Why is the Universe accelerating?

Beverly Berger, Oakland University
berger@oakland.edu

One of the most important discoveries of the late 20th century was the evidence from Type Ia supernovae (SNe-Ia) that the expansion of the Universe is accelerating [1]. If this result holds up, it will have fundamental significance for gravitation and cosmology.

Of course, the historically honored mechanism for the gravitational
repulsion needed to provide the acceleration is the cosmological
constant, , originally proposed (and then retracted) by
Einstein. A competing explanation, originally proposed by Caldwell * et
al.* [2], is quintessence, a mechanism to obtain a time dependent
cosmological constant with a scalar field and a potential
. However, quintessence models have a large number of
adjustable parameters and are * ad hoc* in the sense that there is no
underlying quantum theory for . Both the cosmological constant
and quintessence are added to ``standard'' cold dark matter (CDM)
Friedmann-Robertson-Walker (FRW) cosmologies. With suitable adjustments
of parameters, both CDM and QCDM models can fit the SNe-Ia
data.

Another important and recently discovered constraint on cosmological models is the angular dependence of the cosmic microwave background fluctuations (CMBF) initially detected by COBE and more accurately extended to smaller angular scales by BOOMERANG and MAXIMA [3]. Assuming the scale invariant spectrum of initial adiabatic fluctuations and flat spatial geometry of inflationary models, a series of peaks are predicted to occur in the CMBF data. The detailed predictions of the CDM and QCDM models are remarkably consistent with the observations.

To many in gravitational physics, however, the cosmological constant is
repulsive in more than one way. We should therefore welcome a scenario
proposed by Leonard Parker and Alpan Raval (PR) of the University of
Wisconsin at Milwaukee. The PR scenario agrees at least equally well with
both the SNe-Ia and CMBF data as the CDM and QCDM models with no
more adjustable parameters than CDM, need not be ``fine-tuned''
to have the desired properties, and is, in several ways, less * ad
hoc* than its competitors. They call their scenario the VCDM model since
the vacuum energy of a quantized scalar field provides negative pressure
to accelerate the Universe.

The PR proposal, described in detail in [4, 5], adds a non-minimally coupled ultra-low-mass free scalar field to (e.g.) the FRW-CDM model. The required mass eV might be reasonable for a pseudo-Nambu-Goldstone boson or even for the graviton. Given such a scalar field, , standard techniques of quantum field theory in curved spacetime may be used to construct the effective action for the scalar field coupled to gravity by integrating out the quantum fluctuations of the scalar field. PR discovered that it is possible to perform a non-perturbative (i.e. infinite number of terms) sum of all terms in the propagator with at least one factor of the scalar curvature . It is the non-perturbative effects which become dynamically important on gigayear timescales when where indicates conformal coupling and is minimal coupling and is the mass of the scalar field. The single parameter replaces in the VCDM scenario. In [5], PR give a solution for the FRW scale factor for a VCDM model containing vacuum energy, nonrelativistic matter, and radiation. It is close to power law in for (as expected for zero cosmological constant) and close to exponential in if (as expected for non-zero cosmological constant). The non-perturbative vacuum energy effects cause the transition at when the pressureless matter density at , . With this solution, it is possible to construct the equation of state for the vacuum from the Einstein tensor. The ultra-low-mass gives transition times corresponding to cosmological redshifts of .

Due to the mass scale, the non-perturbative effects are dynamically negligible in the early universe. (In [4c], PR point out that particle creation--which is part of the non-perturbative effective action--might be used to solve some current problems with the inflationary scenario.) At late times, with the transition time controlled by the mass of the scalar field, the vacuum energy dominates the dynamics. Vacuum energy typically violates the energy conditions--in this case with a negative pressure (but positive energy density). The FRW scale factor responds as if there were a cosmological constant. Since for the present value of the Hubble parameter, the effective cosmological constant turns on very late in the history of the universe.

Things to note about the PR scenario compared to the competition are that
given the existence of such an ultra-low-mass scalar field, there are no
more * ad hoc* assumptions. The VCDM equation of state results
from a wide range of values of . The current fits to SNe-Ia and
CMBF data by PR use preferred values of cosmological parameters
(
,
, and for
the CDM and baryon fractions and the current Hubble parameter in units of
100 km/s/Mpc). Should these change, it is still probable that a value of
can be found to fit the data.

PR compare their VCDM model to a CDM model with the same cosmological parameters. The fits to the CMPF data are almost indistinguishable. This is not true for the SNe-Ia data which has information only back to . The VCDM model predicts significantly fainter supernovae near than does the CDM model. The VCDM prediction seems to follow the trend in the data more closely. However, the current quality of the data does not allow either model to be ruled out. Improved data might be able to choose between these models.

Caldwell has recently argued [6] that the key ingredient in the dependence of luminosity on redshift is , the ratio of vacuum pressure to vacuum energy density. Quintessence models appear to require while PR's VCDM scenario yields . Caldwell demonstrated that the latter behavior yields better fits to the SNe-Ia data but could only generate a very contrived model with that property. The PR scenario yields this behavior naturally.

Detailed properties of the PR scalar field scenario may be found in [4]
and references therein. Graphs showing the comparisons to the SNe-Ia and
CMBF data may be found in a very recent * Physical Review Letter* [5].
An even more recent discussion of this work as a plausible explanation of
the accelerating universe may be found in * Nature Science Update* [7].

Leonard Parker and Alpan Raval have discovered a scenario which might represent the first observation of an effect predicted using quantum field theory in curved spacetime as well as a new quantized field. In their own words [5]: ``If the universe is indeed acting, through its own acceleration, as a detector of this very low mass quantized field, then there would be a wealth of implications for particle physics and cosmology.''

** References:**

[1] S. Perlmutter * et al.*, Nature ** 391**, 51 (1998); A. Riess *
et al.*, Astron. J. ** 116** 1009 (1998).

[2] R.R. Caldwell * et al.*, Phys. Rev. Lett. ** 80**, 1582 (1998).

[3] P. de Bernardis * et al.*, Nature ** 404**, 955 (2000); S. Hanany
* et al.*, Astrophys. J. Lett. * 545*, 5 (2000).

[4] L. Parker and A. Raval, Phys. Rev. D ** 60**, 063512 (1999); **
60**, 123502 (1999); ** 62**, 083503 (2000).

[5] L. Parker and A. Raval, Phys. Rev. Lett. ** 86**, 749 (2001).

[6] R.R. Caldwell, astro-ph/9908168.

[7] Philip Ball, Nature Science Update, http://www.nature.com/nsu/010208/010208-2.html.