Global dynamics and asymptotics for monomial scalar field potentials and perfect fluids
We consider a minimally coupled scalar field with a monomial potential and a perfect fluid in flat FLRW cosmology. We apply local and global dynamical systems techniques to a new three-dimensional dynamical systems reformulation of the field equations on a compact state space. This leads to a visual global description of the solution space and asymptotic behavior. At late times we employ averaging techniques to prove statements about how the relationship between the equation of state of the fluid and the monomial exponent of the scalar field affects asymptotic source dominance and asymptotic manifest self-similarity breaking. We also situate the ‘attractor’ solution in the three-dimensional state space and show that it corresponds to the one-dimensional unstable center manifold of a de Sitter fixed point, located on an unphysical boundary associated with the dynamics at early times. By deriving a center manifold expansion we obtain approximate expressions for the attractor solution. We subsequently improve the accuracy and range of the approximation by means of Padé approximants and compare with the slow-roll approximation.
The present paper investigates general relativistic flat Friedmann-Lemaître-Robertson-Walker (FLRW) models with a minimally coupled scalar field with a monomial potential, (, ), and a perfect fluid. The perfect fluid is assumed to obey a linear equation of state, , where and are the pressure and the energy density, respectively. The adiabatic index is assumed to satisfy , where corresponds to dust and to radiation. When the matter term can be reinterpreted as , where is the spatial 3-curvature of the open FLRW model, i.e., leads to equations that are the same as those for a scalar field in open FLRW cosmology. The case corresponds to a matter content described by a cosmological constant, i.e., , while describes a stiff perfect fluid; both cases are associated with significant bifurcations, and we therefore refrain from discussing them.
The Einstein and matter field equations for these models are given by
Here an overdot signifies the derivative with respect to synchronous proper time, ; is the Hubble variable, which is given by , where is the cosmological scale factor, and throughout we assume an expanding Universe, i.e. , where is related to the expansion according to . We use (reduced Planck) units such that , where is the speed of light and is the gravitational constant (in the inflationary literature the gravitational constant is often replaced by the Planck mass, ).
Heuristically eq. (1c) can be viewed as an equation for an anharmonic oscillator with a friction term . This suggests that toward the future in an oscillatory manner, which is indeed correct. This qualitative picture, however, does not show how this comes about in a quantitative way, nor how the fluid affects the situation via its influence on . Running the time backwards allows one to heuristically interpret as an energy input, which suggests that the scalar field oscillates with increasing amplitude toward the past, but this picture breaks down in the limit . Even though this is beyond the Planck regime, this limit is also needed in order to describe the classical behavior at early times after the Planck epoch. Furthermore, eq. (1d) yields that
where an are constants, and hence at late times while at early times.
Note that the above qualitative considerations say nothing about how e.g.
behaves asymptotically, i.e., if the model is fluid or scalar field dominated, or neither, asymptotically. Nor does the above say anything about the role of the so-called attractor solution in a global solution space setting.
This exemplifies that there is a need for a more careful examination, which is illustrated by some previous heuristic considerations for a scalar field with a monomial potential by e.g. Turner  and Mukhanov  p. 242, which in turn inspired the rigorous work by Rendall ; in addition de la Macorra and Piccinelli introduced a new heuristic approach to study dynamics at late times for a scalar field with a monomial potential and a perfect fluid ; rigorous work in this context was also obtained for the special case by Giambo and Miritzis .111Some further examples of references that describe minimally coupled scalar field cosmology in dynamical systems settings are [6, 7, 8], with additional references therein. Nevertheless, this still leaves room for improvements and extensions, and, as will be shown in this paper, it is possible to shed light on interesting previously neglected physical and mathematical aspects.
The main purpose of this paper in, primarily, mathematical cosmology is two-fold: Firstly, to obtain a global visual picture of the solutions space, thus, e.g., situating the so-called attractor solution in a global solution space context. Secondly, to prove issues concerning asymptotical behavior at late and early times. This includes introducing averaging techniques to determine late time behavior, generalizing and simplifying earlier proofs in the literature, and using center manifold theory to rigorously derive approximations for the attractor solution at early times, as well as clarifying the physically important issue of asymptotic self-similarity.
The outline of the paper is as follows. In the next section we introduce our new three-dimensional dynamical systems reformulation of the field equations on a relatively compact state space. We also present two other complementary dynamical systems formulations of the field equations, which allow us to effectively obtain approximations for the attractor solution. In Section 3 we apply global and local dynamical systems techniques to obtain a complete and illustrative picture of the solution space and its properties, including asymptotics. In particular, we introduce averaging techniques in our global dynamical systems setting, which allows us to prove the following theorem:
If , then for all solutions with , which implies that the solutions are future asymptotically scalar field dominated.
If , then for all solutions with , and thus the solutions in this case are future asymptotically perfect fluid dominated.
If , then when , and there is thus no future scalar field or perfect fluid dominance.
It should be pointed out that similar conclusions have been reached heuristically with quite different arguments in e.g. . Furthermore, Giambo and Miritzis gave a proof for for the cases (i) and (ii) in  (in the case of general relativity). However, apart from that our proof rigorously generalizes previous results, our method can, in principle, be modified to treat even more general situations. Moreover, we tie our results to the global dynamical systems picture and discuss their physical implications, e.g., situating them in the context of future manifest asymptotic self-similarity breaking. In Section 4 we focus on the attractor solution, where we introduce and compare several approximation schemes, such as center manifold and slow-roll based expansions and Padé approximants, in order to describe it quantitatively. Finally, Section 5 contains some general remarks, e.g. about the de Sitter solution on the unphysical boundary of the state space.
2 Dynamical systems formulations
2.1 Global dynamical systems
Our main global (i.e. compact) dynamical systems formulation is based on the dependent variables , which are defined as follows:
In addition it is of interest to define
To introduce a new suitable time variable we note the following: At early times it is natural to use a Hubble-normalized time variable defined by , due to that the expansion provides a natural variable scale when via the Raychaudhuri equation, as further discussed in e.g. , and references therein (in an inflationary context is often interpreted as the number of -folds ). At late times the square root of the second derivative of the potential, (for simplicity we here incorporate into ), provides a natural variable (mass) scale. Due to the Gauss constraint (1a), which relates a scale given in to one given in according to , this scale can be expressed in terms according to which leads to a dimensionless time variable defined by , where the constant have the same dimension as . To incorporate these features in a global dynamical systems setting we introduce a new time variable that interpolates between these two regimes,
where when and , respectively.
The above leads to the following three-dimensional dynamical system for :222The variable has been used ubiquitously in the scalar field literature (often denoted by ), while was used in  where it was denoted by , however, as far as we know, the variable and the independent variable are new. The reason for the name is that mathematically this variable plays a role that is reminiscent to that of Hubble-normalized shear, which is usually denominated by in anisotropic cosmology, for a number of situations (the subscript follows the notation in ). Thus the present nomenclature is designed to pave the way for eventually situating the present problem in a broader context than isotropic scalar field cosmology.
where the deceleration parameter, , defined via , is given by
where the last equation follows from the Gauss constraint (1a), while the inequality is due to . Above we have also introduced an effective equation of state parameter for the scalar field which is defined according to
From the above relations it follows that . In addition it is of interest to give the following auxiliary evolution equation for :
The state space associated with eq. (7) is given by a finite (when deformed) cylinder described by the invariant pure scalar field boundary subset, (i.e. ), and thus , which we denote by , and . From now on, we analytically extend to the state space by including the unphysical invariant submanifold boundaries and . Although these boundaries are unphysical, we stress that it is essential to include them since they describe the past and future asymptotic states, respectively, of all physical solutions.
Note that (i.e. ) and hence is an interior invariant subset, , which is just a straight line in the center of the cylinder, describing the flat FLRW perfect fluid model without a scalar field (this solution appears as a straight line in the present state space due to that it is a self-similar solution, where describes the temporal change in the dimensional variable ). Note also that the dynamical system (7) is invariant under the discrete symmetry , leading to a double representation of the physical solutions when , which is a consequence of that the potential is invariant when .333The system (7) is differentiable for non-integer when , where the differentiability depends on , describing problems with potentials , where is to be replaced with in (7).
To describe the dynamics on the scalar field boundary , where , it is useful to introduce a complementary global formulation, which is based on the following transformation of and :
This leads to the following regular unconstrained two-dimensional dynamical system:
In this case the deceleration parameter is given by
2.2 Complementary non-bounded dynamical systems
We here introduce two complementary dynamical systems on unbounded state spaces that are useful for describing the dynamics at early times. The first system is based on the dependent variables and the independent variable , where we recall that and are defined by
where can be viewed as the number of -folds , i.e., . This leads to the dynamical system:
where is still given by (8), (9).444In the special case and this system coincides with eq. (16) in ; incidentally, this model was also the example discussed by Linde in his paper “Chaotic inflation” . It is also useful to consider auxiliary equations for and :
The second complementary dynamical system concerns the dynamics on the scalar field boundary . Expressed in terms of and the unconstrained system (14) takes the form:
Note that the above systems share the same equations as the previous ones with bounded state spaces on the invariant boundary subset .
3 Global dynamical systems analysis
It follows from (7) that
on (note that since we have assumed that it follows from eq. (8) that only when and ). Due to that , it follows from (7a) and (20) that is a monotonically increasing function on (although eq. (20) shows that solutions on the scalar field boundary have inflection points when ) and hence can be viewed as a time variable if one is so inclined. As a consequence all orbits (i.e. solution trajectories) in originate from the invariant subset boundary , which is associated with the asymptotic (classical) initial state, and end at the invariant subset boundary , which corresponds to the asymptotic future, and therefore all fixed points are located at and .555A fixed point, sometimes called an equilibrium, critical, or stationary point, is a point in the state space of a dynamical system for which . It also follows from the monotonicity of that the past (future) attractor resides on ().
The equations on the subset (or, equivalently ) are given by
as follows from (7), or, equivalently (17) (with replaced with ), where is given by (8) and (9). It follows that the state space on is divided into four sectors defined by the invariant subsets and . The intersection of these subsets with and with each other yield five fixed points on :
where are two equivalent fixed points for which and (and ), i.e., they are associated with a massless scalar field state, while the two equivalent fixed points , for which and (and ) correspond to a de Sitter state.666Note that the present de Sitter fixed points are distinct from de Sitter states that are associated with potentials that admit situations for which for some constant finite value of for which both and have constant, bounded, and positive values. In contrast the present de Sitter states correspond to the limits , , and therefore reside on the unphysical boundary . The fixed point gives and and corresponds to the flat perfect fluid Friedman model.
As shown below, the two fixed points are sources on ; the fixed points are sinks on , but they also have one zero eigenvalue that corresponds to a one-dimensional unstable center submanifold on , i.e., one solution, called an attractor solution, originates from each fixed point into on the pure scalar field boundary subset . The fixed point is a saddle that gives rise to a 1-parameter set of solutions entering (the associated unstable tangent space is given by ), one being the perfect fluid solution given by . The system (22) admits the following conserved quantity when :
which determines the solution trajectories on the subset, see Figure 1.
The equations on the subset are given by
This system has a non-hyperbolic fixed point,777A fixed point is hyperbolic if the linearization of the dynamical system at the fixed point is a matrix that possesses eigenvalues with non-vanishing real parts; if the linearization leads to one or more eigenvalues with vanishing real parts it is said to be non-hyperbolic.
with three zero eigenvalues. Fortunately resides at the intersection of two invariant subsets: the invariant subset and the invariant perfect fluid subset , and thus, since is monotone in , attracts at least this orbit. On is conveniently analyzed by considering eq. (11) on , which yields that
To make further progress as regards the global properties of the solution space we need to consider the asymptotic dynamics at early and late times.
3.1 Asymptotic dynamics at early times
The linearization of the system (7) at the fixed points and is conveniently described as follows:
where the right hand sides constitute the eigenvalues of the fixed points. Hence is a hyperbolic saddle, with an unstable manifold tangential to , i.e., there is a 1-parameter set of solutions that originate from entering the state space tangentially to . The fixed points are hyperbolic sources; it follows from the invariant submanifold structures that there exists a 2-parameter set of solutions entering the interior of the cylinder with from each fixed point , while a 1-parameter set of solutions originate from each fixed point into the boundary subset .888The above results follow from the Hartman-Grobman theorem, which states that in a neighborhood of a hyperbolic fixed point the full nonlinear dynamical system and the linearized system are topologically equivalent, see e.g. [16, 12].
On the subset linearization of the system (22) gives
and hence are hyperbolic sinks on , as illustrated in Figure 1. In the full state space, however, each equivalent fixed point have an additional zero eigenvalue associated with a one-dimensional so-called center manifold. Fortunately, the center direction lies on the subset, and hence we can investigate the center manifold by means of the unconstrained system (14), which will be done in Section 4. There we show that the center manifold of each (equivalent) fixed point corresponds to a single solution that enters the state space (we will even obtain approximate expressions for this solution), and this solution, which hence resides on , is what is often referred to as the ‘attractor’ solution. In the full state space the fixed points are thus center-saddles.
From these considerations, in combination with the monotonicity of , it follows that all solutions are past asymptotically self-similar in the sense that all physical geometrical scale-invariant observables, such as the deceleration parameter , are asymptotically constant. However, there is a twist to this. The geometry of a flat FLRW pure perfect fluid cosmology with a linear equation of state and the geometry of a pure massless scalar field are geometries that admit a proper spacetime transitive homothety group, and such spacetimes are invariant under scalings of the spacetime coordinates. This is the underlying reason why they can be represented by fixed points, but not all fixed points are associated with geometries admitting proper spacetime transitive homothety groups, as exemplified by the de Sitter spacetime. For these spacetimes homothetic scale-invariance is broken by the dimensional cosmological constant, but the 1-parameter set of de Sitter spacetimes (parameterized by ) admits a scaling (self-similar) property that scales (i.e., a scaling that maps one de Sitter spacetime to another with a different ), and it is due to this scaling property de Sitter spacetimes can appear as fixed points.
In the present case the fixed points are not in the interior physical state space, but on the unphysical boundary, but they are nevertheless characterized by e.g. the same value of as the associated physical spacetime. The fact that they in the present context appear on the unphysical boundary has consequences, which we will come back to in a discussion about the de Sitter fixed points in Section 5. Finally we point out that, apart from a set of measure zero, all solutions originate from a massless scalar field state and hence the present models are past generically massless scalar field dominated.
3.2 Asymptotic dynamics at late times
The equations on the subset, i.e. equation (24), are equivalent to that of (1c) when setting , i.e. this problem is exactly that of an anharmonic oscillator (when ; for the problem is that of a harmonic oscillator). This can be seen from eq. (24), which yields
We will now apply the approximate ideas in Mukhanov  to the present exact problem of an anharmonic oscillator. We therefore first multiply the above equation with and rewrite it as
Each periodic orbit is characterized by a constant value of and has an associated time period . The time average of a function over a period for a periodic orbit characterized by is given by
Taking the time average of eq. (30) for a periodic orbit gives
where we for notational convenience from now on drop the subscript . Again, note that in contrast to the result in , this is an exact relation on the subset . Using this result on for a periodic orbit in the definition of yields
which together with (32) leads to
on the subset , i.e., is independent of . It therefore follows that on average, in the above sense, e.g. and on correspond to dust and radiation, respectively. Note that the result (34) coincides with the approximate heuristic results using proper time given in  and ; see also  for a quite different precise definition of , which still yields (34).
Before continuing it is instructive to consider a model that consists of two perfect fluids with constant equation of state parameters and . Then (this expression follows from that , but it can also be obtained from the equation ). Since it is not difficult to show that toward the future it follows that if ; if ; if , i.e., the fluid with the softest equation of state dominates toward the future.
Assuming that asymptotically can be replaced with the asymptotic averaged result in eq. (18b) results in
which suggests Theorem 1.1, given in the Introduction, but this is of course no proof. Next we introduce averaging techniques that are subsequently used to provide the proof of this theorem.
Standard averaging techniques and theorems can be found in Chapter 4 in  (the periodic case) and in  (the general case). In standard averaging theory, a perturbation parameter plays the key role: roughly speaking, a differential equation of the form for is approximated by the averaged equation at , i.e., , where the average is defined in eq. (31). Furthermore, the error has to be controlled. In the problem at hand, we consider the differential equations in the variables and , where the role of the parameter is played by . Therefore, after setting
we have to prove an averaging theorem for the case where is not a parameter, but a variable that slowly goes to zero. The evolution equation of is given in eq. (18), which in terms of takes the form
where . This formulation is problematic due to that is not well-defined when is zero. We therefore use the following formulation:
where solves the system (7), , and
The general idea of averaging is to express as
and prove that the evolution of the variable is approximated at first order by the solution of the averaged equation. For that, consider the average as defined in eq. (31) of the right hand side of eq. (38a). More precisely, considering an equation of the form with -periodic of period , the averaged equation is given by , where . According to (34), we have , where is a constant that does not depend on . Hence the averaged equation reads
while in eq. (40) will be chosen appropriately in the proof.
Proof of Theorem 1.1
To prove Theorem 1.1 we first re-express the theorem in terms of :
If , initial conditions with positive and converge for to the outer periodic orbit with tangentially to the slice .
If , initial conditions with positive and converge for to the center with tangentially to the slice .
If , each periodic orbit on the slice attracts a 1-parameter set of trajectories with positive initial .
Let us first derive a differential equation for by taking the time derivative of eq. (40):
On the other hand,
Let us now set
Note that for large times is well approximated by periodic functions with an average . The right hand side of (44) is for large times almost periodic and has an average that is zero so that the variable is bounded.
As a consequence, the differential equation for the variable takes the form
Using the fact that and that results in the following:
Next we have to prove that the solution of this equation and the solution of the averaged equation (41) have the same asymptotics when . Since the averaged equation (41) is expected to govern the dynamics, we first study the late time behavior of the system
After Euler multiplication by (or equivalently, a singular change of time variable ), this system reads
In cases (i) and (ii), for which , the two fixed points of this system are located at and . The line is a heteroclinic orbit between these two fixed points, whose direction depends on the sign of .
Undoing the Euler multiplication does not affect the trajectories with positive . On the other hand, in the original averaged system (47), the line is a line of fixed points. Solutions with positive initial will approach the line of fixed points at , but slowly slide along this line as in the direction prescribed by the sign of , and go to the left or right fixed point accordingly.
Next we prove that the variables and follow this evolution. First note that the sequences , , defined as follows,
since goes to zero when goes to infinity. For a sufficiently small , eq. (46) guaranties that is monotone (in- or decreasing, depending on the sign of the quantity ) and bounded. Hence must have a limit when . Next we estimate , where and are trajectories with identical ‘initial’ conditions at time :