Dynamical system approach of non-minimal coupling in AdS/CFT cosmology
Our purpose is to develop the anti de Sitter/conformal field theory (AdS/CFT) correspondence as a generalized Randall Sundrum model with a non minimal coupling scalar field in the brane. We derive the modified equations by means of the geometrical approach. The dynamics of this model are studied by rewriting the cosmological field equations in the form of a system of autonomous differential equations. In particular, the analysis is considered by investigating a minimal coupling, a conformal coupling and a non minimal coupling scalar field to the curvature. We present and discuss issues of stability and viability of this model for different behaviors of our universe.
One of the greatest challenges in cosmology is to obtain a simple solution to many longstanding cosmological problems, such as the flatness, the horizon and the primordial monopole problems. The inflationary paradigm makes it possible and leads to a crucial modification of the standard point of view of the large-scale structure of the universe inflation1 ; inflation2 . One of the best ways to set up inflation is to consider scalar fields as important ingredients for inflation models. The idea underlying the inflationary scenario is that there exists a scalar field which is subjected to the slow-roll approximation where the kinetic energy of the scalar field remains sufficiently small compared to its potential energy. Even though the inflationary paradigm successfully describes the aforementioned issues and provides a mechanism of production and evolution of the primordial fluctuations inflation3 ; inflation4 , high energy regime suggests that general relativity should be modified gr . Several models of modification of gravity in which inflation can be realized have been developed jcap ; boer ; mariam ; lidsey ; brane ; Kofinas:2001es ; Deffayet:2000uy ; Deffayet:2002fn ; Kiritsis:2002ca . In this paper, we consider that inflation is realized in the AdS/CFT correspondence.
While previous studies of scalar field cosmology deal with minimally coupled scalar fields minimal , most theories of gravity and scalar fields argued that the coupling of the scalar field must necessarily be non-minimal deals ; mariam ; nm4 ; nm5 ; nm6 . We consider a scenario where the inflaton field is coupled to gravity non-minimally. This kind of non-minimal coupling to gravity has been enough discussed in four dimensions Futamase:1987ua ; Salopek:1988qh ; Fakir:1990eg ; Amendola:1990nn ; Kaiser:1994vs ; Bezrukov:2007ep ; Bauer:2008zj ; Park:2008hz ; Linde:2011nh ; Kallosh:2013maa ; Kallosh:2013tua ; Chiba:2014sva ; Boubekeur:2015xza ; Pieroni:2015cma ; Salvio:2017xul and also in extra dimensions mariam ; Nozari:2012cy ; Bogdanos:2006qw ; Farakos:2006sr . Non-minimal coupling to gravity is a natural ingredient generated by quantum corrections in a curved space-time Birrell:1982ix . So, naturally we should include a coupling between gravity and inflaton field in the action of an inflation scenario. It turns out that in general relativity, the coupling constant is valued to deals . We will analyze this coupling value in the context of AdS/CFT as a particular case.
In order to understand nonlinear gravity in the brane world scenario, Shiromizu et al. proposed a particularly elegant way named the geometrical approach shiro . It had been suggested also that gravity on the brane at low energies can be understood through the AdS/CFT correspondence. This kind of correspondence asserts that there is an equivalence between a gravitational theory in d-dimensional anti de Sitter space-time and conformal field theory living in a ()-dimensional boundary space-time intro2 . This equivalence or duality is best understood in the context of string theory with d=5, where the duality relates type IIB superstring theory on , and supersymmetric Yang Mills theory with gauge group SU(N) in four dimensions intro3 ; intro4 . Following sugumi , both of the geometrical approach and the AdS/CFT correspondence will be subject of our framework in order to derive the modified equations.
Furthermore, The dynamical systems methods are widely used in cosmology. Indeed, these approaches have been applied to study the dynamics of extended theories of gravity ds1 ; ds2 ; ds3 and have been proven to be very successful in providing a simple way of obtaining attractive solutions for a complicated analytic resolution and a description of the global dynamics of the models ds4 . The main analysis is to look for whether the attractors of the system correspond to an inflationary phase and to study other qualitative behaviors.
This paper is organized as follows. In Sec. II, we present the geometrical approach and its relation to the AdS/CFT correspondence. In Sec. III, we present the basic cosmological equations to describe the evolution of a non minimal coupled scalar field such as the modified Friedmann equation, in which we show the existence of two branches of solution as a function of scalar field, and the equation of motion. In Sec. IV, a dynamical systems analysis is developed in the context of this approach by fixing the choice of an exponential potential and a monomial form of the coupling function. Three cases are of interest, the minimal coupling, the conformal coupling and the non minimal coupling. Finally, we present our summary and conclusions in Sec. V.
Ii The setup
where is the 5D gravitational constant, is the Ricci scalar of the five-dimensional metric and is the bulk cosmological constant. is the induced metric on the brane, is the brane tension and , the Lagrangian density of the non minimal scalar field localized on the brane, is defined as
where is the covariant derivative associated with the induced metric on the brane, is the scalar field potential, and is a coupling between the scalar field and the induced gravity .
In order to write the effective Einstein equation on the brane we follow the technical procedure developed by sugumi , in which the relation between the geometrical approach and the AdS/CFT correspondence is revealed. Furthermore, in order that the AdS/CFT correspondence describes the brane world scenario sugumi ; jcap ; boer , we choose the super Yang-Mills theory as the conformal matter. To illustrate our purpose and for the sake of simplicity, we consider the following form of the low energy equations of motion describing gravity on the brane sugumi
where denotes the Newton’s constant () , is the total energy-momentum tensor and denotes the energy-momentum tensor of the cutoff version of conformal field theory (see sugumi for more details).
The total energy-momentum tensor and the effective cosmological constant on the brane are given by mariam
The energy-momentum tensor of the conformal field theory, , cannot be written in the local covariant form, however its trace writes sugumi
where is the conformal anomaly related to the AdS/CFT length.
The total energy-momentum tensor Eq. (4) has been split of into a scalar field energy-momentum tensor,
and into a non-minimal coupling energy momentum tensor
The Bianchi identity, , and the equation of conservation of the energy-momentum, , implies that , which amounts to
where is the Hubble parameter.
Iii Basic Cosmological Equations
In this section we will consider the following spatially flat isotropic and homogeneous FRW brane
where is the scale factor, is a symmetric dimentional metric and , are the comoving spatial coordinates.
From the -component of the field equations (3) together with the equations (4) and (13), the modified Friedmann equation on this spatially flat brane can be obtained as
where is the total energy density and the effective gravitational coupling, , is given by
Following the notation introduced in mariam , we can write the total energy density and the pressure of the universe respectively as
The modified Friedmann equation Eq.(15) can be rewritten as
Furthermore, the modified Raychaudhuri equation can be deduced from Eq. (3) as
Finally, minimising the action (1) with respect to variation of the scalar field, , we obtain the equation of motion in the FRW geometry as
where the prime denotes the derivative with respect to the scalar field .
The intrinsic Ricci scalar for a flat FRW brane is
Iv A Dynamical Systems Approach
In order to simplify the analysis of Eqs (15), (21) and (22), the method taken up is the dynamical systems study. In this section, we present the phase space of the non-minimally coupled scalar field in detail, exact solutions and their stability.
The first step in the implementation of the Dynamical System Approach (DSA) is the introduction of the general dimensionless variables
The dynamical variables Eq. (24) are non-compact, i.e. their values do not have finite bounds as in ds1 ; ds2 ; ds3 ; ds4 . We will come back to this point in the conclusion.
The cosmological equations become equivalent to the following autonomous system:
the derivative is with respect to which is related to the scale factor by , and we define
The prime denotes the derivative with respect to the scalar field .
In order to study inflation we need to understand how the slow-roll parameter is modified in this set up. One can define this parameter as
The sufficient condition for inflation is that . This parameter will be useful in the discussion of inflation in the following paragraphs.
The critical points of any dynamical system can be extracted by setting , while their properties are determined by the eigenvalues of its Jacobian matrix, , which is also called the stability matrix
The critical points are classified according to the sign of their eigenvalues by using linear stability method as:
Attractor critical point, If all eigenvalues have negative real parts. In this case the point would attract all nearby trajectories and is viewed as stable.
Repeller critical point, If all eigenvalues have positive real parts where trajectories are repelled from the fixed point and we speak in this situation of an unstable point.
If there is mixture of both positive and negative real parts of eigenvalues, then the corresponding critical point is called a saddle. This point will attract nearby trajectories in some directions but repels them along others.
However, If at least one of the eigenvalues is zero, the linear stability theory fails to describe the stability of the critical point which is called non-hyperbolic. In this case other techniques have to be employed to study the stability properties, such as the Centre Manifold Theory (CMT) center ; cm1 ; cm2 ; cm3 , the Lyapunov function method method ; lyapunov1 ; lyapunov2 and Kosambi-Cartan-Chern theory Bohmer:2010re .
iv.1 Example of and
Since the dynamical system Eqs. (26a)-(26e) is complicated to analyze in its full generality, we consider a particular case in order to illustrate our purpose. Concerning the scalar field potential, we choose an exponential function which has many implications in cosmological inflation Copeland:1997et ; Leon:2009rc
where corresponds to the maximum value of the potential and is a constant and we choose the following form of the coupling as
where is a constant parameter. If one chooses this monomial form of , the set of phase space variables (24) reduces to a four dimensional by writing the variable as
In the next subsections we will consider first two special values of , namely the minimal coupling for and the conformal coupling for .
iv.1.1 Minimal coupling
To illustrate our purpose, we begin by the simple case, namely the minimal one where . In that case the variable is equal to zero. Using the Friedmann constraint Eq. (25) and Eq.(31), the system (26a)-(26e) reduces to the following autonomous two-dimensional system in terms of the dynamical variables
This nonlinear autonomous system has four critical points A and B. Their properties are given in Table 1 and are summarized below.
|Fixed points||Existence||Eigenvalues||Stability||Physical State|
|A||,||Saddle for||de Sitter|
Critical points A exist for (see Eq. (28)) i.e. the effective cosmological constant on the brane satisfy where, . These points correspond to the case where the kinetic energy density of the scalar field and its potential energy density vanish (as the Hubble rate remains finite). This means that there is no dynamical motion of the the scalar field.
The Freidmann equation of these fixed points writes
Therefore, we conclude that the dynamic of the universe is governed by the effective cosmological constant. we notice also that the critical point A corresponds to an expanding de Sitter universe while A represents a contracting one.
Critical points B exist only for a positive effective cosmological constant, in which the conformal anomaly is given by the condition .
These points correspond to the solution:
Similar to the previous case, there is no dynamical motion of the the scalar field, and we have an expanding de Sitter universe for the point B. The point B represents a contracting de Sitter universe.
The stability of these critical points, A and B111We restrict our analysis to the critical points A and B since we are assuming an expanding universe, i.e. ., is obtained by evaluating the eigenvalues of the Jacobian matrix in the neighborhood of these critical points. We obtain the following eigenvalues and for each critical point. We notice that these points are non hyperbolic.
The stability properties of these points are obtained by applying the CMT to the 2D-system Eqs. (34a) and (34b) (the analysis details are given in the Appendix A).
Around the critical point A the stability depends on the value of the constant .
For (i.e. ), the critical point A is saddle, whereas for both critical points A and B are unstable.
Fig. 1 shows the phase space and the position of the critical points of the system (34a)-(34b).
We conclude that in the case of minimal coupling, the resulting Hubble rate, can be considered as solutions at early times. This means that in the past, each trajectory begins in a de Sitter state as the solution behaves like a cosmological constant of an arbitrary value for .
iv.1.2 Conformal coupling
We now consider the case of a conformally coupled scalar field on the brane mariam ; Faraoni:2000wk , with conformal coupling , and a vanishing potential 222 The Klein-Gordon equation (22) is conformally invariant if or for the conformal coupling mariam ; Faraoni:2000wk ; Birrell ; Wald .. In what follows, we present the results of our dynamical system (26a)-(26e) for the conformal coupling. The system reduces to
In table 2, we present the coordinates of each critical point and the results of their stability analysis by means of the signs of the real parts of the eigenvalues of the Jacobian matrix.
|C||, ,||Stable for||de Sitter|
|D||, ,||Saddle||de Sitter|
Critical points C exist for and amount to assume that the solution is a de Sitter universe
The stability of these two points depend on the value of the constant given by (28).
For (i.e. ), the critical points C are stable since all eigenvalues are negative, whereas for , the two critical points C are saddle since one of the eigenvalues is positive while the others are negative.
Critical points D exist for and represent also a de Sitter universe
Finally, accordingly to their eigenvalues, the critical points D are saddle points.
We notice that in the conformal coupling case, the critical point333We ignore C since we are assuming only expanding universe. C is the future attractor if and only if is negative while the critical point D is always a saddle point meaning that it cannot be the past attractor.
To confirm the stability of the critical point C, we perturb the solutions around this point in order to analyse numerically this property. In Figs. (a)a, (b)b and (c)c, we plot the projection of perturbations of the system (37a)-(37c) along -axis, -axis and -axis respectively with respect to .
From these figures we notice that trajectories of the perturbed solutions approach the coordinates of C for , i.e. , and respectively as . From these behaviours, we can conclude that the critical point C is an attractor solution which is in agreement with our analytical result. Furthermore, this point corresponds to a slow roll parameter which means that this point may sustain inflation. As we can notice from Fig. (d)d, the universe remains eternally in the inflation era even though we perturb it around this attractor solution.
iv.1.3 Non minimal coupling
In what follows we will assume a positif non-minimal coupling constant non equal to and .
In this subsection and due to the complexity of our system, we shall restrict our analysis to the case of , by choosing for in Eq. (5).
The set of the differentiable Eqs. (26a)-(26e) reduces by considering the constraint Eq. (25) to the following autonomous system
The critical point E is formed of a continuous line of critical points, called a critical line or line of non-isolated equilibrium points nonisolated . This critical line exists for an infinite number of critical points for all values of that verify the condition of existence and . The dynamic of the universe for this critical line is dominated by the potential energy density, i.e. and such that . The Friedmann equation and the equation of motion of the scalar field of this critical line write respectively as
The slow-roll parameter (29) is equal to zero (), which means that this critical line corresponds to inflation.
The critical line F exists for . We restrict our analysis to the critical line E since we assume an expanding universe (). Indeed, the critical line F does not correspond to an expanding universe due to the condition of existence for (see table 3).
In order to discus the stability analytically, we use the linear theory. The stability of these lines is shown in444 Note that the stability analysis of these critical lines depends on the value of the variable , and the parameters of our model and which makes the eigenvalues of the jacobian matrix very lengthy this is why we plot the stability region according to the signs of these eigenvalues. Fig. 3. From both figures (a)a and (b)b, we conclude that the critical lines E and F are either stable or saddle. Consequently, the critical line E corresponds to a non-minimally coupled inflation attractor solution for a specific values of our model parameters and in addition to the choice of the value of the dimensionless variable .
However the stability can also be found numerically by perturbing the system around the critical line. We plot in Fig. 4 the projection plots on , and separately for and .
From Fig. (a)a it seems that the trajectories are parallel to an horizontal axis, and that any perturbation of the system near makes it an arbitrary constant as .
We can also see from Fig. (b)b and (d)d, that for each value of , the corresponding trajectories of and also approach the value and respectively as . Some numerical values of any perturbation near , and are also shown in Fig. 4. For example for the corresponding critical point coordinates and are and respectively as .
From Fig. (c)c, we notice that trajectories of the perturbed solutions approach as .
From these behaviours it is evident that the system comes back to the critical point following the perturbation, which means that the critical line E is an attractor line for and . These plots support strongly our analytical findings.
In order to obtain a complete information about the structure of the phase space of the dynamical system (26a)-(26e) it is necessary to investigate the dynamical behavior for . To this aim, we extend the previous study by including negative values of in Eq. (32) to search for any possible attractor inflation solutions.
To keep the definition of the dimensionless variables as in (24), one has to consider a non-minimal function as , where and is a positif constant. It deserves to be mentioned that the constraint equation Eq. (25) reads in this case
For both critical lines G and H the dynamic of the universe is dominated by the potential energy density as vanishes while , with the solution of the Hubble parameter writes as Eq. (45). The parameter Eq. (29) evaluated at these critical lines is also equal to which means that these lines correspond to inflation.
Examination of the stability conditions Fig. 5 indicates that the state can be stable (or saddle) during inflation. The critical line G is always saddle in the region of existence (), while H is an attractor solution for and .
V Summary and conclusions
The present study is devoted to the AdS/CFT correspondence viewpoint of inflation with a non-minimal coupling of the scalar field to the Ricci curvature. The relation between the geometrical approach and the AdS/CFT is used to derive the modified equations. We have studied the impact of this kind of coupling on the inflation dynamics and on the modifications of the slow-roll parameter.
Exact analytic solutions cannot be obtained for the modified Freidmann equation due to the complicated form of the evolution equation. The application of the basic tools of the dynamical systems theory helps us to deeply understand the dynamics of this cosmological model and to determine analytically the global behaviour of the system. The stability of the critical points of the dynamical system is studied by means of linear theory, centre manifold theory in the case of a non hyperbolic critical points and numerically to support our results.
We have started by considering a minimal case where we have shown that the system admits two unstable de Sitter state critical points with no dynamical motion of the scalar field. We have also considered the special case of a conformally coupled scalar field for which we have obtained a future attractor point; this solution corresponds to an inflation era with no exit. For both cases, minimal and conformal coupling, the dynamic of the universe is governed by the effective cosmological constant.
For a positive non-minimal coupling constant, we have found one critical line that corresponds to a future attractor de Sitter inflationary era for specific values of our model parameters and . For a negative non-minimal coupling constant, we have found two critical lines. One of them is saddle while the second one is always a future attractor solution describing a de Sitter inflation scenario.
Finally, one of the interesting results of including non-minimal coupling of the scalar field to the intrinsic curvature on the brane is that we obtain a future attractor solution which corresponds to a scenario where the content of the universe is completely dominated by the exponential potential and a de Sitter inflationary era. Even though the AdS/CFT correspondence implemented by the geometrical approach gives a successful study of non-minimal gravity, the non compactness of our dynamical variables makes the analysis incomplete due to lack of the dynamical analysis at infinity of the phase space. Consequently, there could be missed critical points. This issue will be the subject of the next forthcoming paper.
The authors would like to thank Mariam Bouhmadi-López for many useful discussions and suggestions.
Appendix A Centre Manifold Theory
In Sec. IV, we have mentioned that if the eigenvalues of the Jacobian matrix (30) has one eigenvalue with zero real part while the other one is negative, the critical point is called non-hyperbolic and the linear approach fails to determine the stability properties. Different methods can be employed to study the stability properties in this situation such as the Lyapunov stability method ; lyapunov1 ; lyapunov2 , centre manifold theory (CMT) center ; cm1 ; cm2 ; cm3 and Kosambi-Cartan-Chern theory Bohmer:2010re .
This Appendix is devoted to show how we get the stable conditions of the non-hyperbolic critical points A and B using the CMT.
In what follows, we present the detailed calculus to find the stable conditions of the critical point A. To this purpose and in order to simplify the dynamical system Eqs. (34a)-(34b), we define a new variable as .
We recall that for any dynamical system , the new dynamical system , where is a positive function, has the same critical points with the same stability properties.
For the critical point A, the function
The first step is to consider a specific transformation: and in order to move the critical point A() to the origin of the phase space , where . We obtain the new dynamical system
Our dynamical system has the required form, i.e. the fixed point sits at the origin and the system does not contain any linear term of in the first equation. We rewrite the above system as
The centre manifold (CM) suggests that its geometrical space is tangent at to the eigenspace of the non zero eigenvalue . We may assume from the definition of the CM that with the following conditions:
In this coordinate, the dynamic of the CM, for sufficiently small, can be written as
Assuming that is of the form
and using the Leibnitz rule