Is there Supernova Evidence for Dark Energy Metamorphosis ?
Abstract
We reconstruct the equation of state of dark energy (DE) using a recently released data set containing 172 type Ia supernovae without assuming the prior (in contrast to previous studies). We find that dark energy evolves rapidly and metamorphoses from dustlike behaviour at high ( at ) to a strongly negative equation of state at present ( at ). Dark energy metamorphosis appears to be a robust phenomenon which manifests for a large variety of SNe data samples provided one does not invoke the weak energy prior . Invoking this prior considerably weakens the rate of growth of . These results demonstrate that dark energy with an evolving equation of state provides a compelling alternative to a cosmological constant if data are analysed in a priorfree manner and the weak energy condition is not imposed by hand.
keywords:
cosmology: theory—cosmological parameters—statistics1 Introduction
One of the most tantalizing observational discoveries of the past decade has been that the expansion of the universe is speeding up rather than slowing down. An accelerating universe is strongly suggested by observations of type Ia high redshift supernovae provided these behave as standard candles. The case for an accelerating universe is further strengthened by the discovery of Cosmic Microwave Background (CMB) anisotropies on degree scales (which indicate ) combined with a low value for the density in clustered matter deduced from galaxy redshift surveys. All three sets of observations strongly suggest that the universe is permeated by a relatively smooth distribution of ‘dark energy’ (DE) which dominates the density of the universe () and whose energy momentum tensor violates the strong energy condition () so that .
Although a cosmological constant () provides a plausible answer to the conundrum posed by dark energy, it is well known that the unevolving cosmological constant faces serious ‘fine tuning’ problems since the ratio between and the radiation density, , is already a miniscule at the electroweak scale ( GeV) and even smaller, , at the Planck scale ( GeV). This issue is further exacerbated by the ‘cosmological constant problem’ which arises because the term generated by quantum effects is enormously large , where GeV is the Planck mass (Zeldovich, 1968; Weinberg, 1989).
Although the cosmological constant problem remains unresolved, the issue of fine tuning which plagues has led theorists to explore alternative avenues for DE model building in which either DE or its equation of state are functions of time. (Following Sahni et al. (2003) we shall refer to the former as Quiessence and to the latter as Kinessence.) Inspired by inflation, the first dark energy models were constructed around a minimally coupled scalar field (quintessence) whose equation of state was a function of time and whose density dropped from a large initial value to the small values which are observed today (Peebles & Ratra, 1988; Wetterich, 1988). (‘Tracker’ quintessence models had the advantage of allowing the current accelerating epoch to be reached from a large family of initial conditions (Caldwell, Dave & Steinhardt, 1998).)
Half a decade after SNebased observations pointed to the possibility that we may be living in an accelerating universe, the theoretical landscape concerning dark energy has evolved considerably (see the reviews Sahni & Starobinsky, 2000; Carroll, 2001; Peebles & Ratra, 2002; Sahni, 2002; Padmanabhan, 2003). In addition to the cosmological constant and quintessence, the current paradigm for DE includes the following interesting possibilities:

The Chaplygin Gas whose equation of state drops from at high redshifts to today (Kamenshchik, Moschella & Pasquier, 2001)
Faced with the current plethora of dark energy scenarios the concerned cosmologist is faced with two options:
(i) She can test every single model against observations,
(ii) She can take a more flexible approach and determine the properties of dark energy in a model independent manner.
In this paper we proceed along route (ii) and demonstrate that model independent reconstruction brings us face to face with exciting new properties of dark energy.
Applying the techniques developed in Saini et al. (2000); Sahni et al. (2003) to a new data set consisting of Supernovae from Tonry et al. (2003) and an additional 22 Supernovae from Barris et al. (2003) we show that the DE equation of state which best fits the data evolves from at to An evolving equation of state of DE is favoured by the data over a cosmological constant for a large region in parameter space. today.
2 Model independent reconstruction of dark energy
Supernova observations during the previous decade have been pioneered by two teams: The Highz Supernova Search Team (HZT) (Riess et al. , 1998) and the Supernova Cosmology Project (SCP) (Perlmutter et al. , 1999). The enormous efforts made by these two teams have changed the way cosmologists view their universe. A recent analysis (Tonry et al. , 2003) of 172 type Ia supernovae by HZT gives the following bounds on the cosmic equation of state (at CL)
(1) 
when the 2dFGRS prior is assumed (Percival et al. , 2001). A similar bound
(2) 
is obtained for a new sample of highz supernovae by SCP (Knop et al. , 2003). ^{1}^{1}1It is interesting that, when no priors are set on , the dark energy equation of state becomes virtually unbounded from below and has a confidence limit of being ! (Knop et al. , 2003)
These results clearly rule out several DE contenders including a tangled network of cosmic strings () and domain walls (). However a note of caution must be added before we apply (1) or (2) to the wider class of DE models discussed in the introduction. Impressive as the bounds in (1) & (2) are, they strictly apply only to dark energy having a constant equation of state since this prior was assumed both in the analysis of the supernova data set as well as in the 2dFGRS study (Tonry et al. , 2003; Knop et al. , 2003). Aside from the cosmological constant (), the topological defect models alluded to earlier and the sinehyperbolic scalar field potential (Sahni & Starobinsky, 2000; UrenaLopez & Matos, 2000; Sahni et al. , 2003) no viable DE models exist with the property . Indeed, most models of dark energy (Quintessence, Chaplygin gas, Braneworlds, etc.) can show significant evolution in over sufficiently large look back times.
In this paper we shall reconstruct the properties of dark energy without assuming any priors on the cosmic equation of state. (The dangers of imposing priors on have been highlighted in Maor et al. (2002) and several of our subsequent results will lend support to the conclusions reached in this paper.)
2.1 Cosmological reconstruction of
Cosmological reconstruction is based on the observation that, in a spatially flat universe, the luminosity distance and the Hubble parameter are related through the equation (Starobinsky, 1998; Huterer & Turner, 1999; Nakamura & Chiba, 1999):
(3) 
Thus knowing we can unambiguously determine the Hubble parameter as a function of the cosmological redshift. Next, the Einstein equations
(4) 
are used to determine the energy density and pressure of dark energy:
(5) 
where is the critical density of a FRW universe. The equation of state of DE follows immediately (Saini et al. , 2000)
(6) 
where , . In quintessence models and in CDM, the equation (6) determines the true ‘physical’ equation of state of dark energy. However the subscript ‘eff’ in stresses the fact that this quantity should be interpreted as an ‘effective’ equation of state in DE models in which gravity is nonEinsteinian or in models in which dark energy and dark matter interact. Examples of the former include Braneworld models and scalartensor theories. It is well known that in a large class of Braneworld models the Hubble parameter does not adhere to the Einsteinian prescription (4) since it includes explicit interaction terms between dark matter and dark energy (Deffayet, Dvali & Gabadadze, 2002; Sahni & Shtanov, 2003). In this case the equation of state determined using (6) can still be used to characterize DE, but physical interpretations of need to be treated with caution. ^{2}^{2}2One way around this difficulty is to define observables solely in terms of and its derivatives (called ‘Statefinders’ in Sahni et al. , 2003). A detailed discussion of these issues can be found in Alam et al. (2003).
One route towards the meaningful reconstruction of lies in inventing a sufficiently versatile fitting function for either or . The parameters of this fitting function are determined by matching to Supernova observations and is determined from (3) and (6).^{3}^{3}3Alternatively one could apply an ansatz to itself (Chiba & Nakamura, 2000; Weller & Albrecht, 2002; Corasaniti & Copeland, 2003; Gerke & Efstathiou, 2002; Maor et al. , 2002; Linder, 2003). See Alam et al. (2003) for a summary of different approaches to cosmological reconstruction. Nonparametric approaches are discussed in Wang & Lovelace (2001); Huterer & Starkman (2002); Saini (2003); see also Daly & Djorgovsky (2003); Nunes & Lidsey (2003). Our reconstruction exercise will be based upon the following flexible and model independent ansatz for the Hubble parameter (Sahni et al. , 2003)
(7) 
where . This ansatz for is exact for the cosmological constant () and for DE models with () and (). It has also been found to give excellent results for DE models in which the equation of state varies with time including quintessence, Chaplygin gas, etc. (Sahni et al. , 2003; Alam et al. , 2003). The ansatz (7) is equivalent to the following expansion for DE
(8) 
where is the present day critical density. The condition allows to mimic the properties of dark matter at large redshifts ( follows from large scale structure constraints). From (7) and (8) we find , i.e. the value of in (7) can be slightly larger than in this case.
Substituting (7) into the expression for the luminosity distance we get
(9) 
The parameters are determined by fitting (9) to supernova observations using a maximum likelihood technique. This ansatz has only three free parameters since for a flat universe. A note of caution: since the ansatz (8) is a truncated Taylor expansion in its range of validity is , consequently the ansatzderived and should not be used at higher redshifts.
Note that the weak energy condition for dark energy has the following form for the ansatz (7) :
(10) 
provided we assume that the term in (7) is totally due to nonrelativistic dark matter and does not include any contribution from dark energy. The demand that the WEC (10) be satisfied for all (i.e. in the past as well as in the future) requires to be nonnegative. However, the demand that the WEC (10) be satisfied in the past () but not necessarily in the future, leads to the somewhat weaker constraint
(11) 
(Models in which for and which violate the WEC in the future, have been discussed in Felder et al. (2002); Kallosh et al. (2002); Alam, Sahni & Starobinsky (2003).)
The presence of the term in (7) has two important consequences: (i) It ensures that the the universe transits to a matter dominated regime at early times (), (ii) It allows us to incorporate information (available from other data sets) regarding the current value of the matter density in the universe. This information can be used to perform a maximum likelihood analysis with the introduction of suitable priors on . In further analysis we will assume that the term in (7) does not include any contribution from dark energy.
We have also studied simple extensions of the ansatz (7) by adding new terms and . The term allows to become substantially less than , thereby providing greater leeway to phantom models. The term allows DE to evolve towards equations of state which are more stiff than dust (); its role is therefore complementary to that of . Despite the inclusion of these new terms, our best fit to the supernova data presented below does not change significantly(choosing and ), which points to the robustness of the ansatz (7) for the given data set.
We should add that our reason for choosing an ansatz to fit rather than some other cosmological quantity was motivated by the fact that the Hubble parameter is directly related to a fundamental physical quantity – the Ricci tensor, and is therefore likely to remain meaningful even when other quantities (such as the equation of state) become ‘effective’. (This happens for instance, in the case of the braneworld models of dark energy discussed in Deffayet, Dvali & Gabadadze (2002); Sahni & Shtanov (2003).)
The rationale for choosing a three parameter ansatz for is the following. The observed luminosity distance determined using type Ia supernovae is rather noisy, therefore in order to determine the Hubble parameter from and following that the equation of state, one must take two derivatives of a noisy quantity. This difficulty can be tackled in two possible ways: (i) either one smoothes the data over some interval (binning is one possibility), or (ii) we may choose to smooth ‘implicitly’ by parameterizing through an appropriate fitting function. The number of free parameters in the fit to will be related to the smoothing interval through . Increasing implies decreasing which results in a rapid growth of errors through , and (Tegmark, 2002), therefore in order not to loose too much accuracy in our reconstruction we considered 3 parameter fits for in our paper (these correspond to 2 parameter fits for ).
We now test the usefulness of the ansatz (7) in reconstructing different dark energy models. The ansatz returns exact values for CDM, and , quiessence models. In figure 1 we show the accuracy of the ansatz (7) when applied to several other dark energy models such as tracker quintessence, the Chaplygin gas and supergravity (SUGRA) models. We plot the deviation of (which is the measured quantity for SNe) obtained with the ansatz (7) from the actual model values. Clearly the ansatz performs very well over a significant redshift range for (Also see appendix B). In fact, in the redshift range where SNe data is available, the ansatz recovers these models of dark energy with less than errors. However it would be appropriate to add a note of caution at this point. Although figure 1 clearly demonstrates the usefulness of the ansatz for some DE models, its performance visavis other models of DE is by no means guaranteed. By its very construction the ansatz (7) is expected to have limitations when describing models with a fast phase transition (Bassett et al. , 2002) as well as rapidly oscillating quintessence models (Sahni & Wang, 2000). (The ansatz (7) can give reasonable results even for these models provided the resulting DE behaviour is suitably smoothed.) For this reason, although the bulk of our analysis will be carried out using (7), we shall supplement it when necessary with other fitting functions, which will provide us with an independent means with which to test the robustness of our reconstruction exercise.
Methodology :
For our primary reconstruction, we use a subset of 172 type Ia Supernovae, obtained by imposing constraints and on the 230 SNe sample, as in the primary fit of Tonry et al. (2003). For the ansatz (9), we require to fit four parameters: (). We may use prior information on (, Freedman et al. (2001)) and (, Percival et al. (2001)). ^{4}^{4}4One should note however that the prior on is not model independent since it relies on the CDM model to project from redshift space to real space. Results coming from the use of this prior should therefore not be taken too literally in the present context. See Kunz et al. (2003) for an interesting discussion of related issues.
The measured quantity for this data is , therefore the likelihood function is given by
(12)  
(13) 
where is a normalisation constant. Therefore, the probability distribution function in the fourspace is
(14) 
where refers to the priors applied on the parameters of the system.
Our goal is to reconstruct cosmological parameters such as the equation of state , therefore we marginalise over and obtain the probability distribution function in the space:
(15) 
In order to do this, we have to define the bounds of a fourdimensional volume in . The bounds of are taken at of the HST prior. For , the natural choice is . It is not immediately obvious what the bounds should be for . We choose a sufficiently large rectangular grid for (roughly corresponding to space on which is specified at each point. We may now proceed in two ways. Firstly, we may choose to fix at a suitable constant value (e.g., ) thereby obtaining a grid in the plane with (the probability if is known to be an exact value) defined at each point. For a particular redshift, we may then calculate at each point of the grid. This would yield results that would hold true if were known exactly. Instead of using the exact value of , we may use the prior information about it available to us (), and calculate at each point of a threedimensional grid, the probability at each point being known. Therefore, at any given redshift , can be tagged with a numerical value . Starting from the bestfit (the value at the peak of the probability distribution), we may move down on either side till of the total area is enclosed under the curve, thus obtaining asymmetric bounds on . The bounds can be similarly obtained. ) which includes most known models of dark energy. This bound is merely a matter of convenience and does not affect our results in any way. After marginalisation, we have a grid in
Results :
We first show preliminary results for which the matter density is fixed at a constant value of . A detailed look at the surface in the () plane (figure 2) reveals the existence of two minima in , a shallower one close to CDM (), and a deeper minimum at . We would like to draw the readers attention to the fact that imposing the prior amounts to disallowing a significant region of parameter space (the unshaded region in figure 2). Consequently an analysis which assumes loses all information about the region around the deeper minimum ! Since we have no reason (observational or theoretical) to favour either minimum over the other, we shall always choose the deeper minimum as our bestfit in all the subsequent calculations.
In the figure 3, we plot the deviation of the squared Hubble parameter from CDM over redshift for the bestfit. We note that the quantity has a simple linear relationship with the parameters of the fit (Eq 7), therefore the errors in this quantity increase with redshift. Another quantity of interest is the energy density of dark energy. For this ansatz, (where is the present day critical density). The figure 4 shows the logarithmic variation of with redshift. In this figure too the errors increase with redshift. An interesting point to note is that initially, dark energy density decreases with redshift, showing the phantomlike nature () of dark energy at lower redshifts of , while at higher redshifts, the dark energy density begins to track the matter density. Before moving on to the second derivative of the luminosity distance (e.g., the equation of state) we may obtain more information from the dark energy density by considering a weighted average of the equation of state :
(16) 
where denotes the total change of the variable between integration limits. This quantity can be elegantly expressed in terms of the difference in energy densities over a range of redshift as
(17) 
Thus the variation in the dark energy density depicted in figure 4 is very simply related to the weighted average equation of state !
Bestfit  Confidence levels  CDM  

In table 1 we show the values of obtained using different ranges in redshift for our bestfit with corresponding and errors. We have taken the ranges of integration to be approximately equally spaced in ln, with the upper limit set by the furthest supernova known at present. The values of may be calculated using the equation (16) (which uses the second derivative of the luminosity distance), or they can simply be read off from figure 4 using equation (17). From this table, a “metamorphosis” in the properties of dark energy occurring somewhere between and can be clearly seen (note that, effectively, one needs to differentiate only once to come to this conclusion).
We now reconstruct the equation of state of dark energy which, for the ansatz (7), has the form
(18) 
(Note that, since was derived from the ansatz (7), its domain of validity is .) In figures 5 (a), (b), (c), we show the evolution of the equation of state with redshift for different values of . The and limits are shown in each case. The per degree of freedom for the bestfit for the different cases is given in Table 2. We find that for , the behaviour of the equation of state does not change significantly with change in the matter density. However, for larger values of , a smaller current value of is preferred. In all three cases considered, the present value of the equation of state is for the bestfit, and the equation of state rises steeply from to with redshift. In fact, the behaviour of appears to be extremely different from that in CDM (). We note here that, for this analysis, the errors on appear to decrease with redshift. This may appear counterintuitive, since there are fewer SNe at higher redshifts, but this is merely a construct of the fact that depends nonlinearly on the parameters of the ansatz (see appendix A).
Quintessence models satisfy the weak energy condition (WEC) and it would be interesting to see how the imposition of the WEC as a prior on the equation of state will affect the results of our analysis. We therefore perform the same analysis as above with the added constraint (note that this implies for all for our fitting function of provided ). The results are shown in the figures 6 (a), (b), (c). We see that in this case the errors are larger and the evolution of the equation of state with redshift follows a much gentler slope. Such an equation of state would be largely consistent with the cosmological constant model. (These results are in broad agreement with an earlier analysis of Saini et al. (2000) in which a smaller SNe data set was used and a different ansatz for the luminosity distance was applied.)
In Table 2, we show how the for the bestfit evolving dark energy models compare with that for CDM. We find that always. For , the value of is just within of the best fit , but for , or , is outside the limits of the bestfit. It is also noteworthy that when the prior is used, the bestfit model has a slowly evolving equation of state with and the for the bestfit becomes smaller for a smaller value of the matter density. When no priors are assumed on , the trend reverses, and better fits are obtained for larger values of . From this it appears that at least at the evolving dark energy model is favoured over CDM, and it does as well, if not better at the level, depending upon the value of the presentday matter density.
Using Priors on :
Instead of assuming an exact value for , which is somewhat optimistic given the present observational scenario, we may use the 2dF prior on and calculate as a function of . It should be noted here that the 2dF error bars on have been calculated using 2dF data in conjunction with CMB, and this calculation assumes a CDM model, therefore this prior should be used more as a benchmark for the value of rather than as an absolute when considering evolving dark energy models. The resultant “marginalised” is shown as a function of the redshift in figure 7 (a). The nature of the equation of state for the analysis with the added prior is shown in figure 7 (b). We find that the general nature of evolution of the equation of state is not changed by adding this extra information on the matter density. If no priors are assumed on the equation of state to begin with, still rises sharply from up to at maximum redshift and the analysis appears to favour a fastevolving equation of state of dark energy over the standard CDM model. If a prior is assumed, then the marginalised equation of state is more consistent with the cosmological constant. From this we see that marginalisation over does not lead to any significant change in our results. In the subsequent sections, we will show our results for .
From the above analysis, we find, therefore, that our results change significantly depending upon whether or not the prior is imposed. We saw earlier that in the absence of any prior on , the bestfit equation of state rose from at to at . By imposing a prior on the equation of state, we effectively screen off a sizeable part of the parameter space (see figure 2), and therefore the reconstruction is forced to choose its bestfit away from the true minima of the surface. The effect of imposing a prior on is therefore to make the bestfit grow much more slowly with being preferred at . Our results show that the reconstructed equation of state with the prior is in good agreement with a cosmological constant at the CL. However, if no prior is imposed, then the steeply evolving dark energy models are favoured over the cosmological constant at , and are at least as likely as the cosmological constant at the level.
Age and Deceleration Parameter of the Universe:
We may also use this ansatz to calculate other quantities of interest, such as the age of the universe, , and the deceleration parameter, :
(19)  
(20) 
where .
In figure 8 we plot the evolution of the age of the universe with redshift. We find that the bestfit age of the universe today is Gyrs if the Hubble parameter is taken to be , which is slightly lower than the age of a CDM universe, Gyrs (both values are for ). At the level, the age of the universe today would vary between Gyrs.
Figure 10 shows the evolution of the deceleration parameter with redshift. We find that the behaviour of the deceleration parameter for the bestfit universe is quite different from that in CDM cosmology. Thus, the current value of is significantly lower than for CDM (assuming ). Furthermore the rise of with redshift is much steeper in the case of the bestfit model, with the result that the universe begins to accelerate at a comparatively lower redshift (compared with for CDM) and the matter dominated regime () is reached by .
Bestfit  

Using Priors on Age of the Universe :
Important consistency checks on our bestfit Universe may be provided by observations of the age of the Universe. Unfortunately, estimates of the age of the universe from different methods can produce widely varying results one reason for which is that estimates of the Hubble parameter itself can vary significantly. For instance, the HST key project yields , while studies of the SunyaevZeldovich effect in galaxy clusters give a significantly lower value (Krauss, 2001). Estimates of the ages of the oldest globular clusters suggest Gyrs, at the confidence level (Krauss, 2001; Krauss & Chaboyer, 2001; Gnedin, Lahav & Rees, 2001; Hansen et al. , 2002; Gratton et al. , 2003; Marchi et al. , 2003) and this age estimate is consistent with several other measurements including observations of eclipsing spectroscopic binaries (Thompson et al. , 2001; Chaboyer & Krauss, 2002), results from radioactive dating of a metalpoor star (Cayrel et al. , 2001) and WMAP data (Spergel et al. , 2003) (see also Alcaniz, Jain and Dev (2002)). The results from the WMAP experiment suggest Gyrs with a Hubble parameter , for CDM cosmology (which satisfies the WEC). Adding SDSS and SNe Ia data to WMAP, Tegmark et al. (2003) find an age of Gyrs for a slightly closed CDM universe with . Although these results cannot be carried over to evolving dark energy models including those implied by our bestfit reconstruction (which violate the WEC) they provide an indication of the range within which the age of the universe might vary. Keeping in mind these various results, we use two different priors on the Hubble parameter: ( bound from HST; Freedman et al. (2001)), and (approximate bound from WMAP, SDSS, SNe Ia; Tegmark et al. (2003)). For each case, we choose three different Gaussian priors on the present age of the universe: respectively, and perform the reconstruction for a universe. The results are shown in the figure 9. We find that, for a Hubble parameter of , and with an additional prior on the age of the universe Gyrs, the bestfit remains nearly the same, showing a rapid evolution of the equation of state from at to at , and the errors become narrower. As the age is increased, the bestfit equation of state evolves more slowly, and the also increases (see Table 3). For the prior , we find that the lowest is obtained for the age prior of Gyrs, which once again matches our bestfit. It should be noted that the errors are smaller in all cases, even though the may be larger. We must remember that the addition of a new prior which is consistent with the underlying dataset would lead to a natural reduction in errors. However, the addition of a prior inconsistent with the dataset would lead to a shift of the likelihood maximum as well as a reduction in errors, and the results would then fail to reflect the actual information present in the dataset. That this is happening here for the higher values of age can be seen from the fact that although the errors are reduced, the is actually larger. Therefore priors from other observations should be added prudently to ensure that they do not lead to incorrect representation of the data. Since there is as yet no clear model independent consensus on the age of the universe, the results we obtain in this section should be interpreted with a degree of caution.
Figure 10 shows the evolution of the deceleration parameter with redshift. We find that the behaviour of the deceleration parameter for the bestfit universe is quite different from that in CDM cosmology. Thus, the current value of is significantly lower than for CDM (assuming ). Furthermore the rise of with redshift is much steeper in the case of the bestfit model, with the result that the universe begins to accelerate at a comparatively lower redshift (compared with for CDM) and the matter dominated regime () is reached by .
2.2 Robustness of Results :
Based on the above analysis, it is tempting to conclude that the dominant component of the universe today is dark energy with a steeply evolving equation of state which marginally violates the weak energy condition. (Of course, the less radical possibility of weakly time dependent dark energy satisfying the weak energy condition remains an alternative, too.) However, before any such dramatic claims are made, we need to check if our results are in any fashion a consequence of inherent bias in the statistical analysis itself, or in the sampling of the data. We therefore perform the following simple exercises to satisfy ourselves of the robustness of our results.
Using Different Subsets of Supernova Data :
In an attempt to understand how the nature of the reconstructed equation of state is dependent on the distribution of data, we perform the reconstruction exercise on different samples of data. We have confined ourselves to the case where for these exercises. Firstly, we may exclude the SCP data points from the 172 SNe primary fit, leading to a subsample of 130 SNe. We call this the HZT sample. Figures 11(a) and 12(a) show the result of performing the analysis on this subsample without any constraints. The per degree of freedom for the bestfit is , which is lower than for this sample. In this case we find that, though the error bars are slightly larger, overall the dark energy density behaves in the same way as before (compare figure 11(a) with figure 4), showing phantom like () behaviour at lower redshifts and tracking matter at higher redshifts. The equation of state of dark energy also evolves much in the same way as when the entire sample is used (compare figure 12(a) with figure 5(b)), starting at and evolving rapidly to . We may also use a sample complementary to this sample, where all the SCP data points published till date are considered, along with the low redshift CalanTololo sample. This leads to a sample of 58 SNe (Perlmutter et al. , 1999; Knop et al. , 2003), which we call the SCP sample. Using this sample, we obtain the figures 11(b) and 12(b). The bestfit has a chisquared per degree of freedom: , lower than for this sample. We find that here too, the dark energy density initially decreases and then starts tracking matter. The equation of state shows signs of rising steeply at low redshifts, but since the highest redshift in this sample is , the behaviour of beyond this redshift cannot be predicted, therefore the apparent flattening out of the curve beyond a redshift of one cannot be seen in this case. For both these subsets of data, we may repeat the exercise using the prior . The results obtained for the equation of state, as seen in figures 13(a), (b), are once again commensurate with the results obtained earlier for the full sample (figure 6(b)). We may therefore conclude from this exercise that subsampling the data does not significantly affect our results, and the steep evolution of the equation of state of dark energy is not a construct of the uneven sampling of the supernovae, but rather, reflects the actual nature of dark energy.
Testing our Ansatz against fiducial dark energy models :
The crucial question of course is whether the reconstructed equation of state of dark energy depends upon the ansatz which is used in the exercise, i.e. , whether the behaviour of the equation of state merely reflects a bias in the ansatz itself. In this section we show how the ansatz performs in recovering dark energy models whose equation of state is known, from simulated data. This ansatz was demonstrated to work extremely well when simulations of SNAP data were used (Alam et al. , 2003). However, simulation of SNAPlike data is an optimistic exercise, since data of this quality is unlikely to be available in the near future. We now demonstrate the accuracy with which the ansatz can recover the fiducial background cosmological model if data is simulated using presentday observational standards. In figures 14 (a), (b), (c), we show how well the ansatz recovers the equation of state for three fiducial models (assuming ):
(a) a quiessence dark energy model with a constant equation of state: ,
(b) a generalised Chaplygin gas model with : with and the presentday equation of state , which would give rise to an effective equation of state
(21) 
and
(c) a model with a linearly evolving equation of state: , with .
(For DE models with the ansatz is exact therefore we don’t show the results for these cases.)
We find that in all three cases, the fiducial model lies within the confidence limits around the bestfit . Based on this result, we claim that within the error bars, the reconstructed equation of state represents the true properties of dark energy when we use real data.
Using other Ansatz :
It is also important to check whether the results of our reconstruction can be replicated using other ansatz such as fits to the luminosity distance or the equation of state. Many different fits have been suggested in the literature (see for example Huterer & Turner (1999), Saini et al. (2000), Weller & Albrecht (2002), Gerke & Efstathiou (2002)). Here we choose the fit suggested in Linder (2003) in which the equation of state of dark energy is expanded as
(22) 
The luminosity distance can therefore be expressed as
(23) 
where .
We find that for this fit, the errors in the equation of state get larger with redshift, however this fit too demonstrates that the equation of state of dark energy increases rapidly with redshift (figure 15(a)) when no priors are assumed on the equation of state (EOS). The per degree of freedom at the bestfit is . When the prior is invoked, the bestfit EOS remains very close to the CDM model (figure 15(b)). Therefore, from this ansatz, we may make the statement that at low redshifts, the equation of state of dark energy shows the same signs of rising steeply with redshift if no priors are assumed on the equation of state, thus supporting our earlier results. The large errors in the equation of state at redshifts of however make it difficult to make any definitive statements about the behaviour of dark energy at high redshifts.
A limitation of the fit (22) is that it is unable to describe very rapid variations in the equation of state. An ansatz which accommodates this possibility has been suggested in Bassett et al. (2002)
(24) 
where is the initial equation of state at high redshifts, is a transition redshift at which the equation of state falls to and describes the rate of change of .
The resulting luminosity distance has the form:
(25) 
where .
The results for the analysis using this fit to the equation of state are shown in fig. 16. We find that when the reconstruction is done without any priors on the equation of state (figure 16(a)), the best fit is remarkably close to the result for ansatz (7) (figure 5(b)). The per degree of freedom at the minimum is for this fit. The errors in this case are somewhat larger, especially at high redshift. If we constrain , then as before, the evolution of the equation of state is much slower (figure 16(b)). So the reconstruction using this ansatz appears to confirm our earlier results.
The above exercises lead us to conclude that our results are neither dependent on the nature of the statistical analysis nor on the manner in which the SNe data is sampled. It therefore appears that dark energy with a steeply evolving equation of state provides a compelling alternative to a cosmological constant if data are analysed in a priorfree manner and the weak energy condition is not imposed by hand.
2.3 Reconstructing dark energy using a new Supernova sample
As this paper was nearing completion, a new dataset consisting of 23 type Ia SNe was released by the HZT team (Barris et al. , 2003). It is clearly important to check whether or not these new data points corroborate the findings reported in the previous sections. Accordingly, we use a subset of 200 type Ia SNe with from the 230 SNe sample of Tonry et al. (2003), and 22 SNe with from the new sample to obtain a bestfit for our ansatz with . We then plot the magnitude deviation of our bestfit universe from an empty universe with in order to illustrate how well our model fits the data (figure 17). For clarity, we plot the median values of the data points. We obtain medians in redshift bins by requiring that each bin has a width of at least 0.25 in log and contain at least 20 SNe. For comparison, we also plot an CDM model, as well as OCDM and SCDM models. From figure 17 we see that our dark energy reconstruction is a much better fit to SNe beyond than CDM. At low redshifts () the agreement between data and the two models is rather marginal. We now add 22 of the new supernovae (rejecting one with ) to our existing dataset of 172 supernovae and perform DE reconstruction on this new dataset of 194 SNe, assuming and no other priors. The resultant figure 18 is similar to the figure 5(b), with slightly smaller errors and has a bestfit . The above exercises point to the robustness of results reported in previous sections, and indicate that evolving dark energy agrees well with the full data set containing 194 type Ia SNe.