Cosmic Expansion History from Sne Ia Data Via Information Field Theory: the Charm Code

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 expansion 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 expansion 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, ω ≡ 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 † −1 on potential solutions s(x). We construct this prior from the in- P(s) = G(s − tbg, S ) = √ exp − (s − tbg) S (s − tbg) , | | 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 φ†i φ diation). Translated to the log-log coordinates we advocate this where is the prior covariance matrix = (s|S ) 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 ∞ a†b ≡ 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(s|S ), 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(s|S ), 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(|x − y|), 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 = |x − y|. 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(s|d), given by Bayes’ Theorem, We develop and test charm, so that it is ready for application P(d|s)P(s) P(s|d) = , (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(d|s) 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(s|d) as We present a comparison of charm with previous literature in P(d|s)P(s) 1 Sect. 6. Finally, we present the results of the reconstruction in P(s|d) = = e−H(d,s), (3) Sect. 7 and conclude in Sect. 8. P(d) Z where H is called the information Hamiltonian and Z is the partition function. They are defined as 2. Inference approach H(d, s) = − ln(P(d|s)P(s)), (4) Z Z 2.1. Basic notation Z(d) = Ds e−H(d,s) = Ds P(d|s)P(s) = P(d), (5) First of all, we needed to establish some notations and assumptions. We derived the algorithm of charm in the framework of where Ds is a phase space integral. IFT, following the notation of Enßlin et al.(2009). We have as- The information Hamiltonian comprises all available infor- sumed that we are analyzing a discrete set of data d which may mation and is for this reason the central mathematical object depend on a signal s, which contains the physical quantities of in our method development. Its minimum with respect to s for interest. In this case the signal is a field, s(x), chosen to be the a given dataset d, for example, is the maximum of the pos- logarithm of the cosmic density ρ(z) as a function of the logarith- terior (MAP). Our algorithm calculates the MAP estimation mic scale factor, x = − ln(a) = ln(1 + z), where z is the redshift. (Lemm 1999) of the expansion history. This parametrization is natural, as it deals with dimension- less quantities, represents cosmic periods dominated by a con- 1+ω 3. Database stant equation of state ρ ∝ a as straight lines s = s0 +(1+ω)x, and converts relative variations of O(e) to additive variations SNe Ia have been found to be an excellent probe for studying the of O(1). expansion history of the Universe. Observations of the nearby This coordinate system has the advantage that we can and distant SNe Ia led to the discovery of the accelerating expan- model the signal uncertainties by Gaussian fluctuations around a sion (Perlmutter et al. 1999; Riess et al. 1998). For this reason,

A92, page 2 of9 N. Porqueres et al.: Cosmic expansion history from SNe Ia data – the charm code

have a˙ H = (6) a and

!2 H ρ = , (7) H0 ρcrit0

2 where ρcrit0 = 3H0 /(8πG). The density ρ is usually assumed to be polynomial in a with three contributing terms:

0 −2 −3 −4 ρ(a) = ρΛa + ρka + ρDMa + ρrada . (8)

However, as we aim for a non-parametric reconstruction, we do not assume this polynomial form but just that ρ is smooth with a−1 as well as that power-law like building blocks of ρ(a) are Fig. 1. upper panel Hubble diagram for the Union2.1 compilation ( ) and favored. residuals of data respect to the Planck cosmology (bottom panel). As data we use the distance moduli of SNe from Union2.1, assuming Gaussian noise. As our signal we define we choose supernovae as our initial data set to develop an IFT s(x) = ln(ρ(x)/ρcrit0), (9) based method for reconstructing the cosmic expansion history. Kowalski et al.(2008) described a method to analyze the with SNe Ia data in a homogeneous manner and created a compilation, the Union SN Ia compilation, combining the world’s x = − ln(a) = ln(1 + z). (10) SN data sets. Both new data and literature SNe were fit using a spectral template for lightcurve fitting, also known as SALT The distance modulus is (Guy et al. 2005). Here, we use the Union2.1 compilation, which contains " Z z # 0 1 580 SNe Ia: the 557 data from Union2 and 23 new measurements µ(z) = 5 log (1 + z)dH dz − 5, (11) 10 E(z0) at redshift z > 1 (Suzuki et al. 2012). The data are distributed in a 0 redshift range of 0.015 < z < 1.414 corresponding to an x-range where d is constant in a flat universe, d = c/H , and E = of 0.014 < x < 0.881. Union2.1 provides the redshift, distance H H 0 H/H . In ΛCDM we have modulus and distance modulus error for each supernova, which 0 are shown in Fig.1. The catalog also includes uncertainty co- p 3 variance matrix with systematics. E(z) = Ωm(1 + z) + ΩΛ, (12) In this work, we attempt a tomographic inversion of the SN data into a cosmic expansion history, where the term tomo- while our generic parametrization we have graphic means a reconstruction of a searched object properties distributed along a coordinate for which the allocation does not E(z) = exp(s(x)/2). (13) follow from observation, in this case a reconstruction along the line of sight. Tomography is very sensitive to systematic errors 4.1. Response operator and therefore, we should use a non-diagonal noise covariance matrix N in our reconstruction in order to account for correlated Our cosmology signal s has been imprinted onto the data. The systematic uncertainties. Some systematic errors are common in functional relationship between the noiseless data µ and the sig- all the observations while other sources of systematic errors are nal can be regarded as a non-linear response operator. To be pre- controlled by the individual observers. cise, the response operator describes how the distance moduli Kowalski et al.(2008), identified two categories of system- at different redshifts of the SNe depend on the unknown cosmic atic errors: the ones that affect measurements independently for expansion history encoded in our signal s. each SN Ia (e.g. due to observational method) and systematic Combining our parametrization (Eqs. (9) and (10)) with errors that affect the measurements of SNe at similar redshifts Eq. (11) yields the response as (e.g. due to astrophysics). Since the different sources of system- Z x j atic errors can be considered to be independent, the total error h 1 0 0 i x j 0 − 2 s(x )+x can be well approximated as a Gaussian error. In AppendixA R j(s) = 5 log10 e dH dx e − 5, (14) we briefly discuss the sources of systematics. 0

with x j = ln(1 + z j). 4. Reconstruction The response depends on an intergral over our signal s(x). The goal of our tomographic method is to invert this integration. In order to reconstruct the cosmic expansion history, which is Since the response is not a linear operation of the form R(s) = encoded in the Hubble parameter H(z), from supernovae data R s + const. it cannot be algebraically inverted, which would al- we write H in terms of the scale factor a = (1 + z)−1. Then we low us to minimize the information Hamiltonian directly.

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4.2. Expansion of the information Hamiltonian to a Gaussian joint probability of signal and data, implying a Gaussian posterior, In order to build the information Hamiltonian H(d, s) = − ln P(d|s) − ln P(s), we describe our a priori knowledge on s P(ϕ|d) ≈ G(ϕ − m, D). (23) with a Gaussian P(s) = G(s − tbg, S ) as described in Eq. (1). The background cosmology t and the signal covariance S are to be In this approximation the information propagator is the a poste- bg † specified later. The noise is assumed to be Gaussian and inde- rior uncertainty covariance D = h(ϕ − m)(ϕ − m) i(s|d,S ), very † pendent of the signal, distributed as P(n|s) = G(n, N), since the similar to the a priori covariance S = h(s − tbg)(s − tbg) i(s|S ). sources of uncertainty which operate on the distance modulus µ The covariance D is smaller than S , since the presence of the can be approximately assumed to be independent from µ. data has restricted the set of possible cosmologies. The infor- To perturbatively expand our problem, we can write the in- mation source j contains the data and has excited the increased formation Hamiltonian in terms of s = t + ϕ, where we call t the knowledge on s. This approximative linear solution to the infer- pivot field and ϕ its perturbation. By assuming a signal covari- ence problem is known as Wiener filter solution and D is also ance S , the expanded Hamiltonian reads called the Wiener variance. This Wiener filter solution is not at the minimum of the unap- 1 H(d, ϕ|t) = − log proximated Hamiltonian. To reach it we have to repeat the pro- |2πS |1/2|2πN|1/2 cedure after shifting the pivot cosmology to t ← t + ϕ until t no 1 † longer changes. We initialize the iteration by t = tbg and stop it + d − R(t + ϕ) N−1 d − R(t + ϕ) 2 when t becomes stationary. Our iterating Wiener filtering scheme can be regarded as a Newton method minimizing the full original 1 † −1 + (ϕ + t − tbg) S (ϕ + t − tbg), (15) Hamiltonian, regularized by expanding the response and not the 2 Hamiltonian to ensure convergence of the Newton method, for where the prior background cosmology only affects the terms which we had to simplify our propagator operator (AppendixB). related to our prior. The amplitude of the perturbation is assumed Therefore, the correct MAP solution will be found at the end of to be |ϕ| < 1 and then we can expand the response operator since the iterations. exp(t + ϕ) ≈ exp(t)(1 + ϕ). Our posterior is a non-Gaussian probability distribution 5. Cosmology prior function. In order to minimize the probability in an efficient and numerically robust way, we Gaussianize it by linearizing the re- In this section, we focus on the definition of the prior as a Gaus- sponse R(s) around the pivot field t, sian process characterized by a power spectrum, which varies around a background cosmology (Lemm 1999; Enßlin et al. ∂R 2 2 R(s) = R(t) + |s=tϕ + O(ϕ ) = R(t) + Rtϕ + O(ϕ ). (16) 2009). ∂ϕ First, we consider that the background expansion history tbg is that of the ΛCDM model, which is the currently accepted cos- The linearized response R is t mology. To validate our reconstruction, we take two more ex- q jx pansion histories: (1) a CDM cosmology, without a dark energy, (Rt) jx = −α , (17) in order to check that charm is able to recover the cosmologi- r j cal constant even if the prior does not favor its presence. (2) An with α = 5/(2 ln 10), agnostic cosmology that has little knowledge of the equation of states of radiation, matter and dark energy for maximally model- 1 − tx+x+x j q jx ≡ dH e 2 θ(x j − x), (18) independent cosmic history reconstruction. and 5.1. Prior Z x j 1 0 x j 0 − tx0 +x r j ≡ e dH dx e 2 . (19) As prior knowledge, we assume that the signal s(x) exhibits only 0 limited curvature since all known contributions to s(x) like mat- The resulting approximated Hamiltonian, ter, radiation and dark energy contribute linear segments. We therefore punish curvature as measured by ∇2ϕ and write our 1 † −1 negative log prior as H(d, ϕ|t) = (d − R(t) − Rtϕ) N (d − R(t) − Rtϕ) (20) 2 Z 1 † †2 2 1 † −1 H(s − tbg) = dx (s − tbg) ∇ ∇ (s − tbg). (24) + (ϕ + t − tbg) S (ϕ + t − tbg), 2 2 is then minimized by m = D j with This should control the degree of smoothness of %(a) = sx=ln a %crit0 e in the following fashion. A one-sigma fluctuation in † −1 −1 −1 terms of this prior energy corresponds to a bending of the slope D = (R N Rt + S ) , (21) t of s(x) by one per e-folding of cosmic expansion. Given that in the so-called information propagator operator and the standard cosmological model from radiation domination until today the scale factor changed by eight e-foldings (four orders −1 −1 j = RtN (d − R(t)) − S t − tbg , (22) of magnitude) and the combined equation of state of mass in the Universe changed only by four (from radiation to dark energy) the so-called information source field. we see that that the standard cosmological expansion history is The terms information source j and information propagator well contained within the one sigma contour of our prior. We D may require a brief explanation. In this linear response ap- can quantify the strength also with respect to data consistency: proximation the Hamiltonian is quadratic in s. This corresponds the variance of d2 ln %(a)/(d ln a)2 is punished with a strength,

A92, page 4 of9 N. Porqueres et al.: Cosmic expansion history from SNe Ia data – the charm code which requests that for this quantity being on a level of one it is Thus, the agnostic background expansion history is given by required that the data is under stress by at least one sigma with tbg = 2x. the reconstructed expansion history. The prior for the variation around this background expan- The signal prior covariance matrix S is then given by sion is assumed to be the same prior as before, since we do not S −1 = ∇†2∇2, (25) expect the signal to have any strong curvature. In fact, the agnostic cosmology is the ideal background expansion history for this which is a diagonal matrix in Fourier space3, non-parametric reconstruction, as log-log density expansion s(x) 0 −4 is not curved and therefore any bending of s and ϕ are exactly S 0 = 2πδ(k − k )k . (26) kk equivalent.

5.2. Planck cosmology

The ΛCDM model is based upon a spatially flat and expand- 6. Comparison with other methods ing Universe, in which the dynamic of the cosmic expansion is dominated by the CDM and a cosmological constant (Λ) at late In order to emphasize the advantages of our approach, we com- times. pare charm to previous literature. Ishida & de Souza(2011) de- According to the second Friedman equation, the expansion velop a non-parametric method for the reconstruction of the H(z) rate H(a) is given by based on the principal component analysis (PCA) of the Fisher T 8πGρ kc2 Λc2 matrix F = D ΛD. This Fisher matrix corresponds in our no- H2 − , † −1 = 2 + (27) tation to the term R N R in our propagator operator. While 3 a 3 Ishida & de Souza(2011) regularize their solution by cutting o ff where ρ is the energy density, k is the curvature and G is the higher order PCA components, we can preserve all modes of the gravitational constant. expansion, just weight their contribution to the solution accord- By defining the density parameters ΩX as the ratio between ing to their bending on log-log scale. Furthermore, our method s the density of X and the critical density, the second Friedman enforces the positivity of the density ρ = ρcrit0e . equation reads Shafieloo et al.(2006) reconstructs the expansion history p H(a) = H Ω + Ω a−2 + Ω a−3 + Ω a−4. (28) with a non-parametric method that also enforces smoothness of 0 Λ k m rad the solution. However, in their work the data is smoothed di- In a flat universe, the curvature density parameter is null, rectly to suppress noise (while a background expansion history ΩK = 0. Therefore, from the definition of our signal s(x) = is temporarily subtracted), while we just encourage smoothness ln(ρ(x)/ρcrit0), the background cosmology is given by via a prior without modifying the data. In the end, also our approach averages nearby data to suppress noise, but it does this in −3 −4 ≈ −3 t = ln(ΩΛ + Ωma + Ωrada ) ln(ΩΛ + Ωma ), (29) an adaptive way, having a shorter effective smoothing length in as the Ωrad 1 in the late Universe. regions of dense data, whereas Shafieloo et al.(2006) employ a In terms of the coordinate x = − ln(a) of our signal-space, constant smoothing kernel in redshift space and have to experi- the former equation is written as ment with the kernel size. To summarize, our approach requires less tuning of regu- t ≈ ln(Ω + Ω e3x). (30) Λ m larization parameters compared to previous approaches, as our The values we adopt for the density parameters at present time prior assumption on the problem solution, that power law equa- are the ones found by the Planck mission from the cosmic mi- tion of states within a certain range are more natural, already crowave background (CMB) data, Ωm = 0.314 and ΩΛ = 0.686 fully specifies the necessary regularization. (Planck Collaboration XVI 2014). The same values of the Ωm and H0 are used for the CDM model. We use this unrealistic CDM scenario to test how much our reconstruction depends on the assumed background 7. Results cosmology. Since in this case Ω < 1, we allow for a curvature −2 term Ωk = 1 − Ωm, which evolve with a . We investigate in this section how far the assumption of a prior cosmology affects the reconstruction. 5.3. Agnostic cosmology First of all, in Sect. 7.1, we show that the iterative Wiener filter is able to successfully reconstruct a perturbation of the In order to reconstruct the cosmic expansion history in a way ΛCDM model. For this purpose, a mock data catalog is simu- that is agnostic about the assumed constituents of the Universe, lated from a randomly perturbed cosmology. like matter, radiation and dark energy, we assume another back- Once we have convinced ourselves that our algorithm recon- ground cosmology, which we called agnostic cosmology. structs the cosmology from data accurately, we apply it to real As the ΛCDM model has contributions to the density evolu- data. Now the impact of a prior background cosmology needs to −4 0 tion of terms proportional to a and a , the agnostic background be investigated. For this reason, we start with different expansion −2 cosmology is taken as being proportional to a , which is the histories: the ΛCDM model, for which we expect that the recon- geometric mean between these extremes. The exponent 2 of this structed corrections are nearly a constant around zero. Then, we background slope is then on log-log scale assume the CDM model as our initial guess, for which we ex- ∂s ∂ln ρ pect that the cosmological constant is recovered. Finally we test = = 2. (31) ∂x ∂ln a the agnostic cosmology, in order to see if the iterative Wiener filter is able to reconstruct the standard ΛCDM cosmology from R 3 The following Fourier convention will be used f (k) = f (x) eikx dx. a non-informative prior.

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7.1. Mock data Before applying charm to the real data, we validate it with a simulated database. These data were generated from a perturbed Planck cosmology, where the perturbation was a random ﬁeld. The aim of generating mock data is to show that our mech- anism for the reconstruction of the cosmology is able to ﬁnd a perturbation of the ΛCDM model if it is present in the data, even in case the perturbation amplitude is much smaller than the Planck cosmology amplitude. The perturbation is introduced as an additive term to the ΛCDM model, s = tΛCDM + ϕ. It is generated from Gaussian probability density,

( " !2 !2# ) 1 Z 1 ∂2ϕ 1 ∂ϕ P ϕ ∝ x , ( ) exp 2 2 + 2 d (32) σA 2σc ∂x 2σα ∂x that ensures the smoothness of the perturbed field. Here, σ can α Fig. 2. Reconstruction using mock data generated with a perturbed be considered as a parameter to control the slope of the perturba- Planck cosmology with σA = 0.1 (upper panel). The blue and yellow tion, which we assume to be σα = 2 since this should naturally σ −4 0 regions correspond to the prior and posterior 1 uncertainty limits re- permit slopes of ρ ∝ a to ρ ∝ a bracketing our agnostic back- spectively, which are obtained from the diagonal of the prior and the di- ground cosmology ρ ∝ a−2 in case of σ = 1. The parameter σ 1/2 1/2 A c agonal of the propagator operator via σx = S xx and σx = Dxx . Bottom controls the curvature of the perturbation, which is assume to be panel: reconstruction of the perturbation. 0.5, since we do not expect much curvature, while σA is a nor- malization constant to control the amplitude of the perturbation. As we focus on small perturbations, which are the realistic ones, this parameter is fixed to 10−1. We note that the perturbation is generated from a different probability distribution than the prior used in our inference. This is on purpose to test the robustness of charm. Once the perturbed Planck cosmology and mock data are generated, we can apply charm. The number of simulated SNe Ia is 580 spread in a redshift range of 0.015 < z < 1.414, as in the real database. The amplitude of the noise, σn, is set to the amplitude of the perturbation, σA. In the results shown in Fig.2, the initial background cosmology is assumed to be the Planck model. It is seen that the perturbation is small, since the per- turbated cosmology, used to generate the data, is similar to the Planck cosmology. The prior used for the reconstruction is the one from Eq. (25) and the result follows the perturbed expansion history. In order to see that the perturbation can be recovered, we plot its reconstruction in the bottom panel of Fig.2. Fig. 3. Upper panel: reconstruction assuming Planck cosmology as background cosmology. The blue and yellow regions correspond to the 7.2. Reconstructed cosmology prior and posterior 1σ uncertainty limits respectively. Middle panel: residuals of the reconstruction; bottom panel: deviation of the recon- Now, we can apply charm to real data from Union2.1 compi- struction from Planck cosmology. lation under the various prior background cosmologies as described before. Figure3 displays the results assuming the ΛCDM model residuals are almost the same for the Planck cosmology and for as the cosmology background. The reconstruction is compati- our reconstruction. ble with Planck ΛCDM cosmology. The displayed uncertainty Figure4 shows the reconstruction for the CDM model as limits of the reconstruction are provided by the diagonal of the prior background cosmology. Although this model differs signif- Hessian, considering all the terms in the inverse propagator D−1 icantly from the Planck cosmology, our reconstruction is com- plus the term from Eq. (B.1) that was was omitted during the patible with the ΛCDM model. This shows that the data strongly numerical minimization of the Hamiltonian for stability reasons. favors a cosmological expansion history dominated by CDM and From this, we can conclude that the data agree with the ΛCDM a cosmological constant. In the following subsection, we recover model, as expected. the density of the cosmological constant ΩΛ by fitting this recon- In order to study the agreement of the data with the Planck struction. The residuals in this case are equivalent to the ones in cosmology, we calculate the signal response, Rs, for our recon- the middle panel of Fig.3. struction. The residuals of the real data with respect to this are Finally, we adopt the agnostic cosmology as our prior back- shown in the bottom panel of Fig.1 and they can be compared ground dcosmology. The result is shown in Fig.5 and demon- to the residuals of the Planck cosmology in the middle panel of strates that the Planck cosmology is recovered even in case Fig.3, calculated by applying the response operator to the Planck charm is not informed about the specific equations of state of cosmology and substracting the real data. We can see that the matter, radiation and dark energy.

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Fig. 4. Reconstruction assuming CDM model as background cosmology Fig. 6. Reconstruction of a perturbation of the Planck expansion history (upper panel) and deviation of the reconstruction from Planck cosmol- at high-redshifts. ogy (bottom panel).

and thus, the slope of the linear regression corresponds to the mass density and the independent term is identified as the density of dark energy. The result of the fitting is ΩΛ = 0.670, which is compatible with the value obtained by Planck mission, ΩΛ = 0.686 ± 0.020 (Planck mission, 2014). This parametric fit to our non- parametric reconstruction should just illustrate the consistency of our reconstruction with other measurements. For a proper uncertainty quantification the parametric model should be fit to the data directly, which was already done by many other works.

7.4. Constraints and Union3 Rubin et al.(2016) announced recently that a new update of the Union catalog is in preparation, called Union3 Supernova compilation. This new compilation will include high-redshift (z > 1) supernovae observed with the Hubble Space Telescope. Fig. 5. Reconstruction assuming agnostic cosmology as background cosmology (upper panel) and deviation of the reconstruction from The study of the BAO provides constraints to the evolution Planck cosmology (bottom panel). of dark energy density. Recently, BAO measurements have suggested a change in the density parameter of the dark energy at large redshifts z > 1 (Delubac et al. 2015; Aubourg et al. 2015). From the results in this section, we can conclude that the This early evolution of dark energy could be determined more assumed prior background cosmology does not strongly affect precisely when data of SNe Ia at high-redshift are available. our results. The ΛCDM is better recovered if it is assumed in In order to prepare our reconstruction algorithm for future the prior, but also a CDM or the agnostic cosmology prior yield work, we check that it is able to reconstruct a perturbation that a reconstruction close to the Planck ΛCDM model. Further- exists only for large redshifts (z > 1). We generate a perturbation more, we verified by not here displayed experiments that the ini- at z > 2 by adding a parabola to the Planck cosmology as a test. tial pivot cosmology has no impact on the finally reconstructed As in Sect. 7.1, the number of simulated SNe Ia is 580 but now cosmology. spread in a redshift range of 0.015 < z < 3.0, that is an antici- pation of the future improved datasets. We can see in Fig.6 that this perturbation can be recovered by charm. This reconstruc- 7.3. Fitting ΩΛ from the reconstruction tion assumed a Planck model as initial background cosmology and the prior from Eq. (25). We have seen in Fig.4 that the reconstruction from CDM model recovered the cosmological constant. We can take advantage of charm this to validate by comparing the density of the cosmo- 8. Conclusions logical constant that we obtained with the values in the literature. In order to obtain the value of ΩΛ from our reconstruction, In this work we have developed charm, a Bayesian inference we transform our reconstruction in such a way that we can fit method to reconstruct non-parametrically the cosmic expansion a linear regression model. From the definition of the signal as history from SNe Ia data and applied it to the Union2.1 dataset. s = ln E2(x), this transformation reads as As shown in Sect. 7, the choice of the background prior cosmological expansion history does not significantly affect the final 3x s X ≡ e , Y ≡ e → Y = ΩmX + ΩΛ (33) result of the reconstruction.

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We found that the Planck cosmology is reconstructed inde- Bernal, C., Cardenas, V. H., & Motta, V. 2017, Phys. Lett. B, 765, 163 pendently of a prior assumed background cosmological expan- Delubac, T., Bautista, J. E., Busca, N. G., et al. 2015, A&A, 574, A59 sion history. We have also recovered the density parameter Ω Enßlin, T. A., Frommert, M., & Kitaura, F. S. 2009, Phys. Rev. D, 80, 105005 Λ España-Bonet, C., & Ruiz-Lapuente, P. 2005, Astrophysics, High Energy in Sect. 7.3 and found a value of 0.670 which is compatible with Physics – Phenomenology [arXiv:hep-ph/0503210] the one from Planck Collaboration XVI(2014). España-Bonet, C., & Ruiz-Lapuente, P. 2008, J. Cosmol. Astropart. Phys., 2, Although no evidence for a deviation from the standard cos- 018 mological expansion history has been found, we have tested our Genovese, C., Freeman, P., Wasserman, L., Nichol, R., & Miller, C. 2009, Ann. Appl. Statist., 3, 144 method whether it is able to reconstruct a perturbation if it ex- Guy, J., Astier, P., Nobili, S., Regnault, N., & Pain, R. 2005, A&A, 443, 781 isted. Our method was able to reconstruct a relative perturbation Hee, S., Handley, W. J., Hobson, M. P., & Lasenby, A. N. 2016, MNRAS, 455, to ΛCDM even with an amplitude as low as 10−1. 2461 Since recent analysis of BAO showed new constraints on the Ishida, E. E. O., & de Souza, R. S. 2011, A&A, 527, A49 evolution of dark energy density at early epochs (Delubac et al. Kowalski, M., Rubin, D., Aldering, G., et al. 2008, ApJ, 686, 749 Koyama, K. 2016, Rep. Prog. Phys., 79, 046902 2015), the study of SNe Ia at high-redshift will be crucial in the Lemm, J. C. 1999, Physics [arXiv:physics/9912005] future for constraining any time variation in dark energy. In order Li, Z., Gonzalez, J. E., Yu, H., Zhu, Z.-H., & Alcaniz, J. S. 2016, Phys. Rev. D, to have charm prepared for the release of the database with su- 93, 043014 pernovae at high-redshift, we have checked that the reconstruc- LSST Science Collaboration 2009, ArXiv e-prints [arXiv:0912.0201] Mannucci, F., Della Valle, M., & Panagia, N. 2006, MNRAS, 370, 773 tion method could find a perturbation that existed only at high- Melia, F., & McClintock, T. M. 2015, AJ, 150, 119 redshifts. Thus, charm will be a useful tool to study dark energy Montiel, A., Lazkoz, R., Sendra, I., Escamilla-Rivera, C., & Salzano, V. 2014, in a more model-independent way when the new SNe Ia datasets Phys. Rev. D, 89, 043007 become available. Moresco, M., Pozzetti, L., Cimatti, A., et al. 2016, J. Cosmol. Astropart. Phys., 5, 014 Perlmutter, S., Aldering, G., Goldhaber, G., et al. 1999, ApJ, 517, 565 Acknowledgements. Part of this work was supported by Deutscher Akademis- Planck Collaboration XVI. 2014, A&A, 571, A16 cher Austauschdienst (DAAD). The computations were done using the NIFTy4 Riess, A. G., Filippenko, A. V., Challis, P., et al. 1998, AJ, 116, 1009 package for numerical information field theory (Selig et al. 2013). Riess, A. G., Macri, L. M., Hoffmann, S. L., et al. 2016, ApJ, 826, 56 Rubin, D., Aldering, G. S., Amanullah, R., et al. 2016, in AAS Meeting Abstracts, 227, 139.18 Selig, M., Bell, M. R., Junklewitz, H., et al. 2013, A&A, 554, A26 References Shafieloo, A., Alam, U., Sahni, V., & Starobinsky, A. A. 2006, MNRAS, 366, 1081 Amanullah, R., Lidman, C., Rubin, D., et al. 2010, ApJ, 716, 712 Simon, J., Verde, L., & Jimenez, R. 2005, Phys. Rev. D, 71, 123001 Aubourg, É., Bailey, S., Bautista, J. E., et al. 2015, Phys. Rev. D, 92, 123516 Suzuki, N., Rubin, D., Lidman, C., et al. 2012, ApJ, 746, 85

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Appendix A: Systematic errors A.5. Malmiquist bias A.1. Color corrections Malmiquist bias arises in flux limited surveys. The Union compilation attributes a systematic uncertainty of ∆M = 0.02 in ab- The empirical color corrections account for dust and intrinsic solute magnitude due to this bias. color–magnitude relation. The validity of the color correction, which is an empirical relation, relies on the assumption that nearby and distant SNe Ia have the same color–magnitude rela- A.6. Gravitational lensing tion. This correction could become a source of systematic error Gravitational lensing causes dispersion in the Hubble diagram at if different corrections were required for different SN popula- high redshift (Kowalski et al. 2008). This effect is treated statis- tions or if the distance to the SN affected this magnitude–color tically in the Union compilation. The uncertainty due to grav- relation. itational lensing is larger than the intrinsic dispersion only for The second case seems to be unlikely since the color correc- high-redshift SNe but it causes a bias of magnitudes. This bias is tion relation at high and low redshift agree. Although this agree- not present if fluxes are used instead of magnitudes. ment could arise from a different proportion of reddening and intrinsic color at different redshift, it supports the empirical relation for color correction (Kowalski et al. 2008). A.7. Galactic extinction The presence of two SN populations is supported by two The photometry is corrected for galactic extinction using an ex- types of SN-progenitors timescales argued by Mannucci et al. tinction law that assumes R = 3.1, together with dust maps (2006). If this two populations are present, they might evolve V (Amanullah et al. 2010). The galactic extinction is more impor- in a different way with redshift. If the full sample is divided into tant in nearby SNe, since the distant ones are measured in red- equal subsamples by splitting by color, the color correction is der bands and then, its R is approximately the color correction, significantly different for the two subsamples (Amanullah et al. R without an important effect of the galactic extinction. 2010). This suggests that the color–magnitude relation could be more complex than a linear relation. A.8. Host-mass correction coefficient A.2. Sample contamination The SNe Ia luminosity is related to the mass of the host galaxy, even after color corrections (Suzuki et al. 2012). The host galax- In order to avoid the contamination of the data by non-type ies for low-redshift SNe Ia are more massive on average than the Ia SNe, which are not standard candles, an analysis technique host galaxies for high-redshift SNe Ia. This can bias cosmologi- was developed by Kowalski et al.(2008). This method, which 2 cal results and it can be corrected by fitting a step in the mass of is based on χ minimization, rejects the outliers from the sam- 10 the host-galaxy at mthreshold = 10 M . The problem is that for ple. However, the systematic uncertainties are cast into an un- the Union2.1 compilation the individual host galaxy masses are certainty of the absolute magnitude ∆M. In order to consider the not known. To overcome this problem, a probabilistic method sample contamination, an uncertainty ∆M = 0.015 was added to was defined to determine the host mass correction. This proce- the covariant matrix due to contamination. dure could carry systematic errors, that are taken into account in the covariance matrix as ∆M = 0.02. A.3. Lightcurve model The lightcurve model is a fit with two parameters and it becomes Appendix B: Simplification of the propagator a limit in capturing the diversity of SNe Ia. A problem arises operator when different techniques are used to observe nearby and distant The propagator operator D measures the convex part of cur- supernovae, which implies that the parameters are obtained by vature of the information Hamiltonian, since it is the Hamil- fitting a different part of the curve in high-redshift SNe. tonian second derivative except for terms due to higher order A Monte-Carlo simulation was performed by Kowalski et al. non-linearities in the response. This D operator is mostly used to (2008) in order to quantify this systematic error, obtaining that guide our gradient descent via the regularized Newton method. the difference of nearby and distant SNe is ∆M = 0.02 mag. Since the Newton method is not suited for negative curvatures, the term

A.4. Photometric peak magnitude X 1 qixqiy − riqixδ(x − y) α N−1 d − R(t) , (B.1) 2 2 i j j The uncertainty of the peak magnitude is due to the color cor- i j ri rection. In order to measure the color, the flux is measured in at least two bands. Since the spectra of the SNe at different red- which would be part of the Hessian of the full Hamiltonian is shifts are obtained from different bands, their color is determined dropped by our response linearisation in order to avoid numeri- from different spectral regions. Then, the uncertainty in the dif- cal problems caused by negative Eigenvalues of D. This is justi- ferent regions of the reference Vega spectrum limits the accuracy fied, as D only guides the numerical scheme, while the unmodi- of SNe color measurement. fied j determines where the scheme finally converges to. The Union compilation assumes an uncertainty in the ab- This simplification is allowed because we are iterating the solute magnitude of ∆M = 0.03 for the photometric peak Wiener filter to find the global minimum of the information magnitude. Later, in the Union2 compilation, the numerical ef- Hamiltonian, and for this, it is not necessary to perform an opti- fect of each passband on the distance modulus was computed. mal step in each iteration just that all steps go in the right direc- This was a more efficient way to include this systematic er- tion. Inhibiting negative Eigenvalues of our simplified Hamilto- ror than including a constant magnitude covariance for all SNe nian ensures that the resulting Newton method always descends (Amanullah et al. 2010). toward the global minimum of the Hamiltonian.

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