Possible evolution of a bouncing universe in cosmological models with non-minimally coupled scalar fields
We explore dynamics of cosmological models with bounce solutions evolving on a spatially flat Friedmann–Lemaître–Robertson–Walker background. We consider cosmological models that contain the Hilbert–Einstein curvature term, the induced gravity term with a negative coupled constant, and even polynomial potentials of the scalar field. Bounce solutions with non-monotonic Hubble parameters have been obtained and analyzed. The case when the scalar field has the conformal coupling and the Higgs-like potential with an opposite sign is studied in detail. In this model the evolution of the Hubble parameter of the bounce solution essentially depends on the sign of the cosmological constant.
a]Ekaterina O. Pozdeeva,
\affiliation[a]Skobeltsyn Institute of Nuclear Physics, Lomonosov
Moscow State University, Leninskie Gory 1, 119991, Moscow, Russia
b]Maria A. Skugoreva,
\affiliation[b]Kazan Federal University, Kremlevskaya str. 18, Kazan, 420008, Russia
b,c]Alexey V. Toporensky,
\affiliation[c]Sternberg Astronomical Institute, Lomonosov Moscow State University,
Universitetsky pr. 13, Moscow, 119991, Russia \emailAddatopor@rambler.ru a]Sergey Yu. Vernov \emailAddsvernov@theory.sinp.msu.ru
bounce, scalar field, non-minimal coupling, de Sitter solution
The global evolution of the observable Universe can be separated on four epochs: inflation, radiation domination era, matter domination era and dark energy domination era. All these epochs can be described by General Relativity models with minimally coupled scalar fields, because the assumption that the Hubble parameter is a monotonically decreasing function does not contradict to the observation data. In such type of models the initial period of the Universe evolution with energies above the Planck scale should be described by quantum gravity because the classical evolution includes the initial singularity. Important question of theoretical cosmology is whether the entire Universe evolution can remain classical and has no singularity.
It is possible to avoid this singularity considering modified gravity. Bouncing universe smoothly transits from a period of contraction to a period of expansion and its evolution remains classical. Models of bouncing universes attract a lot of attention [1, 2, 3]. We can mention, in particular, bouncing models with Galileon fields [4, 5, 6, 7, 8, 9, 10], gravity models [11, 12], non-local gravity models [13, 14, 15, 16, 17] and models with non-minimal coupling [18, 19, 20, 21].
The goal of this paper is to explore new mathematical features of cosmological
models with non-minimally coupled scalar fields that admit bounce solutions. We consider models with the spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) metric
Note that the General Relativity models with minimally coupled standard scalar fields have no bounce solution, because in these models the Hubble parameter does not increase. The simplest way to get a bounce solution is to consider a model with a phantom scalar field. Point out that similar models are used to describe dark energy with the state parameter less then minus one [26, 27, 28]. To get non-monotonic behaviour of the Hubble parameter quintom models have been developed. These models can describe both bounce , and dark energy [30, 31]. In models with phantom fields the Null Energy Condition is violated and instability problems arise . At the same time it is easy to get increasing or non-monotonic behavior of the Hubble parameter in models with a standard scalar field non-minimally coupled to gravity , for example, in induced gravity models . Such scalar-tensor models have no ghost [33, 35].
To obtain bounce solutions in cosmological models with standard scalar fields one can use non-minimal coupling between the scalar fields and gravity and consider the following action:
where is the determinant consisting of metric tensor components , is the Ricci scalar, and are differentiable functions of a scalar field .
From the condition that at the bounce point the Hubble parameter , it follows that the potential should be negative: . The characteristic property of models with potentials that are not positive definite is the existence of the unreachable domain on the phase plane, which corresponds to non-real values of the Hubble parameter [36, 37, 38, 39, 40].
In this paper we consider models with bounce solutions that generalize the integrable model proposed in  with the Ricci scalar as an integral of motion. This model has the potential in the form , with and , and the standard quadratic coupling with and a positive constant . As known, in the spatially flat FLRW metric the Ricci scalar is a function of the Hubble parameter and its first derivative. So, the condition that is a constant defines the Hubble parameter as a solution of the first order differential equation. Thereby, we get only monotonic behavior of the Hubble parameter (formulae are presented in Section 3). In particular, this integrable model contains bouncing cosmological solutions with smooth future behavior tending to a de Sitter solution (while past behavior shows emerging of a Universe at a point with zero effective Newtonian constant). It has been shown in  that the integrable cosmological model  with a constant belongs to one-parameter set of integrable models with one and the same function and different potentials . However, the Ricci scalar is not an integral to motion for these integrable models. In this paper we show that the model, proposed in , is unique in the sense that the property of having to be a constant disappears, when one changes coupling function or scalar field potential (or both).
To construct more realistic models with both a bounce, and a non-monotonic behavior of the Hubble parameter one can consider non-integrable models that are close to the above-mention integrable model with a constant . In  the authors got such types of the Hubble parameter considering non-integrable models with the same potential and other values of . In the present paper we mostly vary the potential in searching for conditions of bouncing behavior similar to studied in  to exist in these more general models. In Section 4 we consider the model with and the Higgs-like potential multiplying by a negative constant. In Section 5 we consider models with different positive values of , a cosmological constant and different even degrees of monomial potentials. Using numerical calculations we demonstrate the possibility of different evolutions in the cases of potentials of the second, fourth and sixth degrees. We analyze the reasons of such behavior and find the corresponding conditions on the model parameters.
The paper is organized as follows. In Section 2 we remind a reader the general bounce conditions in non-minimally coupled scalar field theory as well as the conception of the effective potential. In Section 3 we show that the choice of potential in the paper  leading to the property that the curvature is the integral of motion is the unique one. On the contrary, in Sections 4 and 5 we show that bounce behavior admits much larger class of potentials. In Section 4 bounce behavior is studied for potentials with quadratic term in addition to quartic and constant terms and . The models with are considered in Section 5. In Section 6 we show that the obtained bouncing solutions do not suffer from the gradient or ghost instability. Section 7 contains a summary of the result obtained.
2 Bounce solutions in non-minimally coupled models
In the spatially flat FLRW metric with the interval:
we obtain from action (1) the following equations:
where a “dot” means a derivative with respect to the cosmic time , and a “prime” means a derivative with respect to the scalar field . The function is the scale factor, its logarithmic derivative is the Hubble parameter. We note that an effective gravitation constant in the model considered is
The dynamics of the FLRW Universe can be prolonged smoothly into the region of (see, for example [41, 42]), however, any anisotropic or inhomogeneous corrections are expected to diverge while tends to infinity [24, 43]. In this paper we analyze such bounce solutions that for any and conditions of their existence.
If a solution of Eqs. (2)–(4) has such a point that and , then it is a bounce solution. Let us find conditions that are necessary for the existence of a bounce in models with action (1). Using Eq. (2), we get that from it follows . Subtracting equation (2) from equation (3), we obtain
where functions and and their derivatives are taken at the point . The condition gives the restriction on functions and at the bounce point.
To analyze the dynamic it is suitable to introduce a new variable and present equations (2) and (6) in the form similar to the Friedmann equations in the Einstein frame. Following , we introduce the effective potential
If the constant , then coincides with the potential of the corresponding model in the Einstein frame. Note that we do not transform the metric and consider the model in the Jordan frame only. For our purposes any positive value of is suitable (in  it was chosen ). Also in  functions
have been introduced. If , then as well.
As known the Hubble parameter is a monotonically decreasing function in models with a standard scalar field minimally coupled to gravity ( is a constant). For a model with an arbitrary positive the function has the same property.
Equation (2) is a quadratic equation in that has the following solutions:
The value of the function that correspond to is
So, a positive corresponds to and a negative corresponds to .
If the potential is negative at some values of , whereas the function is positive at these points, then one should restrict the domain of absolute values of from below to get a real Hubble parameter. In other words, there exists the unreachable domain on the phase plane . The boundary of this domain is defined by the condition . If , then the sign of the function cannot be changed. If in some domain is negative, then the sign of can be changed from plus to minus, but not vice verse. So, the sign of cannot be changed twice.
If Eq. (2) is satisfied in the initial moment of time, then from system (14) it follows that Eq. (2) is satisfied at any moment of time. So, one can use Eq. (2) to fix initial conditions of system (14). From Eqs. (10) and (11) it is easy to get the following system :
De Sitter solutions correspond to , and hence, . The corresponding Hubble parameter is
We analyze the stability of de Sitter solutions with and only. From (16) it follows that . In this case the Hubble parameter in the neighborhood of is uniquely defined by Eq. (2), so it is enough to consider system (15) to analyze the stability of de Sitter solution. Substituting the de Sitter solution with the first order perturbations:
into (15), we get the following linear system on and :
We find eigenvalues for system (18):
The real part of always, the condition that the real part of is negative is equivalent to . Therefore, for arbitrary differentiable functions and , the model has a stable de Sitter solution with only if the potential has a minimum  and .
Using the obtained values of , it is easy to find  that the de Sitter point is a stable node (the scalar field decreases monotonically) at
and a stable focus (the scalar field oscillations exist) at
3 The choice of the function
In the paper  the cosmological model with a constant Ricci scalar has been considered. In this section we demonstrate how the form of function and the corresponding potential can be found from the requirement that the Ricci scalar is an integral of motion.
From the structure of Eq. (24) it is easy to see that the simplest way to get a constant is to choose such that
where is a constant.
The first two equations of system (25) have the following solution
where is an integration constant. Without loss of generality we put . For such a choice of we get Eq. (24) as follows:
Considering Eq. (27) as a differential equation for , for a constant we get the following solution:
where is an integration constant, for other constants we choose
Thus, requiring that the Ricci scalar is a constant, one can define both functions and . To get a positive for some values of we choose .
Substituting and into Eq. (7), we get that the condition is equivalent to , hence, from it follows . This integrable cosmological model has been considered in , where the behavior of bounce solutions has been studied in detail.
Equation is a differential equation for the Hubble parameter:
Multiplying this equation by , we get
Therefore, there exists the following integral of motion:
Equation (30) with a positive has two possible real solutions in dependence of the initial conditions:
where is an integration constant. Note that the behavior of the Hubble parameter does not depend on the specific dynamics of the scalar field , because two-parametric set of functions corresponds to one-parametric set of .
4 Models with fourth-degree even polynomial potentials
4.1 Equations of motion and restrictions on parameters
The monotonically increasing Hubble parameter is not suitable for construction of a realistic cosmological scenario, whereas is not a bounce solution. In  the authors slightly modify the function to get a bounce solution with non-monotonic behaviour of the Hubble parameters. In this section we modify the potential instead of the function and analyze the obtained bounce solutions. In other words, we choose the function , given by (26), with . As a minimal generalization of the potential we consider the following potential
where are constants. We consider the case only, because the integrable model with the potential has a bounce solution and a stable de Sitter solution only if and .
We plan to consider the evolution of bounce solutions fixing the initial conditions at the bounce point. Without loss of generality we can consider an initial value only. The initial value of is defined by the condition , hence, .
Our first goal is to find conditions on the coefficients of the potential that correspond to the existence of bounce solutions. For , a bounce solution exists only if there exists such a point that the following conditions are satisfied :
So, , and we obtain the following conditions on parameters :
Thus, at least one of the constants or should be positive. A bounce solution has the physical sense only if . The condition means .
The effective potential (8) is
The even potential has an extremum at and at points
We consider such values of parameter that points are real and . Note that for integrable model (, , ) are non-zero real numbers and the condition is equivalent to .
From (36), and we get
So, the model with a bounce solution has real only at
Using these conditions, we get
so, the potential has a minimum at and maxima at .
A few examples of the effective potential with two maxima at nonzero points and a minimum at are presented in Fig. 1.
Stable de Sitter solution at corresponds to . Therefore, such a solution exists for only. It means that the stable de Sitter solution exists only if . In this case .
The potential has the following zeros:
At , one gets . Therefore, and there exist only two real roots:
The bounce point belongs to the interval . We get the condition .
4.2 Analysis of numeric solutions
We integrate this system with numerically.
We consider such a positive that . The evolution of the scalar field starts at the bounce point with a negative velocity, defined by the relation
The field can come to zero passing the maximum of the potential. So, we keep in mind that the following subsequence of inequalities:
In the case there are three possible evolutions of the bounce solutions, depending on whether the solution passes the points of the maximum of or not. In the left and middle pictures of Fig. 2 we present an example of three possible behaviors of the bounce solutions. The solution with less initial value of tends to infinity (blue curve), whereas the bounce solutions with greater initial values tends to zero (cyan curve) or to minus infinity as a monotonically decreasing function (green curve). All trajectories start at bounce points. The corresponding behaviors of the Hubble parameter are presented in the right picture of Fig. 2 (colors the Hubble parameter evolutions on this picture coincide to the colors of the corresponding phase trajectories).
Let us compare two bounce solutions with initial conditions: and and negative initial values of . The corresponding Hubble parameter that is equal to zero at a bounce point is given by (12): . The initial value of for the first solution we denote as . The potential for all , therefore, for any solution that pass . Thus, at some moment of time the solution comes at the point : . The value of its velocity at the bounce point has a minimal absolute value by comparison with any bounce solutions that pass via with a negative velocity. It follows from (2), because both , and are positive at . It proves that if a solution with the bounce point passes through the maximum of the potential and tends to zero, then any solution with the bounce point tends to zero as well.
All bounce solutions with negative initial values of come in the domain where , so, . If a solution does not pass the maximum of the effective potential, then after some moment starts to grow and this solution comes to antigravity domain with (the blue curves in Fig. 2 denote an example of such a solution). Solutions passing the maximum of the potential can be different and need considering in detail.
If for all , then is a monotonic function, because at any point . Similar dynamics is possible even if (see green curves in Fig. 2).
For there exists the stable de Sitter solution and . It is a stable node at and a stable focus in the opposite case . In Fig. 2 (cyan curves), in the left picture of Fig. 3, and Fig. 4 solutions in the case of a stable focus are presented. An example with a stable node at is given in the middle picture of Fig. 3. One can see that in the case of a stable node the presented Hubble parameter (the right picture of Fig. 3) is close to the Hubble parameter obtained in the paper  that has a maximum. Note that in the integrable case and there exists a stable node at .
In the left picture of Fig. 3 the phase trajectories have been constructed for , , and . Let us now change the value of only and consider the model with . In Fig. 4 the corresponding phase trajectory is presented. We see that now the bounce solution that starts at tends to zero and finishes at the point . We can see that the trajectories that revolve around point look similar at and at . Let us consider now the phase trajectory at that is presented in Fig. 5. We see that trajectories are similar at the beginning only. The scalar field tends to infinity and the system comes to antigravity domain with . The form of the effective potential does not depend essentially from the sign value of (see Fig. 1), but the sign of is different. By this reason, the behavior of bounce solutions are essentially different. In the right pictures of Fig. 4 and Fig. 5 one can see that the behavior of the Hubble parameter also essentially depends on the sign of .
The difference between the solutions of system (41) with a positive and a negative is demonstrated in Fig. 6 as well. The cyan curves correspond to , whereas the red curves correspond to . We see that the phase trajectories of the field and behaviours of the Hubble parameter are similar in the beginning, but stand essentially different in the future.
We come to conclusion that the behavior of solutions essentially depends on the sign of . To understand the reason of this dependence let us consider the domain , where . From (12) it follows that the Hubble parameter is real if
If the constants are such that for all , then this condition is always satisfied, and the field tends to a minimum of at . We see such evolutions in Fig. 2, Fig. 3, and Fig. 4. Note that in this case if in the moment when potential stands positive and the function tends to zero, then at any moment in future.
If the constants are such that the potential change the sign and , then the evolution is different (see Fig. 5). The Hubble parameter becomes negative and positive again, so, there are two bounce points. After that the Hubble parameter tends to infinity.
Let us consider this case in detail. If , then there is a restricted domain in the neighborhood of point on phase plane such that the values of the scalar field and its derivative correspond to non-real values of the Hubble parameter. The boundary of this domain is defines by equation . In Fig. 5 we see that the phase trajectory rotates around this domain. The trajectory can not cross the boundary, but can touch it.
We show that all such trajectories touch the boundary at some finite moment of time. Let for some moments of time and we have , then, using and formula (11), we get
where is a negative number. Therefore, this integral has a finite negative value. For any circle value of decreases on some positive value, which doesn’t tend to zero during evolution, when number of circles increase. We come to conclusion that only a finite number of circles is necessary to get the value . At this point as well, so the function changes the sign. When two possible values of the Hubble parameters: and coincide. At this moment the value of the Hubble parameter changes from to . The value of the function continues to decrease, so, the distance between the trajectory and the boundary of unreachable domain increases. We do not say that its increase monotonically but the absolute value of that is a characteristic of this distance increases on a finite quantity after any circle. So, after some finite number of circles the absolute value of becomes more then . After this moment monotonically tends to infinity and at some finite moment we get . Thus, the final of trajectory is in the antigravity domain always.
Note that in the domain with the Hubble parameter is uniquely defined as a function and by (12). If , then whole evolution of bounce solutions is evolution a solution of the second order system, whereas for the third order system (41) with the additional condition (2) is not equivalent to any second order system.
5 Models with monomial potentials and cosmological constant
5.1 Conditions of the bounce existence
In this section we consider the scalar field potential of the form
where is an even natural number, and following non-minimally coupled function
where is positive.
We are interested in cosmological scenarios in the physical region , where the bounce occurs at first and after that stable de Sitter solution is realized. So, the potential should change the sign. The potential is an even function, hence, it has an extremum at . If we additionally suppose that , then we get the condition . Therefore, the bounce condition for the chosen potential gives:
From the condition it follows (see (7)):
(1). If , then .
(2). If , then .
Here is the value of the scalar field at the bounce. We see that there is an additional restriction on the location of points of bounce in comparison with the cases studied earlier in [19, 21], appearing when (the case (i1)). This happens due to condition of the bounce, which can be violated in the case of small enough , causing a recollapse instead of bounce.
The bounce point should corresponds to that gives the following condition
The effective potential has zeros for
They are located in the region only if
Thus, we have received a restriction from above for .
The first derivative of with respect to for the chosen functions and is given by
Therefore, the effective potential has the extremum at the point for any even . Non-zero extrema for are
For values of are roots of the following cubic equation
which has three real roots for . We consider only (see (49)), hence, there are three real roots in this case. Only positive ones have the physical sense ().
Let us study properties of the effective potential to analyse the Lyapunov stability of de Sitter solutions. We calculate for de Sitter solution , . Then it is stable for , namely,
From this it follows that in the case of there is the restriction from below for the coupling constant
On the other hand, for the interval of possible allowing evolution towards de Sitter solution after the bounce is not restricted from below (we remind that and we consider only positive values of in this paper).