A recent model predicting Omori's law giving the number of aftershocks per unit time following an earthquake involves a differential equation analogous to the Friedmann equation of cosmology. The before-shock phase is analogous to an accelerating universe approaching a Big Rip, the main shock to the Big Rip singularity, and the aftershock to a contracting universe. The analogy provides some physical intuition and Lagrangian and Hamiltonian formulations for Omori's law and its generalizations.
We discuss the cosmological reconstruction in modified Gauss-Bonnet (GB) gravity. It is demonstrated that the modified GB gravity may describe the most interesting features of late-time cosmology. We derive explicit form of effective phantom cosmological models ending by the finite-time future singularity (Big Rip) and without singularities in the future (Little Rip).
We have explored cold dark matter and holographic dark energy cosmological models with big rip singularity. To obtain the solution of the field equation, we have supposed that scalar expansion $θ$ is proportional to shear scalar $σ^2 $ which leads to $p_1=(p_2) ^n $, where $A_1 $, $A_2$ are metric potentials and $ n$ is constant. Big bang and Big rip singularity are investigated and analyzed the state-finder parameter. The physical and geometrical parameters of the universe are explained.
The effects of quantized conformally invariant massless fields on the evolution of cosmological models containing a ``Big Rip'' future expansion singularity are examined. Quantized scalar, spinor, and vector fields are found to strengthen the accelerating expansion of such models as they approach the expansion singularity.
In this paper, we have presented the big rip and pseudo rip cosmological models in an extended theory of gravity. The matter field is considered to be that of perfect fluid. The geometrical parameters are adjusted in such a manner that it matches the prescriptions given by cosmological observations, to be specific to the range of Hubble tension $(H_{0})$. The models favor phantom behavior. The violation of strong energy conditions are shown in both the models, as it has become essential in an extended gravity. The representative values of the coupling parameter are significant on the evolution…
For the inverse linear potential, the SO(1,1) field behaves as phantom for late time and the Big Rip will occur. The field approaches zero as time approaches the Big Rip, here. For this potential the phantom equation of state takes the late-time minimum $w_Φ=-3$. We give some discussions that the Big Rip in the SO(1,1) model may be treated as either the transition point of universe from expansion to extract phase or the final state. In the latter picture of the universe, the field has the $T$ symmetry and the scale factor possesses the $CT$ symmetry, for which the SO(1,1) charge $\bar{Q}$ play…
The big rip, the little rip and the little sibling of the big rip are cosmological doomsdays predicted by some phantom dark energy models that could describe the future evolution of our Universe. When the universe evolves towards either of these future cosmic events all bounded structures and, ultimately, space-time itself are ripped apart. Nevertheless, it is commonly belief that quantum gravity effects may smooth or even avoid these classically predicted singularities. In this review, we discuss the classical and quantum occurrence of these riplike events in the scheme of metric $f(R)$ theor…
In this comment we discuss big rip singularities occurring in typical phantom models by violation of the weak energy condition. After that, we compare them with future late-time singularities arising in models where the scale factor ends in a constant value and there is no violation of the strong energy condition. In phantom models the equation of state is well defined along the whole evolution, even at the big rip. However, both the pressure and the energy density of the phantom field diverge. In contrast, in the second kind of model the equation of state is not defined at the big rip because…
It is known that certain quantum cosmological models present quantum behavior for large scale factors. Since quantization can suppress past singularities, it is natural to inquire whether quantum effects can prevent future singularities. To this end, a Friedmann-Robertson-Walker quantum cosmological model dominated by a phantom energy fluid is investigated. The classical model displays accelerated expansion ending in a Big Rip. The quantization is performed in three different ways, which turn out to lead to the same result, namely there is a possibility that quantum gravitational effects could…
It is stated that holonomy corrections in loop quantum cosmology introduce a modification in Friedmann's equation which prevent the big rip singularity. Recently in \cite{h12} it has been proved that this modified Friedmann equation is obtained in an inconsistent way, what means that the results deduced from it, in particular the big rip singularity avoidance, are not justified. The problem is that holonomy corrections modify the gravitational part of the Hamiltonian of the system leading, after Legendre's transformation, to a non covariant Lagrangian which is in contradiction with one of the …
Late time cosmological solutions for scalar field model with kinetic and Gauss Bonnet couplings are considered. The quintom scenario is realized with and without Big Rip singularity. We find that under specific choice of the Gauss Bonnet coupling, the model considerable simplifies, giving rise to solutions where the kinetic term is proportional to the square of the Hubble parameter. This allows to reconstruct the model for a suitable cosmological evolution. We considered a solution that matches the observed behavior of the equation of state, while Big Rip singularity may be present or absent, …
Cardassian models are an alternative to general relativity which have been proposed as an approach to explaining accelerated cosmic expansion while avoiding directly introducing dark energy. They are generally formulated only in the homogeneous and isotropic case. In this paper an extension of the usual formulation to general spatially homogeneous geometries is given. A characteristic feature of many classes of Cardassian models is the occurrence of big rip singularities where the scale factor tends to infinity after a finite time. It is shown that big rip singularities are also widespread in …
The present study deals with dissipative future universe without big rip in context of Eckart formalism. The generalized chaplygin gas, characterized by equation of state $p=-\frac{A}{ρ^\frac{1}α}$, has been considered as a model for dark energy due to its dark-energy-like evolution at late time. It is demonstrated that, if the cosmic dark energy behaves like a fluid with equation of state $p=ωρ$; $ω< -1$, as well as chaplygin gas simultaneously then the big rip problem does not arises and the scale factor is found to be regular for all time.
We present an holographic brane-world model of the Dvali-Gabadadze-Porrati (DGP) scenario with and without a Gauss-Bonnet term (GB) in the bulk. We show that an holographic dark energy component with the Ricci scale as the infra-red cutoff can describe the late-time acceleration of the universe. In addition, we show that the dimensionless holographic parameter is very important in characterising the DGP branches, and in determining the behaviour of the Ricci dark energy as well as the asymptotic behaviour of the brane. On the one hand, in the DGP scenario the Ricci dark energy will exhibit a p…
We show that a generalised phantom Chaplygin gas can present a future singularity in a finite future cosmic time. Unlike the big rip singularity, this singularity happens for a finite scale factor, but like the big rip singularity, it would also take place at a finite future cosmic time. In addition, we define a dual of the generalised phantom Chaplygin gas which satisfies the null energy condition. Then, in a Randall-Sundrum 1 brane-world scenario, we show that the same kind of singularity at a finite scale factor arises for a brane filled with a dual of the generalised phantom Chaplygin gas.
The time evolution of a wormhole in a Friedmann universe approaching the Big Rip is studied. The wormhole is modeled by a thin spherical shell accreting the superquintessence fluid - two different models are presented. Contrary to recent claims that the wormhole overtakes the expansion of the universe and engulfs it before the Big Rip is reached, it is found that the wormhole becomes asymptotically comoving with the cosmic fluid and the future evolution of the universe is fully causal.
Depending on the choice of its potential, the scalar phantom field $ϕ$ (the equation of state parameter $w<-1$) leads to various catastrophic fates of the universe including big rip, little rip and other future singularity. For example, big rip results from the evolution of the phantom field with an exponential potential and little rip stems from a quadratic potential in general relativity (GR). By choosing the same potential as in GR, we suggest a new mechanism to avoid these unexpected fates (big and little rip) in the inverse-\textit{R} gravity. As a pedagogical illustration, we give an exa…
Rip cosmological models have been investigated in the framework of $f(T,\mathcal{T})$ theory of gravity, where $T$ denotes the torsion and $\mathcal{T}$ is the trace of the energy-momentum tensor. These phantom cosmological models revealed that at initial epoch a EoS parameter $ω<-1$ and tends asymptotically at late phase to $-1$ $(ω\rightarrow -1)$. On the other hand, Wormhole Solutions and Big Trip have been subject of an investigation. The wormhole throat radius $R(t)$ and the conditions to be satisfied so that produces the Big Trip phenomenon have been discussed.
It is shown in this paper that minisuperspace quantization of homogeneous and isotropic geometries with phantom scalar fields, when examined in the light of the Bohm-de Broglie interpretation of quantum mechanics, does not eliminate, in general, the classical big rip singularity present in the classical model. For some values of the Hamilton-Jacobi separation constant present in a class of quantum state solutions of the Wheeler-DeWitt equation, the big rip can be either completely eliminated or may still constitute a future attractor for all expanding solutions. This is contrary to the conclus…
I investigate the repercussions of particle production when the Universe is dominated by a hypothetical phantom substance. I show that backreaction due to particle production prevents the density from shooting to infinity at a Big Rip, but instead forces it to stabilise at a large constant value. Afterwards there is a period of de-Sitter inflation. I speculate that this might lead to a cyclic Universe.
The effect of quantum particle production near the big rip singularity has been investigated previously, with the conclusion that the energy of the produced particle decreases as the future singularity is approached. Hence, the effect of particle production would not be effective to avoid the big rip singularity. That calculation was performed by introducing an ultra-violet cut-off. In the present work we consider a renormalization of the energy-momentum tensor, obtainning a different expression for the particle production. The new expression seems to indicate that the effect of particle produ…
Extended theories of gravity have gathered a lot of attention over the last years, for they not only provide an excellent framework to describe the inflationary era but also yields an alternative to the elusive and mysterious dark energy. Among the different extended theories of gravity, on this work we focus on metric $f(R)$ theories. In addition, it is well known that if the late-time acceleration of the universe is stronger than the one induced by a cosmological constant then some future cosmic singularities might arise, being the Big Rip the most virulent one. Following this reasoning, on …
An exact solution is obtained for a massive scalar field conformally coupled with the curvature in a cosmological model with the scale factor $a(t) = a_0 / |t|$, corresponding to background matter with the equation of state $ p = - 5 \varepsilon /3 $. An expression for the number density of created particles is obtained, and its behavior is studies as the model approached the instant of a Big Rip. Renormalization of the energy-momentum tensor is considered, and it is shown that back reaction of the quantum effects of a conformally coupled scalar field on the space-time metric can be neglected …
We discuss exotic singularities in the evolution of the universe motivated by the progress of observations in cosmology. Among them there are: Big-Rip (BR), Sudden Future Singularities (SFS), Generalized Sudden Future Singularities (GSFS), Finite Density Singularities (FD), type III, and type IV singularities. We relate some of these singularities with higher-order characteristics of expansion such as jerk and snap. We also discuss the behaviour of pointlike objects and classical strings on the approach to these singularities.
Aspects of non-linear Schrödinger-type (NLS) formulation of scalar (phantom) field cosmology on slow-roll, acceleration, WKB approximation and Big Rip singularity are presented. Slow-roll parameters for the curvature and barotropic density terms are introduced. We reexpress all slow-roll parameters, slow-roll conditions and acceleration condition in NLS form. WKB approximation in the NLS formulation is also discussed when simplifying to linear case. Most of the Schrödinger potentials in NLS formulation are very slowly-varying, hence WKB approximation is valid in the ranges. In the NLS form of …
We study the quantum avoidance of the big rip singularity in the Eddington-inspired-Born-Infeld (EiBI) phantom model. Instead of considering a simple phantom dark energy component, which is described by a perfect fluid, we consider a more fundamental degree of freedom corresponding to a phantom scalar field with its corresponding potential, which would lead the classical universe to a big rip singularity. We apply a quantum geometrodynamical approach by performing an appropriate Hamiltonian study including an analysis of the constraints of the system. We then derive the Wheeler-DeWitt (WDW) eq…
In this paper we study the behavior of the multi-field in phantom inflation, when massive scalar fields work collectively, in which the scale factor is a power law. We evaluate its parameter values by applying certain constraints on our model parameters, and we investigate that before the Big Rip singularity occurs the universe is in phantom inflationary phase. Furthermore, we calculate these values for this period then compare with current observations of CMB, BAO and observational Hubble data. We find that results may be consistent with observations. This implies that in the dark-energy equa…
We revise the future fate of a group of scalar-tensor theories known as kinetic gravity braiding models. As it is well-known, these theories can safely drive the expansion of the universe towards a future de Sitter state if the corresponding Lagrangian is invariant under constant shifts in the scalar field. However, this is not the only possible future state of these shift-symmetric models as we show in this letter. In fact, future cosmic singularities characterized by a divergence of the energy density can also appear in this framework. We present an explicit example where a big rip singulari…
Motivated by condensed matter physical systems, in which a finite-time singularity indicates that the topology of the system changes, we critically examine the possibility of the Universe's topology change at a finite-time cosmological singularity. We emphasize on Big Rip and Type II and IV cosmological singularities, which we classify to future spacelike and timelike singularities. For the Type IV and Type II singularities, since no geodesics incompleteness occurs, no topological change is allowed, by using Geroch's theorem arguments. However, for the Big Rip case, Tipler's arguments allow a …
Models of the universe with arbitrary (non gravitational) interaction between the components of the cosmic fluid: the phantom energy and the background, are investigated. A general form of the interaction that is inspired in scalar-tensor theories of gravity is considered. No specific model for the phantom fluid is assumed. We concentrate our investigation on solutions that are free of the coincidence problem. We found a wide region in the parameter space where the solutions are free of the big rip singularity also. Physical arguments, together with arguments based on the analysis of the obser…
We analyze late-time evolution of the Universe in the framework of the self-consistent model, in which the dark matter is influenced by the Archimedean-type force proportional to the four-gradient of the dark energy pressure. The dark energy is considered as a fluid with the equation of state of the relaxation type, which takes into account a retardation of the dark energy response to the Universe accelerated expansion. The dark matter is guided by the Archimedean-type force, which redistributes the total energy of the dark fluid between two its constituents, dark energy and dark matter, in th…
We construct phantom energy models with the equation-of-state parameter $w$ such that $w<-1$, but finite-time future singularity does not occur. Such models can be divided into two classes: (i) energy density increases with time ("phantom energy" without "Big Rip" singularity) and (ii) energy density tends to constant value with time ("cosmological constant" with asymptotically de Sitter evolution). The disintegration of bound structure is confirmed in Little Rip cosmology. Surprisingly, we find that such disintegration (on example of Sun-Earth system) may occur even in asymptotically de Sitte…
In the Big Rip cosmological scenario, a FRW universe containing dark energy with $w < -1$ in its equation of state (phantom energy) expands in such way that, in a finite time span, the scale factor diverges to infinity and the size of the cosmic horizon goes to zero. Here we revisit this scenario in light of the fact that Hawking radiation is expected to be generated at the apparent horizon of a FRW universe, and show that the energy density and temperature of that radiation both diverge at the Big Rip. We then use this fact to propose a new variant of Penrose's conformal cyclic cosmology mode…
The Little Sibling of the Big Rip is a cosmological abrupt event predicted by some phantom cosmological models that could describe our Universe. When this event is approached the observable Universe and its expansion rate grow infinitely, but its cosmic derivative remains finite. In this work we have obtained the group of metric $f(R)$ theories of gravity that reproduce this classical cosmological background evolution. Furthermore, we have considered the quantization of some of the resulting models in the framework of quantum geometrodynamics, showing that the DeWitt criterion can be satisfied…
We present a new cosmological event, which we named the little sibling of the big rip. This event is much smoother than the big rip singularity. When the little sibling of the big rip is reached, the Hubble rate and the scale factor blow up but the cosmic derivative of the Hubble rate does not. This abrupt event takes place at an infinite cosmic time where the scalar curvature explodes. We show that a doomsday á la little sibling of the big rip is compatible with an accelerating universe, indeed at present it would mimic perfectly a LCDM scenario. It turns out that eventhough the event seems t…
Analytical properties of the scalar expansion in the cosmic fluid are investigated, especially near the future singularity, when the fluid possesses a constant bulk viscosity ζ. In addition, we assume that there is a Casimir-induced term in the fluid's energy-momentum tensor, in such a way that the Casimir contributions to the energy density and pressure are both proportional to 1/a^4, 'a' being the scale factor. A series expansion is worked out for the scalar expansion under the condition that the Casimir influence is small. Close to the Big Rip singularity the Casimir term has however to fad…
A canonical quantization à la Wheeler-DeWitt is performed for a model of three-form fields in a homogeneous and isotropic universe. We start by carrying out the Hamiltonian formalism of this cosmological model. We then apply this formalism to a Little Sibling of the Big Rip (LSBR), an abrupt event milder than a Big Rip and that is known to be generic to several minimally coupled three-form fields for a variety of potentials. We obtain a set of analytical solutions of the Wheeler-DeWitt equation using different analytical approximations and explore the physical consequences of them. It turns ou…
We consider a cosmological scenario in which the expansion of the Universe is dominated by phantom dark energy and black holes which condense out of the latter component. The mass of black holes decreases via Hawking evaporation and by accretion of phantom fluid but new black holes arise continuously whence the overall evolution can be rather complex. We study the corresponding dynamical system to unravel this evolution and single out scenarios where the big rip singularity does not occur.
In the holographic Ricci dark energy (RDE) model, the parameter $α$ plays an important role in determining the evolutionary behavior of the dark energy. When $α<1/2$, the RDE will exhibit a quintom feature, i.e., the equation of state of dark energy will evolve across the cosmological constant boundary $w=-1$. Observations show that the parameter $α$ is indeed smaller than 1/2, so the late-time evolution of RDE will be really like a phantom energy. Therefore, it seems that the big rip is inevitable in this model. On the other hand, the big rip is actually inconsistent with the theoretical fram…
We introduce a general characterization of sudden cosmological singularities and investigate the classical stability of homogeneous and isotropic cosmological solutions of all curvatures containing these singularities to small scalar, vector, and tensor perturbations using gauge invariant perturbation theory. We establish that sudden singularities at which the scale factor, expansion rate, and density are finite are stable except for a set of special parameter values. We also apply our analysis to the stability of Big Rip singularities and find the conditions for their stability against small …
In the framework of a single scalar field quintom model with higher derivative, we construct in this paper a dark energy model of which the equation of state (EOS) $w$ crosses over the cosmological constant boundary. Interestingly during the evolution of the universe $w<-1$ happens just for a period of time with a distinguished feature that $w$ starts with a value above -1, transits into $w<-1$, then comes back to $w>-1$. This avoids the Big Rip jeopardy induced by $w<-1$.
The effects of the gravitational back reaction of cosmological perturbations are investigated in a cosmological model where the universe is dominated by phantom energy. We assume a COBE normalized spectrum of cosmological fluctuations at the present time and calculate the effective energy-momentum tensor of the gravitational back-reactions of cosmological perturbations whose wavelengths at the time when the back-reactions are evaluated are larger than the Hubble radius. Our results reveal that the effects of gravitational back-reactions will counteract that of phantom energy sooner or later an…
We study the evolutions of 4D universe on the M2-M5 BIon in the thermal background. The BIon consists of a D-brane and a parallel anti-D-brane which a wormhole connects them. When the branes and antibranes are far from each other and the brane's spike and the antibrane's spike are well separated, the wormhole cannot be built. However, when the branes and anti-branes are close to each other, a wormhole can be formed between them. Under this condition, there are many ways for flowing energy from transverse dimensions into our universe. This energy overcomes all other forms of energy, such as the…
By studying a modified Friedmann equation which arises in an extension of general relativity which accommodates a time-dependent fundamental length $L(t)$, we consider cosmological models where the scale factor diverges with an essential singularity at a finite future time. Such models have no dark energy in the conventional sense of energy possessing a truly simple pressure-energy relationship. Data on supernovae restrict the time from the present until the Rip to be generically longer than the current age of the Universe.
We consider a non-minimal coupling of a perfect fluid matter system with geometry, which the coupling function is taken to be an arbitrary function of the Ricci scalar. Due to such a coupling, the matter stress tensor is no longer conserved and there is an energy transfer between the two components. By solving the conservation equation and applying the second law of thermodynamics, we show that direction of the energy transfer depends on the equation of state of the matter fluid. In particular, a phantom fluid should loose energy with expansion of the universe. This energy reduction can avoid …
We analyze from a classical and quantum point of view the behavior of the universe close to a little rip, which can be interpreted as a big rip sent towards the infinite future. Like a big rip singularity, a little rip implies the destruction of all bounded structure in the Universe and is thus an event where quantum effects could be important. We present here a new phantom scalar field model for the little rip. The quantum analysis is performed in quantum geometrodynamics, with the Wheeler-DeWitt equation as its central equation. We find that the little rip can be avoided in the sense of the …
There exists dark energy models that predict the occurrence of "little rip". At the point of little rip the Hubble rate and its cosmic time derivative approach infinity, which is quite similar to the big rip singularity except that the former happens at infinite future while the latter at a finite cosmic time; both events happen in the future and at high energies. In the case of the big rip, a combination of ultra-violet and infra-red effects can smooth its doomsday. We therefore wonder if the little rip can also be smoothed in a similar way. We address the ultra-violet and infra-red effects i…
Effects of quantized free scalar fields in cosmological spacetimes with Big Rip singularities are investigated. The energy densities for these fields are computed at late times when the expansion is very rapid. For the massless minimally coupled field it is shown that an attractor state exists in the sense that, for a large class of states, the energy density of the field asymptotically approaches the energy density it would have if it was in the attractor state. Results of numerical computations of the energy density for the massless minimally coupled field and for massive fields with minimal…
Chevallier-Polarski-Linder (CPL) parametrization for the equation of state of dark energy in terms of cosmic redshift or scale factor have been frequently studied in the literature. In this study, we consider cosmic time based CPL parametrization for the equation of state parameter of the effective cosmic fluid that fills the fabric of spatially flat and homogeneous Robertson-Walker spacetime in general relativity. The model exhibits two worthy features: (i) It fits the observational data from the latest $H(z)$ and Union 2.1 SN Ia compilations matching the success of $Λ$CDM model. (ii) It desc…
Although it has been believed that the models with generalised Chaplygin gas do not contain singularities, in a previous work we have studied how a big freeze could take place in some kinds of phantom generalised Chaplygin gas. In the present work, we study some types of generalised Chaplygin gas in order to show how different sorts of singularities could appears in such models, in the future or in the past. We point out that: (i) singularities may not be originated from the phantom nature of the fluid, and (ii) if initially the tension of the brane in a brane-world Chaplygin model is large en…
We study the conformal structure of exotic (non-big-bang) singularity universes using the hybrid big-bang/exotic singularity/big-bang and big-rip/exotic singularity/big-rip models by investigating their appropriate Penrose diagrams. We show that the diagrams have the standard structure for the big-bang and big-rip and that exotic singularities appear just as the constant time hypersurfaces for the time of a singularity and because of their geodesic completeness are potentially transversable. We also comment on some applications and extensions of the Penrose diagram method in studying exotic si…
We analyse the quantum behaviour of the "Little Sibling" of the Big Rip singularity (LSBR) [1]. The quantisation is carried within the geometrodynamical approach given by the Wheeler--DeWitt (WDW) equation. The classical model is based on a Friedmann--Lemaître--Robertson--Walker Universe filled by a perfect fluid that can be mapped to a scalar field with phantom character. We analyse the WDW equation in two setups. In the first step, we consider the scale factor as the single degree of freedom, which from a classical perspective parametrises both the geometry and the matter content given by th…
Recently, many dark energy models whose equation-of-state parameter can cross the phantom divide $w_{de}=-1$ have been proposed. In a previous paper [Class. Quant. Grav. {\bf 22}, 3189 (2005); hep-th/0501160], we suggest such a model named hessence, in which a non-canonical complex scalar field plays the role of dark energy. In this work, the cosmological evolution of the hessence dark energy is investigated. We consider two cases: one is the hessnece field with an exponential potential, and the other is with a (inverse) power law potential. We separately investigate the dynamical system with …
It is shown that phantom scalar models can be mapped into a mathematically equivalent, modified $F(R)$ gravity, which turns out to be complex, in general. Only for even scalar potentials is the ensuing modified gravity real. It is also demonstrated that, even in this case, modified gravity becomes complex at the region where the original phantom dark energy theory develops a Big Rip singularity. A number of explicit examples are presented which show that these two theories are not completely equivalent, from the physical viewpoint. This basically owes to the fact that the physical metric in bo…
The fate of our universe is an unceasing topic of cosmology and the human being. The discovery of the current accelerated expansion of the universe significantly changed our view of the fate of the universe. Recently, some interesting scenarios concerning the fate of the universe attracted much attention in the community, namely the so-called "Little Rip" and "Pseudo-Rip". It is worth noting that all the Big Rip, Little Rip and Pseudo-Rip arise from the assumption that the dark energy density $ρ(a)$ is monotonically increasing. In the present work, we are interested to investigate what will ha…
Cosmological fluids with a Generalized Equation of State (GEoS) are here considered, whose corresponding EoS parameter $ω$ describes a fluid with phantom behavior, namely $ω<-1$, but leading to universes free of singularities at any past or future, finite time. Thus avoiding, in particular, the Big Bang and the Big Rip singularities, the last one considered to be typical in phantom fluid models. More specifically, such GEoS fluid cosmologies lead to regular Little Rip universes. A remarkable new property of these solutions is proven here, namely that they avoid the initial singularity at early…
