Scalar Fields, Quintessence, and the Cosmological Constant

Why Scalar Fields
Scalar-Tensor Gravitation
Equation of State for Scalar Fields
Quintessence, Dynamical Fields, and Kinky CDM

While observational evidence for a cosmological constant has slowly gathered in the last decade, theory has proceeded apace, and it is now a little passe' to talk about a plain vanilla cosmological constant. There are many related alternatives among the class of scalar fields that both arise naturally in theory and explain well the observational results.

Why scalar fields?

They are ubiquitous. In cosmology we are already used to dealing with two examples of scalar fields - the cosmological constant and the inflaton field (or fields) that drive inflation. In high energy physics unification theory they arise naturally, from theories of dynamical symmetry breaking, higher dimensional Kaluza-Klein theories, supergravity, superstring theories, etc. Two conventional examples from active research are supersymmetry theories which generically predict spin 0 (scalar) partners of spin 1/2 leptons and scalar-tensor theories of gravitation, which pop out of string theory.

A quick bit on scalar-tensor gravity

Newton's theory of gravitation is a scalar theory, so it seems natural to investigate a scalar component to gravity. Einstein showed that a tensor (spin 2) theory, general relativity, fit the observations. One knows that gravitation cannot be an odd spin theory, since those have like charges repelling, and spin 0 (scalar) theories showed no coupling to light - no deflection of starlight by the sun, for example (an important point useful below).

But there is no evidence ruling out a general scalar addition to general relativity (there is for scalar theories with constant coupling: Brans-Dicke-Jordan theories). And they arise naturally in unification theories incorporating gravitation. The Friedmann equations simply have extra terms in them from the scalar part of the action, and one can do the usual cosmological calculations. An interesting recent discovery is the property that a wide class of scalar theories dynamically approach the behavior of general relativity at the present epoch, so that what we perceive as the results of GR may be the results of any of that class of scalar-tensor theories.

An interesting point is that the very property that caused scalar theories to be abandoned in the early part of the century - nondeflection of light - has now been seen to help drive such theories to acceptability by mimicking general relativity. This is because the scalar field couples to the trace of the energy-momentum tensor, \rho - 3p, which is zero for radiation and relativistic particles. One might think therefore that the scalar field has minimal evolution during the radiation dominated era in the universe. But in fact there is a brief epoch, at electron-positron annihilation, when their energy density is nonnegligible yet they are not relativistic. Thus the trace is nonzero and acts as a source term to drive the scalar field to zero coupling, leaving only the tensor field and hence a theory that looks like general relativity.

Equation of State of Scalar Fields

The Lagrangian for a scalar field is very simple and from this one can use Noether's Theorem in the usual way to form the energy-momentum tensor and identify the density \rho and pressure p (equations; also in pdf). One finds the density is just K+V and the pressure K-V, where K = (1/2)\dot\phi^2 is the kinetic energy and V is the potential energy. In general the kinetic and potential energies evolve in time and their ratio has no special value. But it is interesting to look at three special cases.

A free field has no potential energy and hence its equation of state is \rho = p or w = p/\rho = 1. This is like shear energy or superhorizon gravitational waves. In correspondence to inflation, the evolution in this case is sometimes called kination.

In the slow roll case, the kinetic energy is negligible and the equation of state is \rho = -p or w = -1. An example is the inflationary field.

When the field undergoes coherent oscillations, then the time average of the kinetic and potential energies are equal, and the equation of state is p = 0 or w = 0. Then the scalar field acts like ordinary nonrelativistic matter. An example of a coherent oscillating field is the (pseudoscalar) axion.

N.B. Because the field, i.e. the potential and kinetic terms, enters into the Friedmann equations, one can reconstruct the field potential, the high energy physics, by measuring the equation of state of this scalar component, just as one would measure the equation of state of any other component through the cosmological tests. This is considerably easier than the process of recovering the inflationary potential - a similar Holy Grail of cosmology and high energy physics.

Quintessence and Dynamical Fields

One of the problems with the cosmological constant is its seeming unnaturalness. Since it is constant, it defines a fixed energy scale, corresponding to about an meV, vastly lower than other particle physics scales. Also, today its energy density seems comparable to that of matter, but that singles out this as a unique epoch, a cosmic coincidence. Allowing the field to be dynamical alleviates both these fine tunings.

One way is to adopt a scalar field potential that gives rise to a self tuning field, one that follows the evolution (equation of state) of the dominant component. But this fixes the value of \Omega of the scalar field, and so nucleosynthesis constraints that say it had to have been negligible at that early epoch also lead to it being negligible today.

A more flexible potential gives a field that tracks the equation of state, evolving from w = 1/3 in the radiation dominated epoch, to w = 0 about the time of matter-radiation equality, to w < 0 today and approaching w = -1 (behaving as a cosmological constant) in the future. This makes a very attractive form of dark matter (called quintessence or the fifth element after matter [earth], radiation [fire], neutrinos [air], and cold dark matter [water]). Cosmological calculations incorporating this together with cold dark matter (just as the cosmological constant has been) show that it is not ruled out; sometimes this ansatz is called QCDM and sometimes kinky CDM since it doesn't dominate until you want it to (and to match MACHOs and WIMPs).

Such potentials seem to arise naturally and are not ugly to look at: typically involving inverse powers of the field or exponentials. They solve the fine tuning problems, the equation of state today is determined by the matter density today, and observational consequences are easy to calculate, so ongoing precision measurements of the CMB anisotropy or matter power spectrum should probe the theory well.

N.B. Because the field is dynamical, the equation of state is constantly changing and so it is not a good approximation to model the field by taking some particular negative equation of state w < 0. Using the added dynamical field evolution equation of the Friedmann equations, one finds that d\ln w / d\ln a = 2 (1-K/V)-1 d\ln V / d\ln a - 12 KV/(K2-V2). This quantity will only be small in the slow roll regime, where one is back to dealing with a cosmological constant.