Coupled quintessence and the impossibility of an interaction: a dynamical analysis study
Abstract
We analyze the coupled quintessence in the light of the linear dynamical systems theory, with two different interactions: i) proportional to the energy density of the dark energy and ii) proportional to the sum of the energy densities of the dark matter and dark energy. The results presented here enlarge the previous analyses in the literature, wherein the interaction has been only proportional to the energy density of the dark matter. In the first case it is possible to get the wellknown sequence of cosmological eras. For the second interaction only the radiation and the dark energy era can be described by the fixed points. Therefore, from the pointofview of the dynamical system theory, the interaction proportional to the sum of the energy densities of the dark matter and dark energy does not describe the universe we live in.
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1 Introduction
Sixty eight percent of our universe Ade:2015xua consists of a still mysterious component called “dark energy” (DE), which is believed to be responsible for the present acceleration of the universe reiss1998 ; perlmutter1999 . In addition to ordinary matter, the remaining of the energy content of the universe is a form of matter that interacts in principle only gravitationally, known as dark matter (DM). Among a wide range of alternatives for the dark energy, which includes the cosmological constant, scalar or vector fields ArmendarizPicon:2000dh ; Padmanabhan:2002cp ; Bagla:2002yn ; Brax1999 ; Copeland2000 ; Landim:2015upa ; ArmendarizPicon:2004pm ; Koivisto:2008xf ; Bamba:2008ja ; Emelyanov:2011ze ; Emelyanov:2011wn ; Emelyanov:2011kn ; Kouwn:2015cdw , holographic dark energy Hsu:2004ri ; Li:2004rb ; Nojiri:2005pu ; Pavon:2005yx ; Wang:2005jx ; Wang:2005pk ; Wang:2005ph ; Wang:2007ak ; Landim:2015hqa ; Li:2009bn ; Li:2009zs ; Li:2011sd ; Wang:2016och , metastable dark energy Stojkovic:2007dw ; Landim:2016isc ; Greenwood:2008qp ; Abdalla:2012ug ; Shafieloo:2016bpk , modifications of gravity and different kinds of cosmological fluids copeland2006dynamics ; Nojiri:2010wj ; Bamba:2012cp ; dvali2000 ; yin2005 ; Jamali:2016zww ; Capozziello:2013bma , the usage of a canonical scalar field, called “quintessence”, is a viable candidate peebles1988 ; ratra1988 ; Frieman1992 ; Frieman1995 ; Caldwell:1997ii .
In addition, the two components of the dark sector may interact with each other Wetterich:1994bg ; Amendola:1999er ; Farrar:2003uw ; Guo:2004vg ; Cai:2004dk ; Guo:2004xx ; Bi:2004ns ; Gumjudpai:2005ry ; yin2005 ; Wang:2005jx ; Wang:2005pk ; Wang:2005ph ; Wang:2007ak ; micheletti2009 ; Costa:2014pba ; Shahalam:2015sja ; Nunes:2016dlj ; Sola:2016ecz (see Wang:2016lxa for a review) and the interaction can eventually alleviate the coincidence problem Zimdahl:2001ar ; Chimento:2003iea .
When a scalar field is in the presence of a barotropic fluid (with equation of state , where is the pressure and is the energy density of the fluid) the relevant evolution equations can be converted into an autonomous system. Such approach is a good tool to analyze asymptotic states of cosmological models and it has been done for uncoupled dark energy (quintessence, tachyon field, phantom field and vector dark energy, for instance copeland1998 ; ng2001 ; Copeland:2004hq ; Zhai2005 ; DeSantiago:2012nk ; AzregAinou:2013jxa ; Landim:2016dxh ; Alho:2015ila ) and coupled dark energy Amendola:1999er ; Gumjudpai:2005ry ; TsujikawaGeneral ; amendola2006challenges ; ChenPhantom ; Mahata:2015lja ; Landim:2015poa ; Landim:2015uda . The coupling assumed for the quintessence field has been proportional to the energy density of the dark matter . However, there are other possibilities as for instance the coupling proportional to the energy density of the dark energy or the sum of the two energy densities . Similar kernels have been widely studied in the literature Abdalla:2007rd ; He:2008tn ; He:2008si ; Valiviita:2008iv ; Abdalla:2009mt ; Gavela:2009cy ; He:2010im ; Marcondes:2016reb . In particular, the dark energy evolution at high redshifts measured by the BOSSSDSS Collaboration Delubac:2014aqe shows a deviation from the cosmological constant which can be explained assuming interacting dark energy models Abdalla:2014cla .
A dynamical analysis remained to be done for these two kernels. In this paper we use the linear dynamical systems theory to investigate the critical points that come from the evolution equations for the quintessence, assuming the interaction between DE and DM proportional to i) and ii) . We found that in the case i) there are fixed points that can describe the sequence of three cosmological eras. In the second case either radiation era or dark energy era can be described by fixed points, but the matterdominated universe is absent.
The remainder of this paper is structured as follows. In Sect. 2 we present the basics of the interacting dark energy and the dynamical analysis theory. The quintessence dynamics is presented in Sect. 3 and the dynamical system theory is used to study the coupled quintessence in Sect. 4, wherein the critical points are shown. We summarize our results in Sect. 5. We use Planck units () throughout the text.
2 Interacting dark energy and dynamical analysis
We consider that dark energy is described by a scalar field with energy density and pressure , and with an equation of state given by . We assume that the scalar field is coupled with dark matter, in such a way that total energymomentum tensor is still conserved. In the flat Friedmann–Lemaître–Robertson–Walker (FLRW) background with a scale factor , the continuity equations for both components and for radiation are
(1) 
(2) 
(3) 
respectively, where is the Hubble rate, is the coupling and the dot is a derivative with respect to the cosmic time . The indices and stand for matter and radiation, respectively.^{1}^{1}1We could be more economic if we had written the matter and radiation equations in a joint form, as a general barotropic fluid with equation of state . The results would be, of course, unchanged. The case of corresponds to dark energy transformation into dark matter, while is the transformation in the opposite direction. In principle, the coupling can depend on several variables , so we assume for the quintessence the coupling is i) and ii) , where is a positive constant. The case with negative is the same as the case with but with negative fixed point , described in the next section by (11).
To deal with the dynamics of the system, we will define dimensionless variables. The new variables are going to characterize a system of differential equations in the form
(4) 
where is a column vector of dimensionless variables and the prime is the derivative with respect to , where we set the present scale factor to be one. The critical points are those ones that satisfy . In order to study stability of the fixed points we consider linear perturbations around them, thus . At the critical point the perturbations satisfy the following equation
(5) 
where is the Jacobian matrix. The stability around the fixed points depends on the nature of the eigenvalues () of , in such a way that they are stable points if they all have negative values, unstable points if they all have positive values and saddle points if at least one eigenvalue has positive (or negative) value, while the other ones have opposite sign. In addition, if any eigenvalue is a complex number, the fixed point can be stable (Re ) or unstable (Re ) spiral, due to the oscillatory behavior of its imaginary part.
3 Quintessence dynamics
The scalar field is described by the Lagrangian
(6) 
where is the scalar potential given by and and are constants. This choice is motivated by the autonomous system, as we shall see soon. For a homogeneous field in an expanding universe with FLRW metric with scale factor , the equation of motion is
(7) 
where the prime denotes derivative with respect to .
The interaction between the quintessence field with DM enters in the righthand side of Eq. (7).
In the presence of matter and radiation, the Friedmann equations for the scalar field are
(8) 
(9) 
and the equation of state becomes
(10) 
We are now ready to proceed to the dynamical analysis of the system.
4 Autonomous system
The dimensionless variables are defined as
(11)  
The dark energy density parameter is written in terms of these new variables as
(12) 
so that Eq. (8) can be written as
(13) 
where the matter and radiation density parameter are defined by , with . From Eqs. (12) and (13) we have that and are restricted in the phase plane by the relation
(14) 
due to . The equation of state becomes
(15) 
and the total effective equation of state is
(16) 
with an accelerated expansion for .
4.1 Interaction
The dynamical system for the variables , , and with the interaction proportional to is
(17) 
(18) 
(19) 
(20) 
where
(21) 
4.1.1 Critical points
The fixed points of the system are obtained by setting , , and in Eq. (17)–(20). When , is constant the potential is copeland1998 ; ng2001 .^{2}^{2}2The equation for is also equal zero when or , so that should not necessarily be constant, for the fixed points with this value of . However, for the case of dynamical , the correspondent eigenvalue is equal zero, indicating that the fixed points is not hyperbolic. The fixed points are shown in Table 1. Notice that cannot be negative and recall that .
Point  

(a)  
(b)  0  0  0  
(c)  0  
(d)  0  0  0  
(e)  0  
(f)  0  

The point is
(22) 
The eigenvalues of the Jacobian matrix were found for each fixed point in Table 1. The results are shown in Table 2, where the eigenvalues are
(23)  
Point  Stability  
(a)  see the main text  saddle  
(b)  saddle or unstable  
(c)  saddle or unstable  
(d)  saddle  
(e)  saddle or unstable  
(f)  stable  

The fixed point (a) describes a radiationdominated universe and in order to the fixed points be real and satisfy the nucleosynthesis bound bean2001 we should have and . The eigenvalues were found numerically. For and the the upper limit for the interaction () we get the eigenvalues , and , so this critical point is a saddle point. Similar results are found for other values of and .
Both points (b) and (c) also describe the radiation era and are unstable or saddle points. The eigenvalues and of the point (b) can be either positive or negative, depending on the values of and . On the other hand, the first eigenvalue is always positive. The same happens with the eigenvalues of the point (c) and in this case the interaction must be for the fixed points be real and for .
The matterdominated universe is described by the saddle point (d) and also by the point (e), provided that is sufficiently large for the latter case. Whatever the value of the interaction all eigenvalues of the point (e) cannot be simultaneously negative.
The last fixed point (f) is an attractor and it describes the darkenergy dominated universe if either and or and . The real part of the eigenvalues are negative for these range of values, thus the fixed point is stable or stable spiral. Its behavior is illustrated in Fig. 1, where we plot the phase plane with and .
The allowed values of and , for the fixed points (a), (c) and (f), are shown in Fig. 2. From the figure we see that the fixed points (a) and (f) do not have common regions.
Therefore, the sequence of cosmological eras (radiation matter dark energy) is reached considering the transition: (b) or (c) (d) or (e) (f).
4.2 Interaction
The dynamical system for the variables , , and with the interaction proportional to is
(24) 
(25) 
(26) 
(27) 
where
(28) 
All equations above but the first one are identical to the previous case.
4.2.1 Critical points
As before fixed points of the system are obtained by setting , , and in Eq. (24)–(27). The fixed points are shown in Table 3, where
(29) 
Point  

(a)  
(b)  0  0  0  
(c)  0  
(d)  0  0  
(e)  0  

The eigenvalues of the Jacobian matrix were found for each fixed point of the Table 3. The results are shown in Table 4, where
(30) 
and
(32) 
Point  Stability  
(a)  see the main text  saddle  
(b)  saddle  
(c)  saddle  
(d)  see the main text  saddle or unstable  
(e)  stable  

The point (a) describes a radiationdominated universe and in order to the fixed points be real and satisfy the nucleosynthesis bound bean2001 we should have or for any value of positive . The eigenvalues were found numerically and similarly to the case of the previous section, the fixed point is a saddle point for the allowed values of . and
The radiation era is also described by the points (b) and (c). They are saddle points and for (c) the interaction must be in order not to conflict the nucleosynthesis bound.
The matterdominated universe can be described by the point (d) but only if the interaction is zero, which in turn is known in the literature copeland2006dynamics .
The fixed point (e) can describe the current stage of accelerated expansion of the universe for some values of and . The critical points are real with and for and or for and . For these ranges of and the real part of the eigenvalues are negative, so the point is stable or stable spiral. The attractor point has and for .
Therefore, both radiation and darkenergydominated universe can be described by the fixed points, however, none of them represent the matter era.
5 Conclusions
In the light of the linear dynamical systems theory we have studied coupled quintessence with dark matter with two different interactions: i) proportional to the energy density of the dark energy and ii) proportional to the sum of the two energy densities . The results presented here enlarge the previous analysis in the literature, wherein the interaction has been only proportional to the energy density of the dark matter. In the case i) the transition of cosmological eras is fully achieved with a suitable sequence of fixed points. In the second case either radiation era or dark energy era can be described by the fixed points, but not the matterdominated universe. Therefore, the second interaction does not provide the cosmological sequence: radiation matter dark energy. This is not the first time that an interaction proportional to the sum of the energy densities leads to cosmological disasters. A phenomenological model with that coupling suffers earlytime instability for , as shown in Valiviita:2008iv ; He:2008si . Further analysis for high redshifts and different coupling are summarized in Wang:2016lxa .
Acknowledgements.
We thank Elcio Abdalla for comments. This work is supported by CAPES and CNPq.References
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