reconstruction of late-time cosmology using principal

EPJ manuscript No.(will be inserted by the editor)

Reconstruction of latetime cosmology using Principal ComponentAnalysis

Ranbir Sharma1a Ankan Mukherjee2b H. K. Jassal1c

1 Indian Institute of Science Education and Research Mohali, SAS Nagar, Punjab 140306, India2 Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India.

the date of receipt and acceptance should be inserted later

Abstract. We reconstruct late-time cosmology in a model-independent manner using the technique ofPrincipal Component Analysis (PCA). In particular, we focus on the reconstruction of the dark energyequation of state from two different observational data-sets, Supernova type Ia data, and Hubble parameterdata. The analysis is carried out in two different approaches. The first one is a derived approach, where wereconstruct the observable quantity using PCA and subsequently construct the equation of state parameter.The other approach is the direct reconstruction of the equation of state from the data. A combination ofPCA algorithm and calculation of correlation coefficients are used as prime tools of reconstruction. Wecarry out the analysis with simulated data as well as with real data. The derived approach is found tobe statistically preferable over the direct approach. The reconstructed equation of state indicates a slowlyvarying equation of state of dark energy.

PACS. Cosmology, dark energy equation of state reconstruction, Principal Component Analysis, correla-tion coefficient.

1 Introduction

Cosmological parameters are now constrained to muchbetter precision than before [1]. This has been facilitatedwith significant improvements in observational techniquesand the accessibility of various observational data. Datain the last two decades has confirmed that the dynamics ofthe Universe is dominated by a negative pressure compo-nent, known as dark energy. Dark energy is understood tobe the component driving the observed accelerated expan-sion of the present Universe. One of the prime endeavors ofmodern cosmological research is to reveal the fundamentalidentity of dark energy, its exact nature and evolution.

It is not clear from the present observations whetherdark energy is a cosmological constant [2,3,4,5] or a time-evolving entity [6,7]. The dark energy can be describedby equation of state parameter w = −Pde/ρde, whereρde is the energy density and Pde is its pressure contri-bution. The form of the equation of state parameter wof dark energy depends on the theoretical scenario be-ing considered. A constant value w = −1 corresponds tothe ΛCDM (cosmological constant with cold dark matter)model, whereas in case of time-evolving dark energy, theequation of state parameter can have different values [2,5,6,8,9,10,11,12,13,14,15,16,17]. We have little theoretical

a E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]

insight into these models, except for the ΛCDM model,which has a strong theoretical motivation. However, thestandard (ΛCDM) model faces the problem of fine-tuning[2,5,8,9]. The observed value of the cosmological constantis found to be many orders of magnitude smaller than thevalue calculated in quantum field theories. Various alter-native models have been proposed, which are based on flu-ids, canonical and non-canonical scalar fields. These mod-els have the fine-tuning problem of their own, for instance,scalar fields require potentials which are specifically tai-lored to match observations [18,19,20,21,22,23,24,25,26,27,28,29,30,31].

Since the approach to understanding dark energy isprimarily phenomenological, it is necessary to determinethe model parameters to constrain and rule out mod-els that are not consistent with data. Likelihood analy-sis is the most commonly used technique in cosmologi-cal parameter estimation and model fitting [26,27,32,33,34,35,36,37]. It is based on Bayesian statistical inference,where the posterior probability distribution of a parame-ter is determined with a uniform or variant prior functionand the likelihood function. A combination of differentdata-sets, with likelihood regions complementary to eachother allows a very narrow range for cosmological param-eters. Thus the combination of various data-sets tightensthe constraints on the parameters. This parametric recon-struction [31,38,39,40,41,42,43,44,45,46] assumes a func-tional form of the equation of state parameter of dark

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2 Sharma, Mukherjee, Jassal: Reconstruction of late time cosmology using PCA

energy. However, it induces the possibility of bias in theparameter values due to the assumption of the paramet-ric form. One elegant way to constrain the equation ofstate parameter is by the differential age method of cos-mic chronometer technique, which can be used to find outthe evolution of the Universe without any inference froma particular cosmological model [16,47,48,49].

An alternative method is to reconstruct the evolutionof cosmological quantities in a non-parametric fashion.Various statistical techniques have been adopted for non-parametric reconstruction of cosmological quantities [50,51,52,53,54,55]. One of the promising techniques in non-parametric approaches is the Gaussian Process(GP) [49,56,57,58,59,60,61], where we can create a multivariateGaussian function with a determined mean and corre-sponding covariance function from any finite number ofcollections of random variables.

To accomplish the reconstruction of cosmological quan-tities, we use the Principal Component Analysis (PCA). Acomparative study of different model-independent meth-ods can be found in [62]. PCA is a multivariate analysisand is usually employed to predict the form of cosmo-logical quantities in a model-independent, non-parametricmanner [63,64,65,66,67]. In [68] and [69], different vari-ants of PCA techniques have been adopted. Reference [68]uses an error model and then creates a different set ofsimulated Hubble data to construct the covariance ma-trix while [69] use the weighted least square method andcombines it with PCA.

From the viewpoint of the application of PCA, thereexist two distinct methodologies. These methods differmainly in the way the covariance matrix is calculated,which is the first step of any PCA technique. One way toimplement PCA is through the computation of Fisher ma-trix, [62,63,64,65,66,67,70,71,72,73,74,75]. One can binthe redshift range and assume a constant value for thequantity to be reconstructed in that redshift bin. Theseconstant values are the initial parameters of the PCA.Fisher Matrix quantifies the correlation and uncertain-ties of these parameters. Therefore, by deriving these con-stants in different bins using PCA, we can reproduce ourtargeted quantity in terms of redshift [67]. Alternatively, apolynomial expression for the dynamical quantity can beassumed. In this case, the coefficients of the polynomialare the initial parameters for PCA, and the analysis givesthe final values of these coefficients, which eventually givesthe dynamical quantity [64].

PCA is independent of any prior biases and it is alsohelpful in comparing the quality of different data-sets [70,76]. It is an application of linear algebra, which makesthe linearly correlated data points uncorrelated. The cor-related data points of the data-set used in PCA are trans-formed by rotating the axes, where the angle of rotationof these axes is such that linear-correlations between data-points are the smallest compared to any other orientation.The new axes are the principal component(PC)s of thedata points, and these PCs are orthogonal to each other.In terms of information in the data-set, PCA creates a hi-erarchy of priority between these PCs. The first PC con-

tains information of the signal the most and hence hasthe smallest dispersion of data-points about it. The sec-ond PC contains less information than the first PC andtherefore provides a higher dispersion of the data-pointsas compared to the first PC. Higher-order PCs have theleast priority as these correspond to noise and we can dropthem. The reduction of dimensions is a distinctive featureof PCA. Therefore the final reconstructed curve in thelower dimension corresponds predominantly to the signalof the data-set. The PCA method also differs from theregression algorithms, which can not distinguish betweensignal and the noise. PCA can omit the features com-ing from the noise part and can pick the actual trend ofthe data-points [77,78,79]. In our case, the data-points onwhich we apply PCA are created in the process of recon-struction, see sect(2).

We assume polynomial expressions for the observa-tional quantities, namely the Hubble parameter H(z), andthe distance modulus (µ(z)). The polynomial form is thenmodified and predicted solely by PCA reconstruction al-gorithm. The only assumption we make is that it is possi-ble to expand the reconstructed quantity in a polynomial.The dark energy equation of state is constructed in thesubsequent steps after the reconstruction of the Hubbleparameter and distance modulus.

The two-step process of reconstruction is termed as thederived approach. Further, in a direct approach, we applythe PCA technique to the polynomial expression of theequation of state parameter w(z). For the application ofour methodology, we require a tabulated dataset of theobservational quantity, which can schematically be rep-resented as, (independent variable)–(dependent variable)–(error bar of dependent variable). In the direct approach,described below, we do not have the tabulated data-setof w(z) and we derive the functional form of w(z) fromthe tabulated data-set of z − H(z) and z − µ(z). In thecase of the derived approach, PCA first reconstructs thefunctional form of the H(z) and µ(z) for the Hubble pa-rameter data and Supernova data. Application of PCA onpolynomial expressions to reconstruct cosmological quan-tities have been employed earlier in [67,68,69].

For a further check on the robustness of our results;we compute the correlations of the coefficients, presentin the polynomial expressions of the reconstructed quan-tities. We apply PCA in the coefficient space, which iscreated by the polynomial. The PCA rearranges the cor-relation of the coefficients in the polynomial. Calculationof correlation-coefficient helps to identify the linear andnon-linear correlations of the components in the recon-structed quantities. We show that using the correlationcoefficients calculation, we can restrict the allowed termsin the polynomial.

This paper is structured as follows. In sect(2) we de-scribe the reconstruction algorithm and the use of cor-relation tests in those reconstructions. In sect(3), we de-scribe the two approaches of reconstruction we follow. Wepresent our results in sect(4). In sect(5), we conclude bysummarizing the results.

Sharma, Mukherjee, Jassal: Reconstruction of late time cosmology using PCA 3

2 Methodology

We begin with an initial basis, gi = f(x)i−1, where i =1, 2, ...., N through which we can express the quantity tobe reconstructed as,

ξ(x) =

N∑i=1

bif(x)(i−1) (1)

The initial basis can be written in matrix form as,G = (f1(x), f2(x), ..., fN (x)). Coefficients bi create a coef-ficient space of dimension N . Each point in the coefficientspace gives a realization of ξ(x), including a constant ξ(x).PCA modifies the coefficient space and chooses a singlerealization. One can consider different kinds of functionsas well as different combinations of polynomials as initialbases. Choosing one initial basis function over the otheris done by correlation coefficient calculation described be-low.

The Pearson correlation coefficient for two parametersA and B is given by,

ρ =Cov(A,B)

σAσB(2)

where ρ ∈ [−1, 1]. For linearly uncorrelated variables, thecorrelation coefficient, ρ = 0. An exact correlation is iden-tified by ρ = −1 or ρ = +1. The Spearman rank coefficientis, in turn, the Pearson correlation coefficient of the ranksof the parameters; rank being the value assigned to a setof objects and it determines the relation of every object inthe set with the rest of them. We mark the highest numericvalue of a variable A as ranked 1, the second-highest nu-meric value of the variable as ranked 2 and so on. A similarranking is done for the ranks of the parameter B.

To obtain the coefficients for the correlation analysis,we divide the parameter space into n patches. We there-fore have n values associated with one coefficient of thepolynomial; this is the number of columns of the coeffi-cients matrix Y, (eq(5)). After ranking all the values ofA and B, we obtain the table for the ranks of A and B.We then proceed to compute the Spearman correlationcoefficient(r), which is the Pearson Correlation coefficientof rank of A and B. Like in the case of the Pearson Cor-relation coefficients, r ∈ [−1, 1]. Computing Kendall cor-relation coefficient(τ) gives a prescription to calculate thetotal number of concordant and dis-concordant pairs fromthe values of the variables A and B [80].

If we pick two pairs of points from the table of A andB, say (ai, bi) and (aj , bj), for i 6= j if ai > aj when bi > bjor if ai < aj when bi < bj ; then that pair of points aresaid to be in concordance with each other. On the otherhand, for i 6= j, ai > aj when bi < bj or if ai < ajwhen bi > bj , then these two pairs are called to be indis-concordance with each other. Every concordant pairis scored +1 and every dis-concordant pair is scored −1.The Kendall correlation coefficients are defined as,

τ =actual score

maximum possible score(3)

maximum possible score =n(n− 1)

2

Again, if Ncp is the number of concordance pair and Ndpis the number of dis-concordance pair is

actual score = Ncp −Ndp

Hence the expression of τ is,

τ =Ncp −Ndpn(n− 1)/2

(4)

where, τ ∈ [−1, 1].We perform the correlation coefficients calculation twice.

The first time is to select the number of terms in the ini-tial polynomial N(eq(1)), and second time to select thenumber of terms in the final polynomial, (eq(8)). We se-lect that value of N for which the Pearson CorrelationCoefficient is higher than the Spearman and Kendall Cor-relation coefficients. We use the R-package for statisticalcomputing to calculate the correlation coefficients [81].

Values of linear and non-linear correlations depend onthe quantity we want to reconstruct. They are also sensi-tive towards the data-set that we use in reconstruction.For instance, reconstruction of a fast varying function,which has non-zero higher order derivatives will introducemore non-linear contributions to the correlation of bi thanlinear contributions, and we need a greater set of initialbasis, which means a higher value of N . We take differentvalues of N and check the linear and non-linear correla-tion coefficients to fix the value of N . Though a large valuewould help, we can not, however, fix N to any arbitrarilysignificant number as it makes the analysis computation-ally expensive.

To compute the covariance matrix, we define the coef-ficient matrix (Y) by selecting different patches from thecoefficient space,

Y = [[bji]]j=1,N

i=1,n(5)

where n is the number of patches that we have taken intoaccount and N is the total number of initial basis definedin eq(1); therefore, Y is a matrix of dimension N ×n and

b(N)n being the value of Nth coefficient in nth patch.

In the present analysis, we have taken n to be the orderof 103. We estimate the best-fit values of the coefficientsat each patch by χ2 minimization, where χ2 is defined as

χ2 =

k∑j=1

(ξ(x)data − ξ(bi, x))2

σ2j

(6)

k is the total number of points in the data-sets. If theobservational data-set have significant non-diagonal ele-ments in the data covariance matrix(Cdata), we have toincorporate Cdata in eq(6), as χ2 = ∆TC−1data∆, where∆ = ξ(x)data − ξ(bi, x) and x, ξ(x)data vary for eachdata-points. Calculation of χ2 in all the patches gives usthe variation of the N coefficients and finally gives n num-ber of points in the coefficient space, over which we apply


PCA. We calculate the covariance matrix and correlationsof the coefficients for these n points. In this analysis, eachpatch contains the origin of the multi-dimensional coeffi-cient space.

The covariance matrix of the coefficients, C is writtenas,

C =1

nYYT

Eigenvector matrix, E of this covariance matrix will ro-tate the initial basis of the coefficient space to a positionwhere the patch-points will be uncorrelated. We organizethe eigenvectors in the eigenvector matrix E in the increas-ing order of eigenvalues.

If U = (u1(x), u2(x), ..., uN (x)) the final basis is givenby,

U = GE (7)

The final reconstructed form of ξ(x) is,

ξ(x) =

M∑i=1

βiui(x) (8)

where M ≤ N and the βis are the uncorrelated coeffi-cients associated with the final basis. The coefficients βiare re-calculated by using χ2 minimization, for the samen patches considered earlier to create initial coefficientmatrix (5). The value of M can be determined by cor-relation coefficient calculation discussed above. As PCAonly breaks the linear correlation, we select M for whichPCA is able to break Pearson correlation coefficient tothe largest extent. Eigenvalues of the covariance matrix(e1, e2, e3, ..., eM ) indicate how well PCA can pick the bestpatch-point in the coefficient space. We look for the low-est number of final basis which can break the linear cor-relation to the greatest extent. We can start with smallervalue of N which eventually influence the value of M , butit will pose the risk of losing essential features from PCAdata-set.

3 Reconstruction of dark energy equation ofstate

The Hubble parameter (H(z)) for a spatially flat Universe,composed of dark energy and non-relativistic matter isgiven by,

H2(z) = H20 [Ωm(1 + z)3 +Ωxe

3∫ z0

1+w(z′)1+z′ dz′ ] (9)

Here we have assumed that the contributions to the energydensity is only due to the non-relativistic dark matter anddark energy. The density parameters for non-relativisticmatter and dark energy are given by Ωm and Ωx. Thequantity H0 denotes the present-day value of the Hubbleparameter, namely the Hubble constant and w(z) is thedark energy equation of state parameter (w(z)).

We assume no interaction between matter and dark en-ergy in the present analysis. In the following subsections,we discuss the derived and direct approach to reconstructw(z) using PCA.

3.1 Derived Approach

The derived approach is a two-step process in the re-construction of dark energy Equation of State(EoS). Inthe first step, we reconstruct the observable, namely theHubble parameter using the H(z) data and the distancemodulus (µ(z)) using the type Ia supernova data. Subse-quently, we reconstruct w(z) as a derived quantity fromthese two different physical quantities. Similar sequenceof reconstructions have already been discussed in [63,65,67,76]. We follow the approach mentioned in sect(2) toreconstruct the curve of H(z) and µ(z). Differentiatingeq(9) with redshift z as the argument and rearranging theterms we can express w(z) as,

w(z) =3H2 − 2(1 + z)HH ′

3H20 (1 + z)3ΩM − 3H2

(10)

Here, H ′ is the derivative of Hubble parameter with re-spect to redshift z. Since w(z) is related to H(z) througheq(10) by the zeroth and the first order differentiation ofH(z), the small difference in the actual and the recon-structed curve of H(z) is amplified by the H ′ term. Thisprocess of amplification of the deviation from actual na-ture becomes more severe with subsequent higher-orderdifferentiation of the reconstructed quantity.

The luminosity distance dL(z) is given by,

dL(z) =c

H0(1 + z)

∫ z

0

dH(z′)dz′ (11)

where dH , in terms of eq(9) is,

dH(z) =

(Ωm(1 + z)3 +Ωxe

3∫ z0

(1+w(z′))dz′

(1+z′)

)−1/2(12)

and is related to the distance modulus as

µ(z) = 5 log

(dL

1Mpc

)+ 25 (13)

which is the dependent-variable in the type Ia supernovaedata. From PCA, we determine the form of µ(z) directlyfrom data and then from eq(11) and eq(12) find the ex-pression of dL. From eq(13), we trace back to eq(12) andfind an expression which gives the w(z) in terms of thedistance modulus.

Since D(z) = (H0/c)(1 + z)−1dL(z), the equation ofstate parameter is given by

w(z) =2(1 + z)D′′ + 3D′

3D′3Ωm(1 + z)3 − 3D′(14)

The second order derivative in eq(14) makes the recon-struction of w(z) through that of distance modulus unsta-ble. For instance, if the reconstruction fails to pick someof the minute difference in the observational curve, thenthat difference will be amplified two times in the final cal-culation of the EoS. Therefore, the reconstruction of µ(z)should be more accurate in picking up approximately all


the features of w(z) which may be hidden within the su-pernovae data [47,64,82,83].

In the reconstruction of H(z) and µ(z), we begin withpolynomial expansions in terms of the different variablesz, a and (1 − a) where z is the red-shift and a is thescale factor. The analysis is carried out with seven termsin initial basis, which means creating a coefficient spaceof N = 7 dimensions (eq(1)). We test our algorithm ona simulated data-set of ΛCDM cosmology. We generatethe data-points of h(z) and µ(z) for w(z) = −1 and thevalues of cosmological parameters Ωm and H0 are fixed atPlanck 2018 values [1]. Fig(3) and table 1 show that ouralgorithm can predict the simulated data.

3.2 Direct Approach

For the direct reconstruction approach, we begin with apolynomial form of w(z) itself. In eq(9), the quantity w(z)is in the exponent, and considering a polynomial form forw(z) implies addition of some non-linear components toour linear analysis in coefficient space. Again we fix thedimension of the initial basis N by computing the corre-lation coefficients. Here we need to balance available com-putational power as well as the non-linearity we introduce,with the accuracy we demand to choose the value of N .

In this case too the independent variables are takento be a, z and 1 − a. Here we introduce non-linear termsin the initial coefficients of PCA; here correlation coef-ficients calculation is not of assistance as in the case ofderived approach 3.1 to select M . Due to the risk of com-promising different features of the data-set and also dueto the complex dynamics of correlation coefficients, we donot reduce any terms in the final basis (N = M) of thedirect approach.

To test the effectiveness of both the approaches de-scribed above, we first work with simulated data-sets forspecific models. We create simulated data-points for w =−1 (ΛCDM) and w(z) = − tanh (1/z) [68] where the val-ues of Ωm and H0 are fixed at Planck 2018 values [1].We use eq(9) to calculate H(z) at the same redshift valueas in the real Hubble parameter vs redshift data-set [84,85,86,87,88]. Similarly, the distance modulus data pointsare simulated using equations (11),(12) and (13). Here wehave evaluated distance modulus µ(z) at the same redshiftvalues as are there in type Ia supernovae (SNe) data [89].We later utilize the observational measurements of Hub-ble parameter at different redshift [84,85,86,87,88] andthe distance modulus measurement of type Ia supernovae(SNe) data [89].

4 Results

4.1 Derived approach

We first reconstructH(z) and µ(z) using simulated dataset.The reconstruction with simulated data is a check on theviability of the method. Fig (1) shows the reconstructedcurves of the reduced Hubble parameter h(z) and distance

Reduced Hubble constant h0

Variable Simulated data(1 − a) 0.67385

a 0.66393z 0.78649

Table 1. The value of reduced Hubble constant estimated forthe simulated date-set using the derived approach. For gener-ation of the simulated data, h0 is fixed at 0.685.

modulus µ(z) obtained for the simulated data. The recon-structed curves are for three different reconstruction vari-ables (1 − a), the scale factor a and the redshift z. It isclear from these plots that the PCA reconstruction pro-duces a consistent result when (1 − a) is chosen as theindependent variable. We also plot the difference betweenthe fiducial model and the reconstructed curves along fora comparison. The plots of the residues clearly show thatthe reconstruction of both h(z) and µ(z) validates the re-construction appraoch. The number of terms on the ini-tial basis is fixed at N = 7. We also check our results forN = 8 and N = 10, and find that the results do not varysignificantly

From table 1 we see that both (1− a) and a variablesare able to predict the value of h0 closer to the assumedvalue of h0 in simulated data. Further, the correlation coef-ficient calculation suggests that the best choice of numberof terms for variable (1 − a) in the final polynomial(8) isM = N − 1, reduction of one term from the initial poly-nomial expression(1). For variable a and z, it is M = N ,hence there is no reduction of terms.

The correlation coefficients of the first three coeffi-cients of the polynomial expression eq(1) and eq(8) of thereconstructed quantity is shown in table 2. PCA breaksthe linear correlation of the coefficients, which is evidentfrom table[2]. Presence of the non-linear correlation in theinitial coefficients complicates the process. As the firstthree terms of the ultimate expression of the reconstructedquantity contains the dominant trend of the data-points,we only mention the correlation coefficients only for thefirst three parameters. The reconstruction which can breakPearson Correlation to a greater extent as well as havelesser Spearman and Kendall Correlation coefficients isselected. We can see from table 2 that the reconstruc-tion by (1− a), breaks the correlation to a greater extentthan in the case of a and z. Variation of correlation coef-ficients before and after the application of PCA is similarfor both the simulated as well as the real data-set. Differ-ence of Pearson correlation coefficients for simulated andreal data-set in case of (1 − a) and a is of the order of10−6 or less. For the variable z, the difference of Pearsoncorrelation coefficients for real and simulated data is ofthe order of 10−4 or less.

The left panel of fig(2) shows that the variable (1− a)reproduces w = −1 behaviour when the simulated data isused, which is also evident from fig(3). Shaded region offig(3) represents all possible w(z) vs z curves produced byPCA for simulated ΛCDM data with a variation of Ωmfrom 0.1 to 0.5. Ωm is one of the free parameters in our


0.25 0.75 1.25 1.75 2.25z

0.5

1.0

1.5

2.0

2.5

h(z)

0.25 0.75 1.25 1.75 2.25z

0.5

1.0

1.5

2.0

2.5

h(z)

0.25 0.75 1.25 1.75 2.25z

0.5

1.0

1.5

2.0

2.5

h(z)

0.25 0.75 1.25 1.75 2.25z

0.40.30.20.10.00.10.20.3

h(z)

0.25 0.75 1.25 1.75 2.25z

0.40.30.20.10.00.10.20.3

h(z)

0.25 0.75 1.25 1.75 2.25z

0.40.30.20.10.00.10.20.3

h(z)

0.0 0.25 0.5 0.75 1.0 1.25 1.5z

3436384042444648

(z)

0.0 0.25 0.5 0.75 1.0 1.25 1.5z

3436384042444648

(z)

0.0 0.25 0.5 0.75 1.0 1.25 1.5z

3436384042444648

(z)

0.0 0.25 0.5 0.75 1.0 1.25 1.5z

1

0

1

2

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(z)

0.0 0.25 0.5 0.75 1.0 1.25 1.5z

1

0

1

2

3

(z)

0.0 0.25 0.5 0.75 1.0 1.25 1.5z

1

0

1

2

3

(z)

Fig. 1. The plots in the figure show the reconstructed reduced Hubble parameter h(z) and distance modulus µ(z) for simulatedΛCDM data-set along with their residues. Residues are calculated as the difference between PCA reconstruction and thecorresponding h(z) and µ(z) calculated from the cosmological constant model. The left column is for the independent variable(1−a), the middle column is for a and the right column is for z. For the blue curves, there is no reduction, that is, M = N = 7.The green and red curves are obtained by the reduction of the highest and second-highest Principal Components, respectively.

0.1 0.2 0.3 0.4 0.5z

2.5

2.0

1.5

1.0

0.5

0.0

w(z)

0.1 0.2 0.3 0.4 0.5z

2.5

2.0

1.5

1.0

0.5

0.0

w(z)

0.1 0.2 0.3 0.4 0.5z

2.5

2.0

1.5

1.0

0.5

0.0

w(z)

Fig. 2. The reconstructed w(z) curves, obtained in the derived approach using simulated Hubble parameter data, with variables(1 − a), a and z respectively from left to right. Each colour represents a reconstruction with a fixed value of Ωm. In this Blue-Red-Green color variation the value of Ωm varies from 0.1 to 0.4, with a step of 0.015. We fix the reduced Hubble constant h0

at the value obtained from the PCA algorithm, as mentioned in table 1.


0.0 0.1 0.2 0.3 0.4 0.5z

2.5

2.0

1.5

1.0

0.5

0.0

0.5

w(z)

Fig. 3. The plots in this figure show reconstruction of w(z) for the simulated ΛCDM data. Red, blue and green curves are forthe reconstruction with no reduction and with reduction of one and two terms respectively. In these curves, the value of Ωmis fixed at 0.364, the value of Ωm for which the reconstructed curve with no reduction and reduction of one term of the initialbasis are the closest to the underlying w(z) = −1 for most of the low-redshift range. In the figure there are three patches; grey,yellow and brown, corresponding to all the reconstruction with Ωm vary from 0.1 to 0.5 with no reduction of final terms andwith reduction of one and two terms respectively. Here we fix the reduced Hubble constant at the value predicted by PCA forthe simulated data-set, h0 = 0.674. The black line is for comparison and it corresponds to w(z) = −1.

1.0 1.5 2.0 2.5z

0.5

1.0

1.5

2.0

h(z)

0.25 0.50 0.75 1.00 1.25z

36

38

40

42

44

46

48

(z)

Fig. 4. The plots show the reconstruction of h(z) and µ(z) from the simulated Hubble parameter and supernovae data-set withallowed range ΛCDM cosmology. The brown patch is obtained for the variation of wΛCDM and Ωm in the range [-1.5 to -0.5]and [0.2 to 0.4] for ΛCDM cosmology. Solid blue, green and red line are for PCA reconstruction with no-reduction and withone and two terms respectively.

methodology, and the methodology does not pick any onecurve over another. The polynomial expression of H(z)in (1 − a) is effectively an infinite series in terms of z.When the observational data-set is used in this polyno-mial expression, it indicates a time evolving w(z), fig 8.The other two variables, namely a and z, could not suc-cessfully reproduce the w = −1 nature while using thesimulated data. In the case of simulated supernovae data-set we see that all the w(z) reconstructions for h0 = 0.685and Ωm varying from 0.1 to 0.4 follows a similar trendafter we choose a specific basis function. For all the threereconstruction variable w(z) vs z curves fluctuate betweenphantom and non-phantom regime.

We now construct the equation of state parameter w(z)from the reconstructed H(z) and µ(z). Fig(2) shows thew(z) vs z curves reconstructed for the PCA variables (1−a), a and z, using the simulated Hubble data-sets for theΛCDM cosmology. It is clear from fig(2) that the recon-struction by derived approach for the variable (1 − a)successfully reproduces the w(z) assumed earlier. On theother hand, the reconstruction variable a and z do not

reproduce the w(z) which has been assumed to simulatethe data.

Using the reconstructed analytic form of H(z) fromderived approach, we estimate the present day value ofthe Hubble parameter H0. The EoS parameter w(z) cal-culated by the derived approach has the free parame-ter Ωm, which we vary in the reconstructed w(z) curves,fig 8. We present the estimated values of H0, scaled by100 km sec−1 Mpc−1, for the analysis with simulated andthe observational data in table 1. Fig(3) shows the plot forΩm = 0.364 and where reduced Hubble constant is fixedat h0 = 0.674, the value predicted by PCA.

Fig(4) indicates that the PCA reconstructions of h(z)is well within the range of ΛCDM paramters for the simu-lated data. We vary Ωm and equation of state parameterin ΛCDM cosmology (wΛCDM ) in the range [−1.5, −0.5]and [0.2, 0.4] respectively. The ability of our methodologyto reconstruct h(z) and µ(z) reflects in the reconstructionof w(z) for the simulated data.

To quantify the efficiency of our algorithm in pickingup the underlying theory, we create an error function by


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Fig. 5. Plots of w(z) in the direct approach with simulated data-sets. The left panels are for simulated data with w(z) = −1and the right panels are for w(z) = − tanh (1/z). The first row is the reconstruction by variable (1 − a), second and the thirdrow is the reconstruction by variable a and z, respectively. Black solid line corresponds to the minimum χ2 and the red shadedregion is the reconstruction using the PCA algorithm allowing a deviation of 0.3 from the minimum χ2 curve.

the method of interpolation with the errors of observa-tional data and use χ2 with a data-set constructed fromthe final reconstruction curve of PCA. For this testingpurpose we assume w(z) and Ωm as parameters, thereforeconstants for a particular data-set produced from PCAreconstruction. We find out that PCA reconstruction forboth simulated and observed data-set, if we take ΛCDMas our model, the value Ωm = 0.3 and w(z) = −1 lies wellwithin the 1σ range.

We now do the same analysis using observed data.Fig(6) shows the reconstructed curves of reduced Hub-ble parameter h(z) and µ(z) obtained for the real Hub-ble parameter and supernova data-sets. It is evident fromthe plot that PCA reconstruction with (1 − a) variableproduces consistent results for observational data-set also.

Fig(8) shows the final reconstruction of w(z) from the ob-served Hubble parameter data-set with the reduced Hub-ble constant fixed at the value predicted by PCA. Re-construction of the functional form of h(z) by choosing apoint from coefficient space (eq 8) is the first step of de-rived approach, which eventually gives us the value of h0from the same point. We do not vary the value of h0 in thew(z) reconstruction curves to ensure that both h(z) andh0 are from the same point of coefficient space. The val-ues obtained from the observational data-set are higheras compared to the other model-dependent estimations.Table 3 presents the values of h0, obtained, along withmodel-dependent estimations of h0 from other studies [1,90,91] for a comparison. In the present work, we have notused the latest JLA data, as this data compilation has its


0.25 0.75 1.25 1.75 2.25z

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Fig. 6. The plots in the figure show the reconstructed reduced Hubble parameter h(z) and distance modulus µ(z) for observeddata-sets along with their residues. Residues are calculated as the difference between PCA reconstruction and the correspondingh(z) and µ(z) from the observational data. The left column is for the independent variable (1 − a), the middle column is for aand the right column is for z. For the blue curves, there is no reduction, that is, M = N = 7. The green and red curves areobtained by the reduction of the highest and second-highest Principal Components, respectively.

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Fig. 7. The plots show the reconstruction of h(z) and µ(z) from the observed Hubble parameter and supernovae data-set withallowed range ΛCDM cosmology. The brown patch is obtained for the variation of wΛCDM and Ωm in the range [-1.5 to -0.5]and [0.2 to 0.4] for ΛCDM cosmology. Solid blue, green and red line are for PCA reconstruction with no-reduction and withone and two terms respectively.


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Fig. 8. The figure shows the reconstructed w(z) curves, obtained in the derived approach using real Hubble parameter data,with variables (1 − a), a and z, respectively from left to right panels. Each represents a reconstruction with a fixed value ofΩm. Going from blue to the green color variation the value of Ωm varies from 0.1 to 0.4, in a step of 0.015. We fix the reducedHubble constant h0 at a value obtained from the PCA algorithm, as mentioned in table 1.

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Fig. 9. The plots of w(z) obtained for the real Hubble parameter data in the direct approach. The first column is for thereconstruction by the variable (1 − a) while the second and third column is for reconstruction variables a and z, respectively.In the first row, solid and dashed lines of red, orange, green, blue and violet are for different Ωm, varying from 0.2 to 0.28 andfrom 0.3 to 0.38 respectively, in the interval of 0.02, the long dashed- dot line (red) is for Ωm = 0.4 and the reduced Hubbleconstant is fixed at 0.685. In the second row, solid and dashed lines of red, orange, green, blue and violet are for different valueof reduced Hubble constant h0 from 0.6 to 0.68 and from 0.7 to 0.78 respectively, in the interval of 0.02, the long dashed-dottedline (red) is for h0 = 0.8 and the density parameter is fixed at Ωm = 0.3.

own parameters, which we have to take into account todetermine the distance modulus of Type IA supernovae.This complicates the process of monitoring the correla-tions of the coefficient space, which is crucial to determinethe final number of terms in the reconstructed quantity’spolynomial expression.

4.2 Direct approach

For the direct approach, we have considered only the Hub-ble parameter data-set. Reconstruction is carried out forall the three independent variables (1 − a), a and z. Asin the case of derived approach, we first use simulated

data-set. The results obtained for the simulated data-setsare shown in fig(5). In this case, all three independentvariables reproduce the nature of the equation of stateparameter. In the direct approach, using the correlationtest calculation, we find N = 5 to be the best choice forthe number of terms in the initial polynomial eq(1). Asthe number of terms on initial basis is comparatively lowand we assume M = N as our final basis number, the un-certainties which PCA poses in predicting the best patch-point is relatively low. The tiny non-linearity we intro-duced in the H(z) by considering a polynomial form ofw(z) will be amplified in the case of supernovae data-set.Hence it makes the reconstruction much more unstable in


Variable State Pearson Spearman Kendall

(1-a)

pre-PCA

1 −0.99 0.92∗ 1 −0.96∗ ∗ 1

1 −0.52 0.55∗ 1 −0.94∗ ∗ 1

1 −0.45 0.52∗ 1 −0.85∗ ∗ 1

post PCA

1 0.944 0.219∗ 1 0.506∗ ∗ 1

1 0.92 0.69∗ 1 0.63∗ ∗ 1

1 0.845 0.69∗ 1 0.56∗ ∗ 1

a

pre-PCA

1 −0.98 0.93∗ 1 −0.98∗ ∗ 1

1 −0.68 0.60∗ 1 −0.87∗ ∗ 1

1 −0.62 0.53∗ 1 −0.75∗ ∗ 1

post PCA

1 −0.99 0.99∗ 1 1∗ ∗ 1

1 −0.30 −0.32∗ 1 −0.99∗ ∗ 1

1 −0.21 −0.23∗ 1 0.968∗ ∗ 1

z

pre-PCA

1 −0.92 0.64∗ 1 −0.86∗ ∗ 1

1 −0.31 0.35∗ 1 −0.72∗ ∗ 1

1 −0.23 0.35∗ 1 −0.62∗ ∗ 1

post PCA

1 1 −0.96∗ 1 −0.96∗ ∗ 1

1 0.999 −0.007∗ 1 −0.008∗ ∗ 1

1 0.998 0.0847∗ 1 0.0832∗ ∗ 1

Table 2. This table shows Pearson, Spearman and Kendall correlation coefficients between the coefficients of the first threeterms of the series expansion of the reconstructed quantity for the reconstruction variables (1 − a), a and z respectively. Thisis in the derived approach for simulated Hubble parameter data. The first, third and fifth rows (pre-PCA), in the table, showsthe correlation coefficients of the first three coefficients of the initial polynomial we start with, viz b1,b2 and b3, eq(1) for thevariables (1−a), a and z respectively. The correlation coefficients of the first three coefficients of the final polynomial, viz β1, β2and β3, eq(8) given by the PCA algorithm is given in the second, fourth, sixth rows (post-PCA) for the variables (1−a), a and zrespectively. Since the correlation matrix is symmetric, here we only mention the upper diagonal terms, whereas correspondinglower diagonal terms are replaced by ∗.

the case of reconstruction of w(z) through distance mod-ulus calculation.

For the observed Hubble parameter data-set too wereproduce the curves using the same algorithm. In the caseof direct approach, PCA cannot predict the value of H0

and Ωm; therefore, these two parameters have to be fixedprior to the analysis. w(z) curves obtained for the realdata-set is shown in fig(9). We plot the curves by varyingΩm from 0.2 to 0.4, fixing reduced Hubble constant ath0 = 0.685 and then varying h0 from 0.6 to 0.8 and fixingΩm at 0.30.

In the direct approach, though the reconstruction ofthe fiducial w(z) is consistent (fig(5)), the correlation testcalculation for the direct approach shows that the algo-rithm is not able to break the Pearson Correlation as itbreaks down in the case of the derived approach. In thecase of direct approach, for (1−a) reconstruction, the mag-nitude of Pearson correlation coefficients decreases afterapplying PCA but changes signs for the first two principal

components, whereas Kendall and Spearman correlationcoefficient of (1 − a) for these two principal componentsassume higher negative value. Again, for reconstruction bythe variable a, the Pearson Correlation decreases, thoughboth Spearman and Kendall Correlation coefficients de-crease in magnitude, it changes sign for the first two prin-cipal components. For the variable z, up to the first twoprincipal components, Pearson correlation coefficients de-crease, but Spearman and Kendall correlation coefficientsassume large negative value. From the correlation coeffi-cient calculation the derived approach is selected over thedirect approach and selects the reconstruction by the in-dependent variable (1−a) as compared to variables a andz.

5 Summary and Conclusion

In this paper, we reconstruct late-time cosmology usingthe Principal Component Analysis, which is a model inde-


Variable PCA ΛCDM wCDM Planck(ΛCDM)

(1-a) 78.4001 67.94 ± 5.15 68.07 ± 1.63 67.9 ± 2.6(Plank+WP+SDSS+SNLS) (Plank+WP+JLA) (EE+lowE)

a 73.9271 69.85 ± 4.44 68.19 ± 1.33 67.39 ± 0.54(Plank+WP+JLA) (WMAP9+JLA+BAO) (TT,TE,EE+lowE+lensing)

z 77.0216 76.48 ± 7.36 70.33 ± 2.34 67.66 ± 0.42(Plank+WP+C11) (Plank+WP+C11) (TT,EE,TE+lowE+lensing + BAO)

Table 3. This is the comparison table of the values of Hubble constant in standard units (km s−1 Mpc−1), obtained in thepresent analysis (PCA) and obtained from other model-dependent estimations.

pendent and non-parametric approach. There are very fewprior assumptions about nature and distribution of dif-ferent components contributing to the energy of the Uni-verse. Observational Hubble parameter and distance mod-ulus measurements of type Ia supernovae are the observ-able quantities that are taken into account in the presentanalysis. We proceed in two different ways to do the re-construction. The first one is a derived approach wherethe observable quantities are reconstructed from the datausing PCA, and then w(z) is obtained from the recon-structed quantities using Friedman equation. The otherapproach is a direct one. In this case, w(z) is reconstructeddirectly from the observational data using PCA withoutany intermediate reconstruction. Based on the efficiency ofthe method to break the correlation among the coefficientswe can select one reconstruction curve over the other. Weachieve a better reconstruction in the derived approachas compared to the direct approach, even though the di-rect approach has lesser uncertainties than the derivedapproach in predicting the patch of the best coefficientpoint from the N dimensional coefficient space due to thelower number of initial and final bases.

We have adopted the simulated as well as observeddata-sets for our analysis. Simulated data-sets are usedto check the efficiency. For the reconstruction of w(z) theanalysis produces consistent result only for the Hubble pa-rameter data. Though the reconstruction of µ(z) throughthe derived approach is consistent and reconstruction ofµ(z) is within the error bars of the data-set, the result forthe reconstructed w(z) deviates drastically from the phys-ically acceptable range. The increase in the order of differ-entiation to connect w(z) with µ(z) is a possible reason forthis inconsistency. Here we are focusing on the w(z) recon-struction only from H(z)− z data-set. The reconstructedw(z) by the variable (1 − a), obtained in the derived ap-proach for H(z)−z data-set shows a phantom like nature,that is w(z) < −1 at present and a non-phantom naturein the past for most of the values of Ωm and h0. In thecase of direct approach, w(z) curves show oscillations inthe phantom and non-phantom regime. The calculation ofthe correlation coefficients clearly shows a preference forthe derived approach. PCA lacks the efficiency of break-ing the correlation in the initial basis in case of the direct

approach. This probably causes the inconsistency betweenthe results obtained in the derived and direct approaches.

The other important factor is the variable of recon-struction. In the present analysis, we have adopted threedifferent reconstruction variables, namely (1 − a), a andz, in both direct and derived approaches. Values of cor-relation coefficients after PCA select the reconstructionvariable (1 − a) over the other two. We should empha-size the result obtained for variable (1 − a) by derivedapproach. Since we have a finite number of terms in theinitial polynomial, due to the constraints set by compu-tational power, to some extent the results depends on theassumption of polynomial expression of the observables.One of our future plans is to develop an algorithm whichcan inclusively select the most suitable initial basis formfor a fix computational power and a given observationaldata-set. The reconstructed curves, obtained for (1 − a),show that w(z) shows a phantom nature at present epochand it was in non-phantom nature in the past. The model-independent reconstruction indicates an evolution of thedark energy equation of state parameter.

Acknowledgements

The authors acknowledge the use of High-Performance-Computing facility at IISER Mohali. AM acknowledgesthe financial support from the Science and EngineeringResearch Board (SERB), Department of Science and Tech-nology, Government of India as a National Post-DoctoralFellow (NPDF, File no. PDF/2018/001859). The authorswould like to thank J S Bagla for useful suggestions anddiscussions.

Data Availability

The observational data-sets used in the analysis are pub-licly available and duly referred in the text. The simulateddata-sets can be provided on request with appropriate jus-tification.


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