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The problem of the hierarchy of energy scales in particle physics is exacerbated by this reasoning. Consider the problem of the energy scale of L in another way. In order for the constant energy density to be dominant today, it must have been a negligible fraction of the total energy density of the Universe at all times in the past. It's difficult then to see how such an energy component could have ever been in equilibrium or contact with the rest of the cosmic fluid.

As we will argue later, perhaps this component did not always have a constant, or nearly constant energy density. By the principle of Occam's Razor, the simplest explanation for the missing energy component may be the cosmological constant. After all, there is just one number to specify, L itself.

Occam's Razor, however, is not a law of physics, and if too broadly applied it can obscure the interpretation of experiment. As we argue in the next section, a dynamical component is a logical candidate for the missing energy. The proposal of a cosmic scalar field, quintessence, strikes this author as the most logical way to model the missing energy, at present. The most basic tool employed to build a fundamental theory beyond the standard model of particle physics or Einstein gravity is the scalar field.

The Higgs, inflaton, Brans-Dicke, moduli, and dilaton fields are examples of such scalar fields which play a critical role in models of fundamental physics.

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Furthermore, there is precedent to solving "missing energy" problems with a new particle or field, as was the case with the neutrino a success and dark matter to be determined, but supersymmetric particles are strong candidates. In addition, the wide range of behavior encompassed by a scalar field provides a greater context in which to explore and understand the developing cosmological observations.

As well, with a scalar field, it is simple to model the behavior of other forms of energy, such as a network of frustrated topological defects [27]. An important motivation for considering quintessence models is to address the "coincidence problem," the issue of explaining the initial conditions necessary to yield the near-coincidence of the densities of matter and quintessence today.

Symmetry arguments from particle physics are sometimes invoked to explain why the cosmological constant should be zero [28] but there is no known explanation for a positive, observable vacuum density. For quintessence, because it can couple to other forms of energy either directly or gravitationally, one can envisage the possibility of interactions which cause the quintessence component to naturally adjust itself to be comparable to the matter density today. In fact, recent investigations [9, 29] have introduced the notion of "tracker field" models of quintessence which have attractor like solutions [30, 31] which produce the current quintessence energy density without the fine tuning of initial conditions.

Particle physics theories with dynamical symmetry breaking or non-perturbative effects have been found which generate potentials with ultra-light masses which support negative pressure, and exhibit the "tracker" behavior [13, 32]. These suggestive results lend appeal to a particle physics basis for quintessence, as a logical alternative to an ad hoc invocation of a cosmological constant.

The quintessence plus cold dark matter QCDM cosmological scenario is constructed as follows. The cosmic fluid contains quintessence, CDM, plus all the standard model particles in the form of baryons, radiation, and neutrinos. The Universe evolves from an inflationary phase, during which time a spectrum of adiabatic density perturbations are generated, through radiation- and matter-dominated phases, until the present quintessence-dominated era.

The evolution of a cosmic scalar field is obtained by the following set of equations. Starting from the Lagrangian for a self-interacting scalar field, the field is broken into a background, homogeneous portion Q and an inhomogeneous perturbation d Q. The equations of motion for the background field in an expanding FRW space-time are simply obtained:. Here, the prime indicates the derivatives with respect to conformal time.

One need only specify a potential to evolve the equations and obtain the energy density and pressure,. An equivalent formulation of the scalar field is to specify the evolution of the equation of state as a function of the scale factor, w a. Then the energy density can be reconstructed as. It is possible to reconstruct the potential and field evolution for a given equation of state [34, 33]:. See [35] for details of the perturbation theory.

This equation is amenable to Fourier decomposition of the fluctuations. We see that the quintessence reacts to the external gravitational field of, say, dark matter and baryons through h. Again there is a simplification using the equation of state:. From the above equations, this means that fluctuations d Q on scales smaller than the Hubble scale dissipate, so the field is a smooth, non-clustering component there.

Any initial fluctuations in the quintessence field are damped out rapidly. On scales greater than H -1 , the field is unstable to gravitational collapse, and long wavelength perturbations develop. This means the quintessence responds to the large scale fluctuations in the CDM and baryons. As shown in Fig. Trackers represent a particular class of quintessence models which avoid the problem of fine tuning the initial conditions of the scalar field in order to obtain the desired energy density and equation of state at the present time [9, 29].

The tracker is a scalar field Q which rolls down a potential V Q , as shown in Fig. Tracking has an advantage similar to inflation in that a wide range of initial conditions is funneled into the same final condition. The initial energy density of the quintessence, r Q i , can vary by nearly orders of magnitude without altering the cosmic history. In particular, the acceptable initial conditions include equipartition after inflation - nearly equal energy density in Q as in the other - degrees of freedom e.

Furthermore, the cosmology has desirable properties. The equation of state of Q varies according to the equation of state of the dominant component of the cosmological fluid. As displayed in Fig. When the Universe is matter-dominated, the w is negative and r Q decreases less rapidly than the matter density. The consequence is that eventually r Q surpasses the matter density and becomes the dominant component.

An introduction to quintessence

The fact that W Q is not seen to be completely dominating yet, or that W m is measured to be at least 0. Once tracking begins, the equation of state is given by the handy formula. In each case, M is a free parameter which is fixed by the measured value of W Q. Hence, these models each have one free parameter, just as for the cosmological constant. The tracker, however, has a much more plausible origin in particle physics, as the potentials occur in string and M-theory models associated with moduli fields or fermion condensates perturbative effects make flat direction potentials runaway , and can start from a realistic state in equipartition.

For these reasons the claim is made that quintessence is on equal if not stronger theoretical ground than the cosmological constant. Another species of quintessence closely related to the tracker is the creeper. This corresponds to the case in which the initial energy density in Q after inflation is much greater than the radiation energy density.

For either of the potentials described above, this corresponds to starting at a small value of Q , very high up in V. The field evolution is critically damped; Q still moves, but is now creeping down the potential. We now focus on the observational constraints on QCDM cosmological models. These results were explored in depth in [24], where an exhaustive study of the constraints was presented, obtaining a set of quintessence models in concordance with observation.

In the case of tracker quintessence, the equation of state changes slowly with time, but the observational predictions are well approximated by treating w as a constant, equal to. Through out this discussion, we will evaluate the bounds for a constant w , but the same constraints hold for the equivalent QCDM model with. Hubble Constant: The Hubble constant has been measured through numerous techniques over the years.

Although there has been a marked increase in the precision of extragalactic distance measurements, the accurate determination of H has been slow. Other measures can be listed, but clearly convergence has not been reached, although some methods are more prone to systematic uncertainties. Age of the Universe: Recent progress in the dating of globular clusters and the calibration of the cosmic distance ladder has relaxed the lower bound on the age of the Universe.

Using the observed baryon density from BBN, we obtain the constraint. The interpretation of x-ray cluster data for the case of quintessence models has been carried out in detail by Wang and Steinhardt [44], in which case the constraint is expressed as. This fitting formula is valid for the range of parameters considered here. Perhaps the two most important constraints on the mass power spectrum at this time are the COBE [19] limit on large scale power and the cluster abundance constraint which fixes the power on 8 h -1 Mpc scales.

Together, they fix the spectral index and leave little room to adjust the power spectrum to satisfy other tests. Supernovae: Type 1a supernovae are not standard candles, but empirical calibration of the light curve - luminosity relationship suggests that the objects can be used as distance indicators. There has been much progress in these observations recently, and there promises to be more. Hence, a definitive constraint based on these results would be premature.

There is substantial scatter in the supernovae data; the scatter is so wide that no model we have tested passes a c 2 test with the full SCP data set; using a reduced set, Fit C, argued in [6] as being more reliable, a finite range of models do pass the c 2 test, comparable to the range obtained by the c 2 test using the HZS data set.

To gauge the current situation, we will report both c 2 tests and maximum likelihood tests; to be conservative, we use the largest boundary the c 2 test based on HZS data using MLCS analysis for our concordance constraint. Lensing Statistics: The statistics of multiply imaged quasars, lensed by intervening galaxies or clusters, can be used to determine the luminosity distance - red shift relationship, and thereby constrain quintessence cosmological models. There exists a long literature of estimates of the lensing constraint on L models e.

In one approach, the cumulative lensing probability for a sample of quasars is used to estimate the expected number of lenses and distribution of angular separations. In a series of studies, similar constraints have been obtained using optical [49] and radio lenses [46]. In principle, this test is a sensitive probe of the cosmology; however, it is susceptible to a number of systematic errors for a discussion, see [52, 53].

Taking the above into consideration, none of the present constraints on quintessence due to the statistics of multiply imaged quasars are prohibitive: models in concordance with the low- z constraints are compatible with the lensing constraints. One of the most powerful cosmological probes is the CMB anisotropy, an imprint of the recombination epoch on the celestial sphere. The large angle temperature anisotropy pattern recorded by COBE [19] can be used to place two constraints on cosmological models. COBE norm: The observed amplitude of the CMB power spectrum is used to constrain the amplitude of the underlying density perturbations.

We have verified that this method, originally developed for L and open CDM models, can be applied to the quintessence cosmological models considered in this work [34]. This neglects the baryon-photon acoustic oscillations, which produce a rise in the spectrum, slightly tilting the spectrum observed by COBE. In general, the spectral index determined by fitting the large angular scale CMB anisotropy of a quintessence model, which is also modified by a late-time integrated effect, to the shape of the spectrum tends to overestimate the spectral tilt.

When the measurements are analyzed, we can expect that the best determined cosmological quantities will be the high multipole C l moments, such that any proposed theory must first explain the observed anisotropy spectrum. At present, however, there is ample CMB data which can be used to constrain cosmological models. We take a conservative approach in applying the small angular scale CMB data as a model constraint. Our intention is to simply determine which quintessence models are consistent with the ensemble of CMB experiments, rather than to determine the most likely or best fitting model.

At the time of this lecture, the results from several experiments had either been recently presented or shortly expected, so that a detailed analysis would have been premature. The results are best represented by projecting the viable models onto the W m - h and W m - w planes. The concordance region due to the suite of low red shift constraints, including the COBE normalization and tilt n s , are displayed in Figs. Each point in the shaded region represents at least one model in the remaining three dimensional parameter space which satisfies the observational constraints.

In Fig. The age does not impact the W m - h concordance region, since for the allowed values of W m and h , there is always a model with a sufficiently negative value of w to satisfy the age constraint. Relaxing either the BBN or BF constraint would raise the upper limit on the matter density parameter to allow larger values of W m. This requires a simultaneous reduction in the spectral index, n s , in order to satisfy both the COBE normalization and cluster abundance. The lower bound on W m due to the combination of the BBN and BF constraints can be relaxed if we allow a more conservative range for the baryon density, such as 0.

However, the constraints due to s 8 and the shape of the mass power spectrum take up the slack, and the lower boundary of the concordance region is relatively unaffected. The most potent of the intermediate red shift constraints is due to type 1a supernovae, which we present in Fig. Carrying out a maximum likelihood analysis, all three give approximately the same result for the location of the 2 s bound, favoring concordant models with low W m , and very negative w.

A c 2 analysis of the same data gives a somewhat different result: the Fit C SCP data and the HZS data sets give comparable, although weaker, results to the likelihood analysis. In the spirit of conservativism, we have used the weakest bound which we can reasonably justify. Hence, for the concordance analysis, we use the 2 s contour resulting from a c 2 test. We have evaluated the high red shift constraint due to the select ensemble of CMB anisotropy measurements.

Based on a c 2 test in d T l , the set of concordant models projected down to the W m - h and W m - w planes is unchanged from the low red shift concordance region at even the 1 s level. The results are unchanged if we include additional current CMB results, or use a c 2 test in ln d T l 2 [63]. Rather, we must wait for near-future experiments which have greater l -coverage, e. Since the submission of this manuscript, the data from the MAT [3] see Fig. However, neither significantly changes our results. Thus far we have applied the low red shift constraints in sequence with one of the other intermediate or high red shift constraints.

It is straight forward to see how the combined set of constraints restrict the quintessence parameter space. Taking the low red shift constraint region, which is shaped primarily by the BF, BBN, H, and s 8 constraints, the dominant bounds on the W m - w plane are then due to SNe and lensing. The SNe drives the concordance region towards small W m and negative w ; the lensing restricts low values of W m.

Putting these all together, an ultimate concordance test is presented in Fig. If the present observations are reliable, we may conclude that these models are the most viable among the class of cosmological scenarios considered herein. Figure The dark shaded region is the projection of the concordance region on the W m - w plane with the low, intermediate, and high red shift observational constraints. The dashed curve shows the 2 s boundary as evaluated using maximum likelihood, which is the same as Fig.

To what degree do current uncertainties in the Hubble parameter, the spectral tilt and other cosmic parameters obstruct the resolution in w? In Figs. Note first the long, white concordance region that remains in the W m - w plane, which is only modestly shrunken compared to the concordance region obtained when current observational errors are included.

The region encompasses both L and a substantial range of quintessence. Hence, current uncertainties in other parameters are not critical to the uncertainty in w. The figure further shows how each individual constraint acts to rule out regions of the plane. The color or numbers in each patch represent the number of constraints violated by models in that patch.

It is clear that regions far from the concordance region are ruled out by many constraints. Both figures also show that the boundaries due to the constraints tend to run parallel to the boundary of the concordance region. Hence, shifts in the values or the uncertainties in these measurements are unlikely to resolve the uncertainty in w by ruling out one side or the other - either the constraints will remain as they are, in which case the entire concordance region is allowed, or the constraints will shift to rule out the entire region.

The tracker models are a particularly important class of quintessence models, as discussed earlier, because they avoid the ultra-fine tuning of initial conditions required by models with a cosmological constant or other non-tracking quintessence models. Note that the effective or averaged equation of state as described earlier is about 10 per cent larger than the value of w today.

This region retains the core of our earlier low red shift concordance, and is consistent with the SNe constraints. These models represent the best targets for future analysis. The challenge is to prove or disprove the efficacy of these models and, if proven, to discriminate among them. The current observational data appear to indicate very unusual, interesting phenomena.


If this trend continues, as more experiments measure the CMB, large scale structure, and the like, we will then find the evidence supporting new, very low energy physics. In the following, I have constructed an outline of a logical progression for experiments. The first order of business is to refine the measurements of the basic cosmological parameters. The measurement of the Hubble constant must also be further refined. The experiments most likely to accomplish these goals in the near future are: MAP, which will measure the CMB and extract information about W m h 2 , W b h 2 , and n s ; the wide field surveys of large scale structure by the SDSS and 2dF, and the small field x-ray probes by Chandra and XMM, combined will reveal information about the large scale distribution of matter, giving insight into W m ; strong gravitational lensing systems and S-Z clusters will help pin down the value of h.

These results will be in hand within several years, and should the missing energy problem persist, there will be a number of exciting ideas to test. Given that the missing energy problem is real, the next logical step will be to characterize the equation of state, measuring w and , to determine whether the dark energy is L , Q, or other. For fundamental physics, L or Q represents new, ultra-low energy phenomena beyond the standard model.

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If firmly established by observations, the discovery will go down in history as one of the greatest clues to the ultimate theory. The fact that the dark energy can be probed observationally is an unimaginable gift, since most unified theories entail ultra-high energies, far beyond laboratory access. A test of the tracker quintessence scenario can be made by determining the change in the equation of state.

Probes of cosmic evolution are the most direct way to determine w. Hence, observations of the magnitude - red shift relationship using type 1a supernovae are ideal. Another approach is to use the volume - red shift relationship, as with the rate of strong gravitational lensing or number counts. Once the basic properties of the dark energy are determined, W Q and w , we can begin to ask questions about the microphysics - what is it? What clues can it reveal about the structure of the Universe and the nature of physical laws? Long wavelength fluctuations, manifest in very large scale structure and the CMB, are the clues to the microphysics of quintessence.

The best approach in this case is to make full sky maps that trace cosmic structure on the largest scales. These maps can be cross-correlated to isolate the late time, large scale features unique to quintessence. Although cosmic variance blurs information on large length scales, cross-correlation can sharpen the picture. Taking the CMB for example, a given multipole moment can only be measured to due to cosmic variance, and at low l the uncertainty is worse.

However, cross correlation can dramatically reduce this uncertainty. Consider the cross-correlation coefficient between two fields on the sky, such as CMB temperature anisotropy and the x-ray background, or the weak lensing convergence of the temperature field [66, 67]. Hence, a strong cross-correlation is probably the best tool to pin down the microphysics of the quintessence. The missing energy problem and the quintessence hypothesis, and most current cosmological models, are predicated on the validity of Einstein's general relativity, and the existence of cold dark matter with a spectrum of adiabatic perturbations generated by inflation.

At the same time that an effort is directed towards measuring cosmic parameters, it is necessary to test that GR is valid on the largest scales, and to probe for long range forces associated with the missing energy component. By testing the framework we can hope to make connections to fundamental physics.

Detection of a time or spatial variation in coupling constants, such as a or G , would indicate dramatically new physics. In models of fundamental physics, such as M-theory, these field couplings in four dimensions often appear as moduli fields describing the evolution of higher dimensions. Hence, a measurement of , say, would reinforce quintessential ideas of a dynamical, inhomogeneous energy component. If the quintessence field is coupled to the Ricci scalar, there will be observable consequences if Q is rolling sufficiently fast.

The constraints on scalar-tensor theories of gravity apply, and the cosmic evolution and long wavelength fluctuations will differ from the standard QCDM scenario. For recent work, see [68, 69, 70]. If the quintessence field is coupled to the pseudoscalar F mn mn of electromagnetism as suggested by some effective field theory considerations [71], the polarization vector of a propagating photon will rotate by an angle D q that is proportional to the change of the field value D Q along the path.

If these two observations generate non-zero results, they can provide unique tests for quintessence and the tracker hypothesis, because tracker fields start rolling early say, before matter-radiation equality whereas most non-tracking quintessence fields start rolling just recently at red shift of a few.

Facts & Figures

The prospects for decisively testing the quintessence hypothesis in the immediate future are excellent. Whether these ideas are vindicated or not, we will surely discover exciting, new physics. Carlberg et al. Miller et al. Krauss and M. Turner, Gen. Ostriker and P. Steinhardt, Nature , Perlmutter et al.

Riess et al. Caldwell, R. Dave, and P. Steinhardt, PRL 80 , Zlatev, L. Wang, and P. Steinhardt, PRL 82 , Affleck et al. B , Hill and G. Ross, Nucl. Binetruy, M. Gaillard, and Y. Wu, Nucl. Masiero, M. Pietroni, and F. Rosati, Phys. D 61 Bahcall, L. Lubin, and V. Dorman, ApJ L81 Burles and D. Tytler, ApJ , Knox and L. Kamionkowski, D. Spergel, N. Sugiyama, ApJ , L57 Bennett et al. Tegmark, Phys. D 55 , Netterfield, M.

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