The dynamics of the flat anisotropic models in the Lovelock gravity. I: The even-dimensional case
In this article we give a full description of the dynamics of the flat anisotropic (4+1)-dimensional cosmological model in the presence of both Gauss-Bonnet and Einstein contributions. This is the first complete description of this model with both terms taken into account. Our data is obtained using the numerical analysis, though, we use analytics to explain some features of the results obtained, and the same analytics could be applied to higher-dimensional models in higher-order Lovelock corrections. Firstly, we investigate the vacuum model and give a description of all regimes; then, we add a matter source in the form of a perfect fluid and study the influence the matter exerts upon the dynamics. Thus, we give a description of matter regimes as well. Additionally, we demonstrate that the presence of matter not only “improves” the situation with a smooth transition between the standard singularity and the Kasner regime, but also brings additional regimes and even partially “erases” the boundaries between different regimes inside the same triplet. Finally, we discuss the numerical and analytical results obtained and their generalization to the higher-order models.
pacs:04.20.Dw, 04.25.dc, 04.50.-h, 04.50.Kd, 98.80.-k
The idea of extra dimensions traces back to the beginning of last century, to the papers by Nordström Nord1914 (), Kaluza Kaluza () and Klein Klein (). At that time such ideas were considered as mere mathematical speculations, but with time they have come to hold a firm place in the minds of scientists. The 70s gave rise to an interest in the extradimensional theories due to the development of superstring and supergravity theories (see, e.g., super ()). Later in the 90s the interest has only increased owing to the possibility to solve the hierarchy problem hierarchy (), lower the grand unification scale in the M-theory M-GUT () and in many others (see, e.g.,rec-rev () for recent reviews).
Dealing with extra dimensions in the cosmological context, one usually uses a modified gravity (see the forementioned reviews). One of modifications of this kind is the Lovelock gravity Lovelock (). This theory is equivalent to Einstein’s theory in (3+1), but starting from (4+1) it gives rise to higher-order (in powers of curvature) corrections. The first of these corrections is known as the Gauss-Bonnet term, first found by Lanczos Lanczos () (therefore it is sometimes referred to as the Lanczos term). In the cosmological context the Lovelock (and Gauss-Bonnet in particular) gravity was intensively studied over the past two decades studied ().
In this paper we investigate the dynamics of the flat anisotropic Universe in Lovelock gravity. Since we work in a (4+1)-dimensional case, Lovelock contribution reduces to the Gauss-Bonnet term. We took both the General Relativity (GR) and Gauss-Bonnet (GB) contributions into account for a reason. Namely, with only one of them taken into account, some exact solutions (see, e.g., il1 (); we2 (); we3 (); iv1 (); iv2 ()) could be obtained (and that is true not just for GB, but for any order of Lovelock corrections and in any number of dimensions prd ()). Though when we take into account different contributions, the equations become much more complicated making it almost impossible to obtain any exact solution in general case (for the first time solutions with both terms taken into account were found in deser (); in we3 () a cosmological solution in exponential form was obtained). Yet, the influence of the other (than leading) terms could be significant – for instance, in we2 (), while investigating a (4+1)-dimensional model with matter in the form of a perfect fluid in a “pure” Gauss-Bonnet case, we discovered an unusual behavior in case, and we believe it was caused by neglecting the GR term in the equations of motion.
The (4+1)-dimensional case with both GR and GB contributions was studied in ind (), but a full analysis of all regimes was not performed there. In this paper we are going to do this – we will give a description of all regimes, as well as try to explain some of them analytically.
The second aim of this paper is to investigate the regimes in presence of matter in the form of a perfect fluid. We reported some features of the influence of matter in we3 (), but no actual description was given. Namely, we reported that the presence of matter leads to an increase of the “probability” of smooth transition between the low- and high-energy Kasner regimes. In the current paper we give a full description of all types of transitions as well as describe the influence of the matter in large. Finally, we generalize the results obtained – both for the vacuum and matter cases – to higher even-dimensional models with all possible Lovelock corrections taken into account.
This article is the logical continuation of prd (). Indeed, in prd () we considered the flat anisotropic cosmological models in the Lovelock gravity, derived the equations of motion and investigated the dynamics with only the highest Lovelock correction taken into account. In this paper we investigate the lowest model (, ) with all corrections taken into account and based on the results predict the behavior of even-dimensional models with all possible Lovelock corrections considered.
Ii Equations of motion
Equations of motion can be easily derived from the general ones, obtained in prd (). In terms of Hubble functions they take a form: dynamical equations
(this is the first equation; the rest of them could be obtained via a cyclic indices transmutation) and a constraint equation
The system (1)–(2) in the vacuum and (1)–(3) in the matter case combined with the initial conditions completely determines the evolution of the Universe as a whole. Hence we are going to solve numerically these equations to find out the future and past evolution of the model with particular initial conditions. Scanning over the initial conditions gives us the needed distribution of regimes.
Iii Vacuum model: special cases
Before starting to present the results it would be an asset to consider two special cases that can be found on our transition maps. We call the first of them a 3-equal case with three Hubble parameters equal to each other. In ind () it was noted that only this regime in a (4+1)-dimensional case has a smooth transition between the low- and high-energy Kasner regimes. Using and the dynamical equations reduce to
Note that our equations are different from those in ind (),
and we claim ours to be correct
We call the other case a 2-equal – now only two Hubble parameters are equal to each other. Unlike the 3-equal case, here the denominator of can cross zero so a nonstandard singularity could occur. But what is important, in this case there also exists a smooth transition between the low- and high-energy Kasner regimes, thus, the conditions for a smooth transition between the low- and high-energy Kasner regimes are somehow weakened. But since a 2-equal case still needs exact equality, the measure of this transition is not improved.
Iv Vacuum model: results
As we claimed above, we are going to make a scans over the initial values of Hubble parameters to produce maps of trajectories. Instead of producing 3D maps, as it was done in ind (), we will make several 2D plots with different values for the third Hubble parameter. Thus the 3-equal case would be a point on our 2D plot with coordinates that coincide with the fixed third Hubble parameter, and the 2-equal case would be a diagonal line and two perpendicular lines with the - and -coordinates equal to the fixed third Hubble parameter. So we have three “free” Hubble parameters, the fourth is calculated from the constraint equation (2):
Now let us have a look on the denominator of (8). Later, when dealing with the matter case, we will see that this denominator (more specifically, its sign) is an essential thing in describing the matter regimes. For now let us note that if all three Hubble parameters are positive, the denominator is also positive, but if one of the three Hubble parameters is negative, the denominator could be (but not always) negative. We refer to the (, , ) triplet as a “positive” one if it produces the positive denominator of (8) and as “negative” if it is negative.
Thus we fix the initial value for one of the Hubble parameters (without loss of generality, let it be ) and scan over and in [0…1.5 2] range. In previous studies we2 (); we3 (); we1 () we usually considered [0, 1] range, but now, for the demonstrative reasons we decided to expand it. As for , we vary it roughly in [1.5 …1.5] range. Also, as in all previous studies, we consider only the initially-expanding Universe: . First, we present our results for the positive , then for the negative.
Before giving the results, let us summarize what we would expect. For the past evolution, we expect either a standard singularity (high-energy Kasner regime) or a nonstandard one. For the future, there are three possibilities: a low-energy Kasner regime, a recollapse or a nonstandard singularity. For simplicity we denoted them as summarized in Table. With the sign of triplet taken into account, we denote the trajectories like their types with appropriate signs, e.g., type VI+ corresponds to type VI with a positive triplet.
The results for the case are given in Fig. 1. In there, we plotted the resulting past and future behavior for the fixed , and with vs. as coordinates. The resulting transitions are denoted according to the Table, the dashed lines are type I trajectories. The value for is increasing from Fig. 1(a) to Fig. 1(f): in (a), in (b), in (c), in (d), in (e) and in (f).
From Fig. 1 one can understand how the transitions evolve with changing . At some value – in our case with it occured at – two type-IV regions “detach” from each other; with further growth of the bottom-left region shrinks; the upper-right one also decreases in size and moves by diagonal towards the growth of and . The value this “detachment” occurs at is dependent on and governed by a high-order equation, so it is impossible to give it an exact analytical expression.
The case is more complicated than the
previous one. Indeed, with all three “free” initial Hubble
parameters being positive, the last Hubble parameter is always
negative while (one can simply
verify this from (8)). But with being
The results are presented in Fig. 2. Therein we gave 2D scans for four different values of : in (a) panel, in (b), in (c) and in (d). At type-VI trajectories completely “disappear” from the scene, hence we decided to skip the plots of that kind. A white region corresponds to the initial conditions with ; hyperbola-like regions originate from zeros of the denominator in the expression: . Thus, the upper-right “half” of the 2D plots in Fig. 2 has a negative denominator of (8) while the others have positive denominators.
In addition to the type II, IV, and VI trajectories at low values of , one can find type-I trajectories as well. There as well exists the transition of I type, that we can call “the rarest” trajectory. Indeed, it exists only at , only at 2-equal lines, at low enough (type-IV should not disappear yet – ), and on the upper “half” of the plot; the I trajectories are denoted by a dotted line in Fig. 2(a). The I+ transitions are denoted as solid lines in Fig. 2(a) and (b). One can see that type-I trajectories exist between the lower boundary of the and some point that we denoted as diamonds in Fig. 2(a) and (b). These points correspond to the 3-equal situation: indeed, from Sec. III we remember that in the 3-equal regime the calculated Hubble parameter is always negative. If we assume to be that parameter, then the 3-equal situation occurs at
Two different 3-situations are infinitely separated at , become closer with the growth of and coincide when (with it happened at ; the situation is presented in Fig. 2(b)).
V Vacuum model: discussion
In the first half of the article we dealt with the vacuum (4+1)-dimensional flat anisotropic model in the Einstein-Gauss-Bonnet gravity. One can see that the type-II trajectories are dominating in both and situations; in other words, the type-II trajectories are dominating all other regimes in the model considered. In the case we have IV and VI-type trajectories more or less abundant, their presence is in general lessened with the growth of , but they never vanish. In the case the situation is more dramatic: with small we have I, II, IV, and VI type trajectories, but with the growth of we first “loose” I and IV, then VI, therefore finally at (with ; with a different value for this will happen at another value for the ) we have only type-II transitions.
In Fig. 3 we presented the examples of all the possible transitions. Black curves correspond to the expansion rate in terms of the Hubble parameters , and grey curves represent the value for the denominator of the expression. Time is along -axis and is normalized in a way to set ; zero-point corresponds to the start of integration. Type I is in (a), II – (b), IV – (c), and VI – (d). Hence there are no type-III and V transitions, and now we will try to explain why. One can see that the nonstandard singularity occurs only in the vicinity of the standard one (but it is not necessary that it will happen – see Fig. 3(a) for example). From Figs. 3(a) and (c) one can learn that the trajectories, starting from the positive values of the denominator end up in the Kasner regime. We checked that for the scans in Fig. 1, and found that the type-IV regime completely coincides with an initially positive denominator. The reason behind this link between the positivity of denominator and the late-time Kasner behavior is a bit aside from the goals of this paper and is yet to uncover – we are going to devote a separate paper to the nonstandard singularity in flat anisotropic models in the Lovelock gravity. Anyway, type III and V trajectories cannot originate from the positive initial value of the denominator. As for the negative value, the remaining two possibilities are presented in Figs. 3(b) and (d). One could think of the possibility for the type-III trajectory to occur on the singular analog of the left branch in Fig. 3(b) – on the short “window” between the standard and nonstandard singularities, but our numerical analysis demonstrated that it is not happening. We believe this situation could not even be constructed: crossing the nonstandard singularity changes the sign of the denominator leaving the numerator unchanged; thus at this point the sign of changes, making it useless to try to construct any continuation beyond the nonstandard singularity.
Vi Model with matter
In this section we start our investigation of the models filled with matter in the form of a perfect fluid. In this case the constraint equation gives us a bit different expression for the 4th Hubble parameter:
From (10) one can easily see that the expression for is split into two parts – -dependent and -independent. The latter remains the same with varying while the former changes the value for . At this point the positive and negative triplets start to play an important role. These two have a huge difference – the positive triplets have no boundary on the density while the negative do. Indeed, keeping in mind that and the fact that the increasing density decreases , even if at we have , at some value for we will have . This difference between the positive and negative triplets affects the details of their regimes, so while describing the regimes we will describe them separately.
Vii Model with matter: results
Now let us describe the matter regimes. The first one to describe is I+ case. In the vacuum case, only I+ and I have the smooth transition from the standard singularity to the Kasner expansion, but, as we noted in we3 (), in the model with matter it is no longer the case. In the presence of matter the transition remains unchanged: it is from the standard singularity to the expansion. The difference between the and cases lies in the isotropisation: at the initial singularity becomes isotropic (and the expansion is the GR-dominated Kasner) while at we have a standard GB-dominated singularity and an isotropic expansion.
Trajectories of I type, as we already mentioned, are “the rarest” trajectories. In the presence of matter the regimes changes according to Fig. 4(a); the trajectory types are denoted in the figure. As a regime with the negative triplet, it has the maximal value for density; the -axis of the Fig. 4(a) covers all possible values for density.
The transitions in the map for II+ type trajectory are given in Fig. 4(b). There is a difference between the regimes at and : in the case, the GR-Kasner “transforms” into an isotropic expansion while in the case, the isotropic one is the initial singularity.
From Fig. 4(b) one can easily see what brings an addition of matter to this regime. First of all, new, unknown for the vacuum case, type-III regime is introduced. It doesn’t cover a wide range of and , but its presence is still important. Secondly, the type-I trajectory in this case is unbounded from high densities. Thus, we find that the presence of matter in the form of a perfect fluid makes the type-I transition common in terms of initial parameters in contrast with the vacuum case, where one requires fine-tuning to achieve this regime. One can also notice that the type-I transition covers both and cases, so the expansion can be both Kasner-type and isotropic, respectively. Finally, the region of type-IV trajectories is also unbounded from high densities and lies mostly in the lower half of the map; later we will see that this is more or less typical for the type-IV trajectories.
The II case is more complicated than the others. Namely, it has a different behavior in the and cases. We consider them separately, and now we only note one feature of the regimes: the initial singularity is GR-dominated in contrast with I+ and II+ where in the case the initial singularity was isotropic.
In case almost all the trajectories remain the same – II type. Since this is “standard singularity recollapse”, one can find its “lifetime”. To demonstrate the influence of matter on the dynamics, we plotted the resulting lifetimes on the map (see Fig. 4(c)). The stroked region in the bottom-right corner marks the positions of periodic trajectories – a new type of solution in the vicinity of the stationary one (), described in we3 (). Also one can notice that there is no abrupt difference between the and regimes.
And one last point regarding this regime – in the case of (this is possible in some triplets) we do not have anymore periodical trajectories (and so only type II trajectories remains) as they are all located at which is beyond physical assumptions.
The case with is different from the described above – it has more different regimes. In Fig. 4(d) we presented the map of transitions in coordinates.
The map of trajectories for the IV+ case is given in Fig. 5. There in panel (a) we present a large-scale structure of the transitions; in panel (b) – a detailed structure of the boundary between I and IV at low densities and in panel (c) – the boundary between I and IV at large.
The resulting transition map for the IV case is given in Fig. 6(a). The type IV trajectories are quite bounded in there, but we have another regime, unseen in the vacuum case – type V. With it we have all the six possible regimes in the presence of matter. With high enough we have a relatively large area with periodic trajectories.
Viii Model with matter: discussion
In the second part of the paper we studied what influence the
matter in the form of a perfect fluid exerts upon the dynamics of
flat (4+1)-dimensional anisotropic models in the
Einstein-Gauss-Bonnet gravity. One of the two main changes from
the vacuum case is a substantial increase of the type-I
transitions we previously reported in we3 (). An interesting
feature – only positive triads have type-I transition in the
presence of matter, negative (with an exception of I) do not.
In this paper we demonstrate this increase “quantitatively”.
Another feature is the appearance of type-III and type-V
trajectories. First of them appeared exactly the way we discussed
in the vacuum case: on the singular analog of the left branch of
Fig. 3(b). We presented both III and V in Fig.
One can find different “features” on the transition maps and most of them (those that are unrelated to ) are just triplet-specific features. A feature at the II+ transition map (Fig. 4(b)) where all five regimes “touch” is a good example. If we draw a line from that point to we will notice that the lifetimes for type-II trajectories along this line are the same and equal to that of the vacuum case. It is obvious that the lifetime in the vacuum case is triplet-specific and varies from triplet to triplet, hence the location of features of that type also varies from triplet to triplet. The same situation is with the features of other kinds.
Now we want to draw your attention to the II case. One can see that it is very different from other cases. It has periodical trajectories at an “unusual” (from the other regimes that have periodical trajectories) place and it has much more type-VI transitions than the other regimes do. The reasons behind this behavior are not clear; probably, that is due to crossing the “special solution” , found in we2 () numerically for the GB+matter case (later in prd () we demonstrated that this is a general feature for all even-dimensional flat anisotropic models in the Lovelock gravity with only the higher-order correction taken into account).
One can notice that generally, inside a triplet, the transition maps have a lot in common. For the positive triplet we have IIIIIV “regimes stratification” (over ) on large : in the II+ case (Fig. 4(b)), we have the same situation in IV+ (Fig. 5) and VI+ (Fig. 6(c, d)). In the II+ case type-III trajectory “disappears” at some density, but that is just an effect of a particular triplet: in another triplet with a lower value for at which the feature occurs we will have all the three regimes at high densities. Negative triplets also have some features in common. They all have type-II at the intermediate and high values of density and, if , they have periodical trajectories. They all as well (except II) have type-VI at low–intermediate values of density.
Therefore we see that the positive triplets have a high-density regime stratification over while the negative triplets have it over the density itself. Keeping in mind the difference between the triplets one can explain this: for negative triplets the density is more important – high density can cause , so the value for density determines initial regimes. For positive triplets all values of density are acceptable hence the role of density is less important and the equation of state determines the regimes.
Ix Higher-dimensional cases
Finally, as we claimed at the beginning, we are going to generalize the obtained results on the higher-dimensional flat anisotropic models in the Lovelock gravity. First of all, in prd () we obtained the equations of motion for the flat anisotropic cosmological model in the Lovelock gravity in any number of dimensions with only the highest possible correction taken into account. We also demonstrated how one can obtain the equations for the mixture of corrections. If we take into account all possible corrections, the resulting equations for different orders would have a similar structure, and so would the solutions. Therefore we expect the II-type trajectories to dominate in all the even-dimensional models. The negative multiplets (the analogs of triplets from (4+1)) would as well have a more complicated structure than that in Fig. 2. This is caused by more possible zeros of the denominator of the calculated Hubble parameter. In its turn, it will increase the effective presence of type-I, IV, and VI trajectories. The gain in type-I is also due to the increase of possible ()-equalities, but still the measure will be incomparable with other types of regimes.
We do not expect any changes in the negative multiplets – they are governed by the density – but there might be changes in the positive multiplets. Indeed, they are governed by the equation of state, and its influence is dependent on the number of dimensions. The equation of state that corresponds to the highest possible contribution (-order) is . With the growth of we have ; as one of the consequences one would expect a decrease of the presence of the II type trajectory with – the corresponding range of would shrink eventually decreasing the presence. As another consequence one would expect a decrease of the type-I presence in the positive multiplets (apart from I we do not have type-I in the negative triplets in (4+1)). This is caused by the same “shrinking” while is the “special solution” for the even-dimensional case prd (). Thus the region from (4+1) (where the type-I trajectories are located) with the growth of would shrink resulting in a decrease of presence of the type-I transitions.
Also, the characteristic scale of the Hubble parameters where types I, IV, and VI occur will drop with the growth of . This is due solely to the number of dimensions – indeed, the same amount of matter produces lower density in the higher dimensions. Our calculations for and confirmed that. Thus, purely due to the growth of the effective presence of the type-I, IV, and VI trajectories is decreasing.
In this paper we gave the first complete description of all the regimes in the (4+1)-dimensional flat anisotropic cosmological model in the Einstein-Gauss-Bonnet gravity. All through the paper we hold the condition on the initial values of Hubble parameters: . With the case taken into account, our description will be complete. And the case could be easily obtained from the case: indeed, assuming we effectively only make a transform, so we will have the same, but time-reversed results, such as, “standard singularity Kasner” turns into “Kasner recollapse” and so on. With this remark in mind, our description of the regimes is complete for all the possible values of the initial Hubble parameters.
We have found that the type-II transition (“standard singularity recollapse”) dominates over all the other types of trajectories in the vacuum case, and describe it both qualitatively and quantitatively. We as well found that the type-I transition (“standard singularity Kasner regime”) is spread more widely than it was assumed ind (), but still not enough to compete even with type IV.
If we add matter in the form of a perfect fluid, the situation
changes drastically. Not only the abundance of the type-I
transition becomes comparable with abundances of other
types we3 (), but we also obtain new regimes that are unknown
for the vacuum case
Apart from the description of the matter regimes, we found that at intermediate–high densities all the regimes (inside their triplet) look alike: positive triplets have IIIIIV “regime stratification” (with a decrease of ) while negative ones have VIII “regime sequence” (with an increase of ). Thus, the influence of matter is stronger than we anticipated: it does not just significantly increase the abundance of the type-I transition and adds the two remaining possible trajectories, but also almost “erases the bounds” between different trajectories inside the triplet. But there are exceptions like II with that we discussed in the relevant section.
Finally we generalized the obtained results on the higher-dimensional case. We expect that the vacuum even-dimensional flat anisotropic models in the Lovelock gravity of the order are dominated by the type-II transition. The presence of the other types of transitions is in general more and more suppressed with the growth of . The presence of matter acts the same (as described for (4+1)) way in the case of negative multiplets and the same but with decreases of the presence of type-I trajectories with the growth of in the case of positive multiplets. Therefore, the even-dimensional vacuum models are degenerative from the point of view of the presence of a smooth transition between the standard singularity and the cosmological expansion; the presence of matter in the form of a perfect fluid lifts this degeneration but with the growth of this improvement decreases. Let us note, though, that () in differs from the described above picture 2plus1einst (); hence, the analysis is valid only a t .
- One can easily verify it – indeed, substituting into Eq. (7) in ind () (that is the dynamical equation that corresponds to , so it does not have itself but only , , and ), one would directly obtain (5); it is quite easy to solve it to obtain (6). The difference might be caused by a wrong sign in (in ind () the authors have a plus instead of a minus).
- The same with the previous subsection, we fix and plot 2D figs with and as the - and -coordinates.
- Remarks to Fig. 3 regarding the time are valid to Fig. 7 as well.
- Let us note – considering the case would not bring these two cases for they are time-reversal to each other.
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