Assisted dark energy
Cosmological scaling solutions, which give rise to a scalar-field density proportional to a background fluid density during radiation and matter eras, are attractive to alleviate the energy scale problem of dark energy. In the presence of multiple scalar fields the scaling solution can exit to the epoch of cosmic acceleration through the so-called assisted inflation mechanism. We study cosmological dynamics of a multi-field system in details with a general Lagrangian density , where is the kinetic energy of the -th field , is a constant, and is an arbitrary function in terms of . This covers most of the scalar-field models of dark energy proposed in literature that possess scaling solutions. Using the bound coming from Big-Bang-Nucleosynthesis and the condition under which the each field cannot drive inflation as a single component of the universe, we find the following features: (i) a transient or eternal cosmic acceleration can be realized after the scaling matter era, (ii) a “thawing” property of assisting scalar fields is crucial to determine the evolution of the field equation of state , and (iii) the field equation of state today can be consistent with the observational bound in the presence of multiple scalar fields.
The constantly accumulating observational data continue to confirm the existence of dark energy responsible for cosmic acceleration today review (). The cosmological constant, whose equation of state is , has been favored by the combined data analysis of supernovae Ia SNIa (), cosmic microwave background CMB (), and baryon acoustic oscillations BAO (). Meanwhile, if the cosmological constant originates from a vacuum energy associated with particle physics, its energy scale is enormously larger than the observed value of dark energy (). Hence it is important to pursue an alternative possibility to construct dark energy models consistent with particle physics.
Scalar-field models such as quintessence quin1 (); quin2 () and k-essence kes () have been proposed to alleviate the above mentioned problem. In general the energy density of a scalar field dynamically changes in time, so that its value around the beginning of the radiation-dominated epoch can be much larger than the dark energy density today. One of such models is quintessence with an exponential potential Ferreira (); CLW (), where is a constant and with being gravitational constant (see Ref. Halliwell () for the classification of cosmological dynamics and also Refs. earlyexp () for early related papers). In fact, in higher-dimensional gravitational theories such as superstring and Kaluza-Klein theories, exponential potentials often appear from the curvature of internal spaces associated with the geometry of extra dimensions (so called “modulus” fields) expmoti (). Moreover it is known that exponential potentials can arise in gaugino condensation as a non-perturbative effect gaugino () and in the presence of supergravity corrections to global supersymmetric theories CNR ().
The quintessence with an exponential potential gives rise to two distinct fixed points in the flat Friedmann-Lemaître-Robertson-Walker (FLRW) background CLW (): (a) the scaling solution, and (b) the scalar-field dominated solution. If the slope of the potential satisfies the condition , where is the equation of state of a background fluid, then the solutions approach the scaling attractor characterized by a field density parameter . Even if the field energy density is initially comparable to the background fluid density , the field eventually enters the scaling regime in which is proportional to . This is attractive to alleviate the fine-tuning problem of the energy scale of dark energy. However the scaling solution needs to exit from the matter era to the epoch of a late-time cosmic acceleration. The scalar-field dominated solution () can be an accelerated attractor for , but this is incompatible with the condition required for the existence of scaling solutions. Hence the scaling solution cannot be followed by the scalar-field dominated solution responsible for dark energy.
There are a number of ways to allow a transition from the scaling regime to the epoch of cosmic acceleration. One of them is to introduce a single-field potential that becomes shallow at late times, e.g., with and Barreiro () (see Ref. Jarv () for the classification of dynamics and Refs. otherexp () for related works). For this double exponential potential the field equation of state of the final attractor is given by . In order to satisfy the observational constraint obsercon () today, we require that is smaller than the order of 1. If the exponential potential originates from particle physics models then the slope is typically larger than 1, which is difficult to be compatible with the condition for cosmic acceleration.
Another way is to consider multiple scalar fields with exponential potentials, e.g., Coley (); KLT () (see also Refs. phantom ()). In fact such potentials arise from the compactification of higher dimensional theories to 4-dimensional space-time. It is known that the phenomenon called assisted inflation Liddle98 () occurs for the multi-field exponential potential, even if the individual field has too steep a potential to lead to cosmic acceleration (see also Refs. assistedpapers ()). For the sum of steep potentials satisfying the condition (), the multiple fields evolve to give dynamics matching a single-field model with Liddle98 (). Since the conditions are mostly satisfied for the models motivated by particle physics, this cooperative accelerated expansion is attractive for both inflation and dark energy. If we apply this scenario to dark energy, the scaling radiation and matter eras can be followed by the epoch of assisted acceleration as more fields join the scalar-field dominated attractor with an effective equation of state KLT ().
The scaling solution arises not only for quintessence with an exponential potential but also for more general scalar-field models with the Lagrangian density , where is a kinetic term of the field . Here is a metric tensor with the notation . It was found in Refs. PT04 (); TS04 () that the existence of scaling solutions restricts the form of the Lagrangian density to be , where is a constant and is an arbitrary function in terms of (here we use the unit ). The quintessence with an exponential potential () corresponds to the choice , whereas the choice gives rise to the dilatonic ghost condensate model: TS04 () (which corresponds to the string-theory motivated generalization of the ghost condensate model proposed in Ref. Arkani ()). The tachyon Lagrangian density with Paddy () also follows from the above scaling Lagrangian by a suitable field redefinition CGST ().
For the multi-field scaling Lagrangian density it was shown in Ref. Tsuji06 () that assisted inflation occurs with the effective slope , irrespective of the form of . Hence one can expect that the scaling solution is followed by the assisted acceleration phase for such a general Lagrangian. If we consider loop or higher-order derivative corrections to the tree-level action motivated from string theory (such as ), the constant is typically of the order of unity stringreview (). In the single-field case this is not compatible with the condition for cosmic acceleration. It is of interest to see how the presence of multiple fields changes this situation.
In this paper we shall study cosmological dynamics of multiple scalar fields with the Lagrangian density . We are interested in the case where the scaling radiation and matter eras induced by a field are followed by the dark energy dominated epoch assisted by other scalar fields. For the two-field quintessence with exponential potentials a similar analysis was partially done in Ref. KLT (), but we shall carry out detailed analysis by taking into account bounds coming from Big-Bang-Nucleosynthesis (BBN) and supernovae observations. In particular the evolution of the field equation of state will be clarified in the presence of two and more than two fields. We also investigate cosmological dynamics for the multi-field dilatonic ghost condensate model as an example of k-essence models.
This paper is organized as follows. In Sec. II we present the dynamical equations for our general multi-field Lagrangian density without specifying any form of . In Sec. III we derive the fixed points that correspond to the scaling radiation/matter solutions and the assisted field-dominated attractor. In Secs. IV and V we study the multi-field cosmological dynamics for quintessence with exponential potentials and the dilatonic ghost condensate model, respectively. Sec. VI is devoted to conclusions.
Ii Dynamical system
Let us first briefly review single-field scaling models with the Lagrangian density . The existence of cosmological scaling solutions demands that the field energy density , where , is proportional to the background fluid density . Under this condition the Lagrangian density is restricted to take the following form in the flat FLRW background PT04 (); TS04 ()
where is a constant and is an arbitrary function in terms of . The Lagrangian density (1) is valid even in the presence of a constant coupling between the field and non-relativistic matter and also in the presence of a Gauss-Bonnet (GB) coupling between the field and the GB term111It is also possible to obtain a generalized form of the scaling Lagrangian density even when the coupling between and non-relativistic matter is field-dependent AQTW (). ST06 (). In the following we do not take into account such couplings. Throughout this paper we use the unit .
The field density parameter for scaling solutions is given by Tsuji06 (); AQTW (), where is the fluid equation of state. If the field enters the scaling regime during the radiation era, the BBN places the bound at the confidence level Bean (). This then gives the constraint .
Besides scaling solutions, there is a scalar-field dominated point () with the equation of state Tsuji06 (); AQTW (). This can be used for dark energy provided that , i.e. . Unfortunately this condition is incompatible with the constraint coming from the BBN. Hence the scaling solution does not exit to the scalar-field dominated solution in the single-field scenario.
If we consider multiple scalar fields () with the Lagrangian density
the scaling solution can be followed by the accelerated scalar-field dominated point through the assisted inflation mechanism. Even if the individual field does not satisfy the condition for inflation, the multiple fields evolve cooperatively to give dynamics matching a single-field model with Tsuji06 ()
Since is reduced compared to the individual , this allows a possibility to exit from the scaling matter era to the regime of cosmic acceleration.
In addition to the scalar fields with the Lagrangian density (2) we take into account radiation (energy density ) and non-relativistic matter (energy density ). In the flat FLRW space-time with a scale factor they obey the usual continuity equations and , respectively, where a dot represents a derivative with respect to cosmic time and is the Hubble parameter. The pressure and the energy density for the -th scalar field are given, respectively, by
where a prime represents a derivative with respect to . These satisfy the continuity equation
which corresponds to
The Friedmann equations are
In order to derive autonomous equations we define the following quantities
where the quantity defined in Eq. (2) can be expressed as
We also introduce the field density parameters
where is the density parameter of radiation.
Iii Fixed points of the system
Let us derive fixed points for the autonomous equations (II)-(18). In particular we are interested in the scaling solution and the scalar-field dominated solution. For these solutions the variables do not vanish. Setting in Eq. (18), it follows that or . The former corresponds to the solution in the presence of radiation, whereas the latter to the solution without radiation. In the following we shall discuss these cases separately.
iii.1 Radiation-dominated scaling solution
From Eq. (19) the field equation of state for the -th field is
If all scalar fields are in the scaling regime, are the same for all () from Eq. (23) and hence () take a common value with an effective single-field Lagrangian density . Then the total field density is given by
where is defined in Eq. (3). We are interested in the case where one of the fields, say , is in the scaling regime in the deep radiation era, while the energy densities of other fields are suppressed relative to that of . In the BBN epoch we have the following constraint from Eq. (25):
For a given model, i.e. for a given form of , the variables and are determined by solving Eq. (22). If the scalar fields with join the scaling solution at the late epoch of the radiation era, the total field density tends to increase according to Eq. (26) with the decrease of . If the slope of the second scalar field that joins the scaling solution is of the order of 1, the field density (26) can be as large as -1. It is not preferable for many fields with low to join the scaling solution during the radiation era in order to avoid that exceeds 1. This can be avoided if the field densities () are much smaller than the radiation density.
iii.2 Matter-dominated scaling solution and assisted scalar-field dominated point
In the absence of radiation () the fixed points for the -th field corresponding to and obey the following equations
Since we have
In the case of a single field , this equation gives or . The former corresponds to the scaling solution along which is proportional to the matter density , whereas the latter is the scalar-field dominated solution.
If all scalar fields are on the fixed points characterized by the condition (30), it follows that and hence from Eq. (19). In this case one has either or from Eq. (31). Equation (30), which holds for the each scalar field, reduces to the single-field system
In the following we shall discuss the matter-dominated scaling solution and the assisted field-dominated solution, separately.
iii.2.1 Matter-dominated scaling solution
From Eq. (13) we obtain
More generally the field density parameter in the presence of a perfect fluid with an equation of state is given by Tsuji06 ().
If all scalar fields are in the scaling regime, then they can be described by an effective single-field system with and
This scaling solution is stable for Tsuji06 ().
iii.2.2 Assisted field-dominated point
Besides the matter scaling solution discussed above, there is another fixed point that can be responsible for the late-time acceleration. In the single-field case the solutions do not exit to the accelerated field-dominated point () from the scaling matter era, because the scaling solution is stable for . However the presence of multiple scalar fields allows this transition.
This fixed point can be responsible for the late-time acceleration () for . Moreover it is stable under the condition Tsuji06 () (which is opposite to the stability of the scaling matter solution). Using the relations and , we find
We shall study the case in which one of the fields has a large slope to satisfy the BBN bound (27) and other fields with join the scalar-field dominated attractor (38) at late times. Since the joining of such multiple scalar fields reduces it should be possible to give rise to sufficient cosmic acceleration through the assisted inflation mechanism, even if the individual field cannot be responsible for the acceleration.
For a given model one can derive (for the field ) that corresponds to the scaling solution during radiation and matter eras by solving Eqs. (23) and (35), respectively. The field density parameters in these epochs are given by Eqs. (25) and (36), respectively. The assisted field-dominated solution corresponds to
In subsequent sections we shall consider two models: (i) quintessence with multiple exponential potentials, and (ii) the multi-field dilatonic ghost condensate model (one of k-essence models). In our numerical simulations we identify the present epoch (the redshift ) to be with the radiation density in the region .
Iv Quintessence with multiple exponential potentials
The single-field quintessence with an exponential potential corresponds to the Lagrangian density , i.e. the choice in Eq. (1). In the following we shall consider the Lagrangian density (2) of scalar fields with the choice ().
Since in this model the scaling field density during the radiation and matter eras is given by and , respectively [see Eqs. (25) and (36)]. Below we discuss the case in which one of the scalar fields, , is in the scaling regime during most of the radiation and matter eras and other fields eventually join the assisted scalar-field dominated attractor with given by Eq. (38). Then the BBN bound (27) gives
Under this condition, the scaling field density during the matter-dominated epoch is constrained to be . If other fields join the scaling regime in the radiation (matter) era, the field density increases from () to (). This is possible provided that the slopes of the joining scalar fields satisfy the conditions . Meanwhile, if are of the order of 1, this leads to a large density parameter that is comparable to unity. In what follows we focus on the case in which the fields with slopes () enter the regime of the assisted cosmic acceleration preceded by scaling solutions induced by .
It is convenient to introduce the following variable
From Eq. (22) the radiation-dominated scaling solution for the field corresponds to
This is followed by the matter-dominated scaling solution, satisfying
For the -th field we have that and .
iv.1 Two fields
First let us consider the case of two scalar fields and .
In Fig. 1 we plot the evolution of the background fluid density and the field densities versus the redshift ( is the present value of ) for and . We choose three different initial conditions for . The case (i) corresponds to the exact scaling solution starting from the fixed point (43), along which and during radiation and matter eras, respectively. Finally the system enters the epoch in which the energy density of the second field dominates the dynamics. Figure 1 shows that the field eventually joins the scaling regime both for the initial conditions (ii) and (iii) . Thus the cosmological trajectories converge to a common scaling solution for a wide range of initial conditions.
In Fig. 1 we find that the second field density is almost frozen after the initial transient period. In order to understand this behavior we introduce the ratio between the kinetic energy and the potential energy of the -th field:
which is related to the quantity via . Taking the derivative of with respect to , it follows that tracking ()
For the scaling field one has and CLW (), where is the equation of state of the background fluid. This means that , so that the ratio remains constant. In fact, from Eqs. (43) and (44), one has and during the radiation and matter eras, respectively. This reflects the fact that the scaling field has a kinetic energy with the same order as its potential energy.
The field joining the assisted attractor at late-times satisfies and at the early stage of the radiation era, so that initially (unless is unnaturally close to ). At this stage the ratio decreases rapidly as according to Eq. (47), see Fig. 2. In the region the field is almost frozen with nearly constant . As decreases, grows and approaches . This leads to the growth of . As we see in Fig. 2 the ratio starts to increase after becomes larger than 1. When grows to the order of 1, the field begins to evolve to join the assisted attractor given by Eq. (45).
The mass squared for the -th scalar field is given by . The energy density of starts to dominate around the present epoch, so that (the subscript “0” represents present values). Then the mass of can be estimated as
Recall that needs to be of the order of 1 to realize a stable assisted attractor satisfying the condition . Hence the mass is as small as today. In the numerical simulations of Figs. 1 and 2 the field is almost frozen with the mass (48) for the redshift (during which the condition is fulfilled).
Even if the field is rapidly rolling at the initial stage of the radiation era such that , it enters the regime in which is nearly frozen () prior to the matter-dominated epoch. In Fig. 3 the evolution of is plotted for three different initial conditions of with fixed. The dominance of the field kinetic energy relative to its potential energy corresponds to and , which results in the rapid decrease of to reach the regime . Even if initially, the decrease of in the regime is so fast () that the field eventually enters the frozen regime with . For the initial conditions satisfying the field is almost frozen from the beginning, so that is nearly constant until recently.
If we change the initial conditions of associated with the field potential, this leads to the modification of the epoch at which the field dominates at late times. This comes from the fact that the density during which is nearly frozen is sensitive to the choice of its initial potential energy. Thus the evolution of the field depends on its initial potential energy but not on its initial kinetic energy.
Figure 4 illustrates the variation of , , , and for and with the same initial condition as in the case (i) of Fig. 1. The equations of state and are similar to the effective equation of state during radiation and matter eras, but the deviation appears at low redshifts. The field is almost frozen around after the initial transient period, but it begins to evolve for .
From the definition of in Eq. (20) we have
Note that and around the end of the matter-dominated epoch. After gets larger than , begins to be mainly determined by the field , i.e. . As we see in Fig. 4 takes a minimum before reaching the present epoch (), which is followed by its increase toward the attractor value . For the model parameters used in the numerical simulation of Fig. 4 we have , which gives at the scalar-field dominated attractor. This corresponds to the decelerated expansion of the universe. Meanwhile one has and , which means that the transient acceleration occurs at the present epoch. Interestingly, even without the assisted accelerated attractor, such a temporal acceleration can be realized by the presence of the thawing field .
Under the BBN bound (41) and the condition (i.e. the field cannot be responsible for the accelerated expansion as a single component of the universe), the equation of state for the late-time assisted attractor is not very different from . Meanwhile the present value of is smaller than its asymptotic value. For the marginal case with and we find that numerically. For increasing we obtain larger values of and , as we see in Fig. 5. If we do not impose the condition , then can be smaller than . Note that, when and , the scalar-field dominated point ceases to be the late-time attractor. We have also carried out numerical simulations for different values of satisfying the condition and found that and are insensitive to the change of .
iv.2 More than two fields
For three scalar fields the total field equation of state is given by . If the two fields and with slopes join the assisted attractor for , it is possible to obtain smaller values of and relative to the two-field case.
In Fig. 6 we plot the evolution of , () as well as for , , and . The field is in the scaling regime during the radiation and matter eras, which is followed by the epoch of cosmic acceleration once the energy densities of and are dominant. The fields and have been nearly frozen (except for the initial transient period) by the time they start to evolve for . In the numerical simulation of Fig. 6 the energy densities and are the same order when they begin to dominate over the background fluid density. In the numerical simulation of Fig. 6 the field equation of state today is found to be , which is smaller than the minimum value in the two-field case. This comes from the fact that the third field with close to