Observational constraints on cosmological future singularities
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Abstract
In this work we consider a family of cosmological models featuring future singularities. This type of cosmological evolution is typical of dark energy models with an equation of state violating some of the standard energy conditions (e.g. the null energy condition). Such a kind of behavior, widely studied in the literature, may arise in cosmologies with phantom fields, theories of modified gravity or models with interacting dark matter/dark energy. We briefly review the physical consequences of these cosmological evolution regarding geodesic completeness and the divergence of tidal forces in order to emphasize under which circumstances the singularities in some cosmological quantities correspond to actual singular spacetimes. We then introduce several phenomenological parameterizations of the Hubble expansion rate to model different singularities existing in the literature and use SN Ia, BAO and H(z) data to constrain how far in the future the singularity needs to be (under some reasonable assumptions on the behavior of the Hubble factor). We show that, for our family of parameterizations, the lower bound for the singularity time cannot be smaller than about 1.2 times the age of the universe, what roughly speaking means \({\sim }2.8\) Gyrs from the present time.
Keywords
Dark Energy Dark Energy Model Effective Field Theory Null Energy Condition Future Singularity1 Introduction
The standard model of cosmology together with the inflationary paradigm provide an accurate description of the universe, although it requires the presence of three unknown ingredients, namely: Dark matter, dark energy and the inflaton field. The last two share the property of being introduced in order to support phases of accelerating expansion. Moreover, while the inflaton accounts for the first instants of life of our universe, dark energy should determine its final fate as the component that will eventually dominate. If dark energy turns out to be simply a cosmological constant, then we are doomed to an asymptotically de Sitter universe in the future. The situation is much more subtle when dynamical dark energy or modified gravity is brought in as possible explanations for the latetime accelerated expansion (for a review about dark energy models, see [1, 2, 3, 4, 5, 6, 7, 8, 9]). In some cases, dark energy is ascribed to a socalled phantom fluid, i.e., a fluid satisfying \(\rho +p<0\) and, thus, violating the Null Energy Condition (NEC) [10, 11, 12, 13, 14, 15, 16, 17, 18]. For a set of minimally coupled scalar fields, this condition implies the presence of, at least, a Laplacian instability in the inhomogeneous perturbations, although this can be resolved by allowing nonminimal couplings to occur (see for instance [19, 20]). Moreover, such a kind of behavior can also be a consequence of a modification of General Relativity instead of a fluid with a nonstandard equation of state [3, 4, 5, 21]. In any case, the phantom behavior may affect the background evolution giving rise to a future singularity occurring at a finite time where the scale factor diverges. Nevertheless, note that some models with violations of the null energy condition do not drive the universe to a singularity but to regular scenarios that may affect the local structures, known as little Rip, PseudoRip and Little Sibling [22, 23, 24, 25].
The described singular behavior is actually shared by many dynamical dark energy models and modified gravity scenarios, where divergences in different cosmological parameters at a finite time can appear. The nature of the future singularities may differ among the different scenarios and they can be classified according to the cosmological parameters that diverge. An alternative way of classifying the future singularities is by means of the derivative of the scale factor that diverges. This classification is very useful because it helps understanding the severity of the different types of singularities (for a classification of cosmological singularities, see Refs. [26, 27]). In this respect, it is worth reminding the reader that a singular spacetime is characterized by the incompleteness of the geodesics [28]. Since the geodesic equations are linear in the connection, it will contain, at most, first derivatives of the metric. Thus, the geodesics will be regular as long as the metric is continuous at the singularity. For a cosmological model, this will mean that the scale factor should remain finite at the singularity, even if divergences in the Hubble expansion rate or its derivatives are present. This type of behavior has been recently used in [29] in order to replace the Big Bang singularity with a milder one that can be trespassed by the geodesics.
Another useful equation in order to characterize the strength of a singularity is the geodesics deviation equation. That equation essentially determines the tidal forces suffered by two infinitesimally close geodesics and it depends on the curvature of the spacetime. This means that tidal forces are sensitive to singularities which do not necessarily affect the completeness of the geodesics. Again in a cosmological context, if the scale factor remains regular, but the Hubble rate diverges, it is possible to have a regular geodesic congruence with divergent tidal forces. Some criteria based on the behavior of the Riemann tensor as we approach the singularity exist in the literature to decide whether the singularity is strong or weak, the Tipler [30] and Krolak [31] conditions being two widely used ones.
Regardless of the physical consequences of having a future singularity at a finite time, a natural question to ask is how close a given type of singularity is to us [32]. This is the analogous of asking about the age of the universe, determined by our distance to the original Big Bang singularity. Nevertheless, in the same way as we do not expect the Big Bang singularity to exist actually, but rather being regularized by some quantum effects, by high curvature corrections to Einstein’s gravity or even by varying physical constants [33], we do not expect the future singularities to be physical, at least the strongest types where physical quantities diverge [34, 35, 36, 37]. However, it will be useful to have some estimation on how close to us a given singularity can be and, therefore, have an idea of how far in the future we could extrapolate a model with a certain type of future singularity. It is important to notice that an effective equation of state for dark energy \(w<1\) is within the confidence regions of observational data [38, 39, 40, 41, 42], so the possibility of having a future singularity is plausible. Moreover, such models have also received attention because of some theoretical implications, since possible quantum effects close to the singularity become important. We know that General Relativity is to be regarded as an effective field theory whose strong coupling scale is, in the most optimistic scenario, at the Planck scale. Thus, knowing at which time the singularity is essentially reached will give us also an idea of until when we can keep using General Relativity as an effective field theory.
The purpose of the present work is precisely to draw such an estimation in a fairly model independent framework. An important difficulty arising here with respect to the Big Bang case is that, while in that case we have control on the different phases that the universe has gone through from the initial singularity until today, for the future singularity we cannot know what the future phases will be. Thus, we need to make some assumptions to eventually determine how close the singularity can be. In order to achieve this, we will use some classes of phenomenological parameterizations for the Hubble expansion rate as proxies for a universe with a transition from a matter dominated era to a dark energy phase leading to a future singularity. We will then confront them to SN Ia, BAO and H(z) data to obtain the time of the singularity. Obviously, there could be transient phases that could delay the singularity, but this will not concern us since we are actually interested in obtaining a general lower bound for a future singularity.
The paper is organized as follows: Section 2 is devoted to a brief review about future cosmological singularities. In Sect. 3, the parameterizations of the Hubble rate which are analyzed in the paper are introduced. Then the observational data used to fit the models is described in Sect. 4. Finally, Sect. 5 is devoted to the results and discussions.
2 Future cosmological singularities

Type I (“Big Rip singularity”): For \(t\rightarrow t_s\), \(a\rightarrow \infty \), and \(\rho \rightarrow \infty \), \(p\rightarrow \infty \). Timelike geodesics are incomplete [32, 43].

Type II (“Typical Sudden singularity”): For \(t\rightarrow t_s\), \(a\rightarrow a_s\), and \(\rho \rightarrow \rho _s\), \(p\rightarrow \infty \). Geodesics are not incomplete. This is classified as a weak singularity (see Refs. [44, 45, 46]).

Type III (“Big freeze”): For \(t\rightarrow t_s\), \(a\rightarrow a_s\), and \(\rho \rightarrow \infty \), \(p\rightarrow \infty \). No geodesics incompleteness. They can be weak or strong (see Ref. [47]).

Type IV (“Generalized Sudden singularity”): For \(t\rightarrow t_s\), \(a\rightarrow a_s\), and \(\rho \rightarrow \rho _s\), \(p \rightarrow p_s\) but higher derivatives of Hubble parameter diverge. They are weak singularities [48].

Type V (“wsingularities”): For \(t\rightarrow t_s\), \(a\rightarrow \infty \), and \(\rho \rightarrow 0\), \(p\rightarrow 0\) and \(w=p/\rho \rightarrow \infty \). These singularities are weak (see Refs. [49, 50, 51]).
3 The models
In this section we will describe the parameterizations that we will use for the subsequent confrontation to observational data. We emphasize that we intend to establish a general lower bound for the time of the future singularity \(t_s\). Since we are dealing with future singularities occurring at a finite proper time, it is reasonable to perform our parameterizations in terms of proper time. Moreover, as we have discussed, the severity of the different types of singularities is essentially determined by whether the scale factor or any of its timederivatives presents a divergence. Therefore, the natural cosmological quantity to parameterize is the scale factor. However, for convenience when confronting to SN Ia and BAO, it will be more appropriate to parameterize the Hubble expansion rate directly. By doing this, we also avoid the ambiguity in the normalization of the scale factor.
In this table we summarize the five parameterizations that we propose to describe the different types of future singularities that we consider throughout this work. In the first column we give the label we will use for each case, while the second column indicates the type of singularity according to the classification in [26]. In the columns 3 and 4 we give the analytical expressions for H(t) and a(t) (where, as explained in the main text, the normalization \(a_0\) must be chosen so that \(a(x=1)=1\)). In the last columns we give the behavior of a, H, and some of its derivatives at the singularity. We also give the values of \(\rho \), p, and \(w_\mathrm{eff}\) for the theoretical interpretation discussed in Sect. 3. It is important to keep in mind that those values depend on the underlying theoretical model and we only give them here for illustrative purposes
Label  Type  H(x)  a(x)  a  H  \(\dot{H}\)  \(\ddot{H}\)  \(\rho \)  p  \(w_\mathrm{eff}\) 

A  I  \(\frac{2}{3 x} + \frac{2 n }{3(1x/x_s)}\)  \(a_0x^{\frac{2}{3}}(x_{s}x)^{\frac{2}{3} \cdot n x_{s}}\)  \(\infty \)  \(\infty \)  \(\infty \)  \(\infty \)  \(\infty \)  \(\infty \)  \(w_{s} < 0\) 
B  III  \(\frac{2}{3 x} + \frac{2 n}{3\sqrt{1x/x_{s}}}\)  \(a_0x^{\frac{2}{3}} \exp {[\frac{4}{3} n \sqrt{x_{x} (x_{s}x)}]}\)  \(a_{s}\)  \(\infty \)  \(\infty \)  \(\infty \)  \(\infty \)  \(\infty \)  \(\infty \) 
C  III  \(\frac{2}{3 x}  \frac{2 n}{3} \log (1\frac{x}{x_{s}})\)  \(a_0x^{\frac{2}{3}} \exp {[\frac{2}{3} n ( x  x_{s}) (1+\log [1 x/x_{s}])]}\)  \(a_{s}\)  \(\infty \)  \(\infty \)  \(\infty \)  \(\infty \)  \(\infty \)  \(\infty \) 
D  II  \(\frac{2}{3 x} + \frac{2 n}{3}\sqrt{1\frac{x}{x_{s}}}\)  \(a_0(x/x_{s})^{\frac{2}{3}} \exp {[\frac{4}{9} \cdot n x_{s} ( 1 x/x_{s})^{\frac{3}{2}}]}\)  \(a_{s}\)  \(H_{s} > 0\)  \(\infty \)  \(\infty \)  \(\rho _{s}\)  \(\infty \)  \(\infty \) 
E  IV  \(\frac{2}{3 x} + \frac{2 n}{3} ( 1  \frac{x}{x_{s}})^{3/2}\)  \(a_0(x/x_{s})^{\frac{2}{3}} \exp {[\frac{4}{15} \cdot n x_{s} (1x/x_{s})^{\frac{5}{2}}]}\)  \(a_{s}\)  \(H_{s} > 0\)  \(\dot{H}_{s} < 0\)  \(\infty \)  \(\rho _{s}\)  0  0 
This matter dominated phase will then be matched to an evolution with a future time singularity, i.e., \(F(t,t_s)\) is a function that either itself or some of its timederivatives diverges at \(t=t_s\). We will assume that this divergence originates from the fact that the differential equation of the underlying physical model presents a regular singular point so that the solution near the singularity can be expressed as a Frobenius series. We will further assume that the transition from the matter era is sufficiently fast so that the dominant term of the series rapidly takes over. This will not affect our goal of obtaining a lower bound for the time of the transition since making the transition slower typically delays the appearance of the divergence. Since we are looking for a future divergence where a given derivative of the scale factor diverges while the lower derivatives remain finite, a reasonable Ansatz for \(F(t,t_s)\) is some halfinteger power. With these considerations in mind, we have chosen the specific parameterizations summarized in Table 1 together with their main properties.^{2} All the models contain two parameters characterizing the time of the singularity \(t_s\) and an additional parameter n that regulates the time of the transition from matter domination. Notice that all the parameterizations share the property of containing a latetime de Sitter evolution when the time of the singularity is sent to the asymptotic future^{3} \(t_s\rightarrow \infty \). However, it is important to notice that the existence of a matter phase at early times matching a de Sitter universe in the asymptotic future does not necessarily mean that the evolution mimics that of a \(\Lambda \)CDM model, because the transition era between the two phases may be completely different. In fact, it is not difficult to see that none of our parameterizations contains \(\Lambda \)CDM within its parameter space.
Finally, we can also mention that a given background evolution for the scale factor can be mapped onto a scalar field theory by suitable choice of the action. In the spirit of the effective field theory of dark energy, we can think of the time coordinate as corresponding to a foliation of the spacetime according to the scalar field, where the unitary gauge has been chosen. Then a natural interpretation of the future singularity would be a point where the scalar field meets a pathology in its evolution as dictated by the field equations.
4 Data
The analysis has been performed using three different standard cosmological tools. They are at low redshift \((z \lesssim 2)\), because we are not interested in changing early time evolution and we assume that a possible signature for “future” evolution toward a singularity, if any, is detectable now or, at least, in the recent past only.
4.1 Hubble data from earlytype galaxies
4.2 Type Ia supernovae
4.3 Baryon Acoustic Oscillations
Finally, the total \(\chi ^2\) to be minimized will be \(\chi ^2 = \chi ^{2}_{H}+\chi ^{2}_{H_{0}}+\chi ^{2}_{SN}+\chi ^{2}_{BAO}\). We minimize the total \(\chi ^2\) using the Markov Chain Monte Carlo (MCMC) method and we check its convergence with the method developed in [65]. In order to compare the models in the best statistical way possible, we have calculated the Bayesian evidence for each of them. The Bayesian evidence is defined as the probability of the data D given the model M with a set of parameters \(\varvec{\theta }\), \({\mathcal {E}}(M) = \int {\mathrm {d}}\varvec{\theta }L(D\varvec{\theta },M)\pi ( \varvec{\theta }M)\): \(\pi (\varvec{\theta }M)\) is the prior on the set of parameters, normalized to unity, and \(L(D\varvec{\theta },M)\) is the likelihood function.
In this table we present the obtained results for the best fit of each parameterization. In column 1 we give the label identifying each parameterization in Table 1. In columns 2–5 we give the \(1\sigma \) confidence levels for our primary model parameters (notice that they are different for the different cases and, in particular, \(\Omega _m\) is not within the fitting parameters of our parameterizations). In column 6 we show the age of the universe. We also show the effective equation of state parameter (as defined in (14)) for each parameterization evaluated at the present. Finally, in columns 8 and 9 we give the Bayesian evidence and ratio with respect to \(\Lambda \)CDM for Jeffreys’ interpretation
id.  \(\Omega _m\)  \(H_{0}\) (km s\(^{1}\) Mpc\(^{1}\))  \(t_{0}\) (Gyr)  \(w_{\mathrm {eff},0}\)  \({\mathcal {B}}_{ij}\)  \(\log {\mathcal {B}}_{ij}\)  

\(\Lambda CDM\)  \(0.30^{+0.02}_{0.02}\)  \(69.6^{+0.7}_{0.7}\)  \(13.54^{+0.31}_{0.29}\)  \(0.70\)  1  0 
id.  \(\Omega _m\)  \(w_{0}\)  \(w_{a}\)  \(H_{0}\) (km s\(^{1}\) Mpc\(^{1}\))  \(t_{0}\) (Gyr)  \(w_{\mathrm {eff},0}\)  \({\mathcal {B}}_{ij}\)  \(\log {\mathcal {B}}_{ij}\) 

CPL  \(0.36^{+0.05}_{0.09}\)  \(1.00^{+0.23}_{0.24}\)  \(1.14^{+1.87}_{3.08}\)  \(69.5^{+0.7}_{0.7}\)  \(13.25^{+0.38}_{0.32}\)  \(0.64\)  0.56  \(\)0.63 
id.  n  \(\alpha _{s}\)  \(t_{s}/t_{0}\)  \(1/t_{0}\) (km s\(^{1}\) Mpc\(^{1}\))  \(t_{0}\) (Gyr)  \(w_{\mathrm {eff},0}\)  \({\mathcal {B}}_{ij}\)  \(\log {\mathcal {B}}_{ij}\) 

Uniform prior  
A  \(0.27^{+0.07}_{0.06}\)  \(0.31^{+0.16}_{0.18}\)  \(2.17^{+0.85}_{0.42}\)  \(69.9^{+1.1}_{1.1}\)  \(14.00^{+0.23}_{0.21}\)  \(0.74\)  0.62  \(\)0.48 
B  \(0.33^{+0.06}_{0.06}\)  \(0.46^{+0.19}_{0.24}\)  \(1.78^{+0.74}_{0.34}\)  \(69.6^{+1.1}_{1.2}\)  \(14.06^{+0.24}_{0.22}\)  \(0.70\)  0.57  \(\)0.56 
C  \(0.99^{+0.55}_{0.38}\)  \({<}0.29\)  \({>}2.22\)  \(71.1^{+1.1}_{1.0}\)  \(13.76^{+0.20}_{0.21}\)  \(0.92\)  0.34  \(\)1.07 
D  \(0.66^{+0.08}_{0.06}\)  \({<}0.24\)  \({>}2.43\)  \(68.0^{+1.1}_{1.2}\)  \(14.39^{+0.25}_{0.24}\)  \(0.48\)  0.04  \(\)3.23 
E  \(0.81^{+0.08}_{0.07}\)  \({<}0.05\)  \({>}3.92\)  \(67.4^{+1.1}_{1.1}\)  \(14.52^{+0.24}_{0.23}\)  \(0.45\)  0.01  \(\)4.37 
Logarithmic prior  
A  \(0.30^{+0.08}_{0.07}\)  \({<}0.32\)  \({>}2.15\)  \(69.8^{+2.0}_{2.2}\)  \(14.02^{+0.46}_{0.39}\)  \(0.78\)  1.01  0.007 
B  \(0.34^{+0.07}_{0.07}\)  \(0.42^{+0.21}_{0.24}\)  \(1.86^{+0.84}_{0.40}\)  \(69.4^{+1.7}_{1.4}\)  \(14.10^{+0.37}_{0.35}\)  \(0.69\)  1.21  0.19 
C  \(1.44^{+0.67}_{0.47}\)  \({<}0.12\)  \({>}3.16\)  \(71.2^{+0.7}_{1.0}\)  \(13.75^{+0.20}_{0.14}\)  \(0.86\)  0.49  \(\)0.71 
D  \(0.60^{+0.04}_{0.03}\)  \({<}0.05\)  \({>}3.92\)  \(68.2^{+1.0}_{1.0}\)  \(14.34^{+0.21}_{0.21}\)  \(0.53\)  0.08  \(\)2.47 
E  \(0.65^{+0.06}_{0.04}\)  \({<}0.002\)  \({>}7.09\)  \(68.0^{+1.0}_{1.0}\)  \(14.38^{+0.20}_{0.21}\)  \(0.51\)  0.06  \(\)2.82 
5 Results
After the datasets introduced in the previous section and the discussed considerations, we have proceeded to run the MCMC chains in order to obtain the confidence regions of each parameterization and, therefore, achieving the main goal of this work, namely, obtaining a lower bound for the time of a future singularity. The results corresponding to our different cases are shown in Fig. 1, where we display the marginalized contours of the parameters for each model, and in Table 2. In order to have a further criterion to judge the statistical validity of our models, we have also analyzed, using the same datasets we have described in the previous section, the \(\Lambda \)CDM model and the Chevallier–Polarski–Linder (CPL) parametrization [69, 70], which is widely used as the most basic generalization of a constant dark energy to a dynamical fluid.
When considering the combination of all the datasets, as can be visually checked from Fig. 1, we find that the lowest value for the singularity time is achieved for model B and turns out to be \(t_{s,\mathrm{min}}\simeq 1.2\) (at the 2\(\sigma \) level), which corresponds to 2.8 Gyrs from today. Remarkably, this lower bound is priorindependent, i.e., it is for both the flat and the logarithmic priors on \(\alpha \) that we have used; this helps us to state that such limit is not statistically biased, but physically compelling. In that regard, it is advantageous to report that the minimum in the \(\chi ^2\) for models A and B is located, respectively, at \(x_{s} \sim 2.18\) and \({\sim } 1.55\). This does not happen for all other models, whose minima are located at \(\alpha \rightarrow 0\), and are only limited by numerical resolution at \({\sim } 10^{4}\). Interestingly, the obtained lower limit for the occurrence of the singularity is shorter than the expected time for the Sun to burn all its fuel (estimated to be 5–7 Gyrs).
An interesting feature of models A and B is that having the singularity at infinity is excluded at the 1\(\sigma \) level, when a uniform prior is assumed; when using a logarithmic prior, model A only lies within the 1\(\sigma \) region, while model B still excludes the singularity at infinity at the \(1\sigma \) level. We should remember that \(t_s=\infty \) corresponds to having a de Sitter universe in the asymptotic future, so for those two models, such a scenario seems to be disfavored. This highlights that having an asymptotically de Sitter universe in our parameterizations does not necessarily imply being close to a \(\Lambda \)CDM model. And we also have to point out that when we use a logarithmic prior, which is going to give a better sampling than the uniform prior in the range of very small \(\alpha \), such higher bound disappears for model A, but not for model B. Such feature makes the latter model the most interesting among all those we have considered. Remarkably, these two models present a Bayesian evidence which make them equivalent to \(\Lambda \)CDM from a statistical point of view, i.e., they provide fits as good as those of \(\Lambda \)CDM, and they are also even better than the widely used CPL parameterization. Notice, moreover, that the effective equation of state parameter today is close to the one of \(\Lambda \)CDM. For a more direct comparison of our models with \(\Lambda \)CDM, in Fig. 2 we plot the effective equation of state for a total fluid as introduced in Eq. (14) which influences the background dynamics of the universe and the expansion history H(a). However, we need to note that this only happens for the restricted dataset considered in our analysis, while \(\Lambda \)CDM give a good fit to a much wider variety of cosmological observations, while it is not clear whether the models with singularities will fit all those observations as well as \(\Lambda \)CDM.
For model C, the possibility of having the singularity at infinity is within the 1\(\sigma \) region. The Bayesian evidence in this case is worse than that for models A and B, but it is still not strongly disfavoring \(\Lambda \)CDM. Interestingly the effective equation of state parameter today for this case is substantially lower than for \(\Lambda \)CDM. Models D and E are strongly disfavored with respect to the baseline \(\Lambda \)CDM. Again, these models allow one to write \(t_s=\infty \) at the 1\(\sigma \) level. In these cases, we find that \(w_{\mathrm {eff},0}\) is higher than in the \(\Lambda \)CDM case. Finally, in Fig. 3, we plot the interaction term given by Eq. (16), assuming dark energy equation of state equal to \(1\) and a dynamical one given by the CPL best fit we have found in our analysis, and reported in Table 2.
6 Conclusion
In this work we have reconsidered the subject of future cosmological singularities occurring at a finite time. The aim of the work has been to establish a general lower bound for the time of a potential future singularity by using SN Ia, BAO and H(z) data. We have briefly reviewed the cosmological singularities emphasizing the fact that a divergence in a given cosmological parameter does not necessarily implies a singular spacetime. We have then discussed under which conditions a given cosmological singularity actually corresponds to a singular spacetime so that we can discern the severity of the different cosmological singularities. Our discussion focused on the geodesic completeness of the spacetime as well as the presence of divergent tidal forces when approaching the singularity.
After this brief theoretical review, we have constructed a set of parameterizations comprising different types of singularities. These parameterizations have been designed so that we recover an early time matter dominated phase that transits to a phase with a future singularity where a given timederivative of the scale factor diverges, but not the lower ones. We have then run a series of MCMC chains to confront our parameterizations to SN Ia, BAO and H(z) data. The obtained results are then summarized in Table 2. Our main conclusion is that within our family of parameterizations, a potential future singularity cannot be closer to the present time than \({\sim } 0.2 t_0\), which roughly corresponds to 2.8 Gyr. We found that the proximity of the singularity to the present time has a mild dependence on the type of singularity for our parameterizations, but we can conclude that in all cases there is a consistent lower bound around \(1.21.5 t_0\).
Another interesting conclusion that we have found is that, following results from the Bayesian evidence, our parameterizations A and B provide fits which are not significantly worse than \(\Lambda \)CDM for the considered datasets. This was not obvious a priori, since none of our parameterizations contain \(\Lambda \)CDM in its parameter space. Hence, as shown in previous references [38, 39, 40, 41, 42], a singular scenario cannot be discarded right away from tests of the background evolution and the time remaining for the occurrence of a future singularity may be shorter than expected. However, we need to stress that \(\Lambda \)CDM has become the standard model of cosmology because of its outstanding performance in fitting most cosmological observations, not only the ones considered in our analysis, so that in order to be able to establish a compelling scenario with a future singularity on equal footing as \(\Lambda \)CDM, we would need to show its ability to fit the rest of cosmological observations, including those sensitive to the perturbations.
Footnotes
 1.
Here we will focus on spatially flat universes. For the general case see [55].
 2.
We have tested other parameterizations for each type of singularity and found similar results, so we only report here the results for these representative parameterizations.
 3.
For the model C we need to simultaneously send n to infinity so that the product \(n\log (1t/t_s)\) remains finite.
 4.
Data for SDSS DR12 release are available for download at https://sdss3.org/science/boss_publications.php.
Notes
Acknowledgements
We thank Antonio L. Maroto and Diego RubieraGarcia for fruitful discussions and comments. J.B.J. acknowledges the financial support of A*MIDEX project (n. ANR11IDEX000102) funded by the “Investissements d’Avenir” French Government program, managed by the French National Research Agency (ANR), MINECO (Spain) projects FIS201123000, FIS201452837P, and ConsoliderIngenio MULTIDARK CSD200900064. R.L. was supported by the Spanish Ministry of Economy and Competitiveness through research projects FIS201457956P (comprising FEDER funds) and Consolider EPI CSD201000064. D.S.G. acknowledges support from a postdoctoral fellowship Ref. SFRH/BPD/95939/2013 by Fundação para a Ciência e a Tecnologia (FCT, Portugal) and the support through the research grant UID/FIS/04434/2013 (FCT, Portugal). V.S. is financed by the Polish National Science Center Grant DEC2012/06/A/ST2/00395.
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