Dodging the dark matter degeneracy while determining the dynamics of dark energy
One of the key issues in cosmology is to establish the nature of dark energy, and to determine whether the equation of state evolves with time. When estimating this from distance measurements there is a degeneracy with the matter density. We show that there exists a simple function of the dark energy equation of state and its first derivative which is independent of this degeneracy at all redshifts, and so is a much more robust determinant of the evolution of dark energy than just its derivative. We show that this function can be well determined at low redshift from supernovae using Gaussian Processes, and that this method is far superior to a variety of parameterisations which are also subject to priors on the matter density. This shows that parametrised models give very biased constraints on the evolution of dark energy.
Since the discovery in 1998 that the Universe is undergoing a period of accelerated expansion riess1998 (); perlmutter1999 (), considerable work has been done trying to understand what it is driving the phenomenon. The standard explanation beyond a simple cosmological constant relies on introducing a new cosmic fluid, dubbed dark energy, which possesses negative pressure in order to accelerate the expansion. Dark energy is parametrised via an equation of state , where is its pressure, is its energy density and stands for a possible dependence of the equation of state with the redshift . The standard case is given by , which represents the cosmological constant , where together with cold dark matter (CDM) form the cosmic concordance model CDM.
Extensive observational efforts have been carried out in order to falsify through several observables, as type Ia supernovae (SNe Ia), temperature anisotropies of the cosmic microwave background (CMB) and baryon acoustic oscillations (BAOs). As there is no compelling model to explain the acceleration, it is often simplest to use simple parameterisations for , where a dependence with the redshift can be accommodated by adding a new parameter. On the other hand, an interesting and independent way to test for deviations of the CDM model is to use non-parametric methods, which are suitable for describing how evolves not restricted by simple parametric models. Examples considered in the literature include the principal component analysis method (PCAs) pca (), gaussian processes (GPs) gp1 (); gp2 () and local regression smoothing lrs ().
Reconstructions have been limited by degeneracies involving the matter density parameter and the curvature density parameter . From one side, can be inferred due its geometrical structure omega_k (), whereas for there is a so-called dark degeneracy kunz2009 () whereby dark energy and dark matter are hard to separate.
Here we show that there is a simple function of and its first derivative which is completely independent of . This function can be used to look for a time evolution of the dark energy equation of state without any dark matter degeneracy. We use data from SNIa and measurements from cosmic chronometers and BAOs to constrain this new function, along with the evolution of . We compare parametric reconstructions to a GP reconstruction and show that GPs provide much stronger constraints than parameterisations at low redshifts and are broadly consistent with CDM.
Ii The evolution of dark energy and the distance
The dimensionless comoving distance can be written as a function of the luminosity distance as
where is the Hubble constant. The effective dark energy equation of state is
where a prime denotes differentiation with respect to the redshift. Throughout the paper we assume , although we note a degeneracy between and can be important at high redshifts degeneracy_w_om_k (). Given distance-redshift data, we can determine from this relation. However, the value of must be provided a priori from independent methods in order to test for deviations of om-w (). Similarly, one can write the effective adiabatic sound speed for dark energy as
Combining with we define
Note that is completely independent of and is a function of the observable quantity and its derivatives. Model-independent constraints on – and consequently – can be derived through non-parametric methods. Any deviation from would rule out the flat CDM model, and so forms a useful null test of concordance cosmology foot () (see sahni2014 () for another null test independent of ). Another way to think of the function is to say that in the possible function space of , selects the subspace which is independent of , while giving a measure of the first derivative of .
ii.1 Constraining and
We shall now constrain in 2 different ways. Given SNe Ia and data, we need to find and its first 3 derivatives. First we will use Gaussian Processes (GPs) which are a robust smoothing technique which allows for differentiation of data – this technique is summarized in the Appendix. We will use the third line in Eq. II to obtain constraints on and Eq. 2 with a suitable prior on to derive constraints for . Then we will compare these results with two parametric models. We choose a standard one and a new one which produces similar constraints on but rather different constraints on the evolution.
A standard series expansion in : , generally called the CPL (Chevalier-Polarski-Linder) parameterisation cpl ().
Constant : . This model is constructed such that is constant. Note that gives , which means that although rules out flat CDM, is also compatible with different models.
Although the second of these may look rather unusual, these are actually rather similar in terms of their basic assumptions because the CPL parameterisation is constructed so that is constant.
For both the non-parametric and parametric analyses we use 580 luminosity distance measurements from SNe Ia union2.1 (), where we included all the systematic errors, and 26 measurements from cosmic chronometers hz_cc () and BAOs hz_bao (). For the Hubble constant we adopt km s Mpc Komatsu:2010fb (), which is in agreement with the revisited local value based on the NGC 4258 maser distance obtained by Efstathiou efs (): km s Mpc .
As we assume spatial flatness (), the measurements can be directly used to obtain the first derivative of : , which allows us to incorporate the derivative of as data in our GPs analysis.
For the parametric analyses, we obtain the posterior probabilities (see e.g. ratra ()) using emcee emcee (), an affine-invariant ensemble sampler for Markov Chain Monte Carlo (MCMC) affine (). For example, in the CPL case we use the posteriors of to derive constraints to sampling from these parameters, the same for using the second line of Eq. II.
An important issue in our analysis is that distance moduli of SNe Ia were used already fitted for some nuisance parameters in a given cosmological model union2.1 (). Therefore, a more model-independent procedure would be to introduce these nuisance parameters in our GP regression. On the other hand, such approximation was also taken in our parametric analyses, which makes a comparison between them consistent.
In Fig. 1 we show constraints on , and for the different methods. The parametrised models show broadly similar behaviour for the constraints on and give similar errors compared to those found using GPs, at least for low to moderate (strong constraints at high only come from the choice of parameterisation). In this case, the good constraints with GPs are obtained because of the prior adopted on , where much weaker ones would be derived considering a broad flat prior on – this arises simply from (2). For the situation is rather different, with very different constraints found for the different approaches. Constraints at all redshifts between parametrised models versus GPs are very different. With GPs is well constrained at small redshifts and weakly constrained for redshifts above that. Also it is interesting to point out that for GPs the constraints on are weakly dependent on the data, where they are slightly enlarged if we do not use this sample. In redshift regions well sampled by data good constraints are derived, while redshift regions sparsely measured are not able to distinguish between different values for , when combined with the fact that changes in affect at low most strongly. This is in stark contrast with the results from the parametric methods, which provide better results at high redshifts due to the imposed shape of the function a priori.
In Fig. 2 we show the posteriors at for all the different cases. With no prior on only found by GPs has strong constraints, while individual constraints on and are significantly weaker. With a prior on , the constraints on , and vary significantly. In particular, the parametric approaches favour both and in different ways (CPL favours nearer 0 and more negative, and the constant model pulls in the opposite directions), while GPs give tight constraints approaching the union of the parameterised models.
It is important to stress one needs to be careful of not deriving biased results due to a given value for . For example, there is a tension for when using GPs and our (somewhat arbitrary) choice of prior on which completely disappears when using Planck values planck () for and : . The data prefer lower values of as it was shown in H0_bcs (). Therefore, GP reconstructed gives strong constraints favouring a CDM model, with providing an unbiased, more powerful tool to look for deviations of the concordance model as a time-varying dark energy.
ii.4 Validating the method
In order to analyse the robustness of our method, we performed 1000 simulations of SNe Ia data in a CDM model with , km s Mpc and the same redshift distribution of the Union2.1 sample. For the sake of simplicity, we restricted ourselves only to SNe Ia data and we show the results for with an emphasis on .
Fig 3 presents the average posterior for given the 1000 simulations, where the left panel has the result for and the right panel the result for for all redshifts. As we can see, the width of the curve is consistent with the errors obtained, and there is a small bias much smaller than the errors. We also have checked that the results are not influenced by the fiducial cosmological model, since changing for XCDM or a time-varying equation of state for dark energy gave consistent results, as it is shown in Fig. 4. Bigger samples as the ones coming from the ongoing Dark Energy Survey will demand a thorough scrutiny of the bias, although it can be safely neglected for the current size of SNe Ia samples.
We have shown that for a flat model there exists a combination of and its first derivative which can be found purely from Hubble normalised distance data independently of . This provides a route by which we can place constraints on the time evolution of without requiring a prior on . At this corresponds to roughly the lines in the plane where const. We have shown that using GPs strong constraints can be placed on using current data, and with no assumptions on or . This is to be seen in stark contrast to parametric models which have significantly weaker constraints, but an order of magnitude. Furthermore, the constraints on are similarly much stronger than constraints on either or at the origin. Consequently, using as a complimentary function to might be a more useful way to characterize the time evolution of than is currently done.
Of course, if one wants explicitly the degeneracy with is unavoidable. In this sense, have we really dodged the degeneracy? We have presented a new perspective on the question: since is usually a placeholder for whatever is ‘not-’, it may well be a sub-optimal measure of ‘not-’. A set of observables such as , formed from (II) rephrase the question as: do ? but now independently of .
The authors thank M. Seikel for her participation during early stages of this work. VCB thanks Michel Aguena da Silva for his help with MCMC. VCB was supported by CNPq-Brazil, with a fellowship within the program Science without Borders, FAPESP (grant number 2014/21098-1) and CAPES. CC is funded by the National Research Foundation (South Africa).
Appendix A Gaussian processes
A Gaussian process is defined by a collection of random variables, any finite number of which have (consistent) joint gaussian distributions rasmussen (). Basically speaking, while a gaussian variable is a distribution over random variables, a gaussian process is a distributions over random functions, characterized by a mean and a covariance function.
In order to reconstruct a given function, we assume that this function is a realization of a GP with a given mean and covariance function. Assume we have observational data points of a function at redshifts . The errors of the observations are given in the covariance matrix . Additionally we have observations of the first derivative , given by , and . Thus, the goal is to reconstruct the function which underlies the data at redshifts . We denote these function points as and the th derivative of the function at these redshifts as .
Generally, we incorporate the desired properties of the function under study in the covariance function. In this work we adopt the Matérn() covariance function (see sc2013 () for a discussion on the impact of the covariance function)
where we have two hyperparameters and which describe how the function changes in the and axis, respectively.
The probability distribution for the points we want to reconstruct is gaussian with the mean and covariance given by
Here, the covariances are written as matrices with . denotes the th derivative of with respect to the first argument and the th derivative with respect to the second argument.
The hyperparameters are determined by maximizing the likelihood
While the best approach would be to marginalize over the hyperparameters, it was shown in sc2013 () that for a sample of the same size considered in this work, maximization provides basically indistinguishable results at low redshifts, which is the region where good constraints were derived.
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