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In the absence of a symmetry in Nature which would set the value of the cosmological constant to precisely zero, one is forced to either set Lambda = 0 by hand, or else look for mechanisms that can generate Lambda = Lambdaobs > 0, where Lambdaobs ~ 10-29 g cm-3 is the value of the Lambda-term inferred from recent supernovae observations. We have discussed several mechanisms which could, in principle, give rise either to a time independent cosmological constant, or else a time dependent Lambda-term. To the former category primarily belong models which associate Lambda with a property of the vacuum such as the vacuum energy associated with symmetry breaking, or vacuum polarization and particle production effects in curved space-time. Mechanisms predicting a time dependent Lambda take their cue from Inflation and generate a time varying Lambda out of scalar fields rolling down a potential. Models with a fixed Lambda run into fine-tuning problems since the ratio of the energy density in Lambda to that of matter/radiation must be tuned to better than one part in 1060 during the early universe in order that Lambda / 8pi G appeq today. Scalar field models considerably alleviate this problem though some fine-tuning does remain in determining the `correct choice' of parameters in the scalar field potential.

It has been known for several years that the flat FRW LambdaCDM cosmological model with an approximately flat spectrum of initial adiabatic perturbations fits observational data better and has a larger admissible region of parameters (H0, Omegam) than any other cosmological model with both inflationary and non-inflationary initial conditions (see, e.g., [113, 114, 144, 184, 5]). For instance according to a typical expert opinion made several years ago "for H0 > 60 kms-1Mpc-1, this model is probably the only feasable model" [184]). Now, with new data on high redshift type Ia supernovae becoming available, we are closer than ever to concluding that this is the right cosmological model (at least to a first approximation) even if H0 < 60. Moreover, using type 1a supernovae data and with improved data on gravitational clustering at high redshifts soon expected, we may progress further and investigate whether Lambda depends weakly on time.

Turning to the observational situation, constraints on the cosmic equation of state arise from observations at: low redshifts (age of universe, cluster abundances, baryon fraction, velocity fields, etc.), intermediate redshifts (ages of distant galaxies & QSO's, angular size vs. redshift, gravitational lensing, Type 1a supernovae, the Lyman alpha forest etc.) and high redshifts (cosmic microwave background). Each set of observations has its own systematic errors and although considerable progress has been made in trying to understand systematics it is safe to say that at any given time at least one set of observations is likely to be well off the mark !

Of the low redshift tests, the age of the universe, cluster abundances and baryon fraction all appear to favour a low density universe, with Omegam ltapprox 0.3 in clustered matter. A tone of dissonance is however provided by recent observations of the angular size of compact radio sources which seem to suggest a critical density matter dominated universe, although evolutionary effects clearly need to be better understood before a strong case for Omegam appeq 1 is made based on these results alone.

The strongest support for an accelerating universe comes from intermediate redshift results for Type 1a supernovae. At the time of writing close to a hundred supernovae have been analyzed by two teams: The Supernova Cosmology Project and the High-Z Supernova Search Team, both teams getting mutually consistent results for {Omegam, OmegaLambda}. It should be pointed out that the supernovae results do not by themselves pick out a flat universe from other possibilities; a cursory look at fig. (6) shows that a closed universe with Omegam + OmegaLambda > 1 appears preferred although a flat universe is also accommodated by current observations. However the combined likelihood analysis of Sn1a + CMB observations strongly supports a flat universe with Omegam + OmegaLambda appeq 1, primarily due to the presence of a Doppler peak in the CMB data at intermediate angular scales theta ~ 1°. Thus although observations do seem to suggest that the universe may be spatially flat with a large fraction of its density in the form of a cosmological Lambda-term, it may be premature to rule out, on the basis of current data alone, models that are spatially open or even matter dominated and flat.

Great progress is however expected on the observational front in the coming 5 - 10 years. Conservative estimates suggest that one should expect over ~ 50 new Type 1a events to be added to the supernovae inventory every year (including several at significantly higher redshifts than z ~ 1). Thus by the time of the launch of the MAP and PLANCK satellites (during 2001 & 2007 respectively) one would expect our understanding of supernovae related parameter estimation to have improved by over an order of magnitude. Since both MAP and PLANCK missions are expected to pinpoint the location and amplitude of the first Doppler peak at the level of a few percent accuracy, they should provide a decisive answer to the question of whether or not we live in a critical density universe. The definitive answer to the question of whether the universe is flat and accelerating may therefore have to wait just a few more years !


The authors acknowledge stimulating discussions with Somak Raychaudhuri and Tarun Deep Saini. They also thank Neta Bahcall, Richard Ellis, John Peacock, Saul Perlmutter and Max Tegmark for generously supplying some of the figures shown in this manuscript. VS acknowledges support from the Indo-Russian Integrated Long Term Programme of cooperation in science and technology (ILTP). AS was partially supported by the Russian Research Project "Cosmomicrophysics". This review was finished during the visit of AS to the Institute of Theoretical Physics, ETH, Zurich.

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