Acceleration of the Universe from
Heat Kernel Asymptotics

                         Leonard Parker
                 University of Wisconsin-Milwaukee

This talk is mostly about recent work done with Daniel A. T. Vanzella

References and work in progress:
1. L.P. and Alpan Raval, "Nonperturbative effects of vacuum energy on the recent
expansion of the universe," Phys Rev D60, 063512 (Aug 1999), 21 pages.
2. L.P. and Alpan Raval, "Vacuum effects of ultralow mass particle account for the
recent acceleration of the universe," Phys Rev D60, 123502 (Nov 1999),  8 pages.
3. L.P. and Alpan Raval, "New quantum aspects of a vacuum dominated universe,"
Phys Rev D62, 083503 (Sep 2000),  12 pages.
4. L.P. and Alpan Raval, "A new look at the accelerating universe,"
Phys Rev Lett 86(5), 749-752 (29 Jan 2001).
5.  L.P., W. Komp, and D. A. T.Vanzella, Cosmological AccelerationThrough Transition to Constant Scalar Curvature  ApJ. 588, 663-673 (2003); astro-ph/0206488, and paper in preparation.
6.  L. P. and D. A. T. Vanzella, "Acceleration of the universe and the large-time form of the heat kernel," in preparation.
7. R. R. Caldwell, L. P., and D. A. T. Vanzella, work in progress on perturbations.

Motivation

Successes of Quantum Fields in Curved Spacetime

Particle Creation by the Expanding Universe

Mechanism is parametric amplification of vacuum fluctuations of a quantized field. With inflation, this appears to be responsible for the temperature fluctuations now observed in the cosmic microwave background radiation (CMBR).

Particle Creation by Black Holes (Hawking effect)

Black hole radiation is necessary for the second law of thermodynamics to be extended to systems that include black holes and determines their temperature T and entropy S. For a spherical black hole T = 1/(8 π G M), and S = (1/4) A, where M is the mass and A is the area of the black hole.

These successes lead one to ask if quantum field theory in curved spacetime may help us to understand the recent acceleration of the expansion of the universe. Consider the simplest quantum field, namely, a free scalar field in curved spacetime.

Discussion of the Theory

Free Quantized Scalar Field

The field equation is the simplest curved spacetime generalization of the Klein-Gordon equation ,

where R is the scalar curvature of the spacetime and ξ is a dimensionless number that we can take to be zero (minimal coupling). The metric of the gravitational field appears in the covariant derivatives operating on φ. The field φ has no interactions with other fields, apart from that with the gravitational field.

Following Schwinger, DeWitt, Feynman and others, one can write an effective action W that includes the classical Einstein action plus additional terms that embody the quantum effects of the free scalar field. These additional terms involve the curvature, and the vacuum expectation value of the scalar field. In order to avoid introducing additional variables into the action, let us take the vacuum expectation value of φ to be zero.

As in quantum electrodynamics, the infinities that appear in the effective action are absorbed through renormalization of the values of various constants that appear in this action. For example, as Zeldovich pointed out, the infinite zero-point energy of φ gives rise to a term in the effective action that has the same form as a cosmological constant. Therefore, this infinity is absorbed into the value Λ of the cosmological constant, which is then treated as a finite, but arbitrary, constant. In curved spacetime, there are additional infinities that Utiyama and DeWitt showed could be absorbed into the values of the coefficients of terms in the effective action that are quadratic in the Riemann curvature tensor. Again, the values of the resulting finite coefficients are arbitrary. The often stated problem with the value of the cosmological constant that is needed to give the observed recent acceleration of the universe is that there is no explanation of its positive sign or of its unexpectedly small magnitude.

In addition to these undetermined renormalized constants, such as Λ, there are well-defined finite terms in the effective action resulting from the quantum effects of the scalar field. These finite terms depend on the curvature tensor, and have values that are determined by , or more precisely, by a single parameter that is formed from and a function of the dimensionless parameter ξ that appeared in the Klein-Gordon equation. In order to see the effects of these well-determined curvature terms in the effective action, let us set the values of Λ and the other renormalized arbitrary constants to zero.

The surprise is that these well-determined finite curvature terms necessarily give rise to an acceleration of the universe. Changing the value of  affects the magnitude, but not the sign, of the acceleration. In contrast, the renormalized cosmological constant need not cause acceleration because it can have either sign. Not surprisingly, the value of the particle mass m required to give the observed recent acceleration of the universe is quite small, like that of Λ. However, for the free scalar field the value of m, unlike that of Λ, requires no renormalization and therefore retains whatever value is assigned to it in the bare Lagrangian.

How can a free scalar field in its vacuum state accelerate the universe?

In a nutshell, the vacuum fluctuations of the scalar field become large enough to produce a significant negative pressure when the universe has expanded to a size comparable to the Compton wavelength of the particle of mass m associated with the quantum field φ. The negative pressure causes the expansion of the universe to accelerate in such a way as to prevent the scalar curvature of the FRW spacetime from decreasing any further. The acceleration does not begin until the expansion of the universe has slowed down enough for the scalar curvature to fall to a value close to . If the Compton wavelength of the particle is sufficiently large, the acceleration would begin at relatively late cosmological time, when the universe was in its matter-dominated stage of expansion.

Let me describe schematically how this result is obtained. We follow the effective action method developed by Schwinger and others in quantum electrodynamics and generalized to curved spacetime by DeWitt. The action S for an uncharged free scalar field in curved spacetime can be written in the form

with

The quantum effects of the field φ can be found by doing a Feynman integral over the possible configurations of the scalar field that go between the early- and late-time vacuum states. The effective action W that takes into account these possible fluctuations of the scalar field is defined by

The effective action W will not depend on φ because it has been integrated over all configurations. However, it will depend on the metric of the spacetime because the action S of the scalar field contains the metric. Suppose that the initial vacuum state differs from the final vacuum state only by a phase factor, as occurs when one can neglect the production of particles by the expanding universe (which is true in the present stage of the expansion). Then the expectation value of the energy-momentum tensor of the scalar field in a given spacetime metric is obtained by variation of the effective action with respect to the metric:

The metric itself is then determined self-consistently as the solution of the Einstein gravitational field equations:

When we specialize to the metric of an FRW universe, will be the energy-momentum tensor of the radiation and matter (including dark matter) present in the universe. To find we have to calculate the effective action W and then take its variation with respect to the metric. I am skipping over details of renormalization, but we will assume here that the expectation has been renormalized. Furthermore, the value of the cosmological constant Λ will be set to zero, in order to see what effects would occur without the presence of a nonzero Λ.

Solar System Tests

In gravitationally bound regions of the universe, such as the solar system, galaxies and clusters of galaxies, where the average matter density has not fallen to a value for which the scalar curvature R is of the order of , the transition that causes the vacuum energy and (negative) pressure to become large never takes place. Thus, in those regions the vacuum terms are negligible and standard general relativity holds. Also, the theory has a well defined R→0 limit.

Calculation of W

Because the action S of the free scalar field is quadratic in φ, the Feynman integral over configurations of φ reduces to a Gaussian integral, which gives (to within a constant normalization factor that does not affect the variation of W):

Here μ is an arbitrary constant of dimension mass that will not appear in the effective action after renormalization. It follows that

This expression is formally infinite, but it is possible to absorb the infinities by means of renormalization. Here we use the method of zeta-function regularization, developed for curved spacetime by Dowker and Hawking.

The generalized zeta function is defined by

Then

Comparing this with the previous expression for W, one finds that

The generalized ζ function can be regularized by analytic continuation in the parameter ν, so that it has a well-defined finite value at ν = 0. This serves to regularize W. The dependence on the arbitrary constant μ can be absorbed into the definition of the renormalized constants such as G and Λ that appear in Einstein equations. As is well known, additional terms quadratic in the curvature tensor must be added to the Einstein equations in order to absorb all the dependence on μ. (In dimensional regularization there are terms which would be infinite at dimension 4 that are also absorbed through renormalization, giving the same end result).

It is convenient to introduce an integral representation of the generalized ζ function:

where H is regarded as having a small negative imaginary part to make the integrand vanish as s approaches ∞.

The operator, , looks like the Hamiltonian of a nonrelativistic particle moving on a curved spatial hypersurface of 4 dimensions (having coordinates ). The operator may be regarded as the quantum mechanical evolution operator giving the time-development of the state of this fictitious 4-dimensional particle on the hypersurface with s being the "time" in which this 4-dimensional particle evolves.  The parameter s is called the proper time parameter, following Schwinger who introduced this idea in the context of  quantum electrodynamics. Then we can represent the trace as an integral over a complete set of position eigenstates |x> of the particle on the 4-dimensional hypersurface. Thus,

Thus, the problem of finding the effective action for the quantized field φ in an external gravitational field has been reduced to the problem of solving a Schrodinger equation for a fictitious particle moving on a four-dimensional curved surface.

The probability amplitude, <, | , 0>, for the particle to go from a point at proper time 0 to point at proper time , satisfies the Schrodinger equation,

The boundary condition is that <x, s|x', 0> = δ(x, x'). The quantity that appears in the ζ function above, is the coincidence limit:

If one makes the replacement, s → i s, in the Schrodinger equation, it becomes the equation governing heat propagation, with s the time. The amplitude, <x,s|x',0>, is the analytic continuation of the heat kernel.

In a general curved spacetime, DeWitt expanded <x,s|x',0> as a power series in s and used the Schrodinger equation to set up iterative equations giving the coefficients in the power series. These coefficients depend on quantities involving the Riemann tensor and the proper separation of the points x and x'. In the coincidence limit, he was able to calculate explicit expressions for the coefficients up to order The coefficient of in the proper time series for is quadratic in contractions of the Riemann tensor. The coefficients of higher powers of s involve successively higher powers of the Riemann tensor and have only been calculated up to order to date.

In 1981 Beckenstein and I used the Feynman path integral solution of the Schrodinger equation with Fermi coordinates along the dominant path of the fictitious 4-dimensional particle to show that, in Gaussian approximation, a factor of

                                  exp(-i s )

with

appears in the Gaussian approximation for .

Then in 1985, I, David Toms and Ian Jack were able to prove by induction that the proper-time series for has the following form in a general spacetime of arbitrary dimension D:

where the series expansion of in powers of s has no terms that involve any factors of the undifferentiated scalar curvature R(x). We will refer to this expression as the R-summed form of the amplitude (or heat kernel).

Thus, the factor of appears to contain highly nonlinear effects of the gravitational field. Because the series for contains in general no factors of R, it is unlikely that the exponential involving R can be cancelled in general by summing terms in . Therefore, it is reasonable to take seriously the exponential factor, , and to see what physical consequences such a term would have.

Motivated by the unexpected evidence for an accelerating universe that was beginning to come in from observers of supernovae in 1998, I and Raval decided to study this question in a general FRW cosmological spacetime. The relatively recent onset of the acceleration suggests considering a free scalar particle having a mass m that is not too much larger (in units with c = h = 1) than the present value of the Hubble constant, . We used an approximation for the heat kernel or amplitude that was based on the first two terms of the R-summed form of the amplitude. When applied to the FRW universe this implied a transition to constant scalar curvature, R, which would cause an acceleration of the expansion of the universe.

Strenghtening the basis of the theory

The transition to constant R is traceable to the exponential term in the heat kernel that involves R s. Furthermore, it comes from the upper limit of the integration over s in the expression for the effective action. The proper-time parameter s has dimensions of , and the integration over large s is related to integration over long wavelength quantum fluctuations of φ in the original path integral. Thus, the transition is related to the large-time asymptotic form of the heat kernel. This is a manifestly nonperturbative effect. One may therefore object that the argument for the crucial exponential involving R is based on summing a well-defined infinite series of terms in an asymptotic series for the heat kernel, but that series necessarily will not include the contributions of terms in the exact heat kernel that cannot be expanded in powers of s. This objection does not apply to the derivation of the exponential term given by Bekenstein and me, but that is based on a Gaussian approximation to the path integral solution of the Schrodinger equation, so further consideration is justified.

In our paper in preparation, Daniel Vanzella and I have taken a different approach. We have studied the large-s asymptotic form of the heat kernel, or amplitude, for a number of FRW universes where the exact heat kernel is known. The exact solutions we studied were deSitter spacetime, the Einstein static universe, and the spatially flat expansion that is linear in the FRW cosmic time. In all cases, we indeed find an exponential involving R s in the asymptotic form of the heat kernel, so one can expect the kind of behavior of the energy-momentum tensor that was obtained earlier on the basis of the previous approximations. However, in the previous work we had a factor of (ξ - 1/6) R s in the exponential. We find that the factor in front of R in the asymptotic form may differ from (ξ - 1/6), as it does in deSitter spacetime. In addition, there is a factor involving powers of contractions of the Riemann tensor that multiplies the exponential, similar to the factor used by me and Raval in our approximation to the heat kernel for large s. However, we cannot prove, on the basis only of the known exact solutions, that this factor is exactly the one that we used in our previous calculation. Numerical work seems to show that only certain features, such as the signs of various terms quadratic in the Riemann tensor, are important for getting the transition to constant R and for avoiding runaway solutions in the domain of physically relevant initial conditions.

Motivated by the large-s asymptotic forms of the exactly known heat kernels that we considered, we find the following to be a reasonable ansatz for the large-s asymptotic form in a 4-dimensional spacetime:

where is an invariant quadratic in contractions of the Riemann tensor that satisfies certain conditions, and χ is a dimensionless constant that depends on the choice of ξ in the field equation for φ. (In our paper, we also deal with a somewhat more general ansatz.) The effect of cosmological interest comes from the exponential terms that involve the quantity:

Note that χ has replaced (ξ - 1/6) in the definition of M(x).

We show that this asymptotic form for large s is responsible for the terms such as

that cause to become large when χ has the right sign to allow the denominator to become small. The relevant parameter for the cosmological evolution is then the ratio

We also generalized  the previous work by including a small phenomenological dissipative term that has the effect of preventing   from becoming arbitrarily large. To do numerical integration of the Einstein equations we chose one of the possible forms for the quantity , namely:

where α is a dimensionless number of order 1.

from variation of W

Variation of the effective action W gives the following vacuum contribution to the energy-momentum tensor:

where the first term is regular and remains small, while the second term is the one that causes the transition to an accelerating expansion. The quantity that appears above is defined by

where ε is a small finite parameter coming from possible dissipative effects. The parameter χ is written as in the above expression.

The coefficients are expressions quadratic in Riemann tensors, and containing higher derivatives of the metric. For example,

Here is the same as M that was defined earlier, involving χ, and is ξ - .

Solving the Einstein equations in an FRW universe

To see the effect of these curvature terms in , we set Λ = 0 and solve the Einstein equations in the FRW universe containing classical matter (including the dark matter) and radiation, in addition to the vacuum pressure and energy of this quantized scalar field φ. We call this cosmological model the "vacuum cold dark matter model," or VCDM. The transition to constant scalar curvature caused by the change in the magnitude of the vacuum energy-momentum tensor, we refer to as "vacuum metamorphosis."  The very low mass scalar particle causing the vacuum to change its nature and initiate the transition to the accelerating stage of the expansion, we call the acceletron. Because vacuum metamorphosis is triggered in this model when the vacuum energy density rapidly grows to about the magnitude of the ambient matter density, it is no coincidence in this model that the vacuum and matter energy densities are roughly of the same magnitude today, but with the vacuum energy density being the larger of the two.

We assume that the universe follows the standard cosmological scenario in its early history, including an inflationary stage that makes the expanding space nearly flat, then a radiation-dominated stage which passes into a matter-dominated stage. In the analytic work with Raval, we neglected derivatives of curvature tensors, but during the rapid growth of vacuum energy and pressure, the higher derivative terms may be significant. Therefore, in the present work, Vanzella and I face the higher-derivative dragon and carry out a detailed numerical integration. We include the higher-derivative terms and carefully take into account possible runaway solutions.

We find that the transition to an expansion of the universe with constant scalar curvature does occur, much as in the analytic solution. There are runaway solutions if one takes initial conditions that are far from the classical solution before vacuum metamorphosis occurs. However, all solutions that start in the past along the classical trajectory do make a transition to the constant scalar curvature form of the expansion. Thus, none of the physically relevant solutions run away. All physically relevant solutions do make a transition to the expansion having constant scalar curvature R.

Here are some representative graphs. The first is a graph of R versus . The dotted straight line is the classical solution in the matter-dominated stage. The oscillatory behavior that takes place during the transition to constant R would actually be less pronounced. In the numerical integration shown, we used a larger value of than the realistic value dictated by the CMBR and supernovae data. We carried out the integrations for a sequence of decreasing values of and found that the transition to constant R is a robust effect that occurs with smaller oscillations as the parameter is decreased. We also mapped out the solution trajectories in a suitable phase space and found that no runaway solutions occurred for initial conditions starting in the classical matter-dominated stage of the expansion prior to vacuum metamorphosis.

Next are the corresponding graphs for a as a function of t and of H as a function of the redshift z. The numerical solutions are very close to the earlier analytic results in which the transition to constant R is instantaneous, with continuous first and second derivatives of the metric at transition.

Predictions of the VCDM model

During the matter-dominated stage the scalar curvature

is decreasing as the universe expands. Here, a(t) is the cosmological scale factor appearing in the spatially-flat FRW line element

When R approaches the value , the vacuum energy density and pressure of the quantum field φ become comparable to the classical energy density and pressure and have a significant effect on the evolution of a(t). We find that the solution a(t) of the Einstein equations repidly approaches a form in which the scalar curvature R becomes constant at a value slightly larger than . The plots below are given for the analytic solution.

Here is a plot of the scale factor a, total energy density ρ, vacuum energy density , vacuum pressure p, and scalar curvature R as functions of t during a period before and after the transition to constant R. It is the negative pressure of the vacuum that drives the acceleration. You can also see that the vacuum energy density and vacuum pressure are approaching the same magnitude at late times. The transition time in the diagram occurs at t = 1, and the present time is somewhat less than t = 3. The energy density scale is chosen so that the total energy density at the time of transition has the value 1. Notice that the vacuum energy density and pressure are essentially 0 before the transition.

Here is a graph showing the vacuum energy density and the vacuum pressure , followed by a graph of the ratio w of the vacuum pressure to the vacuum energy density.

The value of w is quite negative just after the transition at t = 1 because is much smaller than . However, the ratio approaches -1 as t increases.

Comparison with Observation

Recent CMBR and SNe-Ia Data

This is work in progress with William Komp, me and Vanzella. We find that the predicted curves of the VCDM model agree with recent results on the CMBR from the WMAP project.  Concerning the fit to the CMBR data, one interesting fact is that the best fit to the data using the VCDM model implies a primordial power spectrum of fluctuations that is very close to the predictions of the standard inflationary models. The ΛCDM model is not as close in this regard.

The VCDM model also is in agreement with the recent results from the High-Z Supernova Team and the Supernovae Cosmology Project. Concerning the supenovae data, I will just show the following graph, which includes the recent 2003 observations.

The red curves correspond to the VCDM model with the solid being the most likely value for and the upper and lower dashed curves being the lower and upper limits corresponding to a 2σ uncertainty. The blue solid and dotted curves are the corresponding curves for the ΛCDM model.

Finally, work is in progress with Robert Caldwell on perturbations in the VCDM model with the aim of finding the prediction of the VCDM model for the low l CMBR power spectrum. It is necessary to take into account the perturbations in the vacuum energy and pressure after the transition in doing this. Our preliminary results show a suppression of the CMBR power spectrum at low l. If this preliminary result holds up to further scrutiny, then the VCDM model may explain the observed suppression of the power spectrum at low l.

Created by Mathematica  (November 26, 2003)