A&A 599, A92 (2017) Astronomy DOI: 10.1051/0004-6361/201629527 & c ESO 2017 Astrophysics Cosmic expansion history from SNe Ia data via information field theory: the charm code Natàlia Porqueres1, Torsten A. Enßlin2, Maksim Greiner2, Vanessa Böhm2, Sebastian Dorn2, Pilar Ruiz-Lapuente4; 3, and Alberto Manrique1; 3 1 University of Barcelona, Departament de Física Quàntica i Astrofísica, Martí i Franquès 1, 08028 Barcelona, Spain e-mail: [email protected] 2 Max-Planck-Insitut für Astrophysik (MPA), Karl-Schwarzschild-Strasse 1, 85741 Garching, Germany 3 Institut de Ciències del Cosmos, Martí i Franquès 1, 08028 Barcelona, Spain 4 Instituto de Física Fundamental, CSIC, Serrano 121, 28006 Madrid, Spain Received 13 August 2016 / Accepted 3 December 2016 ABSTRACT We present charm (cosmic history agnostic reconstruction method), a novel inference algorithm that reconstructs the cosmic expan- sion history as encoded in the Hubble parameter H(z) from SNe Ia data. The novelty of the approach lies in the usage of information field theory, a statistical field theory that is very well suited for the construction of optimal signal recovery algorithms. The charm algorithm infers non-parametrically s(a) = ln(ρ(a)/ρcrit0), the density evolution which determines H(z), without assuming an analyt- ical form of ρ(a) but only its smoothness with the scale factor a = (1 + z)−1. The inference problem of recovering the signal s(a) from the data is formulated in a fully Bayesian way. In detail, we have rewritten the signal as the sum of a background cosmology and a perturbation. This allows us to determine the maximum a posteriory estimate of the signal by an iterative Wiener filter method. Applying charm to the Union2.1 supernova compilation, we have recovered a cosmic expansion history that is fully compatible with the standard ΛCDM cosmological expansion history with parameter values consistent with the results of the Planck mission. Key words. methods: statistical – methods: data analysis – supernovae: general – distance scale 1. Introduction The aim of this work is to reconstruct the cosmic expan- sion history, encoded in the Hubble parameter H(z), from super- Combined observations of nearby and distant type Ia Super- novae data in the framework of Information Field Theory (IFT; novae (SNe Ia) have demonstrated that the expansion of the Uni- Enßlin et al. 2009). Conceptionally, IFT is a statistical field the- verse is accelerating in the current epoch (Perlmutter et al. 1999; ory that permits the construction of optimal signal recovery al- Riess et al. 1998). Such a Universe can be described by the cold gorithms. To this end, we developed the charm1 code, which is dark matter (ΛCDM) model, in which the cosmic acceleration freely available2. We use the Union2.1 Supernova compilation, is determined by Einstein’s cosmological constant with a time- which is a database that contains 580 SNe Ia in the redshift range independent equation of state, ! ≡ p/ρ = −1. However, this of 0:015 < z < 1:414. is just one of the possible explanations of the expansion that is Deriving the cosmic expansion history is a major goal of consistent with the SNe Ia measurements. Others include a new modern cosmology. To date, the low-redshift evolution of the field component filling the Universe as a quintessence or modi- Hubble parameter H(z) has been studied with different methods. fied gravity (Koyama 2016). Some recent studies present analysis of the cosmic expansion 2 Constraining the cosmic expansion as a function of redshift by χ minimization (Bernal et al. 2017; Melia & McClintock is a task of major interest, since the evolution of the scale factor 2015) while others develop non-parametric methods to solve the allows us to probe properties of the fundamental components of inverse problem of reconstructing the Hubble parameter H(z) the Universe. This may lead to a better understanding of their (Li et al. 2016; Montiel et al. 2014; Ishida & de Souza 2011; nature as well possibly providing evidence for new fundamental Shafieloo et al. 2006) or the equation of state of dark energy physics. (España-Bonet & Ruiz-Lapuente 2005, 2008; Simon et al. 2005; Genovese et al. 2009). Common to all non-parametric recon- Recent studies of the baryonic acoustic oscillations (BAO) structions, the ones cited above and the one we develop here, have suggested different constraints on the density of dark is that a quantity to be reconstructed (Hubble parameter, cosmic energy at high redshifts (Delubac et al. 2015; Hee et al. density, equation of state, etc.) as well a regularization for the 2016). Such a change in the evolution of the dark energy den- otherwise ill-posed inference problem must be chosen. The dis- sity in the early epoch could be determined from SNe Ia cussed methods differ in what regularization is chose. data at high redshifts (z > 1), which will be available shortly Here, we develop a non-parametric reconstruction in natu- (Rubin et al. 2016). In addition, some years from now a sam- ral coordinates for the reconstruction of the logarithm of the ple of 105 SNe Ia is expected to be available from the LSST (LSST Science Collaboration et al. 2009). This upcoming data 1 charm stands for cosmic history agnostic reconstruction method. will open an entirely new chapter in the study of dark energy. 2 https://gitlab.mpcdf.mpg.de/natalia/charm Article published by EDP Sciences A92, page 1 of9 A&A 599, A92 (2017) cosmic density s = ln(%=%crit0) as a function of the logarithm background cosmology, denoted as tbg(x): of the cosmic scale factor x = − ln a and thereby the Hubble parameter H(z). The regularization arises from a Bayesian prior ! 1 1 y −1 on potential solutions s(x). We construct this prior from the in- P(s) = G(s − tbg; S ) = p exp − (s − tbg) S (s − tbg) ; j j 2 sight that constituents of the cosmic density are likely to scale 2πS with the inverse scale factor to some power typically (but not (1) exclusively) between zero (cosmological constant) and four (ra- S S hφ φyi φ diation). Translated to the log-log coordinates we advocate this where is the prior covariance matrix = (sjS ) with = s − tbg. Scalar products of continuous quantities are defined as to be natural, this means that straight lines in s(x) are preferred R 1 ayb ≡ x a x b x over curved ones. We can also motivate the level of expected 0 d ( ) ( ). 2 curvature: a transition from radiation to dark energy domination The diagonal of the prior covariance, S xx = hφxi(sjS ), encodes within a few e-folds of expansion has to be possible if our stan- how much variation of the signal around the a priori background dard cosmological expansion history should be embraced by the cosmology tbg is expected a priori at every location x. The off- prior. diagonal terms of the covariance, S xy = hφxφyi(sjS ), specify how An advantage of the adopted Bayesian methodology lies in correlated such deviations form the background cosmology are the fact that it provides a flexible framework to question data: expected to be between the points x and y. A larger correlation it can reconstruct the cosmic expansion history using different corresponds to smoother structures of the deviations. In Sect.5, priors. For example, it can be asked how much the data re- we will use simple and intuitive arguments about the expected quests a modification of a given cosmology or what the pref- roughness of s to specify S , as well as different choices of the ered expansion history is from a cosmological composition ag- background cosmology tbg. In particular, as no location of cos- nostic point of view. The main assumption is a smooth behavior mic history is singled out a priori on a logarithmic scale, the prior of the logarithm of the density ln ρ with the logarithmic scale covariance structure should be homogeneous, S xy = Cs(jx − yj), factor, ln a, whereas the strength of this assumption can also be with Cs(r) a correlation function that only depends on the dis- varied. tance r = jx − yj. We probe that charm is sensitive to features in expansion history at any low-redshift, z < 1:5. In addition, the algorithm 2.2. Signal inference is easily extendible to include other datasets, such as BAO or Cepheids (Riess et al. 2016), which provide information of a In the inference problem, we are interested in the probability transition epoch between deceleration and acceleration of the of the signal given the data. This is described by the posterior cosmic expansion (Moresco et al. 2016; Hee et al. 2016). P(sjd), given by Bayes’ Theorem, We develop and test charm, so that it is ready for application P(djs)P(s) P(sjd) = ; (2) to the new catalog Union3 compilation, which is expected to pro- P(d) vide information about the dark energy density at high-redshifts. After this introduction, we establish our notation and present which is the product of the likelihood P(djs) and the signal prior our assumptions and the inference problem in Sect. 2. In Sect. 3, P(s) normalized by the evidence P(d). the SN Ia catalog is described and we derive our reconstruction In the framework of IFT, inference problems are formulated method in Sect. 4. In Sect. 5, we specify our prior knowledge and in the language of statistical field theory. To that end we rewrite the cosmological expansion histories that we use to test charm. the posterior P(sjd) as We present a comparison of charm with previous literature in P(djs)P(s) 1 Sect. 6. Finally, we present the results of the reconstruction in P(sjd) = = e−H(d;s); (3) Sect.