Numerical Simulation of Axion Quintessence
Robust, scaled cosmological equations are derived for simulating the evolution of the scalar field, the scale factor, and the Hubble parameter during both expanding and contracting phases of the universe. These scaled equations are applied to both stable (always expanding universe) and unstable axion quintessence (expanding and then collapsing universe). When applied to unstable axion quintessence, these scaled equations allow the simulations presented here to proceed much closer to the singularity at the end of a collapsing universe than any previous simulations.
Typically in quintessence theories with an asymptotically vanishing effective cosmological constant, the energy contrast in dark energy rises from near zero for redshifts to near one for , mimicking a true cosmological constant. At late times, the quintessence field may begin to oscillate about its minimum, behaving like nonrelativistic matter, or the quintessence field may evolve toward infinity—in both cases with vanishing vacuum energy. In such theories, there is a period between roughly 3.5 Gyr and 20 Gyr after the big bang when . However, if the universe continues to expand forever, or even if positive curvature begins to dominate at late times (after the quintessence field has evolved to its minimum) and the universe enters a contracting stage, this period when the energy densities of dark energy and matter are comparable is a small or vanishing fraction of the total lifetime of the universe. This is called the cosmic coincidence problem.
However in axion quintessence (as in other unstable de Sitter quintessence models), the cosmological era with may represent a significant fraction of the universe’s lifetime if the minimum of the axion potential is negative (unstable [de Sitter] axion quintessence), thus resolving  the cosmic coincidence problem. (Negative or and the fate of the universe are discussed in  plus references therein.)
The unstable axion quintessence potential , where and the Planck mass GeV, addresses the main drawbacks of quintessence models, since the facts that the minimum of the potential is at , , and but ≈−1 are interrelated aspects of the model, and they occur for an appreciable range of initial values for (the subscript “0” denotes present values).
For , the initial value of the scalar field need only satisfy to produce a universe like ours  (due to symmetry, we can restrict our attention to ). Thus, there is a significant 23% range of the possible initial values which will produce a universe like ours. (Qualitatively similar results to those presented here are obtained for for .). For these initial values, the contracting universe enters a late time era of kination (during which the scalar field kinetic energy dominates over all other forms of energy)—the negative Hubble parameter acting like a negative friction term in the Klein-Gordon equation—and the axion field makes many transits of (but never remains in) its vacuum state. (The coupling of the quintessence field to other particles must be very small and will for the most part be neglected in this investigation.)
In Section 2, the basic cosmological equations are presented for the evolution of the scalar field, the scale factor, and the Hubble parameter, and cast in the form of a scaled, dimensionless system of first-order equations in the conformal time, appropriate for a contracting (or expanding) universe. These equations allow the simulations of unstable axion quintessence presented in Section 3 (see Figures 6–9) to proceed much closer to the singularity at the end of the collapsing universe than any other simulations presented in the literature and provide the basis for a more detailed analysis of the last stages of the collapsing universe than has appeared before. We also present simulations of stable axion quintessence which produce a universe like ours—and where asymptotically saturates near for .
2. Cosmological Equations
In the quintessence/cold dark matter (QCDM) model, the total energy density , where , , and are the energy densities in (nonrelativistic) matter, radiation, and the axion quintessence scalar field , respectively. Ratios of energy densities to the critical energy density for a flat universe will be denoted by , , and , while ratios of present energy densities , , and to the present critical energy density will be denoted by , , and , respectively. will denote
Using WMAP5  central values, we will set = 0.72, , , and eV, with the present time Gyr after the big bang for CDM.
The homogeneous scalar field obeys the Klein-Gordon equation: The evolution of the universe is described by the Friedmann equations for the Hubble parameter and the scale factor : where the energy density and the pressure , with = 0, , and The curvature signature , 0, for a closed, flat, or open geometry. Equation (2.4) shows that for an accelerating universe.
The conservation of energy equation for matter, radiation, and the scalar field is Equation (2.6) gives the evolution of and , and with (2.5) the Klein-Gordon equation (2.2) for the weakly coupled scalar field. The time rate of change of the Hubble parameter is given by Only two of (2.3), (2.4), (2.6), and (2.7) are independent. We will assume a flat universe after inflation and henceforth set = 0.
The logarithmic time variable (number of e-folds of the scale factor) is defined as . Note that for de Sitter space , where , and that is a natural time variable for the era of -matter domination (see, e.g., ). We will make the simple approximations:
The equation of state parameter for the scalar field is . Since is a natural time variable for the era of -matter domination, we define the recent average of as We will take the upper limit of integration to correspond to = 1.75. The SNe Ia observations  bound the recent average (95% CL).
For numerical simulations, the cosmological equations should be put into a scaled, dimensionless form. Equations (2.2) and (2.3) can be cast  in the form of a system of two first-order equations in plus a scaled version of : where , , , , , , and . This scaling results in a set of equations that is numerically more robust, especially near and before the time of big-bang nucleosynthesis (BBN)—see .
For a contracting phase (in which goes through zero), a different set of equations and a different scaling should be used. Here, we will use the conformal time variable, where corresponds to the big bang.
Equations (2.2) and (2.7) can be cast in the form of a system of three first-order equations in : where , , , , , , , and . This scaling results in a set of numerically more robust equations, especially near the turn-around time between expanding and contracting phases of the universe.
Note that the conformal time is related to the logarithmic time by
3. Simulations of Axion Quintessence
The unstable de Sitter axion potential is based on M/string theory reduced to an effective supergravity theory , with at the maximum of .
Both axion quintessence models are derivable (up to a constant) from string theory as axion monodromy .
The quintessence axion is a pseudo-Nambu-Goldstone boson: at the perturbative level the theory is shift symmetric under with . The shift symmetry is broken—before or during inflation—by nonperturbative instanton effects to a discrete symmetry , generating a potential . In these theories, quantum corrections to the classical axion potential are suppressed. For quintessence (or for natural inflation ), ; we will take . and are the most interesting unstable axion and original axion cases, respectively.
3.1. Unstable Axion Quintessence
For the unstable axion quintessence computations (with an expanding and then contracting universe), we will use (2.12) and (2.14) with initial conditions and specified at matter-radiation equality , which corresponds to The constant in the potential is adjusted so that = 0.72. This involves the usual single fine tuning. Note that the same anthropic arguments that limit the magnitude of a present-day cosmological constant also limit , so the tuning of is no worse than the tuning of a cosmological constant.
Results for the unstable axion potential are presented in Table 1 for various and for in Figures 1–9. (As , classically , but quantum effects destabilize so that the maximum .) For the values in the table, as increases, also increases and starts to move earlier, leading to a decrease in , , , and , and correspondingly to an increase in away from . Note that, for , the coincidence time ratio approaches 50%.
The QCDM universe mimics the CDM model (see Figure 1; for clarity, only the beginnings of the contracting stage are shown in this figure) until about , after which the QCDM universe begins to decelerate and ultimately to rapidly contract to a big crunch (Figures 2 and 3).
During the contracting stage, acts as a negative friction in amplifying the axion field and its kinetic energy to bring about a late-stage kination era during which the scalar field kinetic energy dominates over all other forms of energy.
The quintessence axion is an ultra-light scalar field with , so “sits and waits” during the early evolution of the universe, and it only starts to move when (Figures 4 and 5). In this way, it is easy to satisfy the BBN (–), cosmic microwave background (CMB) (–), and large-scale structure (LSS) (–) bounds on , as in Figure 1. An ultra-light scalar field also reflects the observational evidence that the universe has only recently become dominated by dark energy.
In Figure 2, the Hubble parameter goes through zero at the turn-around time between an expanding and contracting universe. At the beginning of the contacting stage, has yet to reach the minimum of the potential energy (see Figure 6), and thus the negative Hubble parameter amplifies the kinetic energy of the scalar field, bringing about an era of kination with , as seen in Figures 4, 5, 7, and 8. Also, note that Figures 2, 4, and 5 indicate that , , and are approaching a singularity near . In fact, in an era of kination during contraction during which , , and , while (see Figure 3).
Figure 6 shows that as increases without bound, the potential energy , since a periodic function of , oscillates more and more rapidly. Depending on the strength of the coupling between the quintessence axion and other particles, may decay and populate the universe with additional radiation and matter. (The monotonically increasing field in a periodic potential can be interpreted as an oscillating field.) At the very least, there should be gravitational production of particles by during contraction.
After matter-quintessence equality at , the scalar field energy density always dominates over the matter and radiation energy densities. There is a period from until when the scalar field potential energy is comparable to its kinetic energy, and then the kinetic energy (which scales as ) predominates during the rapid contraction to a big crunch (see Figure 9).
3.2. Stable Axion Quintessence
For the stable axion quintessence computations, we use the same equations (2.12) and (2.14) with initial conditions and also specified at matter-radiation equality. Again, the constant in the potential is adjusted so that = 0.72.
Simulations of stable axion quintessence are presented in . As , a transient de Sitter universe is obtained that mimics the CDM model for a long time. Near , is beginning to evolve toward . The initial value of the scalar field needs only satisfy to produce a universe like ours . Thus, there is a significant 52% range of the possible initial values which will produce a universe like ours.
For , stable axion quintessence not only produces a universe like ours, but also asymptotically saturates near (Figure 10).
In the unstable axion quintessence case, at late times, the contracting universe enters an era of kination in which , , and . The density and pressure during this era scale as . The singularity behavior is analogous to the Type III singularity of [11, 12] in which as , , , and , except here (quantum effects though should stabilize near ).
As the universe contracts, density inhomogeneities are amplified and presumably black holes are formed, similarly to the contracting stage of the ekpyrotic universe  with . Depending on the strength of the coupling (which we have neglected here) between the quintessence axion and other particles, may decay and produce radiation and matter. There should at least be gravitational production of particles by during contraction. As the universe reheats during contraction, broken symmetries are restored. It is possible that inflating patches may be generated, spawning new universes from the old.
In summary, stable axion quintessence can produce a universe like ours, in which for , asymptotically saturates near . The unstable axion quintessence potential resolves the main drawbacks of quintessence: the minimum of the potential is not at zero, but at a negative value , with only a single fine tuning (in the anthropic range), and naturally satisfies , for an appreciable 23% range of possible initial values for the quintessence field. And for a universe like ours, the coincidence time when the energy densities of dark energy and matter are comparable varies (as long as is not too small—say, 0.05) from 25%–50% of the lifetime of the universe.
The author's thanks are due to Lawrence Krauss for valuable comments.
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