We consider cosmological models with a dynamical dark energy field, and study the presence of three types of commonly found instabilities, namely ghost (when fields have negative kinetic energy), gradient (negative momentum squared) and tachyon (negative mass squared). In particular, we study the linear scalar perturbations of theories with two interacting scalar fields as a proxy for a dark energy and matter fields, and explicitly show how canonical transformations relate these three types of instabilities with each other. We generically show that low-energy ghosts are equivalent to tachyonic instabilities, and that high-energy ghosts are equivalent to gradient instabilities. Via examples we make evident the fact that whenever one of these fields exhibits an instability then the entire physical system becomes unstable, with an unbounded Hamiltonian. Finally, we discuss the role of interactions between the two fields, and show that whereas most of the time interactions will not determine whether an instability is present or not, they may affect the timescale of the instability. We also find exceptional cases in which the two fields are ghosts and hence the physical system is seemingly unstable, but the presence of interactions actually lead to stable solutions. These results are very important for assessing the viability of dark energy models that may exhibit ghost, gradient or tachyonic modes.
Gravitational instabilities of isothermal spheres are studied in the presence of a positive or negative cosmological constant, in the Newtonian limit. In gravity, the statistical ensembles are not equivalent. We perform the analysis both in the microcanonical and the canonical ensembles, for which the corresponding instabilities are known as `gravothermal catastrophe' and `isothermal collapse', respectively. In the microcanonical ensemble, no equilibria can be found for radii larger than a critical value, which is increasing with increasing cosmological constant. In contrast, in the canonical …
It is shown that in quantum gravity at finite temperature, the effective potential evaluated in the tadpole approximation can have a local minimum below a certain critical temperature. However, when the leading higher order thermal loop corrections are included, one finds that no static solution exists at high temperature.
A number of stability criteria exist for dark energy theories, associated with requiring the absence of ghost, gradient and tachyonic instabilities. Tachyonic instabilities are the least well explored of these in the dark energy context and we here discuss and derive criteria for their presence and size in detail. Our findings suggest that, while the absence of ghost and gradient instabilities is indeed essential for physically viable models and so priors associated with the absence of such instabilities significantly increase the efficiency of parameter estimations without introducing unphysi…
We revisit nonsingular cosmologies in which the limiting curvature hypothesis is realized. We study the cosmological perturbations of the theory and determine the general criteria for stability. For the simplest model, we find generic Ostrogradski instabilities unless the action contains the Weyl tensor squared with the appropriate coefficient. When considering two specific nonsingular cosmological scenarios(one inflationary and one genesis model), we find ghost and gradient instabilities throughout most of the cosmic evolution. Furthermore, we show that the theory is equivalent to a theory of…
We discuss the cosmological implications of nonlocal modifications of general relativity containing tensorial structures. Assuming the presence of standard radiation- and matter-dominated eras, we show that, except in very particular cases, the nonlocal terms contribute a rapidly growing energy density. These models therefore generically do not have a stable cosmological evolution.
We study cosmological applications of extended vector-tensor theories, whose Lagrangians contain up to two derivatives with respect to metric and vector field. We derive background equations under the assumption of homogeneous and isotropic universe and study the nature of cosmological perturbations on top of the background. We found an example of healthy cosmological solutions in non-trivial degenerate theory of gravity for the first time, where those perturbations do not suffer from any instabilities, that is, ghost and gradient instabilities.
Bimetric gravity theories allow for many different types of cosmological solutions, but not all of them are theoretically allowed. In this work we discuss the conditions to satisfy the Higuchi bound and to avoid gradient instabilities in the scalar sector at the linear level. We find that in expanding universes the ratio of the scale factors of the reference and observable metric has to increase at all times. This automatically implies a ghost-free helicity-2 and helicity-0 sector and enforces a phantom dark energy. Furthermore, the condition for the absence of gradient instabilities in the sc…
I consider couplings between the dark energy and dark matter sectors. I describe how the existence of an adiabatic regime, in which the dark energy field instantaneously tracks the minimum of its effective potential, opens the door for a catastrophic instability. This {\it adiabatic instability} tightly constrains a wide class of interacting dark sector models. This talk was presented at, and will appear in the proceedings of the DPF-2009 conference.
We consider class of modified $f(R)$ gravities with the effective cosmological constant epoch at the early and late universe. Such models pass most of solar system tests as well they satisfy to cosmological bounds. Despite their very attractive properties, it is shown that one realistic class of such models may lead to significant Newton law corrections which become small at the future universe only. The model with acceptable Newton law regime shows the matter instability. This suggests that more complicated version of such theory (or extended parameters space) should be investigated.
In this paper we investigate the consequences of phantom crossing considering the perturbative dynamics in models with interaction in their dark sector. By mean of a general study of gauge-invariant variables in comoving gauge, we relate the sources of instabilities in the structure formation process with the phantom crossing. In order to illustrate these relations and its consequences in more detail, we consider a specific case of an holographic dark energy interacting with dark matter. We find that in spite of the model is in excellent agreement with observational data at background level, h…
We point out that dark-energy perturbations may become unstable in the presence of a gravitational wave of sufficiently large amplitude. We study this effect for the cubic Horndeski operator (braiding), proportional to $α_{\rm B}$. The scalar that describes dark-energy fluctuations features ghost and/or gradient instabilities for gravitational-wave amplitudes that are produced by typical binary systems. Taking into account the populations of binary systems, we conclude that the instability is triggered in the whole Universe for $|α_{\rm B} |\gtrsim 10^{-2}$, i.e. when the modification of gravi…
It is widely accepted that moduli in the mass range 10eV - $10^4$GeV which start to oscillate with an amplitude of the order of the Planck scale either jeopardize successful predictions of nucleosynthesis or overclose the Universe. It is shown that the moduli problem can be relaxed by making use of parametric resonance. A new non-perturbative decay channel for moduli oscillations is discussed. This channel becomes effective when the oscillating field results in a net negative mass term for the decay products. This scenario allows for the decay of the moduli much before nucleosynthesis and, the…
We study the evolution of linear density perturbations in the context of interacting scalar field-dark matter cosmologies, where the presence of the coupling acts as a stabilization mechanism for the runaway behavior of the scalar self-interaction potential as in the case of the Chameleon model. We show that in the "adiabatic" background regime of the system the rise of unstable growing modes of the perturbations is suppressed by the slow-roll dynamics of the field. Furthermore the coupled system behaves as an inhomogeneous adiabatic fluid. In contrast instabilities may develop for large value…
We demonstrate that finite time singularities of Type IV can be consistently incorporated in the Universe's cosmological evolution, either appearing in the inflationary era, or in the late-time regime. While using only one scalar field instabilities can in principle occur at the time of the phantom-divide crossing, when two fields are involved we are able to avoid such instabilities. Additionally, the two-field scalar-tensor theories prove to be able to offer a plethora of possible viable cosmological scenarios, at which various types of cosmological singularities can be realized. Amongst othe…
It is well known that the Klein Gordon (KG) equation $\Box Φ+ m^2Φ=0$ has tachyonic unstable modes on large scales ($k^2<\vert m \vert^2$) for $m^2<m_{cr}^2=0$ in a flat Minkowski spacetime with maximum growth rate $Ω_{F}(m)= \vert m \vert$ achieved at $k=0$. We investigate these instabilities in a Reissner-Nordström-deSitter (RN-dS) background spacetime with mass $M$, charge $Q$, cosmological constant $Λ>0$ and multiple horizons. By solving the KG equation in the range between the event and cosmological horizons, using tortoise coordinates $r_*$, we identify the bound states of the emerging S…
We study the stability of a recently proposed model of scalar-field matter called mimetic dark matter or imperfect dark matter. It has been known that mimetic matter with higher derivative terms suffers from gradient instabilities in scalar perturbations. To seek for an instability-free extension of imperfect dark matter, we develop an effective theory of cosmological perturbations subject to the constraint on the scalar field's kinetic term. This is done by using the unifying framework of general scalar-tensor theories based on the ADM formalism. We demonstrate that it is indeed possible to c…
Thermodynamical stability of fluid spheres is studied in the presence of a cosmological constant, both in the Newtonian limit, as well as in General Relativity. In all cases, an increase of the cosmological constant tends to stabilize the system, making asymptotically de Sitter space more thermodynamically stable than anti-de Sitter at the purely classical level. In addition, in the Newtonian case reentrant phase transitions are observed for a positive cosmological constant, due to its repelling property in this case. In General Relativity is studied the case of radiation, for which is found t…
This paper presents the first derivation of the quadratic action for curvature perturbations, $ζ$, within the framework of cuscuton gravity. We study the scalar cosmological perturbations sourced by a canonical single scalar field in the presence of cuscuton field. We identify $ζ$ as comoving curvature with respect to the source field and we show that it retains its conservation characteristic on super horizon scales. The result provides an explicit proof that cuscuton modification of gravity around Friedmann-Lemaitre-Robertson-Walker (FLRW) metric is ghost free. We also investigate the potent…
Cosmological perturbations of FRW solutions in ghost free massive bigravity, including also a second matter sector, are studied in detail. At early time, we find that sub horizon exponential instabilities are unavoidable and they lead to a premature departure from the perturbative regime of cosmological perturbations.
We argue that cubic order interactions between two scalar gravitons and one tensor graviton are ubiquitous in models of dark energy where the strong coupling scale is $Λ_3$. These interactions can potentially provide efficient decay channels for gravitational waves. They can also lead to gradient instabilities of the scalar perturbations in the presence of large amplitude gravitational waves, e.g. those detected by LIGO/Virgo. In contrast with models in scalar-tensor theories, there is an infinite number of higher order interactions in generic $Λ_3$ models, which make it difficult to predict t…
Bimetric theory describes gravitational interactions in the presence of an extra spin-2 field. Previous work has suggested that its cosmological solutions are generically plagued by instabilities. We show that by taking the Planck mass for the second metric, $M_f$, to be small, these instabilities can be pushed back to unobservably early times. In this limit, the theory approaches general relativity with an effective cosmological constant which is, remarkably, determined by the spin-2 interaction scale. This provides a late-time expansion history which is extremely close to $Λ$CDM, but with a …
We study cosmological perturbations in mimetic matter scenario with a general higher derivative function. We calculate the quadratic action and show that both the kinetic term and the gradient term have the wrong sings. We perform the analysis in both comoving and Newtonian gauges and confirm that the Hamiltonians and the associated instabilities are consistent with each other in both gauges. The existence of instabilities is independent of the specific form of higher derivative function which generates gradients for mimetic field perturbations. It is verified that the ghost instability in mim…
This brief contribution is devoted to phenomenological consequences of deviations from Lorentz invariance in gravity and dark matter. We will discuss main effects on cosmological observables and current constraints derived from astrophysical and cosmological data.
Dark energy is investigated from the perspective of quantum cosmology. It is found that, together with an appropriate normal ordering factor $q$, only when there is dark energy then can the cosmological wave function be normalized. This interesting observation may require further attention.
We propose a novel class of degenerate higher-order scalar-tensor theories as an extension of mimetic gravity. By performing a noninvertible conformal transformation on "seed" scalar-tensor theories which may be nondegenerate, we can generate a large class of theories with at most three physical degrees of freedom. We identify a general seed theory for which this is possible. Cosmological perturbations in these extended mimetic theories are also studied. It is shown that either of tensor or scalar perturbations is plagued with gradient instabilities, except for a special case where the scalar …
We discuss ghost free models of the recently suggested mimetic dark matter theory. This theory is shown to be a conformal extension of Einstein general relativity. Dark matter originates from gauging out its local Weyl invariance as an extra degree of freedom which describes a potential flow of the pressureless perfect fluid. For a positive energy density of this fluid the theory is free of ghost instabilities, which gives strong preference to stable configurations with a positive scalar curvature and trace of the matter stress tensor. Instabilities caused by caustics of the geodesic flow, inh…
We consider a cosmological model in which the two major fluid components of the Universe, dark energy and dark matter, flow with distinct four-velocities. This cosmological configuration is equivalent to a single anisotropic fluid, expanding with a four-velocity that is an appropriate combination of the two fluid four-velocities. The energy density of the single cosmological fluid is larger than the sum of the energy densities of the two perfect fluids, i.e., dark energy and dark matter, respectively, and contains a correction term due to the anisotropy generated by the differences in the four…
Models where Dark Matter and Dark Energy interact with each other have been proposed to solve the coincidence problem. We review the motivations underlying the need to introduce such interaction, its influence on the background dynamics and how it modifies the evolution of linear perturbations. We test models using the most recent observational data and we find that the interaction is compatible with the current astronomical and cosmological data. Finally, we describe the forthcoming data sets from current and future facilities that are being constructed or designed that will allow a clearer u…
We discuss a new interacting model for the cosmological dark sector in which the attenuated dilution of cold dark matter scales as $a^{-3}f(a)$, where f(a) is an arbitrary function of the cosmic scale factor $a$. From thermodynamic arguments, we show that f(a) is proportional to entropy source of the particle creation process. In order to investigate the cosmological consequences of this kind of interacting models, we expand f(a) in a power series and viable cosmological solutions are obtained. Finally, we use current observational data to place constraints on the interacting function f(a).
We study the evolution of cosmological perturbations in f(G) gravity, where the Lagrangian is the sum of a Ricci scalar R and an arbitrary function f in terms of a Gauss-Bonnet term G. We derive the equations for perturbations assuming matter to be described by a perfect fluid with a constant equation of state w. We show that density perturbations in perfect fluids exhibit negative instabilities during both the radiation and the matter domination, irrespective of the form of f(G). This growth of perturbations gets stronger on smaller scales, which is difficult to be compatible with the observe…
In this review we consider in detail different theoretical topics associated with interaction in the dark sector. We study linear and nonlinear interactions which depend on the dark matter and dark energy densities. We consider a number of different models (including the holographic dark energy and dark energy in a fractal universe) with interacting dark energy (DE) and dark matter (DM), have done a thorough analysis of these models. The main task of this review was not only to give an idea about the modern set of different models of dark energy, but to show how much can be diverse dynamics of…
Fundamental bosonic fields of arbitrary spin are predicted by generic extensions of the Standard Model and of General Relativity, and are well-motivated candidates to explain the dark components of the Universe. One of most promising channels to look for their presence is through their gravitational interaction with compact objects. Within this context, in this thesis I study several mechanisms by which bosonic fields may affect the dynamics and structure of black holes and neutron stars. The first part of the thesis is devoted to the study of massive spin-2 fields around spherically symmetric…
In the context of Degenerate Higher-Order Scalar-Tensor (DHOST) theories, we study cosmological solutions and their stability properties. In particular, we explicitly illustrate the crucial role of degeneracy by showing how the higher order homogeneous equations in the physical frame (where matter is minimally coupled) can be recast in a system of equations that do not involve higher order derivatives. We study the fixed points of the dynamics, finding the conditions for having a de Sitter attractor at late times. Then we consider the coupling to matter field (described for convenience by a k-…
It is shown that the first law of thermodynamics and the holographic principle applied to an arbitrary large cosmic causal horizon naturally demand the zero cosmological constant and non-zero dynamical dark energy in the form of the holographic dark energy. Semiclassical analysis shows that the holographic dark energy has a parameter $d=1$ and an equation of state comparable to current observational data, if the entropy of the horizon saturates the Bekenstein-Hawking bound. This result indicates that quantum field theory should be modified at large scale to explain dark energy. The relations a…
It is well known that gravitons can convert into photons, and vice versa, in the presence of cosmological magnetic fields. We study this conversion process in the context of atomic dark matter scenario. In this scenario, we can expect cosmological dark magnetic fields, which are free from the stringent constraint from the cosmic microwave observations. We find that gravitons can effectively convert into dark photons in the presence of cosmological dark magnetic fields. The graviton-dark photon conversion effect may open up a new window for ultra high frequency gravitational waves.
We consider the global cosmological evolution and the evolution of the density contrast in the Bose-Einstein condensate dark matter model, in the framework of a Post-Newtonian cosmological approach. In the Bose-Einstein model, dark matter can be described as a non-relativistic, Newtonian gravitational condensate, whose density and pressure are related by a barotropic equation of state. For a condensate with quartic non-linearity, the equation of state is polytropic with index $n=1$. The basic equation describing the evolution of the perturbations of the Bose-Einstein condensate is obtained, an…
The solutions of the field equations of the conformal theory of gravitation with Dirac scalar field in Weyl-Cartan spacetime in very early universe are obtained. In this theory dark energy (describing by an effective cosmological constant) is a function of the Dirac scalar field $β$. These solutions describe the exponential decreasing of $β$ at the first exponential stage of inflation. One of these solutions has a limit to a constant value of the dark energy at large time that can give a way to solving the fundamental cosmological constant problem as a consequence of the fields dynamics in the…
We derive dynamical equations to describe a single 3-brane containing fluid matter and a scalar field coupling to the dilaton and the gravitational field in a five dimensional bulk. First, we show that a scalar field or an arbitrary fluid on the brane cannot evolve to cancel the cosmological constant in the bulk. Then we show that the Randall-Sundrum model is unstable under small deviations from the fine-tuning between the brane tension and the bulk cosmological constant and even under homogeneous gravitational perturbations. Implications for brane world cosmologies are discussed.
In the Horndeski's most general scalar-tensor theories with second-order field equations, we derive the conditions for the avoidance of ghosts and Laplacian instabilities associated with scalar, tensor, and vector perturbations in the presence of two perfect fluids on the flat Friedmann-Lemaitre-Robertson-Walker (FLRW) background. Our general results are useful for the construction of theoretically consistent models of dark energy. We apply our formulas to extended Galileon models in which a tracker solution with an equation of state smaller than -1 is present. We clarify the allowed parameter…
We discuss the modified gravity which may produce the current cosmic aceleration of the universe and eliminates the need for dark energy. It is shown that such models where the action quickly grows with the decrease of the curvature define the FRW universe with the minimal curvature. It is required the infinite time to reach the minimal curvature during the universe evolution. It is demonstrated that quantum effects of conformal fields may strongly suppress the instabilities discovered in modified gravity. We also briefly speculate on the modification of gravity combined with the presence of t…
The behavior of fundamental fields in strong gravity or nontrivial environments is important for our understanding of nature. This problem has interesting applications in the context of dark matter, of dark energy physics or of quantum field theory. The dynamics of fundamental fields has been studied mainly in static or stationary backgrounds, whereas most of our Universe is dynamic. In this paper we investigate "blueshift" and parametric instabilities of scalar fields in dynamical backgrounds, which can be triggered (for instance) by oscillating stars in scalar-tensor theories of gravity. We …
Scalar cosmological perturbations are investigated in the framework of a model with interacting dark energy and dark matter. In addition to these constituents, the inhomogeneous Universe is supposed to be filled with the standard noninteracting constituents corresponding to the conventional $Λ$CDM model. The interaction term is chosen in the form of a linear combination of dark sector energy densities with evolving coefficients. The methods of discrete cosmology are applied, and strong theoretical constraints on the parameters of the model are derived. A brief comparison with observational dat…
A cosmological approach based on considering a cosmic background with non-zero torsion is shown in order to give an option of explaining a possible phantom evolution, not ruled out according to the current observational data. We revise some aspects of the formal schemes on torsion and, according them, we develop a formalism which can be an interesting alternative for exploring Cosmology.
By analysing the cosmological dynamical system with a dark-sector interaction which changes sign during the cosmological evolution, we find a scaling attractor to help to alleviate the cosmic-coincidence problem. This result shows that this interaction can bring new features to the cosmology.
The thorny issue of relating information theory to cosmology is here addressed by assuming a possible connection between quantum entanglement measures and observable universe. In particular, we propose a cosmological toy model, where the equation of state of the cosmological fluid, which drives the today observed cosmic acceleration, can be inferred from quantum entanglement between different cosmological epochs. In such a way the dynamical dark energy results as byproduct of quantum entanglement.
Recently, the so-called Elko spinor field has been proposed to be a candidate of dark energy. It is a non-standard spinor and has unusual properties. When the Elko spinor field is used in cosmology, its unusual properties could bring some interesting consequences. In the present work, we discuss the cosmological coincidence problem in the spinor dark energy models by using the dynamical system method. Our results show that the cosmological coincidence problem should be taken to heart in the investigations of spinor dark energy models.
We investigate the interaction between dark energy and dark matter in the framework of irreversible thermodynamics of open systems with matter creation/annihilation. We consider dark energy and dark matter as an interacting two component (scalar field and "ordinary" dark matter) cosmological fluid in a homogeneous spatially flat and isotropic Friedmann-Robertson-Walker (FRW) Universe. The thermodynamics of open systems as applied together with the gravitational field equations to the two component cosmological fluid leads to a generalization of the elementary dark energy-dark mater interaction…
Static, spherically symmetric solutions with regular origin are investigated of the Einstein-Yang-Mills theory with a negative cosmological constant $Λ$. A combination of numerical and analytical methods leads to a clear picture of the `moduli space' of the solutions. Some issues discussed in the existing literature on the subject are reconsidered and clarified. In particular the stability of the asymptotically AdS solutions is studied. Like for the Bartnik-McKinnon (BK) solutions obtained for $Λ=0$ there are two different types of instabilities -- `topological' and `gravitational'. Regions wi…
We show the way of dark matter and dark energy presentation via ansatzs on the kinetic energies of the fields in the two-component chiral cosmological model. To connect a kinetic interaction of dark matter and dark energy with observational data the reconstruction procedure for the chiral metric component $h_{22}$ and the potential of (self)interaction $V$ has been developed. The reconstruction of $h_{22}$ and $V$ for the early and later inflation have been performed. The proposed model is confronted to $ΛCDM$ model as well.
In this work we extend the first order formalism for cosmological models that present an interaction between a fermionic and a scalar field. Cosmological exact solutions describing universes filled with interacting dark energy and dark matter have been obtained. Viable cosmological solutions with an early period of decelerated expansion followed by late acceleration have been found, notably one which presents a dark matter component dominating in the past and a dark energy component dominating in the future. In another one, the dark energy alone is the responsible for both periods, similar to …
It is found that the evolutions of density perturbations on the super-Hubble scales are unstable in the model with dark-sector interaction $Q$ proportional to the energy density of cold dark matter (CDM) $ρ_m$ and constant equation of state parameter of dark energy $w_d$. In this paper, to avoid the instabilities, we suggest a new covariant model for the energy-momentum transfer between DE and CDM. Then we show that the the large-scale instabilities of curvature perturbations can be avoided in our model in the universe filled only by DE and CDM. Furthermore, by including the additional compone…
Direct detection of gravitational waves from several compact binary coalescences has ushered in a new era of astronomy. It has opened up the possibility of detecting ultralight bosons, predicted by extensions of the Standard Model, from their gravitational signatures. This is of particular interest as some of these hypothetical particles could be components of dark matter that are expected to interact very weakly with Standard Model particles, if at all, but they would gravitate as usual. Ultralight bosons can trigger superradiant instabilities of rotating black holes and form bosonic clouds t…
We investigate the stability of spherically symmetric, purely magnetic, soliton and black hole solutions of four-dimensional ${\mathfrak {su}}(N)$ Einstein-Yang-Mills theory with a negative cosmological constant $Λ$. These solutions are described by $N-1$ magnetic gauge field functions $ω_{j}$. We consider linear, spherically symmetric, perturbations of these solutions. The perturbations decouple into two sectors, known as the sphaleronic and gravitational sectors. For any $N$, there are no instabilities in the sphaleronic sector if all the magnetic gauge field functions $ω_{j}$ have no zeros,…
We study spherically symmetric solutions with a scalar field in the shift-symmetric subclass of the Horndeski theory. Constructing an effective energy-momentum tensor of the scalar field based on the two-fluid model, we decompose the scalar field into two components: dark matter and dark energy.We find the dark-matter fluid is pressure-less, and its distribution of energy density obeys the inverse-square law. We show the scalar field dark matter can explain the galaxy rotation curve and discuss the time evolution of the dark matter in the cosmic background.
The unphysical spin-2 massive degrees of freedom in higher derivative gravity may be either massive unphysical ghosts or tachyonic ghosts. In the last case there is no Planck-scale threshold protecting vacuum cosmological solutions from instabilities. Within the anomaly-induced action formalism the photon-driven IR running of the coefficient of the Weyl-squared term makes the ghost eventually becoming tachyon, that should produce a gravitational explosion of vacuum. This effect is stable under higher loop corrections and takes place also in known versions of perturbative quantum gravity. Howev…
The today estimated value of dark energy can be achieved by the vacuum condensate induced by neutrino mixing phenomenon. Such a tiny value is recovered for a cut-off of the order of Planck scale and it is linked to the sub eV neutrino mass scale. Contributions to dark energy from auxiliary fields or mechanisms are not necessary in this approach.
The role of dynamical cosmological Casimir effect to phantom (constant $w$) and oscillating Universe is discussed. It is shown explicitly that its role is not essential near to Big Rip singularity. However, the account of Casimir fluid makes the scale factor approach to Rip time to be faster. Rip time itself maybe changed too.
