradigmatic one dimensional model, the effect of a simple class of perturbations. We call them "internal local perturbations" in that the particle energies are perturbed at collisions in a way which depends only on the local properties. Our central finding is that the effect of the perturbations is to drive all the very different QSS we consider towards a unique QSS. The latter is thus independent of the initial conditions of the system, but determined instead by both the long-range forces and the details of the perturbations applied. Thus in the presence of such a perturbation the long-range system evolves to a unique non-equilibrium stationary state, completely different to its state in absence of the perturbation, and it remains in this state when the perturbation is removed. We argue that this result may be generic for long-range interacting systems subject to perturbations which are dependent on the local properties (e.g. spatial density or velocity distribution) of the system itself.

### Non-linear dense core formation in the dark cloud L1517

We present a solution for the observed core fragmentation of filaments in the Taurus L1517 dark cloud which previously could not be explained \citep{hacar2011}. Core fragmentation is a vital step for the formation of stars. Observations suggest a connection to the filamentary structure of the cloud gas but it remains unclear which process is responsible. We show that the gravitational instability process of an isothermal cylinder can account for the exhibited fragmentation under the assumption that the perturbation grows on the dominant wavelength. We use numerical simulations with the code RAMSES, estimate observed column densities and line-of-sight velocities and compare them to the observations. A critical factor for the observed fragmentation is that cores grow by redistributing mass within the filament and thus the density between the cores decreases over the fragmentation process. This often leads to wrong dominant wavelength estimates as it is strongly dependent on the initial central density. We argue that non-linear effects also play an important role on the evolution of the fragmentation. Once the density perturbation grows above the critical line-mass, non-linearity leads to an enhancement of the central core density in comparison to the analytical prediction. Choosing the correct initial conditions with perturbation strengths of around 20\%, leads to inclination corrected line-of-sight velocities and central core densities within the observational measurement error in a realistic evolution time.

### Cosmological perturbations in mimetic Horndeski gravity [Cross-Listing]

We study linear scalar perturbations around a flat FLRW background in mimetic Horndeski gravity. In the absence of matter, we show that the Newtonian potential satisfies a second-order differential equation with no spatial derivatives. This implies that the sound speed for scalar perturbations is exactly zero on this background. We also show that in mimetic $G^3$ theories the sound speed is equally zero. We obtain the equation of motion for the comoving curvature perturbation (first order differential equation) and solve it to find that the comoving curvature perturbation is constant on all scales in mimetic Horndeski gravity. We find solutions for the Newtonian potential evolution equation in two simple models. Finally we show that the sound speed is zero on all backgrounds and therefore the system does not have any wave-like scalar degrees of freedom.

### Cosmological perturbations in mimetic Horndeski gravity

We study linear scalar perturbations around a flat FLRW background in mimetic Horndeski gravity. In the absence of matter, we show that the Newtonian potential satisfies a second-order differential equation with no spatial derivatives. This implies that the sound speed for scalar perturbations is exactly zero on this background. We also show that in mimetic $G^3$ theories the sound speed is equally zero. We obtain the equation of motion for the comoving curvature perturbation (first order differential equation) and solve it to find that the comoving curvature perturbation is constant on all scales in mimetic Horndeski gravity. We find solutions for the Newtonian potential evolution equation in two simple models. Finally we show that the sound speed is zero on all backgrounds and therefore the system does not have any wave-like scalar degrees of freedom.

### Transient dynamics of perturbations in astrophysical disks

This paper reviews some aspects of one of the major unsolved problems in understanding astrophysical (in particular, accretion) disks: whether the disk interiors may be effectively viscous in spite of the absence of marnetorotational instability? In this case a rotational homogeneous inviscid flow with a Keplerian angular velocity profile is spectrally stable, making the transient growth of perturbations a candidate mechanism for energy transfer from the regular motion to perturbations. Transient perturbations differ qualitatively from perturbation modes and can grow substantially in shear flows due to the nonnormality of their dynamical evolution operator. Since the eigenvectors of this operator, alias perturbation modes, are mutually nonorthogonal, they can mutually interfere, resulting in the transient growth of their linear combinations. Physically, a growing transient perturbation is a leading spiral whose branches are shrunk as a result of the differential rotation of the flow. This paper discusses in detail the transient growth of vortex shear harmonics in the spatially local limit as well as methods for identifying the optimal (fastest growth) perturbations. Special attention is given to obtaining such solutions variationally, by integrating the direct and adjoint equations forward and backward in time, respectively. The material is presented in a newcomer-friendly style.

### Asteroid flux towards circumprimary habitable zones in binary star systems: II. Dynamics

Secular and mean motion resonances (hearafter MMR) are effective perturbations to shape planetary systems. In binary star systems, they play a key role during the early and late phases of planetary formation as well as the dynamical stability of a planetary system. In this study, we aim to correlate the presence of orbital resonances with the rate of icy asteroids crossing the habitable zone (hearafter HZ), from a circumprimary disk of planetesimals in various binary star systems. We modelled a belt of small bodies in the inner and outer regions, respectively below and beyond the orbit of a gas giant planet. The planetesimals are equally placed around a primary G-type star and move under the gravitational influence of the two stars and the gas giant. We numerically integrated the system for 50 Myr considering various parameters for the secondary star. Its stellar type varies from a M- to F-type; its semimajor axis is either 50 au or 100 au and its eccentricity is either 0.1 or 0.3. Our simulations highlight that a disk of planetesimals will suffer from perturbations due to a perturbed gas giant, mean motion and secular perturbations. We show that a secular perturbation -- which location and width vary according to the secondary star's characteristics -- can exist in the region of the icy asteroid belt region and overlap with MMRs which will have an impact on the dynamical lifetime of the disk. In addition, we point out that in any case, the 2:1 MMR, the 5:3 MMR and the secular perturbed area are powerful perturbations for the transport of icy material into the HZ.

### Thermal Fluctuations of Dark Matter in Bouncing Cosmology

We investigate the statistical nature of the dark matter particles produced in bouncing cosmology, including its total energy and the evolution of its sub-horizon and super-horizon thermal fluctuations. We find that the super-horizon modes of the dark matter thermal perturbations are developing during the generic bouncing universe scenario--in contrast to the case that no significant super-horizon thermal perturbations of dark matter appear in the inflation scenario such as WIMP(-less) miracles. By explicitly deriving and solving the equation of motion of super-horizon mode, we fully determine the evolution of thermal perturbation of dark matter in a generic bouncing background. And we also prove that the evolution of super-horizon modes is stable and will not ruin out the background evolution till the Planck scale.

### Adiabaticity and gravity theory independent conservation laws for cosmological perturbations [Cross-Listing]

We carefully study the implications of adiabaticity for the behavior of cosmological perturbations. There are essentially three similar but different definitions of non-adiabaticity: one is appropriate for a thermodynamic fluid $\delta P_{nad}$, another is for a general matter field $\delta P_{c,nad}$, and the last one is valid only on superhorizon scales. The first two definitions coincide if $c_s^2=c_w^2$ where $c_s$ is the propagation speed of the perturbation, while $c_w^2=\dot P/\dot\rho$. Assuming the adiabaticity in the general sense, $\delta P_{c,nad}=0$, we derive a relation between the lapse function in the comoving slicing $A_c$ and $\delta P_{nad}$ valid for arbitrary matter field in any theory of gravity, by using only momentum conservation. The relation implies that as long as $c_s\neq c_w$, the uniform density, comoving and the proper-time slicings coincide approximately for any gravity theory and for any matter field if $\delta P_{nad}=0$ approximately. In the case of general relativity this gives the equivalence between the comoving curvature perturbation $R_c$ and the uniform density curvature perturbation $\zeta$ on superhorizon scales, and their conservation. This is realized on superhorizon scales in standard slow-roll inflation. We then consider an example in which $c_w=c_s$, where $\delta P_{nad}=\delta P_{c,nad}=0$ exactly, but the equivalence between $R_c$ and $\zeta$ no longer holds. Namely we consider the so-called ultra slow-roll inflation. In this case both $R_c$ and $\zeta$ are not conserved. In particular, as for $\zeta$, we find that it is crucial to take into account the next-to-leading order term in $\zeta$'s spatial gradient expansion to show its non-conservation, even on superhorizon scales. This is an example of the fact that adiabaticity is not always enough to ensure the conservation of $R_c$ or $\zeta$.

### Adiabaticity and gravity theory independent conservation laws for cosmological perturbations

We carefully study the implications of adiabaticity for the behavior of cosmological perturbations. There are essentially three similar but different definitions of non-adiabaticity: one is appropriate for a thermodynamic fluid $\delta P_{nad}$, another is for a general matter field $\delta P_{c,nad}$, and the last one is valid only on superhorizon scales. The first two definitions coincide if $c_s^2=c_w^2$ where $c_s$ is the propagation speed of the perturbation, while $c_w^2=\dot P/\dot\rho$. Assuming the adiabaticity in the general sense, $\delta P_{c,nad}=0$, we derive a relation between the lapse function in the comoving slicing $A_c$ and $\delta P_{nad}$ valid for arbitrary matter field in any theory of gravity, by using only momentum conservation. The relation implies that as long as $c_s\neq c_w$, the uniform density, comoving and the proper-time slicings coincide approximately for any gravity theory and for any matter field if $\delta P_{nad}=0$ approximately. In the case of general relativity this gives the equivalence between the comoving curvature perturbation $R_c$ and the uniform density curvature perturbation $\zeta$ on superhorizon scales, and their conservation. This is realized on superhorizon scales in standard slow-roll inflation. We then consider an example in which $c_w=c_s$, where $\delta P_{nad}=\delta P_{c,nad}=0$ exactly, but the equivalence between $R_c$ and $\zeta$ no longer holds. Namely we consider the so-called ultra slow-roll inflation. In this case both $R_c$ and $\zeta$ are not conserved. In particular, as for $\zeta$, we find that it is crucial to take into account the next-to-leading order term in $\zeta$'s spatial gradient expansion to show its non-conservation, even on superhorizon scales. This is an example of the fact that adiabaticity is not always enough to ensure the conservation of $R_c$ or $\zeta$.

### Quark Condensate from Renormalization Group Optimized Spectral Density

Our renormalization group consistent variant of optimized perturbation, RGOPT, is used to calculate the nonperturbative QCD spectral density of the Dirac operator and the related chiral quark condensate $\langle \bar q q \rangle$, for $n_f=2$ and $n_f=3$ massless quarks. Sequences of approximations at two-, three-, and four-loop orders are very stable and give $\langle \bar q q \rangle^{1/3}_{n_f=2}(2\, {\rm GeV}) = -(0.833-0.845) \bar\Lambda_2$, and $\langle \bar q q \rangle^{1/3}_{n_f=3}(2\, {\rm GeV}) = -(0.814-0.838) \bar\Lambda_3$ where the range is our estimated theoretical error and $\bar\Lambda_{n_f}$ the basic QCD scale in the $\rm \bar{MS}$-scheme. We compare those results with other recent determinations (from lattice calculations and spectral sum rules).

### Quark Condensate from Renormalization Group Optimized Spectral Density [Cross-Listing]

Our renormalization group consistent variant of optimized perturbation, RGOPT, is used to calculate the nonperturbative QCD spectral density of the Dirac operator and the related chiral quark condensate $\langle \bar q q \rangle$, for $n_f=2$ and $n_f=3$ massless quarks. Sequences of approximations at two-, three-, and four-loop orders are very stable and give $\langle \bar q q \rangle^{1/3}_{n_f=2}(2\, {\rm GeV}) = -(0.833-0.845) \bar\Lambda_2$, and $\langle \bar q q \rangle^{1/3}_{n_f=3}(2\, {\rm GeV}) = -(0.814-0.838) \bar\Lambda_3$ where the range is our estimated theoretical error and $\bar\Lambda_{n_f}$ the basic QCD scale in the $\rm \bar{MS}$-scheme. We compare those results with other recent determinations (from lattice calculations and spectral sum rules).

### Quark Condensate from Renormalization Group Optimized Spectral Density [Cross-Listing]

Our renormalization group consistent variant of optimized perturbation, RGOPT, is used to calculate the nonperturbative QCD spectral density of the Dirac operator and the related chiral quark condensate $\langle \bar q q \rangle$, for $n_f=2$ and $n_f=3$ massless quarks. Sequences of approximations at two-, three-, and four-loop orders are very stable and give $\langle \bar q q \rangle^{1/3}_{n_f=2}(2\, {\rm GeV}) = -(0.833-0.845) \bar\Lambda_2$, and $\langle \bar q q \rangle^{1/3}_{n_f=3}(2\, {\rm GeV}) = -(0.814-0.838) \bar\Lambda_3$ where the range is our estimated theoretical error and $\bar\Lambda_{n_f}$ the basic QCD scale in the $\rm \bar{MS}$-scheme. We compare those results with other recent determinations (from lattice calculations and spectral sum rules).

### Complete Hamiltonian analysis of cosmological perturbations at all orders [Cross-Listing]

In this work, we present a consistent Hamiltonian analysis of cosmological perturbations at all orders. To make the procedure transparent, we consider a simple model and resolve the `gauge-fixing' issues and extend the analysis to scalar field models and show that our approach can be applied to any order of perturbation for any first order derivative fields. In the case of Galilean scalar fields, our procedure can extract constrained relations at all orders in perturbations leading to the fact that there is no extra degrees of freedom due to the presence of higher time derivatives of the field in the Lagrangian. We compare and contrast our approach to the Lagrangian approach (Chen et al [2006]) for extracting higher order correlations and show that our approach is quick and robust and can be applied to any model of gravity and matter fields.

### Analytic self-force calculations in the post-Newtonian regime: eccentric orbits on a Schwarzschild background [Replacement]

We present a method for solving the first-order field equations in a post-Newtonian (PN) expansion. Our calculations generalize work of Bini and Damour and subsequently Kavanagh et al., to consider eccentric orbits on a Schwarzschild background. We derive expressions for the retarded metric perturbation at the location of the particle for all $\ell$-modes. We find that, despite first appearances, the Regge-Wheeler gauge metric perturbation is $C^0$ at the particle for all $\ell$. As a first use of our solutions, we compute the gauge-invariant quantity $\langle U \rangle$ through 4PN while simultaneously expanding in eccentricity through $e^{10}$. By anticipating the $e\to 1$ singular behavior at each PN order, we greatly improve the accuracy of our results for large $e$. We use $\langle U \rangle$ to find 4PN contributions to the effective one body potential $\hat Q$ through $e^{10}$ and at linear order in the mass-ratio.

### Rotation of the polarization vector from distant radio galaxies in the perturbed FRW metric

Analysis of the correlation between the angular positions of distant radio galaxies on the spherical sky and the orientations of their polarization vectors with respect to their major axes indicates a dipolar anisotropy in the large scale. We have considered FRW metric with a single mode of large-scale scalar perturbation which includes both anisotropy and inhomogeneity. Using Newman-Penrose formalism, we have determined the effect of the perturbation on the change of the angle between the galaxy major axis and its polarization vector as the radiation propagates. We argue that this perturbation can lead to the observed dipole anisotropy in the distribution of radio polarization.

### Rotation of the polarization vector from distant radio galaxies in the perturbed FRW metric [Replacement]

Analysis of the correlation between the angular positions of distant radio galaxies on the spherical sky and the orientations of their polarization vectors with respect to their major axes, indicates a dipolar anisotropy in the large scale. We have considered FRW metric with a single mode of large-scale scalar perturbation which includes both anisotropy and inhomogeneity. Using Newman-Penrose formalism, we have determined the effect of the perturbation on the change in the angle between the galaxy major axis and its polarization vector as the radiation propagates. We argue that this perturbation can lead to the observed dipole anisotropy in the distribution of radio polarization.

### Marginal and Irrelevant Disorder in Einstein-Maxwell backgrounds [Replacement]

We study analytically the effect of a weak random chemical potential of zero average in an Einstein-Maxwell background. For uncorrelated disorder this perturbation is relevant however we show that it can become marginal or even irrelevant by tuning disorder correlations. At zero temperature we find that, to leading order in the disorder strength, the correction to the conductivity for irrelevant perturbations vanishes. In the marginal case, in order to renormalize a logarithmic divergence, we carry out a resummation of the perturbative expansion of the metric that leads to a Lifshitz-like geometry in the infrared. Disorder in this case also induces a positive correction to the conductivity. At finite temperature the black hole acquires an effective charge and the thermal conductivity has the expected Drude peak that signals the breaking of translational invariance. However the electric conductivity is not affected by the random chemical potential to leading order in the disorder strength.

### Marginal and Irrelevant Disorder in Einstein-Maxwell backgrounds [Cross-Listing]

We study analytically the effect of a weak random chemical potential of zero average in an Einstein-Maxwell background. For uncorrelated disorder this perturbation is relevant however we show that it can become marginal or even irrelevant by tuning disorder correlations. At zero temperature we find that the correction to the conductivity for irrelevant perturbations is always negative. In the marginal case, in order to renormalize a logarithmic divergence, we carry out a resummation of the perturbative expansion of the metric that leads to a Lifshitz-like geometry in the infrared. Disorder in this case also induces a logarithmic correction to the conductivity that diverges in the limit of infinite system size. This is reminiscent of the situation in two dimensional weakly disordered metals which suggests an instability of the $AdS_4$ metric and therefore the possibility of a metal-insulator transition. At finite temperature the black hole acquires an effective charge and the thermal conductivity has the expected Drude peak that signals the breaking of translational invariance. However the electric conductivity is not affected by the random chemical potential to leading order in the disorder strength.

### Marginal and Irrelevant Disorder in Einstein-Maxwell backgrounds

We study analytically the effect of a weak random chemical potential of zero average in an Einstein-Maxwell background. For uncorrelated disorder this perturbation is relevant however we show that it can become marginal or even irrelevant by tuning disorder correlations. At zero temperature we find that the correction to the conductivity for irrelevant perturbations is always negative. In the marginal case, in order to renormalize a logarithmic divergence, we carry out a resummation of the perturbative expansion of the metric that leads to a Lifshitz-like geometry in the infrared. Disorder in this case also induces a logarithmic correction to the conductivity that diverges in the limit of infinite system size. This is reminiscent of the situation in two dimensional weakly disordered metals which suggests an instability of the $AdS_4$ metric and therefore the possibility of a metal-insulator transition. At finite temperature the black hole acquires an effective charge and the thermal conductivity has the expected Drude peak that signals the breaking of translational invariance. However the electric conductivity is not affected by the random chemical potential to leading order in the disorder strength.

### 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 [Replacement]

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 homogeneous and isotropic reference model, comparing local quantities to the predictions of linear perturbation theory. We find that collapsing perturbations reach the turnaround point much earlier than expected from the reference spherical top-hat collapse model and that the local deviation of an underdensity from the homogeneous expansion can be as high as $28\%$ for an initial density contrast of $10^{-2}$. We then study, for the first time, the exact behavior of the backreaction term ${\cal Q}_{\cal D}$. We find that this term scales as the second-order perturbative prediction for small values of the initial perturbations, and that it is negative with a linearly-growing absolute value for larger perturbation amplitudes, thereby contributing to an overall deceleration of the expansion. Its magnitude, on the other hand, remains very small even for relatively large perturbations.

### Effects of nonlinear inhomogeneity on the cosmic expansion with numerical relativity [Replacement]

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 homogeneous and isotropic reference model, comparing local quantities to the predictions of linear perturbation theory. We find that collapsing perturbations reach the turnaround point much earlier than expected from the reference spherical top-hat collapse model and that the local deviation of an underdensity from the homogeneous expansion can be as high as $28\%$ for an initial density contrast of $10^{-2}$. We then study, for the first time, the exact behavior of the backreaction term ${\cal Q}_{\cal D}$. We find that this term scales as the second-order perturbative prediction for small values of the initial perturbations, and that it is negative with a linearly-growing absolute value for larger perturbation amplitudes, thereby contributing to an overall deceleration of the expansion. Its magnitude, on the other hand, remains very small even for relatively large perturbations.

### 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 [Replacement]

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.

### 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 [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

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.

### 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, $\gamma_{\Lambda}\approx 6/11$.

### Testing an Inflation Model with Nonminimal Derivative Coupling in the Light of PLANCK 2015 Data [Cross-Listing]

We study the dynamics of a generalized inflationary model in which both the scalar field and its derivatives are coupled to the gravity. We consider a general form of the nonminimal derivative coupling in order to have a complete treatment of the model. By expanding the action up to the second order in perturbation, we study the spectrum of the primordial modes of the perturbations. Also, by expanding the action up to the third order and considering the three point correlation functions, the amplitude of the non-Gaussianity of the primordial perturbations is studied both in equilateral and orthogonal configurations. Finally, by adopting some sort of potentials, we compare the model at hand with the Planck 2015 released observational data and obtain some constraints on the model's parameters space. As an important result, we show that the nonminimal couplings help to make models of chaotic inflation, that would otherwise be in tension with Planck data, in better agreement with the data. This model is consistent with observation at weak coupling limit.

### 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.