Adiabatic perturbations in coupled scalar field cosmologies

Adiabatic perturbations in coupled scalar field cosmologies

J. Beyer
Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

We present a comprehensive and gauge invariant treatment of perturbations around cosmological scaling solutions for two canonical scalar fields coupled through a common potential in the early universe, in the presence of neutrinos, photons and baryons, but excluding cold dark matter. This setup is relevant for analyzing cosmic perturbations in scalar field models of dark matter with a coupling to a quintessence field. We put strong restrictions on the shape of the common potential and adopt a matrix-eigensystem approach to determine the dominant perturbations modes in such models. Similar to recent results in scenarios where standard cold dark matter couples to quintessence, we show that the stability of the adiabatic perturbation mode can be an issue for this class of scalar field dark matter models, but only for specific choices of the common potential. For an exponential coupling potential, a rather common shape arising naturally in many instances, this problem can be avoided. We explicitly calculate the dominant perturbation modes in such scenarios.


I Introduction

Scalar fields can appear in modern cosmology in many forms. They are typically associated with phenomena which can not be attributed to ordinary forms of matter, most commonly with an accelerated expansion of the universe. Thus they are particularly important in the very early universe for modeling the era of inflation Sato:1981ds ; Guth:1981uk ; Guth:1980zm ; Linde:1981mu ; Albrecht:1982wi ; Lyth:1998xn , and the very late universe, where scalar quintessence models are one of the most promising candidates for explaining dark energy Wetterich:1987fk ; Wetterich:1987fm ; Ratra:1987rm ; Peebles:1987ek ; Wetterich:1994bg ; Peebles:2002gy ; Copeland:2006wr .

For the long time between these eras, scalar fields typically play only a supporting role in the cosmic evolution. From reheating onwards, throughout the entire radiation dominated era, their energy densities only contribute a small percentage to the total energy density of the universe Bean:2001wt ; Calabrese:2011hg ; Reichardt:2011fv ; Xia:2013dea ; Pettorino:2013ia . The same holds true for the matter-dominated era up to redshifts of about 5, with the exception of scalar field models of dark matter UrenaLopez:2000aj ; Matos:1999et ; Matos:2000ng ; Hu:2000ke ; Matos:2000ss ; Alcubierre:2001ea ; Beyer:2010mt . Still, much effort has ben put into analyzing scalar field dynamics in these eras. The reason for this lies in the so called coincidence problem, which adresses the question why the energy densities of dark energy and matter are roughly of the same order of magnitude only in the recent past. In the CDM model this seems like a very precise finetuning of the cosmological constant. Scalar field models of dark energy provide a possible solution to this problem, since they can exhibit a behavior often called tracking or scaling Copeland:1997et ; Liddle:1998xm ; Steinhardt:1999nw ; Zlatev:1998tr , where the field equations have solutions for which the dynamics follow a specific trajectory largely irrespective of the initial conditions chosen, thereby explaining why dark energy became dominant only recently.

In this paper we are interested in such scaling solutions involving multiple scalar fields which couple through their common potential and the evolution of perturbations in such scenarios. Similar models have been investigated before in the context of inflation Gordon:2000hv ; Malik:2002jb , i.e. without any additional matter content. We consider the dynamics of multiple scalar fields in the presence of ordinary matter, i.e baryons, neutrinos and photons, but exclude cold dark matter or a cosmological constant. This setting is relevant for models where the entire dark sector, i.e. both dark energy and dark matter is described by scalar fields.

If one looks beyond the homogeneous and isotropic background cosmology, one quickly finds that the the inhomogeneities of scalar fields in such models are of great importance for the study of cosmological perturbations in general. This starts with the fact that all structures seen in the universe today are now widely believed to have been sourced by quantum fluctuations of scalar fields during the era of inflation, but does not end there. Scalar quintessence models can influence late-time structure growth Bartelmann:2005fc ; Doran:2001rw , but even during the era of radiation domination, where scalar fields are subdominant, their presence can impact how perturbations in the baryonic or dark matter sector evolve. While standard minimally coupled one-field scaling quintessence models seem to be unproblematic in this respect Doran:2003xq , recent studies have shown that a coupling between a scalar field and dark matter can result in an instability of the adiabatic perturbation mode Majerotto:2009zz ; Majerotto:2009np , potentially rendering such scenarios difficult to reconcile with observational bounds, which clearly demand largely adiabatic perturbations Enqvist:2000hp ; Enqvist:2001fu ; Beltran:2005xd ; Seljak:2006bg ; Keskitalo:2006qv ; Kawasaki:2007mb ; Castro:2009ej ; Valiviita:2009bp ; Li:2010yb ; Ade:2013uln . It is therefore interesting to investigate whether such problems can also arise in models where multiple scalar fields couple to each other, e.g. in coupled scalar field dark matter models, and we do so in our analysis below.

This paper is organized as follows:

In section II we derive the generic shape of the potential necessary for multiple canonical scalar fields to exhibit scaling solutions in the presence of a background fluid. In section III we study the evolution of linear perturbations in the superhorizon regime in such models, in the presence of photons, baryons and neutrinos. We restrict ourselves to the case of two scalar fields here for simplicity. In section IV, we apply the results to one particular choice of scalar potential with an exponential coupling. This is the relevant potential for the coupled scalar dark matter model investigated in an accompanying paper CB_LinPers . Finally, in section V we investigate alternative potentials and construct an explicit example for which the adiabatic perturbation mode is unstable. We present our conclusions in section VI.

Ii Coupled canonical scalar fields and scaling solutions

In this section we investigate exact scaling solutions in FLRW-cosmologies involving multiple coupled canonical scalar fields and a background matter fluid, defined by an energy density and a constant equation of state . In the literature the definition of a scaling solution is not completely unambiguous. In the case of a single scalar field it can either mean a solution for which the scalar energy density scales like the scale factor to some constant power, or, and this can be considered a stricter definition, a solution for which the scalar energy density scales exactly like the background fluid. In this work we adopt the stricter definition and consider an exact scaling solution to be a scenario in which all scalar energy densities scale like the background fluid.

It is well known that the only potential providing an exact scaling solution for a single canonical scalar field is an exponential potential. This even holds true in the presence of a coupling to the background fluid Amendola:1999qq ; Uzan:1999ch ; Copeland:2004qe ; Tsujikawa:2004dp . Furthermore, even in the presence of non-canonical kinetic terms - however still restricted to those yielding second order field equations - the form of possible Lagrangians yielding scaling solutions can be strongly restricted Tsujikawa:2004dp .

We will now show that in the case of multiple canonical coupled scalar fields similar restrictions for the common potential can be found. To do this we employ an approach already used in the earlier papers cited above, slightly adjusted to fit our scenario. We start by considering the following action


where , denotes the number of scalar fields present and denotes the matter action. The scalar field equations for this action in the FLRW-background read


where a prime denotes a derivative with respect to conformal time and is the conformal hubble parameter . The common potential can in principal be identified with any one of the scalar field energy densities or even split in some arbitrary fashion, but we choose to assign it to for simplicity and define scalar energy densities and pressure densities as follows:


As we assume no direct coupling between the background fluid and the scalar sector, the equations governing the background-evolution can be written as


where and conservation of the total energy-momentum tensor implies that the scalar couplings have to satisfy


The definition of the couplings in this manner is of course a matter of convention, here we remain consistent with the general formulas given in appendix A. In an exact scaling scenario we demand


Note that the scalar equations of state can differ from the background equation of state even in an exact scaling scenario if a coupling between the scalar fields is present, only the combined scalar equation of state has to fulfill


if the scalar density parameters are non-zero. From equations (5) and (8) we can directly conclude that




Since , equations (10) and (11) directly give


Furthermore all couplings have to be constant by virtue ofequations (6) and (10):


Now we can easily derive a differential equation for the potential by again employing equations (6) and (10):


which directly gives


where and is some arbitrary (smooth) function. One can quickly check that this solution is not in conflict with the the demand of constant couplings simply by noting that by virtue of equation (12) all have to be constant for a scaling solution and by convention (3) we therefore have


for and dynamics corresponding to a scaling solution.

One comment might be in order here: In the derivation of equation (15) we made some implicit assumptions. The most obvious one is that , i.e. the scalar potential energy density should be non-vanishing. As we will see in section IV, purely kinetic fixed points with do exist, and we expect them to be present for a large class of potentials, including many not of the shape derived here. The only restriction here is that the common potential should become zero somewhere, at least asymptotically. Furthermore one might suggest that we have excluded solutions for which cosmological expansion is purely scalar field dominated. This is however not the case, as one can simply replace all appearing in the above equations with a constant effective equation of state and reach the same conclusions concerning the potential shape.

To clarify this result: If you want to construct a potential allowing for exact scaling solution with non-vanishing potential energy density involving multiple canonical scalar fields coupled through their potential, you first have to check if your potential fulfills equation (15) for some set of constants . If so, a scaling solution can exist.

Iii Early Universe Perturbations

We will now turn our attention to the evolution of perturbations in the early universe in models containing multiple non-minimally coupled canonical scalar fields. In order to keep things simple, we restrict our attention to the case of two fields and call them and , deviating from the conventions used in section II. We also split the potential differently, this time assigning a -dependent part to and a common part to , i.e.


By ”early universe” we mean an era during which the cosmic expansion proceeds as if dominated by radiation, i.e. , with being conformal time. We will consider a background-evolution during which both fields follow an exact scaling bahaviour, in particular the equations of state are assumed to be constant. Note that this does not imply a strongly subdominant role of the scalar fields or equations of state of exactly . The coupling between the two fields enables a wide range of possible ’s and ’s while still allowing for a radiation-like expansion () and non-negligible scalar field contributions to the energy-densities already in very simple models, as we will see below.

In addition to the two coupled scalar fields we include neutrinos, photons and baryons into our model, but exclude a possible cold dark matter contribution. This sets our study apart from former systematic studies of early universe perturbation modes, which have usually been limited to cosmologies containing either no scalar fields Ma:1995ey , only scalar fields (usually in the context of inflation) Gordon:2000hv or a single quintessence field Malik:2002jb ; Malquarti:2002iu ; Bartolo:2003ad ; Doran:2003xq ; Majerotto:2009np . Leaving out a possible cold dark matter contribution is realistic, if at least one of the two scalar fields changes its dynamics away from a scaling solution and acts like a dark matter component during the later stages of its evolution. If this is the case, the coupling arising from the common potential will result in a coupled quintessence model in the late universe Wetterich:1994bg ; Amendola:1999er ; Pourtsidou:2013nha , more specifically a coupled quintessence-scalar-dark-matter model Beyer:2010mt . It is precisely these kinds of models that require the analysis presented here to draw the initial conditions for numerical evolutions of cosmological perturbations, a study we performed in an accompanying work CB_LinPers .

Before moving on to our analysis, let us recap some relevant results from the previous studies. As was shown in Malik:2002jb , even in generic non-minimally coupled models (which of course includes ours), purely adiabatic perturbations remain purely adiabatic on superhorizon scales, no matter what the interaction is. Here a mode is called adiabatic if all entropy perturbations vanish. This includes relative entropy perturbations between different components of the cosmic fluid as well as internal entropy perturbations for a single component - these can exist for non-perfect fluids (see section 4 in appendix A for more details). However, as was noted in ref. Doran:2003xq , the existence of such an adiabatic mode is non-trivial, since demanding all entropy perturbations to vanish typically introduces more constraints than we have degrees of freedom in the equations, in particular if the number of species considered is large. One should understand that this is not in conflict with the theorem found by Weinberg in Weinberg:2003sw ; Weinberg:2003ur , which states that an adiabatic superhorizon mode always exists, no matter the content of the universe. The apparent contradiction stems from Weinberg’s definition of an adiabatic superhorizon-mode, which he takes to mean a mode for which the total curvature perturbation remains constant, which is the case if the total entropy perturbation (see appendix A). This is a weaker condition than demanding all entropy perturbations to vanish, which is what we take adiabatic to mean in this work.

Furthermore this result is not sufficient to explain a suppression non-adiabatic modes, since any, presumably initially small, admixture of such modes could in principle eventually outgrow the adiabatic perturbations. Such a growth can happen for example during the ’adjustment’ phases in minimally coupled tracking quintessence models when the quintessence field does not yet follow the tracker trajectory. However, a sufficiently long tracking regime typically erases these modes in this case Malquarti:2002iu .

The situation in non-minimally coupled models of quintessence appears to be more complicated. We are aware of only one study which systematically analyzes the evolution of superhorizon-perturbations in a coupled quintessence model Majerotto:2009np . It appears that in such scenarios there are certain sections of parameter space for which the adiabatic mode becomes unstable, i.e. other, faster growing perturbation modes exist. Since this analysis includes a conventional cold dark matter component and uses a specific commonly used form of the coupling given by , the question if similar issues will arise in coupled scalar field models of dark matter remains open, and we will address it below.

iii.1 Basic Formalism

The basic idea we employ in our study of early universe superhorizon perturbations is one already used in earlier works Malquarti:2002iu ; Bartolo:2003ad ; Doran:2003xq . We write the differential equations governing the evolution of linear perturbations in a convenient matrix form, i.e.


Here we introduced a convenient new time variable and combined all relevant perturbative quantities into a single perturbation vector . Which quantities these are depends on the approximations one choses to use in the neutrino-, photon- and baryonic sectors. Here we employ the simplest (lowest order) version of the tight-coupling approximation for photons and baryons, which yields a common velocity potential for both components and excludes all higher momenta of the Boltzmann hierarchy for photons. In the case of neutrinos we truncate the Boltzmann-expansion after the quadrupole, leaving an additional anisotropic stress contribution. More details and precise definitions of the perturbative quantities used in this chapter can be found in appendix A.

Since the scalar fields only have two degrees of freedom, the components of the perturbation vector can be chosen to be the energy density contrasts and velocity potentials for all components of the cosmic fluid as well as anisotropic stress for neutrinos, i.e.


The entries of the perturbation matrix can now be read off from the differential equations governing the evolution of these variables. For the photon-, baryon- and neutrino-sector they read


These equations can be derived from the generic equations given in appendix A by setting . Note that the equation for the velocity potential is different from the one usually employed, e.g. in refs. Doran:2003xq ; Majerotto:2009np . This version is the correct one if one makes no assumption about the ratio . Both version agree in the limit .

The gravitational potentials are then given by


The generic equations of motion for the scalar fields are quite complicated (see appendix A), but they can be simplified considerably in an exact scaling scenario. The background quantities appearing in the scalar sector are the equations of state and , the adiabatic sound speeds and , the couplings and , their time-derivatives and the second derivative of the common potential .

In an exact scaling scenario we assume constant equations of state, i.e. we set . This implies that the adiabatic sound speeds are given by and . Furthermore the couplings and have to be constant by equation (13) and are of course related by equation (7). Finally the scalar equations of state are related by equation (9) and we only have to deal with the second derivative of the common potential. For a generic scalar field evolution this can of course take any, in principle time-dependent, form, but in a scaling scenario it merely introduces another constant into the equations, which can be seen as follows.

We start by investigating the generic potential for exact scaling solutions given by equation (15). In order to split our potential according to equation (17) we have to rewrite as


where and , are some suitable constants. Note that this already shows that has to be an exponential potential. We can take the second derivative of this potential with respect to and obtain (for )


Since is constant during a scaling solution, it is obvious that in such scenarios and we can thus define


with some constant . Note that is in principle a new independent constant depending on the functional form of and can generally generally not be related to , as it depends on the second derivative , whereas only the first derivative enters into .

Now we can finally write down the equation for the scalar sector in a form where the only remaining constant parameters are , and the density parameters. They read


The coefficients of the matrix can be read off from equations, (21) - (26) and (III.1) - (III.1) after replacing the gravitational potentials.

Before analyzing the solutions for the perturbations we have to investigate the background quantities appearing in . Those are generally -dependent, but in this analysis we are interested only in the superhorizon-limit defined by and we can therefore work with Taylor-expansions in . To see that, first note that we assume a radiation-like expansion with a scalar scaling solution in the early universe, a scenario only slightly disturbed by the baryonic density parameter, which is small, but grows linearly in :


This implies that all the other density parameters are constant to leading order, but decrease slowly at linear order in . Evaluating Friedmanns equations for a flat universe in the form order by order in then yields to leading order


Using this, it becomes obvious that we can expand the matrix as a type of Taylor-series in x and recover a constant matrix at leading order:


With this approximation we can immediately write down the leading order solution for equation (19), which reads


where are eigenvectors of the matrix and the corresponding eigenvalues. With this as a starting point, one can now easily determine higher order corrections to the found solution by expanding the eigenvectors as follows:


where the first order correction is then given by


and the other corrections can be calculated in a similar fashion, but we will not need them here. Of course, the general solution is given by


A short comment concerning the different approximations used here might be in order. As explained in appendix A we use the simplest version of the tight coupling approximation for photons and baryons, i.e. we only include coupling terms to leading order (that is zeroth order) of , where is the Thomson interaction timescale. One might question whether it is consistent to go to next to leading order in the -expansion of the matrix , but not include the next order in the TCA, i.e. terms suppressed by . As it turns out, for the relevant wavenumbers we have during radiation domination. This breaks down for very small , but for realistic cosmologies the boundary lies at roughly Mpc, which is already far below what is currently observable.

iii.2 Dominant perturbation modes

iii.2.1 Eigenvalues

In order to determine the dominant perturbation modes in the early universe in coupled two scalar field models one simply has to determine the eigenvalues of the matrix with the largest real parts and the corresponding eigenvectors. In principle these depend on the parameters determining the scaling solution, which can be taken to be , , and . The remaining density parameters for neutrinos and photons are fixed by the fact that we assume a FLRW metric without curvature, i.e. and the well known relation , which is valid after electron-positron annihilation Bartelmann:2009te . However, four of the eigenvalues are always given by , irrespective of the values of these parameters. The remaining 6 eigenvalues are very complicated functions of the 4 free parameters and quoting them is not very enlightening. We will however present numerical results below, obtained simply by running through a grid of the four independent parameters .

Figure 1: Real part of one critical eigenvalue for and . The solid red line represents a model with an exponential potential discussed in section IV, the dashed red line a power-law potential discussed in section V.

Figure 2: Real part of one critical eigenvalue for and . The solid red line represents a model with an exponential potential discussed in section IV, the dashed red line a power-law potential discussed in section V.

For the wide range of parameters we investigated, 4 of the 6 remaining eigenvalues had real parts which were bound from above by . Only two eigenvalues can have a real part bigger than zero, a case will will call strongly growing from now on. As we will see below, the nullspace of always contains an adiabatic mode. It is precisely the two potentially strongly growing modes which can render the adiabatic mode unstable, which would make the corresponding scenario difficult to reconcile with observations Enqvist:2000hp ; Enqvist:2001fu ; Beltran:2005xd ; Seljak:2006bg ; Keskitalo:2006qv ; Kawasaki:2007mb ; Castro:2009ej ; Valiviita:2009bp ; Li:2010yb ; Ade:2013uln . On could of course conceivably come up with reasons why these modes are initially strongly suppressed compared to the adiabatic mode, so that it does not become dominant during the radiation dominated era, but such a fine-tuning of initial conditions is often not very natural and generally undesirable.

To visualize the effect of the coupling between the two scalar fields we have plotted the real part of one of the two critical eigenvalues in Figures 1 and 2. As can be seen, the values become bigger than zero for large deviations of from the uncoupled value of , i.e. for big couplings. Furthermore, as one would expect, this effect becomes stronger for larger scalar field density parameters. In the uncoupled case, even the critical eigenvalues have real parts bound by from above and the corresponding modes are thus subdominant. This shows that a strong coupling can be relevant for early universe perturbations, because it can potentially change the dominant perturbation mode, rendering the adiabatic mode unstable.

One should note however, that just because we can find a set of parameters which yield a strongly growing eigenvalue in this numerical treatment does not necessarily mean that we can construct a model actually yielding a scaling solution with such a feature. The problem is of course, that for any given scaling solution some of the parameters treated here as free might well be related, or scaling solutions with a radiation like expansion might not even exist or be confined to specific regions of parameter-space. Later in this work we will examine one specific potential where the relation is fixed to the solid red line in Figures 1 and 2, thus excluding the region of strongly growing eigenvalues, and a second one represented by the dashed red line, for which the adiabatic mode becomes unstable for strong couplings.

iii.2.2 Eigenvectors

Since the generic eigenvalues have very complicated forms in general, so do the corresponding eigenvectors. This also holds for the eigenvectors belonging to the two critical eigenvalues, representing potentially dominant modes. Things look a lot better for the nullspace of , which of course includes all dominant perturbations modes in the absence of strongly growing modes. One is free to choose any suitable basis of this three-dimensional space, we choose to categorize the modes according following ref. Doran:2003xq , i.e. by means of the total curvature and relative entropy perturbations.

In this categorization a mode is called an isocurvature mode if the total curvature perturbation vanishes, i.e.


and adiabatic if all relative and internal entropy perturbations, i.e.


There are some deviating definitions of the total curvature pertubation in the literature Mukhanov:1990me , but they agree in the superhorizon limit in a flat universe.

This categorization splits the nullspace into a two-dimensional isocurvature space and one adiabatic mode. The adiabatic eigenvector is given by


where . This is very similar to the adiabatic mode in minimally coupled quintessence scenarios Doran:2003xq .

The isocurvature-subspace can be further decomposed into one baryon-isocurvature and one neutrino/scalar-isocurvature mode.

The baryon isocurvature mode is characterized by the demand that all species except for baryons should be adiabatic, i.e. for all , but for . For our set of scaling scenarios this necessarily implies that the scalar internal entropy perturbations also vanish, i.e. . In the corresponding eigenvector all entries except for vanish at leading order, we therefore quote the result to subleading order in here:


where we defined the quantities and . This vector differs considerably from the ones found in previous studies, which is due not only to the presence of two coupled scalar fields, but also to the improved treatment of the tight coupling approximation, which changes the subleading order contributions to the perturbation matrix .

Finally, the neutrino/scalar isocurvature mode is characterized by which , as this is the only relative entropy perturbation not involving neutrinos or scalar fields. Enforcing this necessarily implies , with all other relative entropy perturbations non-vanishing. The corresponding eigenvector reads at leading order:


Note that due to the convenient choice of perturbation variables, the dependence on the coupling only appears at subleading order in the baryon isocurvature mode through . At leading order, the nullspace of the leading order perturbation matrix is completely independent of or .

Iv Coupled exponential potentials

In this section we investigate one specific example of two coupled canonical scalar fields given by


We chose this particular shape for two reasons. First, it arises in the cosmon-bolon model for quintessence and dark matter Beyer:2010mt , which gets investigated more deeply in an accompanying paper CB_LinPers . Furthermore, as we will show in the next section, this simple potential is already the most generic case, in the sense that all exact scaling solutions existing for other potentials are already present for this exponential form.

One can easily check that this potential fulfills equation (15) with , and


For the subsequent analysis we split the potential as follows




and assign energy- and pressure-densities accordingly:

Point x y u v z
Table 1: Fixed points for coupled exponential potentials. Here , and the sign change for the point should be taken simultaneously in and .

iv.1 Exact scaling solutions

We start by finding all exact scaling solutions for the case of two coupled canonical scalar fields and in the presence of a radiation fluid, which is the relevant case for studying physics in the early universe. The equations of motion governing this system can be directly read off from equations (2) and (5). As is common when trying to find exact scaling solutions, we now employ the dynamical systems approach Copeland:1997et ; Gumjudpai:2005ry and introduce the following new variables:


where these variables are subject to the constraint


since are assuming a flat universe. The equations of motion can now be rewritten as


and a fixed point is characterized by . When solving the resulting algebraic equations we can reduce the number of fixed points by eliminating some redundancies. First we assume that and , both can be easily achieved by a suitable sign-change in the fields and . Furthermore we have and by definition. Restricting ourselves to these ranges, the complete set of all fixed points is given in Table 1.

These fixed points can be split up into the purely scalar field dominated fixed points to and radiation-like fixed points to . We expect the kinetically dominated fixed points and to always exist, irrespective of the exact shape of the scalar potential, as long as it vanishes asymptotically. Since the kinetic energies are non-zero whereas the potential energies vanish, this requires an ’extended region of a zero potential’ where the fields can roll freely, a scenario reached asymptotically in the limit for our potential. Furthermore the point corresponds to both scalar fields sitting at a root of the potential, in the case of at infinity. This point is present even more generically, all that is required of the potential is a (possibly asymptotic) root. However, these points are always unstable, as we will see below.

The remaining points to and to are the ones only existing for potentials of the form derived in section II and among these, only the ones exhibiting a radiation-like expansion, i.e. to , can result in a realistic early cosmology.

Point Existence Stability
K unstable
K unstable
S ,
R always unstable
R , ,
Table 2: Conditions for existence and stability of the fixed points.

The stability of the fixed points can be analyzed by investigating the Jacobian matrix for and for each fixed point. The typical classification distinguishes between nodes, spirals and saddle-points as well as stable and unstable points Copeland:1997et ; Gumjudpai:2005ry . Here we are not interested in the precise details and simply call points for which the Jacobian matrix has only positive (or zero) eigenvalues unstable, and points for which all eigenvalues are negative stable. The conditions for existence and stability of the fixed points are given in Table 2.

As is common in scenarios such as ours, the stable fixed points split up the parameter space into disjunct sections. For a given set of parameters and we therefore have a unique attractive scaling solution towards which the cosmological evolution will adjust itself relatively quickly in the early universe. To avoid this fixed point for the prolonged period of radiation domination, one would necessarily have to start with initial conditions very far away from the fixed point configuration. The splitting of the parameter space can be seen in Figures 3 and 4.

iv.2 Dominant perturbation modes

The dominant perturbation modes in this potential can now be obtained in the same fashion as the general results in section III, but this time with the simplifications




The generic results for the eigenvalues do of course still hold, but as is shown in Figures 1 and 2, the particular shape of the potential leads to a form of the coupling term that excludes the region of strongly growing eigenvalues. This holds for all stable fixed points, with the exception of , where and which therefore requires some additional treatment. However, a vanishing clearly requires by virtue of equation (9), effectively reducing the scenario to an uncoupled one, where all eigenvalues are non-positive.

Thus for the coupled exponential potential there is a huge region of the parameter space which is not only viable as an early cosmology at the background level, but for which no strongly growing perturbation modes exist and the adiabatic mode is therefore stable. With the results presented here one can therefore easily write down the early universe background evolution for any set of parameters and and also the dominant perturbation modes, given by the nullspace of , with the density parameters replaced by the ones given in Table 1.

Figure 3: Parameter space of stable fixed points for the coupled exponential potential with , the structure remains valid for all . The shaded region shows the points which allow for a realistic early universe cosmology, i.e. a radiation-like expansion.

V Non exponential models

We now move on to provide short arguments why all exact scaling solutions for coupled two scalar field cosmologies are effectively the ones found for the coupled exponential potential in section IV. We start by recalling the generic shape of the common scalar potential required for the existence of scaling solutions given in equation (15) together with the splitting (17), which implies via equation (30) that the common potential is given by


where with and . Our aim is to rewrite the field equations in the same form used in section IV, in order to recover a set of algebraic equations whose fixed points determine the existing scaling solutions. For non-exponential potentials we can of course not express the potential derivatives in terms of the potential, and we therefore need a new set of variables, defined by


The evolution equations for these variables are very simple and read


The field equations can now be rewritten as in section IV. They read: