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Quasi-Rip: A New Type of Rip Model without Cosmic Doomsday

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 $\rho(a)$ is monotonically increasing. In the present work, we are interested to investigate what will happen if this assumption is broken, and then propose a so-called"Quasi-Rip"scenario, which is driven by a type of quintom dark energy. In this work, we consider an explicit model of Quasi-Rip in detail. We show that Quasi-Rip has an unique feature different from Big Rip, Little Rip and Pseudo-Rip. Our universe has a chance to be rebuilt from the ashes after the terrible rip. This might be the last hope in the"hopeless"rip.


I. INTRODUCTION
Since its discovery in 1998, the current accelerated expansion of our universe [1] has been one of the most active fields in modern cosmology. As is well known, it could be due to an unknown energy component (dark energy) or a modification to general relativity (modified gravity) [1,2]. Before 1998, it was commonly believed that in the future our universe will either expand forever or contract again into a final Big Crunch. However, the discovery of the current accelerated expansion of the universe significantly changed our view of the fate of the universe. In fact, many novel possibilities are under the active consideration in the community nowadays.
Today, there are many dark energy candidates in the market. Among them, the famous phantom dark energy [3,4] is very interesting. Its equation-of-state parameter (EoS) is smaller than −1. Although phantom dark energy is consistent with the current observational data [1,3], it violates all the energy conditions. One of the consequences is that our universe will encounter a singularity at a finite time, namely the so-called Big Rip [4]. At this singularity, the scale factor a, energy density and pressure of our universe are all divergent. In fact, besides the traditional Big Bang, Big Crunch, and the Big Rip, many novel singularities have been considered in the literature, such as Sudden singularities, Generalized sudden singularities, Quiescent singularities, Big Boost, Big Brake, Big Freeze, w singularities, Inaccessible singularities, Directional singularities (see e.g. [5,6] and references therein for some brief reviews). These singularities arise at the price of violation of one or several energy conditions. As is well known, in [6] (see also e.g. [5,7,8]) the future singularities have been classified into four types, namely • Type I (Big Rip): a → ∞, ρ → ∞, H → ∞, |p| → ∞, when t → t s < ∞ ; • Type II (Sudden singularity): a → a s , ρ → ρ s , H → H s , |p| → ∞,Ḣ → ∞, when t → t s < ∞ ; where t s , a s , ρ s , H s , p s ,Ḣ s are all finite constants (a s = 0); ρ and p are energy density and pressure respectively; H ≡ȧ/a is the Hubble parameter, and a dot denotes a derivative with respect to the cosmic time t. In fact, the above four types can include almost all known future singularities. Of course, singularities usually are not desirable in physics. Therefore, other possible fates of our universe are also considered in the literature, such as the cyclic/oscillatory cosmology. Recently, some interesting scenarios concerning the fate of the universe attracted much attention in the community, namely the so-called "Little Rip" [9] and "Pseudo-Rip" [10]. In [9][10][11], their authors showed that if the cosmic energy density will remain constant or monotonically increase in the future, then all the possible fates of our universe can be divided into four categories based on the time asymptotics of the Hubble parameter H(t) [10], namely where H ∞ is a constant. Obviously, the Big Rip singularity is not the only fate of our universe with the phantom-like dark energy. Both Little Rip and Pseudo-Rip are non-singular, and hence fall outside of the four categories in [6]. Similar to the Big Rip, the Little Rip dissociates all bound structures, but the strength of dark energy is not enough to rip apart spacetime (unlike the Big Rip) [9,10]. On the other hand, the Pseudo-Rip dissociates the bound structures which are held together by a binding force at or below a particular threshold, and hence it is possible that only some bound structures are dissociated while the others are not dissociated (depending on the model parameters) [10]. In fact, the Little Rip is an intermediate case between the cosmological constant and the Big Rip [9], while the Pseudo-Rip is an intermediate case between the cosmological constant and the Little Rip [10].
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 [9][10][11], i.e., the dark energy is phantom-like (its EoS w < −1). In the present work, we are interested to investigate what will happen if this assumption is broken. Obviously, in the case of the dark energy density ρ(a) is monotonically decreasing (i.e., the dark energy is quintessence-like with an EoS w > −1), no rip will happen and no bound structures will be dissociated. On the other hand, in the case of the dark energy density ρ(a) monotonically decreases (namely w > −1) in the first stage and then monotonically increases (namely w < −1) in the second stage (this is the case of the so-called "quintom A" dark energy in terminology of e.g. [12] and references therein), the fate of our universe is the Big Rip, which is trivial in some sense. The third case is that the dark energy density ρ(a) monotonically increases (namely w < −1) in the first stage and then monotonically decreases (namely w > −1) in the second stage (this is the case of the so-called "quintom B" dark energy in terminology of e.g. [12] and references therein). It can be expected that in the first stage some or all bound structures will be dissociated (similar to the case of Pseudo-Rip), but then the disintegration process will stop, and the already disintegrated structures have the possibility to be recombined in the second stage. We dub it "Quasi-Rip", which is the subject of the present work (here we temporarily do not consider the case of oscillatory quintom dark energy). Since the quintom-like dark energy [13] (whose EoS can cross the so-called phantom divide w = −1) is slightly favored by the observational data [1] (see e.g. [12] for a comprehensive review), we note that the Quasi-Rip is well-motivated in fact.
This paper is organized as follow. In Sec. II, we discuss the disintegration of bound structures. In Sec. III, we present an explicit model of Quasi-Rip. We constrain this model with the observational data, and then clearly show the Quasi-Rip in this model. In Sec. IV, some concluding remarks are given.

II. THE DISINTEGRATION OF BOUND STRUCTURES
It is useful to introduce the concept of "inertial force" when we discuss the disintegration of bound structures. According to [10,11,23], in a flat Friedmann-Robertson-Walker (FRW) universe dominated by dark energy, (motivated by the well-known Newton's second law) the inertial force F inert on a mass m as seen by a gravitational source separated by a comoving distance r 0 is given by in which we have used the energy conservation equationρ + 3H(ρ + p) = 0. A bound structure dissociates when the inertial force F inert (dominated by dark energy) is equal to the force F bound holding together this bound structure [10,11]. For the gravitationally bound structure whose mass is M , it dissociates when or equivalently where ω 0 is the angular velocity; ρ 0 is the present density of dark energy; Ω 0 ≡ (8πGρ 0 )/(3H 2 0 ) is the present fractional density of dark energy. Note that in this paper, the subscript "0" usually indicates the present value of corresponding quantity, and we have set a 0 = 1. In fact, one can check that the results Eqs. (3) and (4) are coincident with the ones in the case of phantom dark energy whose EoS w is a constant [4] (we refer to e.g. [14] for its detailed derivation; note that the result of [4] is invalid for a non-constant w, while our results Eqs. (3) and (4) are still valid).
However, if the bound structure is massive enough to significantly affect the local spacetime metric, it is not accurate to express F inert in terms of FRW metric [9,10]. A more accurate method was presented in [14]. Following [14], in the Newtonian limit, the interpolating metric of the local spacetime is given by where η is the comoving radial coordinate. The radial equation of motion for a test particle in the Newtonian limit readsr where L = r 2φ = r 2 ω = const. = r 2 0 ω 0 is the constant angular momentum per unit mass. Noting that (6) can be recast as where the time-dependent effective potential is given by The bound structure dissociates when the minimum of the time-dependent effective potential (including the centrifugal term) disappears. Note that the corresponding radius r V min at which the effective potential reaches its minimum is determined bÿ One can find (with the help of e.g. Mathematica) that Eq. (9) Therefore, the minimum of the time-dependent effective potential disappears when which is also the time when the bound structure dissociates. Similarly, we can also recast Eq. (11) as As is shown in [14], the accurate disintegration time obtained here is usually earlier than the qualitative one obtained in [4] (this point can also be seen by comparing Eqs. (11), (12) with Eqs. (3), (4)). It is worth noting that most of the results in [14] are valid for dark energy whose EoS w is a constant, while our results obtained here are also valid for dynamical dark energy whose EoS w is not a constant.
Since the current observational data are in the epoch a < 1, we cannot ignore the contribution from pressureless matter in this stage, although it can be safely ignored in the above discussions when the universe is dominated by dark energy. From the Friedmann equation 3H 2 = 8πGρ tot = 8πG(ρ + ρ m ) and ρ m = ρ m0 a −3 , we have where Ω m0 ≡ (8πGρ m0 )/(3H 2 0 ) is the present fractional density of pressureless matter. If ρ(a) is given, we can use Eq. (13) to constrain the model parameters with the current observational data. On the other hand, from Eqs. (12) or (4), one can find the corresponding scale factor a * at which the bound structure dissociates. Then, noting H =ȧ/a, we can evaluate the disintegration time measuring from today (a = 1), namely where H −1 0 = 9.7776 h −1 Gyr.

III. AN EXPLICIT MODEL OF QUASI-RIP
Here, we consider an explicit model of Quasi-Rip. In some sense, constructing a suitable model is a smart task. There are various ways to this end. For example, one can specify the scale factor a as a function of the cosmic time t (see e.g. [5,15,16]), the pressure p as a function of the energy density ρ (see e.g. [6,17,18]), the energy density ρ as a function of the scale factor a (see e.g. [9,10]), or the Hubble parameter H as a function of the cosmic time t (see e.g. [11]). It is shown in [9][10][11] that these ways are equivalent in fact. In our case, for convenience, we choose to specify the energy density ρ as a function of the scale factor a, similar to the cases of Little Rip [9] and Pseudo-Rip [10]. As mentioned in Sec. I, here our task is to construct an explicit function ρ(a), which monotonically increases (namely w < −1) in the first stage and then monotonically decreases (namely w > −1) in the second stage. A naive idea is to use a piecewise function, which is phantom-like in the first sector (namely ρ(a) ∝ a −3(1+w1) with a constant EoS w 1 < −1) and is quintessence-like in the second sector (namely ρ(a) ∝ a −3(1+w2) with a constant EoS w 2 > −1). Then, let us refine this naive idea with a smooth function ρ(a) ∝ a µ(a) , in which µ(a < a t ) > 0 and µ(a > a t ) < 0, where a t is the transition point. Noting that it is more convenient to use ln a as the variable in Eqs. (12), (4), and (14), as well as the fact a = e ln a , it is appropriate to write µ as a function of ln a instead. Therefore, the simplest function ρ(a) can be given by where α and β are both constants. Let us have a closer observation. By requiring dρ/d ln a = 0, we find that its extremum locates at ln a = α/(2β). To ensure this extremum is a maximum, d 2 ρ/d(ln a) 2 should be negative here, therefore β > 0 is required. Substituting Eq. (15) into Eq. (13), we have where z = 1/a − 1 is the redshift. There are three free model parameters, namely Ω 0 , α and β. With this E(z), following the methodology in e.g. [19][20][21], we can constrain this model with the latest Union2.1 Type Ia supernovae (SNIa) dataset [22] which consists of 580 SNIa. In fact, we find that the best fit has χ 2 min = 562.224, and the best-fit parameters are Ω 0 = 0.718859, α = 0.0294816 and β = 0.0002525, whereas the corresponding h = 0.698862. In Fig. 1, we present the corresponding 68.3% and 95.4% confidence level (C.L.) contours in the α − β parameter space. Although their best-fit values are small, α and β can still be consistent with the observational data in the large 2σ C.L. region, i.e., −1.2 ∼ < α ∼ < 0.5 and 0 ≤ β ∼ < 3.2. Next, we show the Quasi-Rip in this model clearly. Substituting Eq. (15) into the reduced inertial force f (a) defined in Eqs. (12) and (4), we have f (a) = ρ ρ 0 (2 + α − 2β ln a) = (2 + α − 2β ln a) a α−β ln a = (2 + α − 2β ln a) e ln a (α−β ln a) .
In Fig. 2, we plot ln (ρ/ρ 0 ) and ln f as functions of ln a for various model parameters. For convenience, we fix β = 0.0001 which is at the same order of its best-fit value, while the various values of α are taken from the 1σ C.L. region in Fig. 1. From the left panel of Fig. 2, it is easy to see that our ρ(a) in Eq. (15) is the desirable one, which monotonically increases in the first stage and then monotonically decreases in the second stage. It is not surprising that the plot of ln f is very similar to the one of ln (ρ/ρ 0 ), since their difference is given by ln (2 + α − 2β ln a) which is a small quantity in fact (cf. Eq. (17)). Depending on the model parameters α and β, the inertial force can successively disintegrate the Coma Cluster, the Milky Way galaxy, the Solar System, the Earth and the Hydrogen atom. In different cases, some bound structures will be disintegrated while the other bound structures will not. If none of the bound structures is disintegrated, the corresponding Quasi-Rip is a failed rip. From the right panel of Fig. 2, we find that the innermost one is a failed rip, the outermost one is the strongest Quasi-Rip which can disintegrate all the five bound structures, and the others are moderate ones which can only disintegrate one or several of the five bound structures. From Eqs. (12) or (4), we can find the corresponding ln a * at which the bound structure dissociates. Then, we can evaluate the disintegration time measuring from today t * − t 0 by using Eq. (14) with E(ln a) given in Eq. (16), and present the results in Tab. I. For a fixed β, a bound structure dissociates earlier with a larger α, since the corresponding inertial force is stronger. In the case of α = 0.27, the difference between the disintegration time of Earth and Solar System is 4349 years, while the difference between the disintegration time of Hydrogen atom and Earth is only 15 years. Finally, it is easy to see that the most distinct feature of Quasi-Rip is that the inertial force monotonically decreases in the second stage. Eventually, it will become lower than all the thresholds to disintegrate the bound structure. Therefore, the already disintegrated structures have the possibility to be recombined in the second stage. This is the unique feature of Quasi-Rip different from Big Rip, Little Rip and Pseudo-Rip. Our universe has a chance to be rebuilt from the ashes after the terrible rip. This might be the last hope in the "hopeless" rip.

IV. CONCLUDING REMARKS
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 happen if this assumption is broken, and then propose a so-called "Quasi-Rip" scenario, which is driven by a type of quintom dark energy. In this work, we consider an explicit model of Quasi-Rip in detail. We show that Quasi-Rip has an unique feature different from Big Rip, Little Rip and Pseudo-Rip. Our universe has a chance to be rebuilt from the ashes after the terrible rip. This might be the last hope in the "hopeless" rip.
Some remarks are in order. Firstly, as is shown in Sec. III, our Quasi-Rip model is well consistent with the current observational data. However, even in the 1σ C.L. region of α − β parameter space, the future behavior of our universe can be different enough (depending on the particular model parameters α and β). In fact, as is well known, the current observational data can be consistent with all the phantomlike, quintessence-like and quintom-like dark energy models. Therefore, the current observational data cannot tightly tell what is the true fate of our universe. Most of the possibilities (including Big Rip, Little Rip, Pseudo-Rip, Quasi-Rip, de Sitter expansion, other future singularities and so on) are still living.
Secondly, the explicit model of Quasi-Rip considered in the present work is the simplest case. One can construct other more complicated ρ(a) to implement the Quasi-Rip. For example, one might construct an EoS w(a) as a function of scale factor a, which is smaller than −1 when a < a t and larger than −1 when a > a t . Then the corresponding ρ(a) can be found from the energy conservation equatioṅ ρ + 3Hρ(1 + w(a)) = 0. Of course, other smart methods to construct the desirable ρ(a) are awaiting us.
Thirdly, as mentioned in [10], in its second Pseudo-Rip model, the reduced inertial force f (a) can also have a peak, similar to our Quasi-Rip model. However, we note that in the Pseudo-Rip model, after the peak, the reduced inertial force f (a) → const. which is still higher than the corresponding threshold to disintegrate the bound structure. Therefore, the already disintegrated structures have no possibility to be recombined in the Pseudo-Rip models. On the contrary, in the Quasi-Rip models, the reduced inertial force f (a) monotonically decreases in the second stage. Eventually, it will become lower than all the thresholds to disintegrate the bound structure. Therefore, the already disintegrated structures have the possibility to be recombined in the second stage.
Fourthly, it is well known that phantom is unstable at quantum level and hence the perturbations grow large. Noting that in the present work our discussions are at classical level instead, this problem could be set aside. In fact, the quantum stability of a phantom phase has been considered in [24]. The authors of [24] studied the perturbations in the quantum-corrected effective field equation at one-and two-loop order, and they found that the system is stable. On the other hand, it is claimed in [25] that scalar perturbations can grow during a phantom phase if EoS w < −5/3. However, from Eq. (15) and Fig. 1, it is easy to see that the corresponding w is only slightly smaller than −1 in the phantom phase for our particular α and β (n.b. Fig. 1). So, w > −5/3 instead and hence our Quasi-Rip model can avoid the problem raised in [25]. Further, in fact the dark energy considered in this work is not necessarily a scalar field. It could even be an effective dark energy from modified gravity, namely the so-called "geometric dark energy". So, it might avoid the corresponding problems in the phantom phase.
Finally, in the present work, we consider only the quintom dark energy which crosses the phantom divide w = −1 once. In fact, we can also consider the quintom dark energy which can cross the phantom divide for many times. The most attractive Quasi-Rip model might be the one driven by the oscillatory quintom dark energy. In this oscillatory Quasi-Rip model, our universe will be destroyed and then be rebuilt again and again.