# Reconstructing the history of dark energy using maximum entropy

###### Abstract

We present a Bayesian technique based on a maximum entropy method to reconstruct the dark energy equation of state in a non–parametric way. This MaxEnt technique allows to incorporate relevant prior information while adjusting the degree of smoothing of the reconstruction in response to the structure present in the data.

After demonstrating the method on synthetic data, we apply it to current cosmological data, separately analysing type Ia supernovae measurement from the HST/GOODS program and the first year Supernovae Legacy Survey (SNLS), complemented by cosmic microwave background and baryonic acoustic oscillations data. We find that the SNLS data are compatible with at all redshifts

Our method highlights the danger of employing parametric fits for the unknown equation of state, that can potentially miss or underestimate real structure in the data.

###### keywords:

Cosmology: dark energy – methods: data analysis, Bayesian techniques## 1 Introduction

With the confirmation of the accelerated expansion of the universe
(Perlmutter
et al., 1999; Lange
et al., 2001; Hoekstra
et al., 2002; Riess
et al., 2004; Cole
et al., 2005; Astier
et al., 2006; Spergel
et al., 2006; Riess
et al., 2006) comes the inference that the cosmic
dynamics are today dominated by a component that competes
*against* gravitational collapse of matter and thus must have
negative pressure (Frieman
et al., 1995; Peebles &
Ratra, 2003): this has
been dubbed dark energy. All observations are presently compatible
with dark energy being in the form of Einstein’s cosmological
constant , a new form of matter–energy with equation of
state (EOS) . However, it has been shown that an
equation of state which changes with redshift, , can mimic a
cosmological constant and fit the current data *if the
parameterization of is assumed to be a constant*
(Linder, 2004; Simpson &
Bridle, 2006). At the same time, an
explanation based on the cosmological constant still suffers from
the so–called “coincidence” and “fine tuning” problems, and
it remains unclear whether selection effects of the kind embodied
by anthropic arguments can offer a solution
(Starkman &
Trotta, 2006).

Determining whether dark energy is constant in time or has dynamical properties is one of the most pressing outstanding questions in cosmology, as witnessed by the multiplication of observational proposals trying to elucidate the question (see e.g. Trotta & Bower (2006) for an overview). We are thus led to question the simple solution of a constant , especially in view of the fact that several recent works have highlighted the dangers of fitting current data by assuming a specific parameterization for the dark energy EOS (Bassett et al., 2004; Linder, 2004).

The purpose of this paper is to explore the use of Bayesian statistical techniques based on a maximum entropy method to investigate the time dependence of dark energy by imposing minimal assumptions on the functional form of . Whenever external (prior) information is used, its impact is clearly expressed by our formalism, making the reconstruction totally transparent. To do so we require information about the expansion of the universe. The quality and quantity of observational data of cosmological relevance is rapidly increasing: type 1a supernova (SNIa) (see e.g. Riess et al. (2004); Astier et al. (2006)) can be calibrated to serve as standard candles (see Nugent et al. (2006) for a recent proposal of using type II-P supernovae instead). Supernovae type Ia observations can thus measure the luminosity distance as a function of redshift, ,

(1) |

where the present–day Hubble constant is km/s/Mpc, is the speed of light in km/s and the redshift dependent Hubble function can be expressed in terms of the present–day matter–energy content of the Universe as (here and in the following we assume a spatially flat Universe)

(2) | |||||

Here, is the the density of matter in units of the critical density today and is the (present–day) dark energy density in units of the critical density. The dark energy density time evolution is determined by its equation of state (EOS) ,

(3) |

Observations of the luminosity–distance find a powerful geometric complement in the use of “standard rulers” such as the position of the acoustic peaks in the cosmic microwave background (CMB) power spectrum (Bennett et al., 2003; Spergel et al., 2006) and the (transversal) baryon acoustic oscillation (BAO) signature recently measured in the galaxy matter power spectrum (Eisenstein et al. (2005); Cole et al. (2005)). Such data can be used to constrain the angular diameter distance

(4) |

Apart from its geometrical impact on the angular diameter and luminosity distance relations, the properties of dark energy also influence the growth of structures and can therefore be constrained through weak lensing (see e.g. Hu (2002); Hoekstra et al. (2005); Jarvis et al. (2006)) and cluster counts data. In order to constrain from measurements of either or , we need to perform two derivatives, as it is evident from (2) and (3). This is problematic if we consider the increase in the noise that accompanies each derivative. In addition to information loss through this indirect determination of , the current data tends to be sparse with a large sample variance.

For these reasons, it seems timely and relevant to shift attention to establishing more powerful statistical methods to extract in the most efficient and faithful way the information on contained in present and upcoming large data sets. The development of a technique that can cope with the patchy distribution in redshift space while making minimal assumptions on the time properties of dark energy is the logical next step towards improving our understanding of dark energy. In this paper we apply a modified Maximum Entropy (MaxEnt) technique that has been diversely used to successfully reconstruct images and spectra under unfavourable conditions (for applications to astrophysical problems, see e.g. Bridle et al. (1998); Marshall et al. (2002); Maisinger et al. (2004)). With a firm basis in probability theory, the method can be tailored to the needs of dark energy reconstruction from present data. Our application of MaxEnt aims at reconstructing the dark energy EOS while minimizing our assumptions regarding the form of .

## 2 Maximum entropy reconstruction technique

### 2.1 Motivation

When attempting to constrain the nature of dark energy, a procedure common in the literature is to Taylor expand the quantity or around , then constraining the expansion coefficients through the data (Efstathiou, 1999; Huterer & Turner, 2001; Weller & Albrecht, 2002). An example of such a parameterization that is commonly employed is where (Chevallier & Polarski, 2001; Linder, 2003). Alternatively, one might prefer to parameterize the time–dependence of the EOS using some smooth function (such as the ones used in Dick et al. (2006)), that is hoped will encapsulate the essential features of the dynamics one wishes to constrain. Both procedures are not free from the risk of giving misleading results, since they impose artificial assumptions on the form of the EOS, which often have no basis in any physical mechanism. Bassett et al. (2004) highlight the dangers of implementing such parameterizations. Moreover, this will only be sensitive to departures from a constant density within a restricted set of models (Dick et al., 2006).

Huterer & Starkman (2003) introduced a principle component analysis (PCA) of the function , with the aim of adopting a parameterization appropriate to the data sets used (see also Dick et al. (2006); Simpson & Bridle (2006) for related issues). These PCA modes are argued to form a natural basis in which to characterize dark energy evolution and by using only the first few well–determined eigenvectors in the reconstruction one tries to exclude noisy modes and thereby gain accuracy in the reconstruction. However, the method has the disadvantage of introducing an ill–controlled bias at high redshifts, i.e. the removal of strongly oscillating (and noisy) modes may mislead one to the conclusion that the EOS reverts to the fiducial model at large redshift with artificially small error bars. While we recognize the merits of the PCA method, we wish to improve on it in this last respect by making fully explicit the assumptions that will control the behaviour of the reconstructed at large redshifts.

The MaxEnt technique we employ has parallels with the well known
maximum likelihood (ML) approach, but introduces new features
ensuring that in the case where insufficient information is
available the most likely distribution is the most uniform, i.e.
the one with *maximum entropy* (or minimum information
content. For an overview of the connection between entropy and
information content, see e.g. Trotta (2005); Kunz
et al. (2006)).
In other words, where ML merely maximizes the likelihood, often
unnecessarily overfitting the noise in the data, MaxEnt seeks the
optimum trade–off between a smooth, maximally entropic
distribution and the rough distribution mapped out by the data.
The most important characteristic is that the MaxEnt method is
auto–regulating, i.e. the amount of smoothness (or raggedness) in
the reconstruction is consistently determined through the data
themselves (see section 2.2.4 below). In our Bayesian
perspective, extra information coming from prior beliefs or
theoretical prejudice can be naturally incorporated in the
reconstruction via the MaxEnt prior. As we show below, this gives
MaxEnt the power to cope with situations where the dimensionality
of the parameter space potentially exceeds the number of data
points, a difficult reconstruction problem that is ill–defined
under ML techniques. This feature clearly makes MaxEnt highly
applicable to the case of dark energy reconstruction.

### 2.2 The MaxEnt formalism

The task at hand is to determine the EOS of dark energy from sparse data on , and, to a limited extent, on (as encapsulated by today’s detection of BAO). We consider a piece–wise constant in bins out to a maximum redshift . Let be the value of the EOS in the –th bin, . In analogy with the treatment given in Skilling (1989) for the case of image reconstruction, we gather all the EOS bins values in an “image vector” . We seek to determine the posterior distribution of given the observed data , . This is obtained through Bayes’ theorem as

(5) |

The quantity is the prior probability representing all the information about the distribution before the data has been collected; is the likelihood and describes the underlying statistical process and is the model likelihood (“evidence”), which is relevant for model selection questions but that is unimportant in this case. We shall therefore neglect this proportionality constant in the following.

#### 2.2.1 The MaxEnt prior

The principle of MaxEnt is employed to determine a prior that encapsulates all the external information about we wish to specify in the absence of the data. Following Skilling (1989), we adopt the principle that the least biased model that encodes any given prior information is the one which maximizes the entropy of the distribution while remaining consistent with the information. This prior is appealing for its characteristic of maximising the uncertainty (entropy) of the distribution thus making minimal assumptions. The MaxEnt prior (Skilling, 1989) takes the form

(6) |

where is the entropy of relative to the model and is a regularizing constant. The model defines the image vector to which reverts in the absence of any data, and as such it defines a measure on the DE parameter space. The information entropy is analogous to the thermodynamic entropy in statistical mechanics, which is given by the logarithm of the number of states by which one can arrive at a given macroscopic constraint. In the same way, the information entropy can be described as the logarithm of the numbers of ways in which one can arrive at a particular in a Poisson process when starting with the model (Silver et al., 1990). The entropy for an –dimensional discrete parameter space is (Skilling, 1989)

(7) |

The log term is reminiscent of the Kullback–Leibler divergence between and , encoding the amount of information present in with respect to the model . In our case, we do not apply the entropy to the values of directly, but rather to the space of coefficients of an expansion of in a series of basis functions (that we choose to be top–hat functions in redshift space, see section 2.3 for details). We can think of the coefficients of the expansion as a series of weights that encode how much each basis function contributes to the total . We can then apply the MaxEnt prior on the space of these weights, by thinking of them as expressing the relative contribution of each basis function – in other words, in a phenomenological way we take the weights to represent relative probabilities for the presence of each basis function in the final . Below, we will use the notation as a shortcut to indicate the vector of weights of the dark energy expansion. The same applies to the model , that in the entropy term is represented by its expansion coefficients in the chosen basis functions.

Evidently, (for a fixed ) is a convex function which reaches a maximum for with a value . Thus in the absence of any information from the data, the entropy term reverts to the model. The normalizing partition function for the entropy is given by

(8) |

The measure is defined as the invariant volume of the metric defined on the space where and for (also known as the Fisher information matrix). By expanding to second order around the model (at the maximum ), we obtain the partition function in the Laplace approximation:

(9) |

where is the number of parameters, in our case the number of expansion coefficients for .

#### 2.2.2 The likelihood

The likelihood is defined as
of the probability of the *data given the parameters*:

(10) |

and is the probability that the observed data could have been generated from a given . For data subject to Gaussian noise the likelihood function is

(11) |

where is the data covariance matrix and denotes the functional dependence of the observable on the DE parameters in our case, or ( and being simply related through the redshift, see (4)). In the case of independent data points with uncorrelated noise, the covariance matrix is diagonal with the non–zero elements being the variances of each measurement, denoted by , . The normalizing partition function for is

(12) |

Using the identity for the normalized probability distribution we obtain:

(13) |

#### 2.2.3 The posterior probability

From the likelihood in Eq. (10) and the prior in (6), we obtain from Bayes’ theorem the posterior probability for the DE parameters :

(14) |

Given that is quadratic in and is a convex function, the above is well–constrained with the peak of the posterior for being determined by a competition between and , mediated by the value of . We thus see that the MaxEnt prior will be useful in the case where the parameter space dimensionality exceeds the size of the dataset in that the entropy is incorporated as a regularization to avoid over–fitting, while retaining maximum flexibility in the underlying parameterization of . penalizes the excess “structure” in the data, with the regularizing parameter dictating the degree of this smoothing. The choice of is thus very important: a small value of produces little smoothing and the likelihood will dominate, resulting in a reconstructed distribution where the noise might be mistaken as real structure. Alternatively, too large a value for leads to information loss, with the entropic prior overriding the information coming from the data.

#### 2.2.4 The regularization parameter

In order to select the correct value of we add it to the hypothesis space as an additional parameter and let the data select the optimum value. Using once more Bayes’ theorem we have

(15) |

where is the prior on . The joint posterior probability is then (up to irrelevant constants)

(16) | |||||

We adopt a Jeffreys’ prior on , which is flat in , reflecting ignorance of the scale of the variable, within the range . This corresponds to choosing . In the final inference on we marginalize over the nuisance parameter , even though the distribution of is usually fairly strongly peaked and thus marginalization is almost equivalent to maximisation (i.e., simply fixing to the value of the peak of the posterior).

#### 2.2.5 Model specification

The joint posterior in Eq. (16) is conditional on
the specific choice of model , to which the reconstructed
defaults in the absence of data. The entropic prior
distribution is introduced to penalize the posterior for
unwarranted complexity. Given that the model is the vector to
which should revert in the absence of data, it must
represent maximal smoothness. We need to establish what
distribution will encompass this in the context of the
EOS of dark energy. In image reconstruction the default model is
usually taken to be a flat surface equal to the mean of the data.
When the data is then included via the likelihood, variation about
this mean is introduced. In our case, this means choosing a
contant model, that is *uniform* in
redshift space. There is however no obvious choice for the
magnitude of this constant. There are various possible choice of
, reflecting different prior beliefs about the dark energy
EOS. One can thus usefully think of as encapsulating a
fiducial, reference model we want to test the observations
against. One possibility is to set at all redshifts,
thus representing a cosmological constant. This is recommended if
we are testing for deviations from : if significant
deviations from the model are found in the reconstructed EOS, this
is an indication that the data are informative enough to override
the entropic pull towards the model, and thus that such deviations
are likely to reflect real structure in the data. A more skeptical
attitude towards dark energy might be encapsulated by choosing a
constant model , which correspond to a pressureless,
dust–like fluid. In this case, if the reconstructed
assumes values below 0, this can be interpreted as a strong
indication for the presence of a fluid with negative pressure,
with data being strong enough to dominate the entropic tendency
for a pure matter Universe. In principle, a theoretical prejudice
in the form of a particular redshift–dependence of could
also be implemented easily in the same fashion.

Finally, one can also employ Bayes theorem to take into the joint posterior, by writing

(17) |

and marginalizing over in the left–hand–side, after specifying a prior over the model space, . In this work we take all the model vectors to be constant over the whole redshift range, thus the specification of the model amounts to the choice of the value of the constant. We restrict our considerations to the range , and when performing a marginalization over the model we will take a flat prior in this range, i.e. .

### 2.3 Dark energy parameterization and reconstruction

As motivated above, we decompose into a weighted sum of orthogonal functions in redshift space, with the parameters being given by the weights encoding the amount that each function contributes to the overall . There are of course several different meaningful expansion functions such (for example, principal components, Chebychev functions, etc) but we make use of the simplest option, decomposing is into a series of step–functions :

(18) |

where for and everywhere else. Since the least stringent limits we will impose on are , the above ensures that the expansion coefficient () are positive numbers, a necessary requirement for our entropic prior. The parameter space is thus constructed from the co-efficients themselves, which are allowed to vary within the range . An advantage of this piece-wise parameterization of is that it will be possible to capture a sharp change in the EOS, provided the binning is sufficient. In order to capture different features of the time evolution of the EOS, other expansion functions may be more appropriate. We experimented with Chebychev functions and found that their oscillatory behaviour was not helpful in reconstructing sharp changes in the EOS. Such smooth functions might be more useful if one wants to test quintessence models exhibiting a gentle evolution of .

In the bulk of recent analyses the limit is imposed; the lower limit stemming from the null energy condition which must be satisfied for dark energy to be stable (Alcaniz, 2004). Although models of dark energy that allow violate the weak energy condition in the context of general relativity, these “phantom” components have been studied by many authors (Caldwell, 2002). There have been claims that such phantom behaviour is unstable when regarded as a quantum field theory (Carroll et al., 2003). From a phenomenological perspective, it makes sense to both restrict the range of our reconstruction to lie within , but also to extend the parameter space to values below to check the stability of the reconstruction. We will thus presents results also for the case where the equation of state can attain values as low as .

Parameter | Prior | |
---|---|---|

Top–hat | ||

0 | Flatness imposed | |

Top–hat | ||

Top–hat | ||

Top–hat |

In this work we assume flat spatial sections, and thus . The parameters included in the hypothesis space are summarized in Table 1. These are , the Hubble parameter today, in km/s/Mpc, and the coefficients of the DE decomposition, , as described by Eq. (18). This generic characterization requires the number of expansion functions (which can be effectively characterized as top–hat bins) to be sufficiently large for this to be a suitable description of . As described above, we also include the nuisance parameters and the value of the model EOS, , whenever this is marginalized over.

Assuming uncorrelated Gaussian noise, the log–likelihood of a point in parameter space is given by

(19) |

where for each datum at redshift we have for future radial baryonic oscillation measurements, for present and future transversal baryonic oscillation data and CMB data and for SNIa data. Furthermore, is the measurement variance. The Hubble parameter as a function of redshift is obtained via Eq. (2), where the energy density is calculated using, for falling within the -th bin;

(20) | |||||

From the above, the angular diameter distance can then be computed using equations (1-2). For piecewise constant we employ the trapezoid rule to approximate the integral, obtaining

(21) | |||||

The binning of the integral, defined by is determined based on a fixed level of fractional accuracy for the integration, that we set to as determined by the extrapolation error estimate. Finally, the entropy of a vector with respect to a model is given by

(22) |

where the , are the coefficient of the model in the expansion (18).

To sample the posterior probability distribution efficiently we use a Monte Carlo Markov Chain (MCMC) which employs a Metropolis algorithm. For more details about MCMC, see e.g. Neal (1992); Lewis & Bridle (2002). Since the MaxEnt method is designed to achieve the optimal reconstruction independently of the number of degrees of freedom in the parameterization of , we expect that the number of basis functions will not affect greatly the reconstruction, as long as is chosen large enough to capture the possible structure in the data. In the following we choose but we have checked that the results do not vary much if one uses or instead.

When using actual data, we divide the redshift range spanned by either the SNLS or the HST/GOODS supernova measurements into equally spaced bins, corresponding to the basis functions for . We then extend the last bin to cover all of the redshift range to last scattering when computing the angular diameter distance to the CMB. In other words, we take to be constant (but not fixed to ) between the redshift of the highest supernova in the samples and . This extrapolation is weaker than the ’strong’ prior used in the analysis of Riess et al. (2006), which assumed that at .

## 3 Demonstration of the MaxEnt method

We now proceed to test our MaxEnt reconstruction method with synthetic data. Our benchmark dataset consists of measurements of and of (or equivalently, ) distributed uniformly in the redshift range . Although the actual measurements will in reality be less homogeneous, this does not represent a problem for our reconstruction algorithm, as we show below. Existing measurements of out to will be vastly improved when future surveys such as DES or LSST will be able to observe thousands of SNIa per year (Abbott et al., 2005; Tyson, 2006) and space–based projects such as SNAP (Aldering et al., 2004), ADEPT or DUNE will provide observations at redshift beyond . Future spectrographic surveys such as the Wide–Field—-Multi–Object Spectrograph (WFMOS) or HETDEX ought to deliver constraints on of at and at () and to (Glazebrook et al., 2005; Kelz et al., 2006), with better performance still to be expected when the Square Kilometer Array will come online (Blake et al., 2004).

We add Gaussian noise to our synthetic data as a fixed percentage of the true value of the measured quantity. We use an optimistic noise level of . Although it is beyond the scope of this paper to make quantitative predictions about the performance of future surveys in reconstructing the EOS, our benchmark data set roughly reflects the potentiality of future observations. We also test the performance of the method when the signal–to–noise level is degraded by a factor of 10, in order to check for bias in the reconstruction due to our entropic prior when the quality of the data is poor. In this case we use a noise level of in the luminosity distance and Hubble function measurements.

We show in Figure 1 the reconstructed EOS for our benchmark scenario with high quality observations (, observations). In all three panels, we have marginalized over the prior model . We notice that the reconstruction is satisfactory in all three cases. We have checked that the extracted when marginalizing over the prior model has comparable accuracy to the case of a fixed model . The bottom panel shows how the method deals with gaps in the redshift range of the observations: the smoothing effect of the entropic term enlarges the errors in the region where no data are available, while the reconstructed EOS tracks the true value at small and large redshifts, where the data are clustered. Here we have employed a 10–dimensional , but we have checked that increasing the number of elements to does not lead to any significant change in the reconstruction. As expected, the error bars in the regions where is well constrained by observations are considerably smaller.

We now turn to the case where the data are noisy, and hence we expect our entropic prior to play a more important role in the reconstruction. The top panel of Figure 2 shows the reconstructed EOS for a noisy data set of measurements with noise and a slowly evolving true EOS, . In the top panel, we show the result when employing as prior model values the two liming cases of with strong theoretical prejudice: (cosmological constant, red errorbars) and (Einstein-de Sitter Universe, blue errorbars). In the high bins the reconstruction becomes increasingly mismatched with the underlying true EOS. Because the dependence of the data on the EOS requires integrating the EOS over redshift, any error in the reconstruction at low redshift is accumulated as increases. As a result the entropy tends to dominate over the likelihood and the mean parameter values collapses towards the model at higher redshift, especially for the case where . Even though the reconstruction has appeared to degrade for the case, it is encouraging that the mean values of the parameters in the lower redshift bins (

Evidently the choice of prior model does have some bearing on the reconstructed value of the parameters at high redshift (given that the entropy dominates the posterior for poorly informative data) and this must be kept in mind when interpreting the results when dealing with noisy data. This example highlights the problem of distinguishing a genuine affinity for a certain function that happens to closely resemble the model from a strong default towards the model on account of noisy observations, i.e. how will we interpret a result very close to ? In this case, there are two options: the observational evidence is noisy and uninformative, leading to entropy domination, or the data is good and favours an actual value close to . In the latter case, attempting a reconstruction with will allow to test the strength of the data in pulling towards the cosmological constant value. An alternative mean of recovering in a truly model–independent way is to include the elements of the model vector in the hypothesis space, as discussed in section 2.2.5. Marginalizing over then amounts to selecting the optimum distribution at each sampled point. The model still represents the most entropic distribution given that it is uniform in –space. The result of this procedure is shown in the bottom panel of Figure 2, where marginalization over the model has cured the skew in the reconstruction observed above, albeit at the price of delivering larger error bars.

## 4 Application to present-day data

We now apply our reconstruction procedure to actual data, encompassing cosmic microwave background observations, baryonic acoustic oscillations measurements, determination of the present value of the Hubble parameter and two different supernovae type Ia samples.

### 4.1 Data sets

#### 4.1.1 Baryon Acoustic Oscillations

The comoving sizes along the line of sight, (in redshift space), and in the traverse directions, , of a feature sitting at a redshift are related to the redshift range covered and the angle subtended , respectively, by

(23) |

If the absolute values of and are known, they become standard rulers giving us a handle on and . If only the relative sizes are known, then the standard rulers are expressed in units of . If only the ratio is known, this becomes the Alcock–Paczynski test. The baryon acoustic oscillation phenomenon can be used as such a standard ruler.

After recombination, when the Universe becomes neutral and photons free–stream from the cosmic plasma, the driving force of the harmonic oscillation is removed and the sound speed of the now-neutral medium essentially falls off to zero, ending wave propagation. The spatial distribution of the baryons at this stage will then reflect the characteristic scale of the acoustic waves. Seeing as the perturbations in the baryon and CDM distributions seed the formation of large scale structure, we expect to see acoustic peaks in the late-time matter power spectrum (Eisenstein et al., 2005). This becomes a standard ruler because the scale of these acoustic oscillations is self-calibrated under standard recombination (Hu, 2005). It depends solely on the photon–baryon ratio and radiation–matter ratio at recombination which are determined with excellent precision in the CMB power spectrum from the CMB peak morphology (Eisenstein & White, 2004). The change in the apparent size of this scale from recombination to the present will depend on the expansion history of the universe through projection effects. These acoustic features appear as rings in angular and redshift space (Hu & Haiman, 2003). The actual measurement from the SDSS LRG sample is of the dilation factor, defined as

(24) |

where the comoving angular diameter distance is taken as the transverse dilation and the line of sight measurement of this scale is taken to be the radial dilation. The observed correlation scale constrains a function of the dilation factor, and a single data point is reported in Eisenstein et al. (2005):

(25) |

(see also Cole et al. (2005) for a similar detection of the acoustic feature in the 2dF catalogue). The above assumes . The log–likelihood for the baryon acoustic oscillation data is given by

(26) |

#### 4.1.2 The Supernova (SNIa) data

Type Ia supernovae are good candidates for standard candles and are useful in determining distances on extragalactic scales. Due to the complexity of the physics involved, the SNIa are not perfect standard candles, having a dispersion of mag in their peak magnitudes (Straumann, 2006). However the peak brightnesses appear to be tightly correlated to the time-scale of their brightening and fading and one can extract an empirical relation between absolute peak luminosity and the morphology of the light curves to constrain the absolute brightnesses, and thus obtain measurements of .

The first group of supernovae, termed the ‘gold’ set, is from the
HST/GOODS programme (Riess
et al., 2004), complemented by the
recently discovered higher redshift supernovae, reported in
Riess
et al. (2006), while the second sample is taken from the
Supernovae Legacy Survey (SNLS) (Astier
et al., 2006). As
discussed in e.g. Wang &
Mukherjee (2006), it appears that there are
systematic differences between these two data sets that arise from
differences in the data processing. It is therefore necessary to
consider the two datasets separately, and compare the results as a
consistency check.
*The ’gold’ sample:*

The distance modulus is defined as the difference between
the apparent magnitude and the absolute magnitude :

(27) |

Given that the absolute magnitude is a unknown, we consider the following distance modulus :

(28) |

Here is the
difference between the mean of the true absolute magnitudes and
the estimated absolute magnitude, while the are the
observed magnitudes after dust corrections and recalibration
through the shape of the luminosity evolution function. The
quantity is the difference between the true mean
absolute magnitude and the estimated absolute magnitude of the
supernovae and is marginalized over.
*The SNLS sample:*

The SNIa data from the Supernovae Legacy Survey (SNLS) are reduced
in a different manner in that the light curves provide constraints
on various parameters which are then used to calculate the
effective apparent magnitude. For a description of the calculation
of see Dick
et al. (2006). From the observed with
variances for each set of SNIa, we perform an
analytical marginalization over the absolute magnitude .
Defining the quantities

(29) |

(30) |

and

(31) |

the M-independent log-likelihood for the SNIa is calculated as

(32) |

#### 4.1.3 CMB and HST data

The WMAP 3-year measurement of the CMB shift parameter describing the location of peaks in the CMB power spectrum serves to constrain the angular diameter distance to last scattering (Spergel et al., 2006). This is independent of most assumptions of the form of dark energy, and is given by (Wang & Mukherjee, 2006)

(33) |

where is the redshift to last scattering, taken in our case to be .

We also include the constraint on the present value of the Hubble constant obtained by the Hubble Space Telescope Key Project Freedman et al. (2001), by using a Gaussian likelihood with mean and standard deviation given by km Mpc s.

### 4.2 Results

We plot in Figure 3 the results of our reconstruction from the SN type Ia data from the SNLS and from the HST/GOODS programmes. In both cases we have added the CMB, HST and BAO measurements, and we have marginalized over the model, in order to be as conservative as possible. Furthermore, the maximum range for the EOS has been taken to be . We plot regions encompassing of posterior probability for each bin – notice that since these are marginalized values, their magnitude is independent of the correlations between reconstructed points (the issue of correlations is addressed in detail below, see Figure 4).

We do not find any significant deviation from a cosmological constant behaviour from the SNLS data (see top panel of Figure 3) at all redshifts. Our posterior constraints in this case are rather conservative, as a consequence of the assumptions made in the reconstruction (i.e., large range and marginalization over the model). When using the HST/GOODS data, the recovered in the first bin agrees with that found for the SNLS data. Given that a number of the SNIa in this bin are common to both surveys, this provides a consistency check. The reconstructed EOS from the HST/GOODS data is however found to prefer slightly higher values in the third bin (), excluding the cosmological constant value to a little bit more than significance. The significance of this rise however has to be assessed with care, especially if we recall that the constraining power of the SNIa data degrades in this region (Simpson & Bridle, 2006). At redshifts above , the mean of the reconstruction settles around , although we notice that the best fitting points remain very close to (red triangles in Figure 3). The implied early-time behaviour of dark energy is consistent with the result of found in Riess et al. (2006) for , using what they call their ’strong’ prior. If instead the EOS is assumed to be constant over the entire redshift space (i.e., if we reduce the number of components to ), then we obtain from the HST/GOODS data , in agreement with usual results (see e.g. Riess et al. (2006)). This clearly demonstrates the danger of assuming to be time–independent, as one would miss in this way possible features in the data.

The use of the entropic prior introduces correlations among the reconstructed points (see Huterer & Cooray (2005) for a technique to extract uncorrelated band power estimates of the EOS). The correlation coefficients from the posterior distribution over the parameters are shown for both data sets in Figure 4. We notice that correlations are in general relatively mild, flattening around the level of for correlations with bins at larger redshifts, where the entropic prior becomes more important. The strongest correlations (at the level of ) are observed among parameters in the 2nd and 3rd redshift bins, where the BAO measurement strongly constrains the EOS and due to the fact that the observables are integrated over redshift, we expect a negative correlation among the well–constrained value at the position of the BAO measurement and the values at lower redshift.

We now investigate the case where we impose that on our parameter space. The results are shown in Figure 5, where the reconstructed EOS using the SNLS tracks the cosmological constant value at all redshifts, with tail errors of order at all values. Because of the reduced freedom in , the reconstruction collapses to the lower limit of the allowed range, even after marginalization over the model values. In the case of the HST/GOODS data, a gentle rise of away from is again observed. The larger error bars suggest that the entropy becomes important and that the value to which the reconstruction tends at redshifts

We can increase the amount of prior information by considering the case where a constant prior model value is used, see Figure 6. This is helpful in assessing whether the structure observed in the HST/GOODS sample is strong enough to override our entropic prior. The reconstruction from the SNLS data remains close to with errorbars of order at all redshift. One has however to keep in mind that the tightness of the errors is partially helped by the supplementary information provided by the entropic prior. This demonstrates how the use of cosmological constant as the model can be problematic as one can not say conclusively whether this indicates that the data is very strong or alternatively overridden by the entropy if no significant deviations from the model are observed. The result from the HST/GOODS data shows again a high value of being favoured in the third bin and subsequent collapse towards to the model at higher redshifts. The persistence of a deviation towards at redshift in the presence of the strong prior favouring the cosmological constant suggests it is a real feature of the data.

Finally, we also investigated the stability of the reconstruction against a change in the number of reconstructed components. Since the MaxEnt technique is designed to automatically deal with the structure in the data by adjusting the degree of smoothness of the reconstruction, we do not expect that a change in the number of bins would make a large difference in the reconstructed EOS. This is demonstrated in Figure 7, where the reconstruction analogous to the bottom panel of Figure 3 has been performed by halving the number of bins to , without appreciable differences in the end result.

Lastly, we investigate the dependency of our results on the model one chooses for the entropic prior. Up until this point we have assumed the prior model to be constant with redshift such that it introduces a suitable degree of smoothing of time–dependent noisy features in the data. This choice reflects a specific belief in the true form of the EOS and given the large range of dark energy models on the market, it is important to assess the impact of a different prior model . Another popular class of models is given by an equation of state that is a smooth varying function of redshift, such as

(34) |

Here the assumption is that the true EOS is a function of time and evolves sufficiently slowly such that it may be effectively characterized in a phenomenological way by the two parameters and (Sahni & Starobinsky, 2006). This particular function is a good approximation to many dark energy models but clearly it is limited to how well it can cope with a rapidly evolving EOS (Liddle et al.,, 2006). Following e.g. Ichikawa & Takahashi (2006), we impose the further constraint that the early Universe is matter–dominated, i.e. we impose the condition on the prior model. In order to be as conservative as possible, we again marginalize over both prior model parameters, and , as follows. The expression for the entropy is again given by Eq. (22), but the model coefficients are now given by

(35) |

Here and are the coefficient representing the prior model parameters. They are included in the hypothesis space, then marginalized over.

We display our results using this slowly–evolving prior model in Figure 8 (with the model parameters marginalized over). This is to be contrasted with the analogous case of Figure 3, where the entropic model is constant in redshift space. At low redshifts the reconstructions for different prior models do not differ appreciably, with the most notable difference being a shift towards slightly higher values of for the case of the SNLS data set. For the GOODS/HST programme data set, the peak in the third bin is recovered but is lower than in the previous constant case. Again, the reconstructed EOS prefers higher values in the last few bins. At high redshifts, due to the accumulation of error, we expect more entropic domination and hence a stronger weight of the prior model choice. However in this region dark energy becomes progressively less important and the ability of data to constrain the time–depedence of the EOS thus degrades considerably. It is encouraging however that the low redshift behaviour of the reconstructed EOS agrees well for both marginalized prior models. This further strengthens our conclusions regarding the power and robustness of our technique.

## 5 Conclusions

Given that the dark energy models on the market are predominantly phenomenological, a reconstruction technique that does not require specifying a parametric form for would be an advantage. We presented such a method based on maximum entropy to reconstruct the equation of state of dark energy within a Bayesian framework. The principle of maximum entropy is invoked when assigning the Bayesian prior. This means that in the absence of genuine signal, the model that is most smooth, or that maximizes the information entropy is favoured, with the extent of this bias being determined by a regularizing constant that is automatically adjusted to the data. In our analysis we decompose into a sum of weighted orthogonal step–functions to facilitate the reconstruction of sharp features (provided binning is sufficient). Extensions of our analysis could easily include using alterative expansion functions to see whether other properties of the time–evolution of dark energy are detected or constrained.

We find that the reconstruction of a dynamical using artificial datasets of and is very promising at low redshifts but suffers from a bias towards the chosen default model at higher . To combat this effect, the prior model was incorporated into our hypothesis space allowing the reconstruction of a model–independent distribution of , with a manageable loss of accuracy. Once the technique was established and demonstrated, it was applied to a combinations of the current cosmological datasets and two popular choices of prior models, namely a constant EOS and a mildly evolving .

Using a dataset including the current WMAP3 measurement of the CMB shift parameter, the baryon acoustic oscillation measurement and the HST Key project measurement of the Hubble parameter in conjunction with the SN type Ia data from the SNLS project, we found that in the redshift range , with errorbars depending on the prior and model assumptions. In the most optimistic case, where the data are supplemented by an entropic prior around , the error is of order at all redshifts. When the same dataset was instead supplemented by the SN sample from the HST/GOODS program, the results agree at low (

The MaxEnt technique presented here improves on other methods designed to minimize noise artifacts in that the amount of information taken from the data is not determined by the analyst but rather dictated by the data themselves. The presence of real structure rather than noise-induced complexity is indicated by the size of the error bars. The entropic prior adjusts the error bars when the information provided by the data is unreliable. Secondly, in the absence of real information the reconstruction tends toward our most intuitive estimate of with suitably large variance. In conclusion the merits of this technique are that it is self–regulating in the sense that it allows the data to determine the amount of structure that is included. More importantly it does not require an inherent assumption of the functional form of the true equation of state.

We hope that this technique will prove useful in deriving even stronger, model–independent constraints on the dark energy history from future, high–quality data.

Acknowledgments The authors wish to thank Bruce Bassett, Pedro Ferreira, Scott Dodelson, Josh Frieman, Dragan Huterer, Joe Silk and Glenn Starkman for stimulating discussions and useful suggestions. We are grateful to Sarah Bridle for suggesting to include the model in the parameter space and to an anonymous referee for many valuable comments. CZ is supported by a Domus A scholarship awarded by Merton College. RT is supported by the Royal Astronomical Society through the Sir Norman Lockyer Fellowship and by St Anne’s College, Oxford. The use of the Glamdring cluster of Oxford University is acknowledged.

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