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, $\Lambda$, 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 $\phi_Q$ and a potential $V(\phi_Q)$. 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 $\phi_Q$. 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 $\Lambda$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 $\Lambda$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 $\Lambda$CDM and QCDM models with no more adjustable parameters than $\Lambda$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 $m \approx 10^{-33}$ eV might be reasonable for a pseudo-Nambu-Goldstone boson or even for the graviton. Given such a scalar field, $\varphi$, 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 $R$. It is the non-perturbative effects which become dynamically important on gigayear timescales when $R \approx m^2/(- \bar
\xi)$ where $\bar \xi = 0$ indicates conformal coupling and $\bar \xi =
1/6$ is minimal coupling and $m$ is the mass of the scalar field. The single parameter $\bar m \equiv
m/\sqrt{\bar \xi}$ replaces $\Lambda$ in the VCDM scenario. In [5], PR give a solution for the FRW scale factor $a(t)/a(t_j)$ for a VCDM model containing vacuum energy, nonrelativistic matter, and radiation. It is close to power law in $t$ for $t < t_j$ (as expected for zero cosmological constant) and close to exponential in $t$ if $t > t_j$ (as expected for non-zero cosmological constant). The non-perturbative vacuum energy effects cause the transition at $t_j$ when the pressureless matter density at $t_j$, $\rho_j = \bar m^2 / (8 \pi G)$. 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 $z \approx 1$.

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 $t_j \approx H_0^{-1}/2$ for $H_0$ 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 $\bar m$. The current fits to SNe-Ia and CMBF data by PR use preferred values of cosmological parameters ( $\Omega_{\rm CDM} = 0.50$, $\Omega_{\rm B} = 0.06$, and $h = 0.7$ 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 $\bar m$ can be found to fit the data.

PR compare their VCDM model to a $\Lambda$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 $z \approx 1$. The VCDM model predicts significantly fainter supernovae near $z \approx 1$ than does the $\Lambda$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 $w$, the ratio of vacuum pressure to vacuum energy density. Quintessence models appear to require $-1 < w <
0$ while PR's VCDM scenario yields $w < -1$. 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.''


[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.

Jorge Pullin