# Posts Tagged perturbation

## Recent Postings from perturbation

### Higher derivatives and power spectrum in effective single field inflation [Cross-Listing]

We study next-to-leading corrections to the effective action of the curvature perturbation obtained by integrating out the coupled heavy isocurvature perturbation. These corrections result from applying higher order derivative operators of the effective theory expansion with respect to the mass scale of the heavy modes. We find that the correction terms are suppressed by the ratio of the Hubble parameter to the heavy mass scale. The corresponding corrections to the power spectrum of the curvature perturbation are presented for a simple illustrative example.

### Higher derivatives and power spectrum in effective single field inflation [Cross-Listing]

We study next-to-leading corrections to the effective action of the curvature perturbation obtained by integrating out the coupled heavy isocurvature perturbation. These corrections result from applying higher order derivative operators of the effective theory expansion with respect to the mass scale of the heavy modes. We find that the correction terms are suppressed by the ratio of the Hubble parameter to the heavy mass scale. The corresponding corrections to the power spectrum of the curvature perturbation are presented for a simple illustrative example.

### Higher derivatives and power spectrum in effective single field inflation

We study next-to-leading corrections to the effective action of the curvature perturbation obtained by integrating out the coupled heavy isocurvature perturbation. These corrections result from applying higher order derivative operators of the effective theory expansion with respect to the mass scale of the heavy modes. We find that the correction terms are suppressed by the ratio of the Hubble parameter to the heavy mass scale. The corresponding corrections to the power spectrum of the curvature perturbation are presented for a simple illustrative example.

### On the thermal sensitivity of binary formation in collapsing molecular clouds

We report the results of a numerical study on the initial formation stages of low-mass protostellar binary systems. We determine the separation of protostellar binaries formed as a function of the initial thermal state by varying the initial temperature in a slightly modified version of the Burkert and Bodenheimer collapse test. We find that the outcome is highly sensitive to both the initial temperature of the cloud and the initial amplitude of azimuthal density perturbation A. For A=10 %, variations of only 1 unit Kelvin below 10 K lead to changes of up to 100 AU ( i.e. of order 30 %) in the instantaneous separation, whereas for this small A the initial temperatures above 10 K yield, instead of a binary, a single low-mass fragment that never reaches protostellar densities. Protostellar binaries, however, do emerge when the perturbation amplitude is increased from 10 % to 25 %. We also investigate the impact of the critical density which governs the transition from isothermal to adiabatic thermodynamic behaviour of the collapsing gas. We find that the critical density not only affects the overall structural evolution of the gas envelope, but also the size of the rotating disk structures formed during collapse as well as the number of protostellar fragments resulting from the final fragmentation of the disks. This mechanism can give rise to young protostellar objects constituting bound multiple stellar systems.

### Entropy of conformal perturbation defects

We consider perturbation defects obtained by perturbing a 2D conformal field theory (CFT) by a relevant operator on a half-plane. If the perturbed bulk theory flows to an infrared fixed point described by another CFT, the defect flows to a conformal defect between the ultraviolet and infrared fixed point CFTs. For short bulk renormalization group flows connecting two fixed points which are close in theory space we find a universal perturbative formula for the boundary entropy of the corresponding conformal perturbation defect. We compare the value of the boundary entropy that our formula gives for the flows between nearby Virasoro minimal models Mm with the boundary entropy of the defect constructed by Gaiotto in [1] and find a match at the first two orders in the 1/m expansion.

### Testable constraint on near-tribimaximal neutrino mixing

General lowest order perturbations to hermitian squared mass matrices of leptons are considered away from the tribimaximal (TBM) limit in which a weak flavor basis with mass diagonal charged leptons is chosen. The three measurable TBM-deviants are expressed linearly in terms of perturbation induced dimensionless coefficients appearing in the charged lepton and neutrino flavor eigenstates. With unnatural cancellations assumed to be absent and the charged lepton perturbation contributions to their flavor eigenstates argued to be small, we analytically derive that a deviation from maximal atmospheric neutrino mixing and CP violation in neutrino oscillations cannot both be observably large, posing the challenge of verification to forthcoming experiments at the intensity frontier.

### Thermal Effects and Sudden Decay Approximation in the Curvaton Scenario [Cross-Listing]

We study the impact of a temperature-dependent curvaton decay rate on the primordial curvature perturbation generated in the curvaton scenario. Using the familiar sudden decay approximation, we obtain an analytical expression for the curvature perturbation after the decay of the curvaton. We then investigate numerically the evolution of the background and of the perturbations during the decay. We first show that the instantaneous transfer coefficient, related to the curvaton energy fraction at the decay, can be extended into a more general parameter, which depends on the net transfer of the curvaton energy into radiation energy or, equivalently, on the total entropy ratio after the complete curvaton decay. We then compute the curvature perturbation and compare this result with the sudden decay approximation prediction.

### Thermal Effects and Sudden Decay Approximation in the Curvaton Scenario

We study the impact of a temperature-dependent curvaton decay rate on the primordial curvature perturbation generated in the curvaton scenario. Using the familiar sudden decay approximation, we obtain an analytical expression for the curvature perturbation after the decay of the curvaton. We then investigate numerically the evolution of the background and of the perturbations during the decay. We first show that the instantaneous transfer coefficient, related to the curvaton energy fraction at the decay, can be extended into a more general parameter, which depends on the net transfer of the curvaton energy into radiation energy or, equivalently, on the total entropy ratio after the complete curvaton decay. We then compute the curvature perturbation and compare this result with the sudden decay approximation prediction.

### Thermal Effects and Sudden Decay Approximation in the Curvaton Scenario [Cross-Listing]

We study the impact of a temperature-dependent curvaton decay rate on the primordial curvature perturbation generated in the curvaton scenario. Using the familiar sudden decay approximation, we obtain an analytical expression for the curvature perturbation after the decay of the curvaton. We then investigate numerically the evolution of the background and of the perturbations during the decay. We first show that the instantaneous transfer coefficient, related to the curvaton energy fraction at the decay, can be extended into a more general parameter, which depends on the net transfer of the curvaton energy into radiation energy or, equivalently, on the total entropy ratio after the complete curvaton decay. We then compute the curvature perturbation and compare this result with the sudden decay approximation prediction.

### Comparison of two approximation schemes for solving perturbations in a LTB cosmological model

Recently, the present authors studied perturbations in the Lemaitre-Tolman-Bondi cosmological model by applying the second-order perturbation theory in the dust Friedmann-Lemaitre-Robertson-Walker universe model. Before this work, the same subject was studied in some papers by analyzing linear perturbations in the Lemaitre-Tolman-Bondi cosmological model under the assumption proposed by Clarkson, Clifton and February, in which two of perturbation variables are negligible. However, it is a non-trivial issue in what situation the Clarkson-Clifton-February assumption is valid. In this paper, we investigate differences between these two approaches. It is shown that, in general, these two approaches are not compatible with each other. That is, in our perturbative procedure, the Clarkson-Clifton-February assumption is not valid at the order of our interest.

### A Nonminimal Coupling Model and its Short-Range Solar System Impact

The objective of this work is to present the effects of a nonminimally coupled model of gravity on a Solar System short range regime. For this reason, this study is only valid when the cosmological contribution is considered irrelevant. The action functional of the model involves two functions $f^1(R)$ and $f^2(R)$ of the Ricci scalar curvature $R$, where the last one multiplies the matter Lagrangian. Using a Taylor expansion around $R=0$ for both functions $f^1(R)$ and $f^2(R)$, it was found that the metric around a spherical object is a perturbation of the weak-field Schwarzschild metric. The $tt$ component of the metric, a Newtonian plus a Yukawa perturbation term, is constrained using the available observational results. First it is shown that this effect is null when the characteristic mass scales of each function $f^1(R)$ and $f^2(R)$ are identical. Besides, the conclusion is that the nonminimal coupling only affects the Yukawa contribution strength and not its range and that the Starobinsky model for inflation is not experimentally constrained. Moreover, the geodetic precession effect, obtained also from the radial perturbation of the metric, reveals to be of no relevance for the constraints.

### Acoustic geometry through perturbation of accretion rate : II - radial flow in stationary axisymmetric spacetime

We introduce a novel perturbation scheme to study the stability properties of the stationary transonic integral solutions for low angular momentum axisymmetric accretion flow in a general stationary axisymmetric space time around non rotating black holes for generalized geometric configuration of infalling matter. We discuss the emergence of the relativistic acoustic geometry as a consequence of such stability analysis and establish the fact that the space time structure of such acoustic geometry is independent of the matter geometry as well as the perturbation scheme. The acoustic metric elements obtained by perturbing the velocity potential of the background fluid flow remains the same as of that obtained by perturbing the mass accretion rate.

### Near Horizon Extremal Geometry Perturbations: Dynamical Field Perturbations vs. Parametric Variations

In arXiv:1310.2737 we formulated and derived the three universal laws governing Near Horizon Extremal Geometries (NHEG). In this work we focus on the Entropy Perturbation Law (EPL) which, similarly to the first law of black hole thermodynamics, relates perturbations of the charges labeling perturbations around a given NHEG to the corresponding entropy perturbation. We show that field perturbations governed by the linearized equations of motion and symmetry conditions which we carefully specify, satisfy the EPL. We also show that these perturbations are limited to those coming from difference of two NHEG solutions (i.e. variations on the NHEG solution parameter space). Our analysis and discussions shed light on the "no-dynamics" statements of arXiv:0906.2380 and arXiv:0906.2376.

### Acoustic geometry through perturbation of accretion rate : I - radial flow in static spacetime

We propose a novel perturbation scheme to perform the linear stability analysis of stationary transonic integral accretion solutions corresponding to the hydrodynamic non self-gravitating spherically symmetric flow of matter in a generalized static black hole spacetime. We demonstrate that a metric independent perturbation scheme, which also does not depend on the mode of perturbation, can be developed, which, not only ensures the stability of the aforementioned stationary solutions, it rather leads to the emergence of a relativistic acoustic geometry represented by a pseudo-Riemannian curved manifold as well. The acoustic metric and the corresponding line elements remains the same irrespective of whether they are obtained by perturbing the velocity potential or the mass accretion rate. This conclusion holds, as we will show in our forthcoming papers, for axisymmetric flow in any arbitrary black hole metric as well.

### On the realizability of relativistic acoustic geometry under a generalized perturbation scheme for axi-symmetric matter flow onto black holes

We propose a novel linear perturbation scheme to study the stability properties of the stationary transonic integral solutions for axisymmetric matter flow around astrophysical black holes for the Schwarzschild as well as for rotating Rindler space time. We discuss the emergence of the relativistic acoustic geometry as a consequence of such stability analysis. Our work thus makes a crucial connection between two apparently non-overlapping fields of research – the accretion astrophysics and the analogue gravity phenomena.

### Calculating the mass spectrum of primordial black holes [Cross-Listing]

We reinspect the calculation for the mass fraction of primordial black holes (PBHs) which are formed from primordial perturbations, finding that performing the calculation using the comoving curvature perturbation $\mathcal{R}_{c}$ in the standard way vastly overestimates the number of PBHs, by many orders of magnitude. This is because PBHs form shortly after horizon entry, meaning modes significantly larger than the PBH are unobservable and should not affect whether a PBH forms or not – this important effect is not taken into account by smoothing the distribution in the standard fashion. We discuss alternative methods and argue that the density contrast, $\Delta$, should be used instead as super-horizon modes are damped by a factor $k^{2}$. We make a comparison between using a Press-Schechter approach and peaks theory, finding that the two are in close agreement in the region of interest. We also investigate the effect of varying the spectral index, and the running of the spectral index, on the abundance of primordial black holes.

### Stability analysis of (a class of) anisotropic spacetimes [Replacement]

We consider spherically symmetric spacetimes sourced by a fluid with pressure anisotropy in the radial direction. We use gauge-invariant perturbation theory to study the stability of this class of spacetimes under axial perturbations. We apply our results to three diverse examples. Two examples arise as endpoints of collapse of a ball of fluid — one describes a well-behaved stellar interior and the other has a naked singularity. We prove the stability of the stellar interior both with respect to Dirichlet and quasinormal mode boundary conditions on the perturbation. Surprisingly, the naked singularity is also stable under axial perturbations. Lastly, we take the example of anisotropic cosmology to show that in this case, the relevant perturbations are those in which the direction of anisotropy is also perturbed.

### Stability analysis of (a class of) anisotropic spacetimes [Replacement]

We consider spherically symmetric spacetimes sourced by a fluid with pressure anisotropy in the radial direction. We use gauge-invariant perturbation theory to study the stability of this class of spacetimes under axial perturbations. We apply our results to three diverse examples. Two examples arise as endpoints of collapse of a ball of fluid — one describes a well-behaved stellar interior and the other has a naked singularity. We prove the stability of the stellar interior both with respect to Dirichlet and quasinormal mode boundary conditions on the perturbation. Surprisingly, the naked singularity is also stable under axial perturbations. Lastly, we take the example of anisotropic cosmology to show that in this case, the relevant perturbations are those in which the direction of anisotropy is also perturbed.

### Stability analysis of (a class of) anisotropic spacetimes [Cross-Listing]

We consider spherically symmetric spacetimes sourced by a fluid with pressure anisotropy in the radial direction. We use gauge-invariant perturbation theory to study the stability of this class of spacetimes under axial perturbations. We apply our results to three diverse examples. Two examples arise as endpoints of collapse of a ball of fluid — one describes a well-behaved stellar interior and the other has a naked singularity. We prove the stability of the stellar interior both with respect to Dirichlet and quasinormal mode boundary conditions on the perturbation. Surprisingly, the naked singularity is also stable under axial perturbations. Lastly, we take the example of anisotropic cosmology to show that in this case, the relevant perturbations are those in which the direction of anisotropy is also perturbed.

### Reanalyzing the upper limit on the tensor-to-scalar perturbation ratio r_T in a quartic potential inflationary model [Replacement]

We study the polynomial chaotic inflation model with a single scalar field in a double well quartic potential which has recently been shown to be consistent with Planck data. In particular, we study the effects of lifting the degeneracy the between two vacua on the inflationary observables, i.e. spectral index and tensor-to-scalar perturbation ratio r_T. We find that removing the degeneracy allows the model to satisfy the upper limit constraints on r_T from Planck data. We also calculate the scalar power spectrum and non-Gaussianity parameter f_NL for the primordial scalar perturbations in this model.

### Reanalyzing the upper limit on the tensor-to-scalar perturbation ratio r_T in a quartic potential inflationary model

We study the polynomial chaotic inflation model with a single scalar field in a double well quartic potential which has recently been shown to be consistent with Planck data. In particular, we study the effects of lifting the degeneracy the between two vacua on the inflationary observables, i.e. spectral index and tensor-to-scalar perturbation ratio r_T. We find that removing the degeneracy allows the model to satisfy the upper limit constraints on r_T from Planck data. We also calculate the scalar power spectrum and non-Gaussianity parameter f_{NL} for the primordial scalar perturbations in this model.

### Relevant Perturbation of Entanglement Entropy and Stationarity [Replacement]

A relevant perturbation of the entanglement entropy of a sphere is examined holographically near the UV fixed point. Varying the conformal dimension of the relevant operator, we obtain three different sectors: 1) the entanglement entropy is stationary and the perturbative expansion is well-defined with respect to the relevant coupling, 2) the entropy is stationary, but the perturbation fails, 3) the entropy is neither stationary nor perturbative. We compare our holographic results with the numerical calculation for a free massive scalar field in three-dimensions, and find a qualitative agreement between them. We speculate that these statements hold for any relevant perturbation in any quantum field theory invariant under the Poincare symmetry.

### Relevant Perturbation of Entanglement Entropy and Stationarity

A relevant perturbation of the entanglement entropy of a sphere is examined holographically near the UV fixed point. Varying the conformal dimension of the relevant operator, we obtain three different sectors: 1) the entanglement entropy is stationary and the perturbative expansion is well-defined with respect to the relevant coupling, 2) the entropy is stationary, but the perturbation fails, 3) the entropy is neither stationary nor perturbative. We compare our holographic results with the numerical calculation for a free massive scalar field in three-dimensions, and find a qualitative agreement between them. We argue that these statements hold for any relevant perturbation in any quantum field theory invariant under the Poincare symmetry.

### Relevant Perturbation of Entanglement Entropy and Stationarity [Replacement]

A relevant perturbation of the entanglement entropy of a sphere is examined holographically near the UV fixed point. Varying the conformal dimension of the relevant operator, we obtain three different sectors: 1) the entanglement entropy is stationary and the perturbative expansion is well-defined with respect to the relevant coupling, 2) the entropy is stationary, but the perturbation fails, 3) the entropy is neither stationary nor perturbative. We compare our holographic results with the numerical calculation for a free massive scalar field in three-dimensions, and find a qualitative agreement between them. We speculate that these statements hold for any relevant perturbation in any quantum field theory invariant under the Poincare symmetry.

### 3D MHD simulation of linearly polarised Alfven wave dynamics in Arnold-Beltrami-Childress magnetic field

Previous studies [Malara et al ApJ, 533, 523 (2000)] considered small-amplitude Alfven wave (AW) packets in Arnold-Beltrami-Childress (ABC) magnetic field using WKB approximation. In this work linearly polarised Alfven wave dynamics in ABC magnetic field via direct 3D MHD numerical simulation is studied for the first time. Gaussian AW pulse with length-scale much shorter than ABC domain length and harmonic AW with wavelength equal to ABC domain length are studied for four different resistivities. While it is found that AWs dissipate quickly in the ABC field, surprisingly, AW perturbation energy increases in time. In the case of the harmonic AW perturbation energy growth is transient in time, attaining peaks in both velocity and magnetic perturbation energies within timescales much smaller than resistive time. In the case of the Gaussian AW pulse velocity perturbation energy growth is still transient in time, attaining a peak within few resistive times, while magnetic perturbation energy continues to grow. It is also shown that the total magnetic energy decreases in time and this is governed by the resistive evolution of the background ABC magnetic field rather than AW damping. On contrary, when background magnetic field is uniform, the total magnetic energy decrease is prescribed by AW damping, because there is no resistive evolution of the background. By considering runs with different amplitudes and by analysing perturbation spectra, possible dynamo action by AW perturbation-induced peristaltic flow and inverse cascade of magnetic energy have been excluded. Therefore, the perturbation energy growth is attributed to a new instability. The growth rate appears to be dependent on the value of the resistivity and spatial scale of the AW disturbance. Thus, when going beyond WKB approximation, AW damping, described by full MHD equations, does not guarantee decrease of perturbation energy.

### Inflationary Tensor Perturbation in Eddington-inspired Born-Infeld gravity [Cross-Listing]

We investigate the tensor perturbation in the inflation model driven by a massive-scalar field in Eddington-inspired Born-Infeld gravity. For short wave-length modes, the perturbation feature is very similar to that of the usual chaotic inflation. For long wave-length modes, the perturbation exhibits a peculiar rise in the power spectrum which may leave a signature in the cosmic microwave background radiation.

### Inflationary Tensor Perturbation in Eddington-inspired Born-Infeld gravity

We investigate the tensor perturbation in the inflation model driven by a massive-scalar field in Eddington-inspired Born-Infeld gravity. For short wave-length modes, the perturbation feature is very similar to that of the usual chaotic inflation. For long wave-length modes, the perturbation exhibits a peculiar rise in the power spectrum which may leave a signature in the cosmic microwave background radiation.

### Inducing chaos by breaking axial symmetry in a black hole magnetosphere

While the motion of particles near a rotating, electrically neutral (Kerr) and a charged (Kerr-Newman) black hole is always strictly regular, a perturbation to the gravitational or the electromagnetic field generally leads to chaos. Transition from regular to chaotic dynamics is relatively gradual if the system preserves axial symmetry, whereas non-axisymmetry induces chaos more efficiently. Here we study the development of chaos in an oblique (electro-vacuum) magnetosphere of a magnetized black hole. Besides the strong gravity of the massive source represented by the Kerr metric we consider the presence of a weak, ordered large-scale magnetic field. An axially symmetric model consisting of a rotating black hole embedded in an aligned magnetic field is generalized by allowing an oblique direction of the field having a general inclination with respect to the rotation axis of the system. Inclination of the field acts as an additional perturbation to the motion of charged particles as it breaks the axial symmetry of the system and cancels the related integral of motion. The axial component of angular momentum is no longer conserved and the resulting system thus has three degrees of freedom. Our primary concern within this contribution is to find out how sensitive the system of bound particles is to the inclination of the field. We employ the method of the maximal Lyapunov exponent to distinguish between regular and chaotic orbits and to quantify their chaoticity. We find that even a small misalignment induces chaotic motion.

### Inducing chaos by breaking axial symmetry in a black hole magnetosphere [Replacement]

While the motion of particles near a rotating, electrically-neutral (Kerr), and charged (Kerr–Newman) black hole is always strictly regular, a perturbation in the gravitational or the electromagnetic field generally leads to chaos. The transition from regular to chaotic dynamics is relatively gradual if the system preserves axial symmetry, whereas non-axisymmetry induces chaos more efficiently. Here we study the development of chaos in an oblique (electro-vacuum) magnetosphere of a magnetized black hole. Besides the strong gravity of the massive source represented by the Kerr metric we consider the presence of a weak, ordered, large-scale magnetic field. An axially-symmetric model consisting of a rotating black hole embedded in an aligned magnetic field is generalized by allowing an oblique direction of the field having a general inclination, with respect to the rotation axis of the system. The inclination of the field acts as an additional perturbation to the motion of charged particles as it breaks the axial symmetry of the system and cancels the related integral of motion. The axial component of angular momentum is no longer conserved and the resulting system thus has three degrees of freedom. Our primary concern within this contribution is to find out how sensitive the system of bound particles is to the inclination of the field. We employ the method of the maximal Lyapunov exponent to distinguish between regular and chaotic orbits and to quantify their chaoticity. We find that even a small misalignment induces chaotic motion.

### New framework for calculating cosmological perturbations in a scenario of dark energy interacting with cold dark matter [Replacement]

Dark energy might directly interact with cold dark matter. However, in such a scenario, an early-time large-scale instability occurs occasionally, which may be due to the incorrect treatment for the pressure perturbation of dark energy as a nonadiabatic fluid. To avoid this nonphysical instability, we establish a new framework to correctly calculate the cosmological perturbations in the interacting dark energy models. Inspired by the well-known parameterized post-Friedmann approach, the condition of the dark energy pressure perturbation is replaced with the relationship between the momentum density of dark energy and that of other components on large scales. By reconciling the perturbation evolutions on the large and small scales, one can complete the perturbation equations system. The large-scale instability can be successfully avoided and the well-behaved density and metric perturbations are obtained within this framework. Our test results show that this new framework works very well and is applicable to all the interacting dark energy models.

### New framework for calculating cosmological perturbations in a scenario of dark energy interacting with cold dark matter [Replacement]

Dark energy might directly interact with cold dark matter. However, in such a scenario, an early-time large-scale instability occurs occasionally, which may be due to the incorrect treatment for the pressure perturbation of dark energy as a nonadiabatic fluid. To avoid this nonphysical instability, we establish a new framework to correctly calculate the cosmological perturbations in the interacting dark energy models. Inspired by the well-known parameterized post-Friedmann approach, the condition of the dark energy pressure perturbation is replaced with the relationship between the momentum density of dark energy and that of other components on large scales. By reconciling the perturbation evolutions on the large and small scales, one can complete the perturbation equations system. The large-scale instability can be successfully avoided and the well-behaved density and metric perturbations are obtained within this framework. Our test results show that this new framework works very well and is applicable to all the interacting dark energy models.

### New framework for calculating cosmological perturbations in a scenario of dark energy interacting with cold dark matter [Replacement]

Dark energy might directly interact with cold dark matter. However, in such a scenario, an early-time large-scale instability occurs occasionally, which may be due to the incorrect treatment for the pressure perturbation of dark energy as a nonadiabatic fluid. To avoid this nonphysical instability, we establish a new framework to correctly calculate the cosmological perturbations in the interacting dark energy models. Inspired by the well-known parameterized post-Friedmann approach, the condition of the dark energy pressure perturbation is replaced with the relationship between the momentum density of dark energy and that of other components on large scales. By reconciling the perturbation evolutions on the large and small scales, one can complete the perturbation equations system. The large-scale instability can be successfully avoided and the well-behaved density and metric perturbations are obtained within this framework. Our test results show that this new framework works very well and is applicable to all the interacting dark energy models.

### New framework for calculating cosmological perturbations in a scenario of dark energy interacting with cold dark matter

Dark energy might directly interact with cold dark matter. However, in such a scenario, an early-time large-scale instability occurs occasionally, which may be due to the incorrect treatment for the pressure perturbation of dark energy as a nonadiabatic fluid. To avoid this nonphysical instability, we establish a new framework to correctly calculate the cosmological perturbations in the interacting dark energy models. Inspired by the well-known parameterized post-Friedmann approach, the condition of the dark energy pressure perturbation is replaced with the relationship between the momentum density of dark energy and that of the other components on large scales. By reconciling the perturbation evolutions on both the large and small scales with the help of the energy-momentum conservation laws, we can complete the perturbation equations system. The large-scale instability can be successful avoided and the well-behaved density and metric perturbations are obtained within this framework. Our test results show that this new framework works very well and is applicable to all the interacting dark energy models.

### New framework for calculating cosmological perturbations in a scenario of dark energy interacting with cold dark matter [Cross-Listing]

Dark energy might directly interact with cold dark matter. However, in such a scenario, an early-time large-scale instability occurs occasionally, which may be due to the incorrect treatment for the pressure perturbation of dark energy as a nonadiabatic fluid. To avoid this nonphysical instability, we establish a new framework to correctly calculate the cosmological perturbations in the interacting dark energy models. Inspired by the well-known parameterized post-Friedmann approach, the condition of the dark energy pressure perturbation is replaced with the relationship between the momentum density of dark energy and that of the other components on large scales. By reconciling the perturbation evolutions on both the large and small scales with the help of the energy-momentum conservation laws, we can complete the perturbation equations system. The large-scale instability can be successful avoided and the well-behaved density and metric perturbations are obtained within this framework. Our test results show that this new framework works very well and is applicable to all the interacting dark energy models.

### New framework for calculating cosmological perturbations in a scenario of dark energy interacting with cold dark matter [Cross-Listing]

Dark energy might directly interact with cold dark matter. However, in such a scenario, an early-time large-scale instability occurs occasionally, which may be due to the incorrect treatment for the pressure perturbation of dark energy as a nonadiabatic fluid. To avoid this nonphysical instability, we establish a new framework to correctly calculate the cosmological perturbations in the interacting dark energy models. Inspired by the well-known parameterized post-Friedmann approach, the condition of the dark energy pressure perturbation is replaced with the relationship between the momentum density of dark energy and that of the other components on large scales. By reconciling the perturbation evolutions on both the large and small scales with the help of the energy-momentum conservation laws, we can complete the perturbation equations system. The large-scale instability can be successful avoided and the well-behaved density and metric perturbations are obtained within this framework. Our test results show that this new framework works very well and is applicable to all the interacting dark energy models.

### Parameterized Post-Friedmann Framework for Interacting Dark Energy [Replacement]

Dark energy might directly interact with cold dark matter. However, in such a scenario, an early-time large-scale instability occurs occasionally, which may be due to the incorrect treatment for the pressure perturbation of dark energy as a nonadiabatic fluid. To avoid this nonphysical instability, we establish a new framework to correctly calculate the cosmological perturbations in the interacting dark energy models. Inspired by the well-known parameterized post-Friedmann approach, the condition of the dark energy pressure perturbation is replaced with the relationship between the momentum density of dark energy and that of other components on large scales. By reconciling the perturbation evolutions on the large and small scales, one can complete the perturbation equations system. The large-scale instability can be successfully avoided and the well-behaved density and metric perturbations are obtained within this framework. Our test results show that this new framework works very well and is applicable to all the interacting dark energy models.

### New framework for calculating cosmological perturbations in a scenario of dark energy interacting with cold dark matter [Cross-Listing]

Dark energy might directly interact with cold dark matter. However, in such a scenario, an early-time large-scale instability occurs occasionally, which may be due to the incorrect treatment for the pressure perturbation of dark energy as a nonadiabatic fluid. To avoid this nonphysical instability, we establish a new framework to correctly calculate the cosmological perturbations in the interacting dark energy models. Inspired by the well-known parameterized post-Friedmann approach, the condition of the dark energy pressure perturbation is replaced with the relationship between the momentum density of dark energy and that of the other components on large scales. By reconciling the perturbation evolutions on both the large and small scales with the help of the energy-momentum conservation laws, we can complete the perturbation equations system. The large-scale instability can be successful avoided and the well-behaved density and metric perturbations are obtained within this framework. Our test results show that this new framework works very well and is applicable to all the interacting dark energy models.

### Parameterized Post-Friedmann Framework for Interacting Dark Energy [Replacement]

Dark energy might directly interact with cold dark matter. However, in such a scenario, an early-time large-scale instability occurs occasionally, which may be due to the incorrect treatment for the pressure perturbation of dark energy as a nonadiabatic fluid. To avoid this nonphysical instability, we establish a new framework to correctly calculate the cosmological perturbations in the interacting dark energy models. Inspired by the well-known parameterized post-Friedmann approach, the condition of the dark energy pressure perturbation is replaced with the relationship between the momentum density of dark energy and that of other components on large scales. By reconciling the perturbation evolutions on the large and small scales, one can complete the perturbation equations system. The large-scale instability can be successfully avoided and the well-behaved density and metric perturbations are obtained within this framework. Our test results show that this new framework works very well and is applicable to all the interacting dark energy models.

### Parameterized Post-Friedmann Framework for Interacting Dark Energy [Replacement]

Dark energy might directly interact with cold dark matter. However, in such a scenario, an early-time large-scale instability occurs occasionally, which may be due to the incorrect treatment for the pressure perturbation of dark energy as a nonadiabatic fluid. To avoid this nonphysical instability, we establish a new framework to correctly calculate the cosmological perturbations in the interacting dark energy models. Inspired by the well-known parameterized post-Friedmann approach, the condition of the dark energy pressure perturbation is replaced with the relationship between the momentum density of dark energy and that of other components on large scales. By reconciling the perturbation evolutions on the large and small scales, one can complete the perturbation equations system. The large-scale instability can be successfully avoided and the well-behaved density and metric perturbations are obtained within this framework. Our test results show that this new framework works very well and is applicable to all the interacting dark energy models.

### New framework for calculating cosmological perturbations in a scenario of dark energy interacting with cold dark matter [Replacement]

Dark energy might directly interact with cold dark matter. However, in such a scenario, an early-time large-scale instability occurs occasionally, which may be due to the incorrect treatment for the pressure perturbation of dark energy as a nonadiabatic fluid. To avoid this nonphysical instability, we establish a new framework to correctly calculate the cosmological perturbations in the interacting dark energy models. Inspired by the well-known parameterized post-Friedmann approach, the condition of the dark energy pressure perturbation is replaced with the relationship between the momentum density of dark energy and that of other components on large scales. By reconciling the perturbation evolutions on the large and small scales, one can complete the perturbation equations system. The large-scale instability can be successfully avoided and the well-behaved density and metric perturbations are obtained within this framework. Our test results show that this new framework works very well and is applicable to all the interacting dark energy models.

### Negative running prevents eternal inflation

Current data from the Planck satellite and the BICEP2 telescope favor, at around the $2 \sigma$ level, negative running of the spectral index of curvature perturbations from inflation. We show that for negative running $\alpha < 0$, the curvature perturbation amplitude has a maximum on scales larger than our current horizon size. A condition for the absence of eternal inflation is that the curvature perturbation amplitude always remain below unity on superhorizon scales. For current bounds on $n_{\rm S}$ from Planck, this corresponds to an upper bound of the running $\alpha < – 4 \times 10^{-5}$, so that even tiny running of the scalar spectral index is sufficient to prevent eternal inflation from occurring, as long as the running remains negative on scales outside the horizon. In single-field inflation models, negative running is associated with a finite duration of inflation: we show that eternal inflation may not occur even in cases where inflation lasts as long as $10^4$ e-folds.

### Negative running prevents eternal inflation [Replacement]

Current data from the Planck satellite and the BICEP2 telescope favor, at around the $2 \sigma$ level, negative running of the spectral index of curvature perturbations from inflation. We show that for negative running $\alpha < 0$, the curvature perturbation amplitude has a maximum on scales larger than our current horizon size. A condition for the absence of eternal inflation is that the curvature perturbation amplitude always remain below unity on superhorizon scales. For current bounds on $n_{\rm S}$ from Planck, this corresponds to an upper bound of the running $\alpha < – 4 \times 10^{-5}$, so that even tiny running of the scalar spectral index is sufficient to prevent eternal inflation from occurring, as long as the running remains negative on scales outside the horizon. In single-field inflation models, negative running is associated with a finite duration of inflation: we show that eternal inflation may not occur even in cases where inflation lasts as long as $10^4$ e-folds.

### Conserved Quantities in Lemaitre-Tolman-Bondi Cosmology [Replacement]

We study linear perturbations to a Lema{\^\i}tre-Tolman-Bondi (LTB) background spacetime. Studying the transformation behaviour of the perturbations under gauge transformations, we construct gauge invariant quantities. We show, using the perturbed energy conservation equation, that there are conserved quantities in LTB, in particular a spatial metric trace perturbation, \zeta_{SMTP}, which is conserved on all scales. We then briefly extend our discussion to the Lema{\^\i}tre spacetime, and construct gauge-invariant perturbations in this extension of LTB spacetime.

### Conserved Quantities in Lemaitre-Tolman-Bondi Cosmology

We study linear perturbations to a Lemaitre-Tolman-Bondi (LTB) background spacetime. Studying the transformation behaviour of the perturbations under gauge transformations, we construct gauge invariant quantities. We show, using the perturbed energy conservation equation, that there are conserved quantities in LTB, in particular a spatial metric trace perturbation, {\zeta}, which is conserved on all scales. We then briefly extend our discussion to the Lemaitre spacetime, and construct gauge-invariant perturbations in this extension of LTB spacetime.

### Conserved Quantities in Lemaitre-Tolman-Bondi Cosmology [Replacement]

We study linear perturbations to a Lema{\^\i}tre-Tolman-Bondi (LTB) background spacetime. Studying the transformation behaviour of the perturbations under gauge transformations, we construct gauge invariant quantities. We show, using the perturbed energy conservation equation, that there are conserved quantities in LTB, in particular a spatial metric trace perturbation, \zeta_{SMTP}, which is conserved on all scales. We then briefly extend our discussion to the Lema{\^\i}tre spacetime, and construct gauge-invariant perturbations in this extension of LTB spacetime.

### Conserved Quantities in Lemaitre-Tolman-Bondi Cosmology [Replacement]

We study linear perturbations to a Lemaitre-Tolman-Bondi (LTB) background spacetime. Studying the transformation behaviour of the perturbations under gauge transformations, we construct gauge invariant quantities. We show, using the perturbed energy conservation equation, that there are conserved quantities in LTB, in particular a spatial metric trace perturbation, {\zeta}, which is conserved on all scales. We then briefly extend our discussion to the Lemaitre spacetime, and construct gauge-invariant perturbations in this extension of LTB spacetime.

### BICEP2, the curvature perturbation and supersymmetry [Replacement]

The tensor fraction $r\simeq 0.16$ found by BICEP2 corresponds to a Hubble parameter $H\simeq 1.0\times 10^{14}\GeV$ during inflation. This has two implications for the (single-field) slow-roll inflation hypothesis. First, the inflaton perturbation must account for much more than $10\%$ of the curvature perturbation $\zeta$, which barring fine-tuning means that it accounts for practically all of it. It follows that a curvaton-like mechanism for generating $\zeta$ requires an alternative to slow roll such as k-inflation. Second, accepting slow-roll inflation, the excursion of the inflaton field is at least of order Planck scale. As a result, the flatness of the inflaton presumably requires a shift symmetry. I point out that if such is the case, the resulting potential is likely to have at least approximately the quadratic form suggested in 1983 by Linde, which is known to be compatible with the observed $r$ as well as the observed spectral index $\ns$. The shift symmetry does not require supersymmetry. Also, the big $H$ may rule out a GUT by restoring the symmetry and producing fatal cosmic strings. The absence of a GUT would correspond to the absence of superpartners for the Standard Model particles, which indeed have yet to be found at the LHC. It therefore seems quite possible that the quantum field theory chosen by Nature is not supersymmetric.

### BICEP2, the curvature perturbation and supersymmetry

The tensor fraction $r\simeq 0.16$ found by BICEP2 corresponds to a Hubble parameter $H\simeq 1.0\times 10^{14}\GeV$ during inflation. This has two implications for the (single-field) slow-roll inflation hypothesis. First, the inflaton perturbation must account for much more than $10\%$ of the curvature perturbation $\zeta$, which barring fine-tuning means that it accounts for practically all of it. It follows that a curvaton-like mechanism for generating $\zeta$ requires an alternative to slow roll such as k-inflation. Second, accepting slow-roll inflation, the excursion of the inflaton field is at least of order Planck scale. As a result, the flatness of the inflaton presumably requires a shift symmetry. I point out that if such is the case, the resulting potential is likely to have at least approximately the quadratic form suggested in 1983 by Linde, which is known to be compatible with the observed $r$ as well as the observed spectral index $\ns$. The shift symmetry does not require supersymmetry. Also, the big $H$ may rule out a GUT by restoring the symmetry and producing fatal cosmic strings. The absence of a GUT would correspond to the absence of superpartners for the Standard Model particles, which indeed have yet to be found at the LHC. It therefore seems quite possible that the quantum field theory chosen by Nature is not supersymmetric.

### BICEP2, the curvature perturbation and supersymmetry [Cross-Listing]

The tensor fraction $r\simeq 0.16$ found by BICEP2 corresponds to a Hubble parameter $H\simeq 1.0\times 10^{14}\GeV$ during inflation. This has two implications for the (single-field) slow-roll inflation hypothesis. First, the inflaton perturbation must account for much more than $10\%$ of the curvature perturbation $\zeta$, which barring fine-tuning means that it accounts for practically all of it. It follows that a curvaton-like mechanism for generating $\zeta$ requires an alternative to slow roll such as k-inflation. Second, accepting slow-roll inflation, the excursion of the inflaton field is at least of order Planck scale. As a result, the flatness of the inflaton presumably requires a shift symmetry. I point out that if such is the case, the resulting potential is likely to have at least approximately the quadratic form suggested in 1983 by Linde, which is known to be compatible with the observed $r$ as well as the observed spectral index $\ns$. The shift symmetry does not require supersymmetry. Also, the big $H$ may rule out a GUT by restoring the symmetry and producing fatal cosmic strings. The absence of a GUT would correspond to the absence of superpartners for the Standard Model particles, which indeed have yet to be found at the LHC. It therefore seems quite possible that the quantum field theory chosen by Nature is not supersymmetric.

### BICEP2, the curvature perturbation and supersymmetry [Cross-Listing]

The tensor fraction $r\simeq 0.16$ found by BICEP2 corresponds to a Hubble parameter $H\simeq 1.0\times 10^{14}\GeV$ during inflation. This has two implications for the (single-field) slow-roll inflation hypothesis. First, the inflaton perturbation must account for much more than $10\%$ of the curvature perturbation $\zeta$, which barring fine-tuning means that it accounts for practically all of it. It follows that a curvaton-like mechanism for generating $\zeta$ requires an alternative to slow roll such as k-inflation. Second, accepting slow-roll inflation, the excursion of the inflaton field is at least of order Planck scale. As a result, the flatness of the inflaton presumably requires a shift symmetry. I point out that if such is the case, the resulting potential is likely to have at least approximately the quadratic form suggested in 1983 by Linde, which is known to be compatible with the observed $r$ as well as the observed spectral index $\ns$. The shift symmetry does not require supersymmetry. Also, the big $H$ may rule out a GUT by restoring the symmetry and producing fatal cosmic strings. The absence of a GUT would correspond to the absence of superpartners for the Standard Model particles, which indeed have yet to be found at the LHC. It therefore seems quite possible that the quantum field theory chosen by Nature is not supersymmetric.