Modi ed Statistical Analysis of Type 1a Supernovae … Projects...Modi ed Statistical Analysis of Type 1a Supernovae Data Lisa Goh Wan Khee A0144115U Abstract In this thesis, we review

NATIONAL UNIVERSITY OF SINGAPORE

Faculty of Science

Department of Physics

Modified Statistical Analysis of Type 1aSupernovae Data

Submitted By:

Lisa Goh Wan Khee

A0144115U

Supervisor:

Dr Cindy Ng

A thesis submitted in partial fulfillment

for the Degree of Bachelor of Science with Honours in Physics

AY 2018/2019

Modified Statistical Analysis of Type 1a Supernovae Data

Lisa Goh Wan Khee

A0144115U

Abstract

In this thesis, we review a new statistical method to analyse Type 1a Supernovae data. We estimate

the maximum likelihood by conducting a parameter sweep across 8 SNe1a parameters in the Ωm

and ΩΛ parameter space, usinga Markov Chain Monte Carlo optimization algorithm. This method

has the advantage of being bias-free as compared to the least χ2 method, which allows an arbitrary

value of uncertainty to be added to the calculation. We use the Joint Lightcurve Analysis (JLA)

dataset, containing 740 SNe1a data samples for our study, and compare it against 5 different

models: the concordance ΛCDM model, the flat wCDM model, its non-flat generalization, as well

as two of its other simplest and most studied parametrizations. We find that the ΛCDM model

is favoured over the other models, and the best fit values based on this model are Ωm=0.40,

ΩΛ=0.55. Interestingly, the contour plots we obtain mostly cross the line of no acceleration at

2 ∼ 3σ confidence levels, which is similar to the results published by Nielsen et al, the original

authors who introduced the Maximum Likelihood Estimation method. When we generalize this

model to other parametrizations, the evidence for a constant expansion becomes even stronger.

This raises the question of how secure we can be of an accelerating expansion of the Universe.

Acknowledgments

I would like to thank Dr. Cindy for being my supervisor, and for being ever so supportive during the

course of my FYP. Without her guidance and encouragement, all this would not have been possible.

I would also like to thank my parents for supporting my education and for just supporting and

loving me in general. I hope that I did them proud, or at least if I have not, that I will someday.

Next, I would like to thank all my friends and professors in NUS who have made my undergraduate

life so fun and fulfilling. Lastly, I would like to thank Chung Sern for always believing in me, and

for the occasional fruitful discussions in coding.

List of Figures

1 Geometry of a two-dimensional surface for different values of curvature k. . . . . . . 5

2 Contour plot obtained using the MLCS lightcurve fitting method of R98. . . . . . . 13

3 Contour plot obtained using the ∆m15 template fitting method of R98. . . . . . . . 14

4 Histograms of x1 and c, with a Gaussian distribution overlaid. . . . . . . . . . . . . . 17

5 Contour plot obtained by N16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Area estimation using the Monte Carlo method. . . . . . . . . . . . . . . . . . . . . . 21

7 A schematic diagram of a Markov chain. . . . . . . . . . . . . . . . . . . . . . . . . . 22

8 Pictorial explanation of a stretch (left) and a walk (right) of an MCMC algorithm. . 25

9 Values of M0,x10,c0,σM0

,σx10,σc0 ,α,β sampled, as well the χ2 values for each run

when Nmcmc=50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

10 Values of M0,x10 ,c0,σM0 ,σx10,σc0 ,α,β sampled, as well the χ2 values for each run

when Nmcmc=1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

11 Values of M0,x10,c0,σM0

,σx10,σc0 ,α,β sampled, as well the χ2 values for each run

when Nmcmc=10000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

12 Resultant posterior distribution of the 8 parameters after discarding the first 3000

runs as burn in. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

13 Comparison of x1 and c distributions between the actual data and the values obtained

through MCMC method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

14 Best fit values for the 8 parameters as presented by N16, compared to the results we

obtain with our MCMC code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

15 Contour plot and scatter plot of Ωm against ΩΛ for the ΛCDM model. . . . . . . . . 33

16 Plot of Ωm against w for Ωk = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

17 Plot of w against Ωm for ΩΛ = 0.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

18 Plots of w against Ωm for ΩΛ = 0 to 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . 36

19 Plots of w against Ωm for ΩΛ = 0.5 and 0.7 < ΩΛ < 1.0. . . . . . . . . . . . . . . . . 37

20 Plots of w0 against Ωm for values of −2 < w1 < 1 for Parametrization B. . . . . . . . 38

21 Plots of w0 against Ωm for values of −2 < w1 < 1 for Parametrization C. . . . . . . 39

Contents

1 Introduction 1

2 Theory 3

2.1 Type 1a Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 The Friedmann-Lemaitre-Robertson-Walker Metric . . . . . . . . . . . . . . . . . . . 4

2.2.1 The Friedmann Equations and the Hubble Constant . . . . . . . . . . . . . . 5

2.2.2 Measuring the Components of the Universe . . . . . . . . . . . . . . . . . . . 6

2.3 Models of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 ΛCDM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.2 wCDM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Calculating Luminosity Distance dL . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Deceleration parameter q0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Methodology 11

3.1 The Least χ2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 SALT2 Lightcurve Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 The Joint Lightcurve Analysis Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 The Maximum Likelihood Estimation Method . . . . . . . . . . . . . . . . . . . . . . 16

3.4.1 Bayesian Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4.2 Calculating Likelihood L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4.3 Comparison to the Least χ2 Method . . . . . . . . . . . . . . . . . . . . . . . 19

3.5 Markov Chain Monte Carlo Optimization . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5.1 Principles of MCMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.5.2 Implementation of MCMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.5.3 Testing our MCMC code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Results 33

4.1 ΛCDM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 wCDM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2.1 Parametrization A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2.2 Parametrization B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2.3 Parametrization C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Discussion 40

5.1 ΛCDM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2 wCDM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2.1 Parametrization A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2.2 Parametrizations B and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.3 Model Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.4 Sources of Error and Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.5 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 Conclusion 43

A Derivation of Friedmann Equations 44

B MATLAB Codes 46

B.1 χ2 Calculation (based on R98) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

B.2 MLE Calculation (ΛCDM Model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

B.3 MCMC Sampler Code (ΛCDM Model) . . . . . . . . . . . . . . . . . . . . . . . . . . 49

B.4 Log Likelihood Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

B.5 Distance Modulus Calculation (ΛCDM Model) . . . . . . . . . . . . . . . . . . . . . 54

B.6 MLE Calculation (wCDM Model Parametrization A) . . . . . . . . . . . . . . . . . . 55

B.7 MCMC Sampler Code (wCDM Model Parametrization A) . . . . . . . . . . . . . . . 56

B.8 Distance Modulus Calculation (wCDM Model Parametrization A) . . . . . . . . . . 58

B.9 MLE Calculation (wCDM Model Parametrization B) . . . . . . . . . . . . . . . . . . 60

B.10 MCMC Sampler Code (wCDM Model Parametrization B) . . . . . . . . . . . . . . . 60

B.11 Distance Modulus Calculation (wCDM Model Parametrization B) . . . . . . . . . . 63

B.12 MLE Calculation (wCDM Model Parametrization C) . . . . . . . . . . . . . . . . . . 63

B.13 MCMC Sampler Code (wCDM Model Parametrization C) . . . . . . . . . . . . . . . 64

B.14 Distance Modulus Calculation (wCDM Model Parametrization C) . . . . . . . . . . 67

1 Introduction

In 2011, Perlmutter, Schmidt and Riess were awarded the Nobel Prize in Physics for their discov-

ery of the accelerating expansion of the Universe [1][2], by studying distant Type 1a Supernovae

(SNe1a). They noticed that these supernovae were dimmer than expected of a constantly expanding

Universe, and after some data analysis, concluded that the cosmological term Λ in Einstein’s field

equations had to be non-zero, or there existed an additional dark energy term.

This discovery made waves across the cosmological world, and forced cosmologists to question

their current knowledge of the Universe, as they now had to reconstruct their model of the Universe

to account for this accelerating expansion. 8 years on, much research has been done to study the

nature of this expansion—to measure its rate and attempt to explain how it came about [3][4].Many

surveys of the night sky have been conducted to gather data of not just supernovae, but also the

Cosmic Microwave Background (CMB), Baryonic Acoustic Oscillations (BAO) and so forth.

Surveys such as the Sloan Digital Sky Survey [5], the Supernova Legacy Survey [6], the Hubble

Space Telescope [7], ESSENCE [8], LOSS [9], Harvard-Smithsonian Centre for Astrophysics CfA1-4

survey [10], and the Carnegie Supernova Project [11] have collected large amounts of data in the

past two decades covering a wide range of redshifts from 0.01 < z < 0.1. In the last one year alone,

there has also been new surveys conducted, such as the PAN-STARRS 1[12] and Dark Energy

Survey[13], which has brought the total number of supernova samples to an unprecedented number

of over 1000 samples.

Along CMB data taken from studies such as the Wilkinson Microwave Anisotropy Probe (WMAP)

[14] and BAO data from surveys like SDSS-III’s BOSS [15], the accuracy and precision of our mea-

surements with regards to the expansion of the Universe have improved by leaps and bounds since

2008. With an increasing number of datasets, we can place further constraints on the exact quan-

tities of dark energy and matter in the Universe, consequently allowing us to improve or debunk

our current theories and models.

However, data will only be useful if we can analyse them accurately in order to obtain mean-

ingful results, and draw correct conclusions. Many different statistical data analysis methods have

been proposed as means to analyse SNe1a data, based on both Bayesian [16][17] and frequentist

[18] approaches. Particularly in a paper that we will be looking at in more detail, the methods they

have used led them to the conclusion that a constant expansion cannot be ruled out. Their results

have been verified by some [19][17], which is interesting to note. Therefore this thesis aims to study

1

their method more in-depth and put forth some modifications to it, by exploiting the advantages of

both Bayesian and frequentist statistics. We will then use our new method on an expanded range

of cosmological models to test their goodness of fit, and perhaps give additional insight towards the

probability of a constant rate of expansion.

This thesis has been arranged as such: Section 2 introduces the cosmological theories behind Uni-

verse expansion, Section 3 will explain and justify our proposed methodology, in Section 4 we

present our results, Section 5 will provide a discussion of our results and propose possible future

works and extensions, and finally we give conclude in Section 6.

2

2 Theory

2.1 Type 1a Supernovae

The main source of data being used to investigate the expansion of the Universe are SNe1a. SNe1a

are the result of the collapse of a white dwarf, a dense (on the order of 109kg/m3) celestial object

which is the remnant of a main sequence star that has evolved to the end stage of its life. As the

core of a star progressively fuses hydrogen to helium, then from helium to carbon and oxygen, a

star of mass 0.5M to 8M will not have a sufficiently hot core to continue this fusion process.

It will then expand into a red giant, expelling most of its gaseous envelope to leave behind its

carbon-oxygen core — a white dwarf [20].

A white dwarf is supported by electron degeneracy pressure, a phenomenon dictated by Pauli’s

Exclusion Principle. No two fermions can occupy the same quantum state simultaneously, there-

fore the electrons, being so densely packed, will progressively occupy each quantum state starting

from the ground state, creating a pressure that counters the inward gravitational pull. However,

when a white dwarf accretes mass and exceeds the Chandrashekar mass limit (∼ 1.4M), the grav-

itational force becomes larger than the electron degeneracy pressure and the white dwarf collapses

on itself.

SNe1a occur when a white dwarf is in a binary system, either with another white dwarf, or a

giant star. Due to its intense gravitational field, the white dwarf will accrete matter from its binary

partner, and its mass increases. Its temperature will subsequently increase to the point where car-

bon and oxygen begin fusing into nickel and iron. Once the Chandreshekar mass limit is reached,

the white dwarf will explode through a mechanism which exact details are still a topic of investi-

gation.

SNe1a are characterised by their lack of hydrogen lines, unlike Type II Supernovae. Since all SNe1a

occur at the same mass through the same mechanisms, their peak luminosities are roughly constant

across all SNe1a, thus making them good standard candles with which to measure distances in the

Universe. Using the equations

µ = m−M (2.1a)

µ = 25 + 5 log(dL) (2.1b)

where m is the apparent magnitude and M is the absolute magnitude (≈ −19),the distance modu-

lus µ and luminosity distance dL can be known.

3

The graph of luminosity against time (known as the lightcurve) of every SNe1a show very sim-

ilar shapes: they rapidly increase in luminosity leading up to the supernova event and peak at a

roughly uniform magnitude, then slowly decrease in magnitude over a period of 1 to 2 weeks [21].

Therefore researchers make use of the lightcurve data in order to obtain distance estimates of µ.

The apparent magnitude m can be measured directly, and the value of the absolute magnitude M

can be obtained from the lightcurve.

2.2 The Friedmann-Lemaitre-Robertson-Walker Metric

In order for any insights to be gained from the observational data, a model of the Universe should

be proposed with which to compare how well the observations adhere to it. In 1915, Einstein put

forth the famous General Theory of Relativity [22], which has provided us with an elegant formula

to probe the geometry of the Universe:

Rµν −1

2Rgµν + Λgµν =

8πG

c4Tµν (2.2)

Solving the Einstein Field Equation by finding the components of the metric tensor gµν will al-

low us to understand the behaviour of the Universe. The left hand side of the equation describes

the curvature of the Universe through the Ricci tensor, Rµν . On the right hand side, the energy-

momentum tensor Tµν describes the components of the Universe. G is the gravitational constant

and c is the speed of light. There are many exact solutions to this equation; however, obtaining a

physical interpretation from them is not always an easy task.

The cosmological principle states that on a large enough scale, the Universe is homogeneous and

isotropic—it looks the same from every point and in every direction; there is no one special obser-

vational viewpoint. In the 1920s and 1930s, 4 scientists—Alexander Friedmann, Georges Lemaitre,

Howard P. Robertson and Arthur Geoffrey Walker—independently worked out an exact solution to

the Einstein Field Equation describing an expanding Universe of such a nature [23]. The Friedmann-

Lemaitre-Robertson-Walker (FLRW) metric tensor is given by

ds2 = −c2dt2 + a(t)2(dx2 + dy2 + dz2) (2.3)

where a(t) is the scale factor that describes the expansion rate of the Universe, and is a function

of time. It is taken to be 1 in the present day, with a(t0) = 1. With this metric, a model of the

Universe can be developed.

4

2.2.1 The Friedmann Equations and the Hubble Constant

In the polar coordinates, the FLRW metric can be expressed as

ds2 = −c2dt2 + a(t)2(dr2

1− kr2+ r2dθ2 + r2sin2θdφ2) (2.4)

where k is the curvature constant which defines the spatial curvature of the Universe, and can take

on the values of +1,0 or −1.

If k = +1, the Universe is closed, and the expansion will slow down and eventually succumb

to gravitational collapse, also known as the ‘Big Crunch’. If k = 0, the Universe is flat, and will

expand forever at a decelerating rate. If k = −1, the Universe is open, and it will expand forever

[24]. Therefore the value of k depends on the components of the Universe—dark energy will speed

up its expansion, whereas matter will stop this expansion.

Figure 1: Geometry of a two-dimensional surface for different values of curvature k (from left to

right): a sphere (k = +1), a saddle (k = −1) and a plane (k = 0). In a closed geometry, the

sum of the angles of a triangle drawn on its surface is more that 180, in a flat geometry it is

exactly 180, while in an open geometry it is less than 180. Image credit: University of Oregon,

http://abyss.uoregon.edu/~js/lectures/cosmo_101.html

To better understand the dynamics of the Universe expansion, we derive the Friedmann equa-

tions from the (00) components and the (ij) components of the Einstein Field Equations, using the

FLRW metric. The full derivation can be found in the Appendix A. After some calculations, we

5

http://abyss.uoregon.edu/~js/lectures/cosmo_101.html

get the two Friedmann equations

a(t)2 + k

a(t)2− Λ

3=

8πG

3ρ(t) (2.5a)

a(t)

a(t)− Λ

3= −4πG

3(ρ(t) + 3p(t)) (2.5b)

The expression a(t)a(t) is known as the Hubble constant H(t), which gives the rate of expansion of the

Universe as a function of time. Various survey such as the HST [25] and Planck Collaboration [14]

have endeavored to measure the value of H0, the value of H(t) at the present time t = t0, and for

the purpose of this thesis we will be using the widely adopted value of 70km/s/Mpc.

2.2.2 Measuring the Components of the Universe

When we rewrite equation (2.5a) into the form:

a(t)2

a(t)2=

8πG

3ρ(t) +

Λ

3− k

a(t)2(2.6)

and substitute H(t) = a(t)a(t) it is easy to see that there are three factors which govern the expansion

of the Universe— the mass term, the dark energy term and the curvature term. In the current

epoch (H(t) = H0), they can be expressed as the matter density parameter Ωm, the dark energy

density parameter ΩΛ and curvature density parameter Ωk as follows [26]:

Ωm =8πG

3H20

ρ0 (2.7a)

ΩΛ =Λ

3H20

(2.7b)

Ωk = − k

a(t0)2H20

(2.7c)

where the present day scale factor a(t0) is defined to be 1. Therefore (2.6) can be rewritten as

H20 =

8πG

3ρ0 +

Λ

3− k

a(t0)2(2.8)

Dividing (2.8) by H20 ,

Ωm + ΩΛ + Ωk = 1 (2.9)

In this thesis, we will be analysing data from SNe1a to calculate the best fit values of these three

parameters.

2.3 Models of the Universe

The models of the Universe which we will be testing our data against are the concordance model,

the ΛCDM model, and other variations of it, namely the wCDM model and its parametrizations.

6

2.3.1 ΛCDM Model

The current concordance model of the Universe, developed as a response to the discovery of accel-

erating expansion, is known as the ΛCDM, or Λ Cold Dark Matter model. Here, Λ represents dark

energy. It has a negative pressure p = −ρc2 which, when substituted into the (ii) components of

the momentum-energy tensor Tµν , (where i =1,2 and 3), gives an accelerating expansion. Therefore

it is now widely accepted to exist in our Universe, although the exact way in which it manifests

itself is still shrouded in mystery.

Cold Dark Matter refers to slow-moving, non-interacting and non-baryonic dark matter, there-

fore excluding all baryons, electrons and neutrinos. It is cold as it moves much slower than the

speed of light, and only interacts through gravity and the weak force [27]. From gravitational lens-

ing surveys, its existence is confirmed [28], however it has yet to be directly detected, as it does not

interact with normal matter.

The ΛCDM model postulates an origin of the Universe, the Big Bang, the event whereby it ex-

panded from a singularity to form spacetime some 13.8 billion years ago. Observations of the CMB

have affirmed the existence of such an event [29], further validating the accuracy of this model and

solidifying its status as the standard model of the Universe.

Based on this model, we wish to obtain an expression for the evolution of H as a function of

redshift z, ie. in the form H(z) = H0E(z). This is known as the Hubble function. From the

conservation of energy and momentum (∇νTµν = 0) we obtain the cosmological fluid equation.

ρ+ 3a

a(ρ+

P

c2) = 0 (2.10)

From this we get the relationship

ρ ∝ a(t)−3(1+w) (2.11)

Where w is a parameter known as the equation of state (EoS) parameter, given by w = pρ . In the

ΛCDM model, w is taken to be −1 for dark energy (since pΛ is negative), and 0 for matter (since

pm = 0). The evolution of the scale factor is related to the redshift by

a(t0)

a(t)= 1 + z (2.12)

Therefore, ρm(t) evolves as ρm(t) = ρ0(1 + z)3. We can write H(z) as such:

H(z)2 =8πG

3ρ0(1 + z)3 +

Λ

3− k

1(1+z)2

(2.13)

7

substituting the expressions of (2.7a), (2.7b) and (2.7c), we get the relation

H(z)2 = H20 [Ωm(1 + z)3 + Ωk(1 + z)2 + ΩΛ] (2.14)

where E(z) =√

Ωm(1 + z)3 + Ωk(1 + z)2 + ΩΛ.

2.3.2 wCDM Model

An extension of the ΛCDM model is the wCDM model, where we are concerned with the EoS pa-

rameter of dark energy, w. It is not assumed to be −1, and might also evolve with time. We model

this evolution with 2 different parametrizations, and test them against our data. Most wCDM

models assume a flat Universe with Ωk = 0, however we shall generalize this in order to find the

best fit value for Ωk, therefore we keep the Ωk term in the Hubble function.

The first parametrization (hereafter referred to as parametrization A) is to treat w as a scalar

value unchanging with redshift. The equation for H(z) can thus be written as

H(z)2 = H20 [Ωm(1 + z)3 + Ωk(1 + z)2 + ΩΛ(1 + z)3(1+w)] (2.15)

Parametrization B defines the evolution of w as a linear function of redshift [30]:

w(z) = w0 + w1z (2.16a)

H(z)2 = H20 [Ωm(1 + z)3 + Ωk(1 + z)2 + ΩΛ(1 + z)3(1+w0−w1)e3w1z] (2.16b)

Parametrization C defines the evolution of w as a first order Maclaurin expansion of redshift [30],

and is also known as the Chevallier-Porlarski-Linder parametrization [31]:

w(z) = w0 + w1z

1 + z(2.17a)

H(z)2 = H20 [Ωm(1 + z)3 + Ωk(1 + z)2 + ΩΛ(1 + z)3(1+w0+w1)e3w1[1/(1+z)−1]] (2.17b)

The best fit values for w, w0 and w1 for each parametrization will be obtained to see which model

best describes the current Universe.

2.4 Calculating Luminosity Distance dL

To calculate distances in an expanding curved spacetime, we define a quantity known as co-moving

distance r, which is the distance to an object without taking into consideration the expansion of

the Universe. Consider a photon traveling towards us at time t from a distance dM away, in the

radial direction (dθ = dφ = 0). For a light ray, ds2 = 0, therefore the metric equation gives

c2dt2

a(t)2=

dr2

√1− kr2

(2.18)

8

When we take the square root of and integrate the left hand side, this gives us the co-moving

distance, in terms of t.

dC(t) = c

∫ t0

0

dt

a(t)(2.19)

In terms of redshift z, it can be expressed as

dC(z) = dH

∫ z0

0

dz

E(z)(2.20)

where dH = c/H0 is known as the Hubble distance. Next, we integrate the right hand side of (2.18)

from 0 to dM , which is known as the transverse co-moving distance, and equate this to dC(z). From

(2.7c), we get the relationship −k = ΩkH20 . Therefore we obtain

dC =

∫ dM

0

dr√1− kr2

=

dH√Ωk

sinh−1(√

ΩkdM/dH), if Ωk > 0

dM , if Ωk = 0

dH√|Ωk|

sin−1(√|Ωk|dM/dH

), if Ωk < 0

(2.21)

Making dM the subject, we get

dM =

dH√Ωk

sinh(√

ΩkdC/dH), if Ωk > 0

dC , if Ωk = 0

dH√|Ωk|

sin(√|Ωk|dC/dH

), if Ωk < 0

(2.22)

The luminosity distance dL is related to dM through a simple equation: dL = (1+z)dM . We finally

arrive at the theoretical expression for the luminosity distance:

dL =

(1 + z) dH√

Ωksinh

(√Ωk∫ z0

0dzE(z)

), if Ωk > 0

(1 + z)dC , if Ωk = 0

(1 + z) dH√|Ωk|

sin(√|Ωk|

∫ z00

dzE(z)

), if Ωk < 0

(2.23)

We substitute this expression into (2.1b) and specify values of ΩΛ and Ωm to obtain the theoretical

value of µ based on the model.

2.5 Deceleration parameter q0

Next, we define the deceleration parameter q = −a(t)a(t)/a2(t). From (2.5b) of the Friedmann

Equations, we can see that − a(t)a(t) = qH2. Therefore, for q0 ie. the deceleration parameter of the

current epoch in the ΛCDM model,

q0 = − a(t0)

a(t0)H20

=4πG

3H20

(ρm + 3pm)− Λ

3H20

(2.24)

9

Substituting (2.7a), (2.7b) and using the fact that pm = 0, this reduces to

q0 =1

2Ωm − ΩΛ (2.25)

If the Universe is accelerating, a > 0 and so q0 < 0. Therefore the criteria for an accelerating

expansion isΩm2

< ΩΛ (2.26)

which is a linear relationship that we can plot in the Ωm − ΩΛ parameter space.

In the flat wCDM model, we can rewrite (2.5b) as

a(t)

a(t)= −4πG

3

∑i

ρi(1 + 3wi) (2.27)

where i are the components of the Universe, in this case m and Λ. Therefore q0 becomes

q0 =∑i

Ωi(1 + 3wi) =1

2Ωm +

1 + 3wΛ

2ΩΛ (2.28)

because wm = 0. When q0 = 0, the accelerating criteria for a flat wCDM model is

w ≤ − 1

3(1− Ωm)(2.29)

which can be plotted on an Ωm − w graph. For a non-flat case, it becomes

w ≤ −1

3− Ωm

3(1− Ωm − Ωk)(2.30)

10

3 Methodology

Now we introduce the various statistical analysis methods that we will be using to analyze the

SNe1a data, and compare these methods to see which one is the fairest and most appropriate. We

start by introducing some of the earliest methods used by the Nobel prize winning team, and the

subsequent improvements that were made to their analyses.

3.1 The Least χ2 Method

The method used by Riess et al. [1] (now on referred to as R98) in their first paper describing

the accelerating expansion of the Universe, was the least χ2 method. From the confidence ellipse

plot they obtained, they were able to come to the conclusion that the Universe is expanding at an

accelerating rate (q0 < 0 and ΩΛ > 0), within 2.8σ uncertainty.

The least χ2 goodness-of-fit method is a commonly used method to access the suitability of a

model in describing the data obtained, ie. how good of a fit the data are to the proposed model.

The lower the χ2 value, the better the data are fitted to the model. We calculate the χ2 value of

each point in the Ωm −ΩΛ parameter space to find the value of Ωm and ΩΛ which gives the lowest

χ2 value. For n number of SNe1a data samples, the formula for χ2 value is given by equation 4 of

R98:

χ2 =

n∑i

(µp,i(zi; Ωm,ΩΛ)− µ0,i)2

σ2µ0,i

+ σ2ν

(3.1)

where µp,i(zi; Ωm,ΩΛ) refers to the theoretical value of µ, the distance modulus calculated based

on the model, given by (2.1b), for a pair of values of Ωm and ΩΛ. The formula used to calculate

dL will be dependent on the model being studied. In R98, the model used was the ΛCDM model.

µ0,i is the distance modulus calculated based on the experimental data, given by (2.1a). σ2µ0,i

is

the uncertainty of the µ0 measurement, which might be due to both systematic and statistical

errors that arise while collecting and processing the data. The uncertainty in galaxy redshift due

to peculiar velocities is captured by the σν term, and is given by

σν =5∆v

cz log 10(3.2)

where ∆v is the peculiar velocity dispersion and is taken to be a uniform value of 200km s−1. The

fraction in (3.1) is evaluated for each SNe1a data sample and summed up across all n samples

to give the total χ2 value for a point in the Ωm-ΩΛ parameter space. We then plot a confidence

ellipse joining the points with constant χ2 value, at 1σ (0.683), 2σ (0.954) and 3σ (0.997) values

of the χ2 distribution. From the χ2 distribution curve, we have to find the values of χ2 that cover

11

68.3%, 95.4% and 99.7% of the area under the curve. From the χ2 distribution function

χ2(k) =1

2k/2Γ(k/2)xk/2−1 exp(−x/2) (3.3)

where kis the number of parameters and Γ(x) is the gamma function, given by

Γ(x) = (x− 1)! (3.4)

For 2 parameters, these three values correspond to a χ2 value of 2.30, 6.18 and 11.83 respectively

from the minimum χ2 value. To judge if the data are a good fit to the model, we divide the mini-

mum χ2 value by the degrees of freedom (d.o.f.), which is given by (n−1). The closer it is to unity,

the better the fit.

We use this method and the data provided by R98 in an attempt to reproduce their results. They

used two different lightcurve fitting methods: the Multi-Colour Light Curve Shape (MLCS) method

and the ∆m15 template fitting method, and published results for both. There were incomplete

lightcurves for some SNe1a data samples, for which they used a snapshot method [32] to obtain µ0

and σµ0 . As we can see, the best fit values of Ωm and ΩΛ are well above the line of no acceleration

for both the MLCS and ∆m15 fitting methods, therefore we can come to the same conclusion as

R98 that the Universe is expanding at an accelerating rate. For the MLCS method, the best fit

values of Ωm and ΩΛ that we obtain are Ωm = 0.2551 and ΩΛ = 0.7143. For the ∆m15 template

fitting method, the best fit values are Ωm = 0.6633 and ΩΛ = 1.4490.

12

Figure 2: Contour plot obtained using the MLCS lightcurve fitting method for 1σ, 2σ and 3σ

contour values. The dashed black line of no acceleration is drawn for reference (Ωm

2 = ΩΛ). Above

the line, q0 < 0. The minimum χ2 value is 51.2746 for 49 d.o.f.

13

Figure 3: Contour plot obtained using the ∆m15 template fitting method for 1σ, 2σ and 3σ contour

values. The dashed black line of no acceleration is drawn for reference. The minimum χ2 value

is 40.2369 for 49 d.o.f. Therefore the MLCS method gives data that is a better fit to the ΛCDM

model.

3.2 SALT2 Lightcurve Fitting

Since the first discovery of accelerating expansion in 1997, many improvements to the method of

statistical analysis have been made. One such modification is to the template fitting method of the

SNe1a lightcurve. The two methods first used by R98 were the MLCS and ∆m15 template fitting

method.

In 2008, Guy et al [33] (hereafter G08) introduced the Spectral Adaptive Lightcurve Template

2 (SALT2) data fitting method which relies on machine learning to fit the lightcurve of an SNe1a

14

to a template model, giving improved distance estimates of µ. They made modifications to address

the fact that the brighter the SNe1a, the slower its decline in luminosity; that is, the brighter the

SNe1a, the wider its lightcurve. Moreover, the brighter the SNe1a, the bluer it will appear to be.

Thus they included an additional two parameters, the stretch x and colour c corrections, when

fitting their lightcurves to the template. The new and improved equation for calculating µ, instead

of (2.1a), becomes, as defined in Section 7.2 of G08,

µB = m∗B −M + α× x1 − β × c (3.5)

where m∗B is the new notation for the apparent magnitude, and the values of α and β are constant

throughout the dataset being studied. Each SNe1a will be stretched along the time axis and colour

corrected by a certain amount to fit the template model. This method has been adopted by many

survey teams when presenting lightcurves of new SN1e data, and is the method we will be adopting

to analyze our SNe1a dataset.

3.3 The Joint Lightcurve Analysis Dataset

The dataset we will be analyzing is the Joint Lightcurve Analysis (JLA) dataset published by Be-

toule et al [18] (hereafter B14). It combines data from the Sloan Digital Sky Survey (SDSS) and

the Supernova Legacy Survey (SNLS) to give 740 SNe1a data samples. B14 also used the SALT2

method to analyse their data and published a complete list of parameters m∗B , x1 and c as well as

their associated uncertainties for each SNe1a data sample.

They present their errors in the form of a 3×740 by 3×740 covariance matrix for the parameters

m∗B , x1 and c. The total covariance matrix C is a sum of 8 covariance matrices that address two

main sources of uncertainties:the systematic uncertainty and statistical uncertainty. It is therefore

given by

C = Ccal + Cmodel + Chost + Cbias + Cpecvel + CnonIa + Cdust + Cstat (3.6)

where Ccal is uncertainty matrix associated with the calibration, Cmodel is the light-curve model

uncertainty, Chost is the mass step uncertainty, Cbias is the bias correction uncertainty, Cpecvel is the

peculiar velocity uncertainty, CnonIa is the contamination due to non-type 1a SN, and Cdust is the

uncertainty due to dust extinction in the Milky Way. These 7 covariance matrices contribute to the

systematic uncertainty of the measurements. Lastly, the statistical error propagation of lightcurve

fit uncertainties is given by Cstat.

The calculation of the χ2 value used by B14 is given by

χ2 = (µobs − µmodel)TC−1η (µobs − µmodel) (3.7)

15

where µobs is the value of µB obtained through (3.5), and µmodel is the value of µ obtained from

the model. Cη is the 740 by 740 covariance matrix of µobs, which is derived from C.

3.4 The Maximum Likelihood Estimation Method

Another statistical method was introduced by Nielsen et al.[16] (hereafter N16), which is based on

calculating the likelihood ratio instead of the χ2 value. Therefore the best fit value is the value that

gives a maximum likelihood value, and not a minimum χ2. Furthermore, they do not just treat Ωm

and ΩΛ as the only parameters to find the best fit values of, but also do a sweep across the set of

8 parameters M0,x10,c0,σM0

,σx10,σc0 ,α,β. Here, we give an introduction of their method.

3.4.1 Bayesian Statistics

In Bayesian statistics, the probability density is given by [34]

P(A|B) ∝ P(B|A)P(B) (3.8)

where P(A) is the prior distribution, P(A|B) is the posterior distribution and the value of P(B|A)

is the likelihood ratio. In this case, A is the model and B is the data. Therefore we are calculating

the probability of the model being a good fit, given the data.

The method used in N16 is based on an extension of Bayesian statistics, known as the Hierar-

chical Bayesian Model. This model should be used, instead of regular Bayesian statistics, when

there are multiple parameters with prior distributions that depend on other parameters [35]. Look-

ing at (3.5), this is indeed true for the data we have, and we shall see that the set of parameters

M0,x10,c0,σM0

,σx10,σc0 ,α,β depend on one another. In this case, the Hierarchical Bayesian prob-

ability density is given by

P(γ, α|β) ∝ P(β|α, γ)P(α|γ)P(γ) (3.9)

Here, P(γ, α|β) is the posterior distribution, P(γ) is the prior distribution and P(β|α, γ)P(α|γ) is

the new likelihood ratio. In the context of SNe1a data analysis, we shall see what α, β and γ

represent, and why we need to include one extra parameter to draw more accurate conclusions

based on the data.

3.4.2 Calculating Likelihood L

The values of m∗B , x1 and c obtained are not the true values due to various sources of noise and

error, both systematic and statistical. We shall denote these observed values as m∗B , x1 and c,

16

and the true values without a hat. Therefore, adopting the form of (3.9), we can express the total

probability density as

P [θ, (M,x1, c)|(m∗B , x1, c)] ∝ P [(m∗B , x1, c)|(M,x1, c), θ]P [(M,x1, c)|θ]P (θ) (3.10)

where the likelihood ratio is L = P [(m∗B , x1, c)|(M,x1, c), θ]P [(M,x1, c)|θ], and θ denotes the model.

In terms of (3.9), θ is γ, and the data we obtain, (m∗B , x1, c), is β. We also wish to infer the

true parameters from the observed parameters, therefore (M,x1, c) is α, where M is the absolute

magnitude which is not taken to be a constant value for every SNe1a here. The likelihood can be

written in terms of an integral

L =

∫P [(m∗B , x1, c)|(M,x1, c), θ]P [(M,x1, c)|θ]dMdx1dc (3.11)

The first factor of the integrand addresses the discrepancy between the obtained values and the

true values, and the second factor addresses the difference between the true values and the model.

We shall now derive an equation to calculate, and subsequently maximize, the value of L.

By plotting a histogram of x1 and c, we see that they roughly follow a Gaussian distribution.

(a) (b)

Figure 4: Histograms of x1 and c, with a Gaussian distribution overlaid. We can see that they

roughly follow a Gaussian distribution, therefore we assume that the true values also follow a

Gaussian distribution.

Assuming a Gaussian distribution of M as well, we model the probability distribution of M , x1

17

and c as a Gaussian function:

P (M) = (2πσ2M0

)−1/2 exp(−[(M −M0)/σM0 ]2/2

)(3.12a)

P (x1) = (2πσ2x10

)−1/2 exp(−[(x1 − x10

)/σx10]2/2

)(3.12b)

P (c) = (2πσ2c0)−1/2 exp

(−[(c− c0)/σc]

2/2)

(3.12c)

where M0, x10 and c0 are their mean values and σM0 , σx10and σc0 are their variances. The to-

tal probability density is a product of these three terms: P (M,x1, c|θ) = P (M)P (x1)P (c). For N

SNe1a data points, we combine the three parameters to form a vector Y = M1, x11, c1, ...,MN , x1N

, cN.The total probability density is then

P (Y |θ) = |2πΣl|−1/2exp(−(Y − Y0)Σ−1

l (Y − Y0)T/2)

(3.13)

where Y0 = M0, x10, c0... is a 3N vector of the mean values, and Σl = diag(σ2

M0, σ2x10, σ2c0) is the

3N×3N diagonal matrix of the variances. The absolute symbol denotes the determinant of the ma-

trix. (3.13) corresponds to the second term that we will substitute into (3.11) in order to calculate L.

Now, we deduce the equation for the first term in (3.11). We define the 3N vector of observed values

X = m∗B1, x11, c1..., m∗BN , x1N , cN and the 3N vector of true valuesX = m∗B1, x11, c1...,m

∗BN , x1N , cN.

The probability density of the observed data X given some true parameters X, P (X|X, θ), is

P (X|X, θ) = |2πΣd|−1/2exp(−(X −X)Σ−1

d (X −X)T/2)

(3.14)

where Σd is the 3N×3N covariance matrix which is referred to as C in B14. However, N16 did not

include the mass step correction Chost, as it does not make a significant difference to the results,

which was also demonstrated by Sharriff et al [16]. To convert m∗B to M , we define the 3N vector

Z = m∗B1−µ1, x11, c1, ..., m∗BN−µN , x1N , cN, where µ is the theoretical value of the distance mod-

ulus based on the model, and the 3N × 3N block diagonal matrix A = 1, 0, 0;−α, 1, 0;β, 0, 1; ...,to get the relation X −X = (ZA−1−Y )A. Then, P (X|X, θ) = P (Z|Y, θ). This is the first term in

(3.11). Combining everything,

L =

∫P (Z|Y, θ)P (Y |θ)dY

= |2πΣd|−1/2|2πΣl|−1/2∫dY exp

(−(Y − Y0)Σ−1

l (Y − Y0)T/2)

exp(−(Y − ZA−1)AΣ−1

d AT(Y − ZA−1)T/2)

=∣∣2π(Σd +ATΣlA)

∣∣−1/2exp[−(Z − Y0A)(Σd +ATΣlA)−1(Z − Y0A)T/2

](3.15)

Just like in R98, we can construct contour plots in the Ωm − ΩΛ parameter space, this time by

finding the set of values for the 8 parameters M0,x10 ,c0,σM0 ,σx10,σc0 ,α,β, that give a maximum

18

likelihood for each pair of (Ωm,ΩΛ). This is known as the profile likelihood LP . Denoting φ as

the 8 parameters, which we treat as the nuisance parameters as we are not plotting them in the

parameter space, and θ as the interesting parameters ΩΛ and Ωm,

LP = maxφL(θ, φ) (3.16)

The likelihood value L is related to χ2 by the formula L = exp(−χ2/2

)—a maximum likelihood

corresponds to a minimum χ2 value. Therefore in terms of χ2, (3.15) becomes

χ2 = 2π∣∣(Σd +ATΣlA)

∣∣− (Z − Y0A)(Σd +ATΣlA)−1(Z − Y0A)T (3.17)

which is a much easier equation to work with. Therefore, we calculate the χ2 values using (3.17)

instead for every (Ωm,ΩΛ), and plot the contour lines of 1σ, 2σ and 3σ.

3.4.3 Comparison to the Least χ2 Method

The Maximum Likelihood Estimation (MLE) method does not only vary Ωm and ΩΛ, but also

the other 8 parameters M0,x10,c0,σM0

,σx10,σc0 ,α,β. An advantage the MLE method has over

the χ2 method is that it does not add an arbitrary uncertainty σµ0(see (3.1)), which N16 argues

can be adjusted to create biased results towards a preferred model, in this case the ΛCDM model.

Therefore the χ2 method only tests for a goodness of fit to the assumed model, but does not check

if it is indeed the correct model. On the other hand, the calculation of L in the MLE method solely

depends on the data obtained without adding uncertainties whose value differs for each SNe1a data

sample.

From the results obtained in N16, the contour plot significantly crosses the line of no accelera-

tion, therefore they argue that a non-accelerating expansion cannot be ruled out.

19

Figure 5: Results obtained by N16, where the dotted lines are the 1σ, 2σ and 3σ contour lines

based on 2 parameter χ2 values. The solid lines are the 1σ and 2σ contour lines for 10 parameters

projected onto this plane. The line of no-acceleration is also shown, and the solid blue contour lines

significantly go beyond it.

Other papers have referenced their method with modifications [36] or additional constraints [19]

to come to the conclusion that current evidence still favour an accelerating expansion. Therefore it

is interesting to expand this new method to explore other Universe models or datasets to see what

results we might obtain from it, and to see if our Universe is indeed expanding at an accelerating

rate, and by how much.

3.5 Markov Chain Monte Carlo Optimization

The method we will be using to find the best fit values of M0,x10,c0,σM0

,σx10,σc0 ,α,β is the

Markov Chain Monte Carlo (MCMC) method. This method, based on Bayesian statistics, is widely

used in many computational fields to approximate the distribution of multiple parameters—here,

we aim to obtain the distributions of all the 8 parameters and find the set of 8 values which gives

the maximum likelihood value.

20

3.5.1 Principles of MCMC

The MCMC algorithm works based on the two concept for which it is named: the Markov Chain

and the Monte Carlo method.

The Monte Carlo method is a powerful method very often used to solve problems in statistics

and physics which require numerical solutions. A basic example of the use of the Monte Carlo

method is to estimate the area of a shape. Random samples are drawn and calculations are done

to determine if they fall within the boundary of the shape. The proportion of the points which fall

within the shape can then be estimated as its area. The more number of samples drawn, the better

the estimation.

Figure 6: Estimating the area of a circle using Monte Carlo. Here the parameter space is represented

by the blue square. The points falling within the circle are black, and the points lying outside are

white. The area of the circle can be approximated by the ratio of black dots to the total number

of dots. When the number of points sampled increases, the approximation becomes more accurate.

A Markov Chain, named after Russian mathematician Andrey Markov, is a chain of events

whereby the probability of moving to any one event only depends on the state of the current event,

and not on the history of the past events traveled [37]. We construct a pictorial representation to

better explain it.

21

Figure 7: A schematic diagram of a Markov chain, with each coloured circle representing a state.

The colour coded arrows and numbers show the possible transitions between the states, and the

probabilities of transitioning to that particular state, respectively. The values from each colour

must add up to 1. In a Markov Chain, the probability of transitioning to a certain state only

depends on where you are right now, and not where you came from.

A Markov chain has the property that, after a long iteration of moving from event to event, the

distribution of past events traveled to becomes independent of its starting point. The distribution

reaches an equilibrium state that will no longer change regardless of how long the iteration continues.

Putting these two concepts together, we get an MCMC algorithm. The algorithm aims to build

up a posterior distribution from the prior distribution, which it is proportional to. From (3.8), we

see that the proportionality constant is nothing but the likelihood ratio. The process of sampling

using MCMC for multiple parameters is roughly summarized in the steps below:

22

1. Choose a random set of points in the specified parameter space, from a given prior distribution

2. Calculate the likelihood ratio of that set of points, and multiply it to the prior to get a new

posterior distribution

3. Based on the posterior probability, choose either to accept or reject that set of points, with

probability p = min(1, P (new position)P (current position) )

4. If the set of points is rejected, repeat from step 1 again

5. If the set of points is accepted, include it into the posterior distribution.

This posterior distribution becomes the new prior. The process repeats from step 1 and the

posterior distribution is progressively updated

The process of random sampling from the prior distribution mirrors a Monte Carlo algorithm.

The MCMC equivalent term is known as ‘random walk’, as the algorithm appears to be randomly

walking within the parameter space. This class of MCMC algorithms described above are known as

the Metropolis-Hastings algorithm and is one of the simplest implementations of MCMC sampling

of multiple variables [38][39].

The array of points being sampled forms a Markov chain, and we exploit its property of reach-

ing an equilibrium distribution no matter the start point to argue that after a sufficient number of

runs, the posterior distribution we obtain through MCMC can reach equilibrium. Moreover, be-

cause of the accept/reject step that we include, the resultant distribution is a good approximation

to the actual distribution, as the MCMC algorithm will tend to sample points in a region of higher

likelihood within the parameter space.

3.5.2 Implementation of MCMC

We adapt an MCMC code by Pitkin [40] and write our own code to calculate the likelihood ratio

and the theoretical value of µ based on the model. The likelihood ratio calculation is based on the

MLE method outlined in Section 3.4.2.

Pitkin implements an affine invariant ensemble method based on Goodman and Weare [41], a

more sophisticated version of the Metropolis-Hastings method that is faster and can handle highly

non-symmetric distributions. The feature of this MCMC algorithm is that it sends out multiple

‘walkers’ (known as an ensemble of walkers) within the parameter space—analogous to creating

multiple Markov chains simultaneously. After one time step, all walkers move by one step. To

determine where each walker travels to next, the algorithm carries out either one of two moves:

A stretch move or a walk move. Pitkin executes a stretch move 75% of the time (stretch option

23

activated with 0.75 probability), with the rest being a walk move that is executed.

In a stretch move, the algorithm randomly chooses another walker with respect to the walker

in question, and proposes its next step to be at a position

Y = Xj + Z(Xk(t)−X(j)) (3.18)

where Xj is the current position of the other walker, Xk(t) is the position of the walker in question

at current time step t, and Z is a scaling factor following a symmetric distribution

g(Z) ∝

1√Z, if Z ⊆ [ 1

a , a]

0, otherwise(3.19)

In Pitkin’s code, the value of a is set to be 3, therefore the code randomly chooses a value and if it

falls between 13 and 3, Z 6= 0.

The proposed position Y , is then accepted with probability

p = min[1, Zn−1 P (Y )

P (Xk(t))] (3.20)

If the position is rejected, the walker’s position will not change, ie. Xk(t+ 1) = Xk.

The second type of move is known as a walk move. For a walker to perform a walk, a set of

more than 2 walkers S is randomly chosen with respect to the current walker in question, Xk. In

Pitkin’s code, the number of walkers chosen is set at 3. Their mean position Xs calculated. The

new proposed position of Xk is Xk(t+ 1) = Xk(t) +W where W has to have the same covariance

as Xs, which gives an equation to calculate it:

W =∑Xj⊆S

Zj(Xj − Xs) (3.21)

where Zj is a random value sampled from a normal distribution. The proposed position is accepted

with probability

p = min[1,P (Xk(t) +W )

P (Xk(t))] (3.22)

which is the same probability used in the Metropolis Hastings algorithm.

24

Figure 8: Pictorial explanation of a stretch (left) and a walk (right). Images taken from [41].

The authors argue that both moves are affine invariant. This gives the advantage that highly

skewed distributions can go through an affine transformation to turn them into much simpler and

manageable distributions to work with, and using an affine invariant MCMC algorithm will not

affect the results.

Pitkin’s MCMC code consists of 6 Matlab functions, of which we use 4: mcmc sampler.m,

draw mcmc sample.m, scale parameters.m and rescale parameters.m.

mcmc sampler is the main MCMC sampler function which takes in 2 functions as inputs: a model

function and a log likelihood calculator function, to run the MCMC sampler. First it resizes the

priors of each parameter such that it ranges from 0 to 1 for flat priors, using the scale parameters

function. It then calls the draw mcmc sample function to propose a new position of the walkers

based on either a stretch or walk move. It calculates the log likelihood of the new point from

the log likelihood function, and adds it to the prior to get the new posterior probability. This is

the probability its uses to choose whether to accept or reject the new point. After the MCMC

run is complete, it will rescale the resultant posterior distributions using rescale parameters.m.

This process of scaling and rescaling is an affine transformation aimed at simplifying the constantly

updating prior distribution it works with. The affine invariant MCMC works particularly well for

our case as we are unsure of the distributions of the 8 parameters we wish to test—they might be

highly non-symmetric therefore using such a method will ensure accuracy in our results.

We write a code to calculate the log likelihood using N16’s method, and another code to cal-

25

culate the distance modulus µ based on the model we wish to test (ΛCDM, wCDM etc.). To run

Pitkin’s MCMC code, we specify the parameters we are running through as well as their initial

priors. For simplicity, we assume flat priors for all 8 parameters, with a range which we will choose.

We also input the data m∗B , x1, c and the covariance matrix C.

Finally, we have to specify the number of steps for the MCMC algorithm to run through, as

well as the number of ensemble walkers. The code will output an array of points sampled for each

parameter, and an array of log likelihood values calculated at each run. Therefore the length of the

vectors will be equal to the number of steps specified.

3.5.3 Testing our MCMC code

In the MCMC implementation, we assume flat priors for all 8 parameters M0,x10,c0,σM0

,σx10,σc0 ,α,β.

First, we must determine the appropriate number of points to sample (Nmcmc). It should be

long enough in order for the resultant posterior distribution to converge and give an accurate esti-

mation of the true posterior distribution, but not too long such that it becomes inefficient.

We vary the number of runs and the plot the values of each parameter. When the MCMC algorithm

has reached equilibrium, the values of each parameter should stabilize after a certain number of

runs. For the test, we use a randomly chosen value of Ωm = 0, ΩΛ = 0.2 and the number of walkers

is set to 20. The burn-in number, the number of points from the start of the first run which will

be discarded, is set to 10 for the time being. The burn-in number allows us to specify the number

of points to discard at the start of the MCMC run, and it usually refers to the points which have

yet to reach equilibrium. This ensures that the resultant array of points that are kept, are part of

a posterior distribution that has reached equilibrium. We will determine the optimum number for

the burn-in in our following tests.

26

Figure 9: Values of M0,x10,c0,σM0

,σx10,σc0 ,α,β sampled, as well the χ2 values for each run when

Nmcmc=50. For every parameter the values still oscillate erratically and have yet to stabilize. The

χ2 value also does not exhibit a downward trend yet, which is what we are looking for to check if

the walkers have reached a region of higher likelihood (ie. low χ2 value).

27

Figure 10: Nmcmc is increased to 1000. Here we start to see decreasing values of χ2, however the

values of the 8 parameters still have not stabilized, which imply that equilibrium is not yet achieved.

28

Figure 11: Nmcmc is further increased to 10000, and the number of ensemble walkers increased to

30. The larger the number of walkers, the faster the algorithm reaches equilibrium. Here, we see

that the values of each parameter start stabilizing around the 3000th run. The values of x1 is still

very erratic. However, we see that this does not really affect the value of χ2: it decreases rapidly

and stabilizes to slightly below −200 (as seen from the inset plot) after the 3000th. Therefore we

can still safely say that the MCMC has reached equilibrium, from approximately the 3000th run

onward Therefore we can set our burn-in value as 3000, and Nmcmc as 7000. This ensures that

7000 values that the code outputs have stabilized and reached equilibrium distribution.

29

Figure 12: Resultant posterior distribution of the 8 parameters after discarding the first 3000

runs as burn in. The blue bar graph represents the histogram, with the red line representing the

distribution fit superimposed onto it. We can see that they mostly follow Gaussian distributions.

The distributions end abruptly for x1 and c as it is confined within the range of values we specify

for our prior. Therefore we subsequently try expanding the range of x1 and c in our subsequent

code so as to explore a wider parameter space and possibly obtain a more symmetric distribution.

30

Figure 13: We find the run with the lowest χ2 value obtained for Nmcmc=10000, and use the values

of c, σc, x1 and σx of that run to plot the Gaussian distribution of x1 and c, following (3.11b) and

(3.11c). We compare it to the distribution curves of Figure 4 (the actual JLA data). We see that our

results (orange curve) match quite well with the data (blue curve), especially for the x1 parameter.

Therefore we argue that our model can accurately describe the data. An even better match might

be obtained when we widen the range of the initial prior, as shown in Figure 12.

However, we still have to ensure that our MCMC code gives accurate results. Therefore we

compare our results with N16 for fixed values of Ωm and ΩΛ. We fix the values of Ωm and ΩΛ, and

test if the best fit values of M0,x10,c0,σM0

,σx10,σc0 ,α,β that the MCMC code obtains (the run

with the lowest χ2 value), match those that were presented by N16.

Figure 14: Best fit values for the 8 parameters as presented by N16 (blue columns), compared to

the results we obtain with our MCMC code (unshaded columns), for 6 special cases of Ωm and ΩΛ

(values in bold). With the exception of one or two cases, almost all values we are within ±0.01 of

N16’s results.

31

We can safely agree that our MCMC code is accurate, with a sufficient number of points sampled

and an appropriate burn-in number. With all our code in place, we run it to obtain results for the

ΛCDM and the various parametrizations of the wCDM models.

To speed up our code, we make use of the Matlab Parallel Computing Toolbox to make our code run

parallel, such as using parfor loops while sweeping through values of ΩΛ and Ωm. Additionally, we

submit our code to NUS’ High Performance Computing (HPC) cluster, which has up to 20 super-

computer CPU cores that can run simultaneously for parallel codes. This cuts short computational

time by a significant amount as compared to running the code on our own computer, which only

has 2 cores.

32

4 Results

4.1 ΛCDM Model

Figure 15: Contour plot (left) and scatter plot (right) of Ωm against ΩΛ for the ΛCDM model,

using the MLE and MCMC method, with 1σ, 2σ and 3σ lines drawn. The dotted line represents

the line of no acceleration. The best fit value of Ωm and ΩΛ is 0.40 and 0.55 respectively, with a

minimum χ2 value of −231.209.

Here, we run through Ωm and ΩΛ from the values of 0 to 1, with a step size of 0.05. The plot we

obtain is comparable to N16 (Fig. 5) where the 1σ and 2σ lines cross the line of no acceleration.

Unlike N16, the contours are not smooth, which is due to the fact that an MCMC method was

used— [17] also obtained ragged contours when they implemented their own MCMC algorithm.

4.2 wCDM Model

4.2.1 Parametrization A

First, we test our code for a flat wCDM model, following [19]. We do a parameter sweep across Ωm

and w for Ωk = 0.

33

Figure 16: Contour plot (left) and scatter plot (right) of Ωm against w for Ωk = 0 with 1σ, 2σ and

3σ lines. The dotted line is the line of no acceleration, given by (2.26). We see that in our plot,

the contour lines do not touch the line of no acceleration. The minimum χ2 value is −229.6220, for

Ωm = 0.4211 and w = −1.

Now, we generalize to a non-flat wCDM model where we do a parameter sweep across Ωm, ΩΛ

and w.

34

Figure 17: Plot of w against Ωm for ΩΛ = 0.6, with 1σ, 2σ and 3σ contour lines corresponding

to a difference in χ2 values for 3 parameters, since we run through 3 parameters instead of just 2.

The line of no acceleration is given by (2.28), with the value of ΩΛ being 0.6. The best fit value is

Ωm = 0.4,w = −1 and ΩΛ = 0.6 with a χ2 of −229.8145.

35

Figure 18: Plots of w against Ωm for ΩΛ = 0 to 0.4, with their respective dotted lines of no

acceleration.

36

Figure 19: Plots of w against Ωm for for ΩΛ = 0.5 and 0.7 < ΩΛ < 1.0, with their respective dotted

lines of no acceleration.

4.2.2 Parametrization B

Following [30], we run through the values of −2 < w0, 0, −2 < w1 < 8 and Ωm for a flat wCDM

parametrization B model. We present plot of w0 against Ωm for interesting values of w1.

37

Figure 20: Plots of w0 against Ωm for values of −2 < w1 < 1, with 1σ, 2σ and 3σ contours. Here,

we still use (2.29) as the equation of the line of no acceleration since q0 does not depend on w1 term

(z = 0). The best fit values are Ωm = 0.4, w0 = −0.8, w1 = −1, with a χ2 value of −229.4125.

4.2.3 Parametrization C

We run through the same range of w0 and w1 as parametrization B for parametrization C and

present our results, again with plots of Ωm against w0 for interesting values of w1.

38

Figure 21: Plots of w0 against Ωm for values of −2 < w1 < 1, with 1σ, 2σ and 3σ contours. The

best fit values are Ωm = 0.4, w0 = −0.8, w1 = −1, with a χ2 value of −227.4537.

39

5 Discussion

5.1 ΛCDM Model

Our results for the ΛCDM model differ slightly from N16 as the contour lines cross the line of no

acceleration to a greater extent. Our minimum χ2 value of −231.209 is also lower than the value

published by N16 (−214.97). The table below shows how our best-fit Ωm and ΩΛ values compare

to other works:

Ωm ΩΛ

Our work 0.40 0.55

N16 0.341 0.569

Sharriff et al 0.340 0.524

R98 (MLCS) 0.2551 0.7143

We see that our best-fit values are within 1σ of N16 and Sharriff et al’s results, further lend-

ing clout to the accuracy of our results. This shows that our MCMC code is sufficiently accurate

in finding the maximum likelihood and optimizing over the 8 parameter space.

5.2 wCDM Model

5.2.1 Parametrization A

We present the best fit values obtained for both the flat and non-flat wCDM models for parametriza-

tion A.

Model Ωm ΩΛ Ωk w

Flat 0.4211 0.5789 0 −1

Non-flat 0.4 0.6 0 −1

We see that the best fit values for the flat and non-flat models are in agreement with each other.

Moreover, for both cases, the contour plot is much wider than that obtained by Haridasu et al,

especially in the non-flat case, where the 3σ contour crosses the line of no acceleration. This differs

from their claim that the results for an accelerating expansion is reinforced in the wCDM model,

particularly when the constraint for a flat Universe is removed since the contours are wider than in

the flat case. Therefore the claim for accelerating expansion should not be made purely based on

assuming a flat Universe

40

5.2.2 Parametrizations B and C

We present the best fit values for Parametrizations B and C below:

Parametrization Ωm w0 w1

B 0.4 −0.8 −1

C 0.4 −0.8 −1

In the best fit plots (ie. w1 = −1) for both parametrizations, we see that the contour lines also cross

the line of no acceleration to an even greater extent than in parametrization A (∼ 2σ level). By

generalizing the evolution of w, the evidence for constant acceleration becomes stronger. Perhaps

if we extend these parametrizations to include the non-flat case, the data will become even more

consistent with a constant acceleration model.

5.3 Model Comparison

To assess which model gives the best fit of our data, we calculate the Akaike Information Criterion

(AIC) and the Bayesian Information Criterion (BIC), two values that are often calculated to test

the goodness of fit of different statistical models. The formulae to calculate AIC and BIC are given

by [42] [43]:

AIC = 2k − 2 ln(L) (5.1a)

BIC = ln(n)k − 2 ln(L) (5.1b)

where k is the number of parameters estimated, n is the number of data samples (in this case, the

number of SNe1a data samples), and L is the maximum likelihood. The best fit model is the model

with the lowest AIC and BIC values. However, the AIC and BIC test does not give an absolute

benchmark on the goodness of fit of a model, only a comparison between models.

We find that the ΛCDM model has the lowest AIC and BIC values, and here, we present the

∆AIC and ∆BIC values for the other cosmological models as compared to the ΛCDM model:

Model ∆AIC ∆BIC

ΛCDM 0 0

Flat wCDM 1.5870 1.5870

Non-flat wCDM (Param. A) 3.3945 8.0012

Flat wCDM (Param. B) 3.7965 8.4032

Flat wCDM (Param. C) 5.7553 16.9686

41

Based on the AIC and BIC values, the ΛCDM model is still the best fit cosmological model given

the data, followed by the flat wCDM model. Moreover, the results from their generalizations serve

to further validate their best fit value results, and the contour plots obtained using the MLE method

provide interesting insights regarding the extent to which an accelerating expansion is certain. In

almost all of our results, an accelerating expansion is only consistent at ∼ 2σ confidence levels. As

such, constant expansion of the Universe cannot be ruled out, especially in the case of generalized

cosmological models.

5.4 Sources of Error and Improvements

We can improve the accuracy of our results by running through the parameters Ωm, ΩΛ, w, w0 and

w1 with smaller step size, as there might be points with smaller χ2 value that we did not manage to

capture with a 0.1 step size. Moreover, our contour plots will become more accurate with a smaller

step size, since the contours are drawn by interpolating between points. With a smaller step size,

different best fit values might be obtained, which will also be more accurate.

To better ensure that the minimum χ2 value we obtain is the global minimum within all pa-

rameters and not just a local minimum, we should also increase the range of the priors of the 8

parameters M0,x10 ,c0,σM0 ,σx10,σc0 ,α,β that we are sampling in our MCMC code. Subsequently,

with a wider prior distribution, we will need to run the MCMC algorithm for a larger number of

iterations to ensure convergence.

5.5 Future Work

This work can be extended by including other SNe1a data sets such as the newest PANTHEON and

DES data sets, and other data sources such as BAO, CMB and gamma ray burst data. Perhaps

with additional data, an accelerating expansion will become more secure, similar to what Haridasu

et al. did by including CMB data to obtain much smaller contour plots which are well below the

line of no acceleration. Naturally, inclusion of these data sets would require modifications to the

MLE equation, since the 8 parameters M0,x10 ,c0,σM0 ,σx10,σc0 ,α,β only apply to SNe1a data.

We can study other parametrizations of the wCDM model and generalize them to the non-flat

case, which might yield results that argue more strongly for a constant acceleration model. This

method can also be extended to other values of the Hubble constant, H0 to study how the contour

plots might shift, similar to how R98 obtained different results when using different values of H0

for other SNe1a light curve fitting methods.

42

6 Conclusion

In this thesis, we have used a modified statistical method to analyze data from the JLA SNe1a

data set against 5 different cosmological models. This statistical method is based on a Maximum

Likelihood Estimation and an MCMC optimization procedure to obtain the best fit values of 8

SNe1a parameters, along with Ωm, ΩΛ, w, w0 and w1. This method has the advantage of being

bias-free as compared to the conventional least χ2 method, as arbitrary values of uncertainty σ can

be added to each data point in the χ2 calculation to make the data fit a desired model.

For the ΛCDM model, we obtain best fit values of Ωm = 0.40, ΩΛ = 0.55, which is within 1σ

of N16. When we do a model comparison based on calculated AIC and BIC values, we see that the

ΛCDM model is still favoured over the wCDM model and its parametrizations.

With the MLE and MCMC method, our contour plots for all 5 models cross the line of no ac-

celeration, which is a different result from that obtained by R98 during their discovery of the

Universe’s accelerating expansion, and is similar to the contour plot published by N16. Moreover,

when we generalize the ΛCDM model to the wCDM model and its parametrizations, we see that

constant expansion becomes increasingly favoured (the contours cross the line of no acceleration at

smaller σ values). The conclusion of an accelerating expansion purely based on the study of SNe1a

data does not seem to be as secure as before, thus more work needs to be done for us to better

understand the Universe we live in.

43

A Derivation of Friedmann Equations

From Einstein’s Field Equation,

Rµν −1

2Rgµν + Λgµν =

8πG

c4Tµν (A.1)

We first solve this equation using the FLRW metric in Cartesian coordinates (in the following

calculations, c = G = 1):

ds2 = −dt2 + a(t)2(dx2 + dy2 + dz2) (A.2)

To evaluate the Ricci tensors and Ricci scalar, we first calculate the Christoffel symbols

Γβµν =1

2gβα(∂νgαµ + ∂µgαν − ∂αgµν) (A.3)

The non-vanishing Christoffel symbols are:

Γ011 = −a(t)a(t) (A.4a)

Γ022 = −a(t)a(t) (A.4b)

Γ033 = −a(t)a(t) (A.4c)

Γ101 =

a(t)

a(t)(A.4d)

Γ202 =

a(t)

a(t)(A.4e)

Γ303 =

a(t)

a(t)(A.4f)

(A.4g)

The Riemann tensor is given by

Rσµγβ = ∂γΓσµβ − ∂βΓσµγ + ΓνµβΓσνγ − ΓνµγΓσνβ (A.5)

44

The non-vanishing Riemann tensors are:

R1010 =

−a(t)a(t)

a(t)2(A.6a)

R2020 =

−a(t)a(t)

a(t)2(A.6b)

R3030 =

−a(t)a(t)

a(t)2(A.6c)

R0101 =

−a(t)a(t)

a(t)2(A.6d)

R2121 = −a(t)2 (A.6e)

R3131 = −a(t)2 (A.6f)

R0202 =

−a(t)a(t)

a(t)2(A.6g)

R1212 = −a(t)2 (A.6h)

R3232 = −a(t)2 (A.6i)

R0303 =

−a(t)a(t)

a(t)2(A.6j)

R1313 = −a(t)2 (A.6k)

R2323 = −a(t)2 (A.6l)

(A.6m)

Therefore the Ricci tensors are:

R00 = g11g11R1010 + g22g22R

2020 + g33g33R

3030 = −3a(t)

a(t)(A.7a)

R11 = g00g00R0101 + g22g22R

2121 + g33g33R

3131 = −a(t)a(t)− 2a(t)2 (A.7b)

R22 = g00g00R0202 + g11g11R

1212 + g33g33R

3232 = −a(t)a(t)− 2a(t)2 (A.7c)

R33 = g00g00R0303 + g11g11R

1313 + g22g22R

2323 = −a(t)a(t)− 2a(t)2 (A.7d)

(A.7e)

The Ricci scalar is:

R = g00R00 + g11R11 + g22R22 + g33R33 =6a(t)a(t) + 6a(t)2

a(t)2(A.8)

45

In spherical coordinates, the metric equation is given by (2.4). The Ricci tensors and Ricci scalar

become

R00 =−3a(t)

a(t)(A.9a)

R11 = R22 = R33 =a(t)a(t) + 2a(t)2 + 2k

a(t)2gii (A.9b)

R =6a(t)a(t) + 6a(t)2 + k

a(t)2(A.9c)

where i = 1, 2, 3. Assuming the components of the energy-momentum tensor Tµν to be as such:

T00 = ρ(t) (A.10a)

T0i = 0 (A.10b)

Tii = p(t)gii (A.10c)

The (00) component of the Field Equation becomes

−3a(t)

a(t)+

1

2· 6a(t)a(t) + 6a(t)2 + k

a(t)2− Λ = 8πGρ(t) (A.11a)

a(t)2 + k

a(t)2− Λ

3=

8πG

3ρ(t) (A.11b)

where we arrive at the first Friedmann equation.

The (ii) component of the Field Equation becomes

a(t)a(t) + 2a(t)2 + 2k

a(t)2gii −

1

2· 6a(t)a(t) + 6a(t)2 + k

a(t)2+ Λgii = 8πGp(t)gii (A.12a)

a(t)

a(t)− Λ

3= −4πG

3(ρ(t) + 3p(t)) (A.12b)

which is the second Friedmann equation.

B MATLAB Codes

Here we publish the codes we have written to produce the results presented above.

B.1 χ2 Calculation (based on R98)

function chi_squared_statistic

%Steps between Omega_m and Omega_Lambda

46

n=50;

%Constants

%H_0=63.8; %template fitting

H_0=65.2; %MLCS

delta_v=200;

c=299792.458;

%Data taken from R98

%data=xlsread(’SN data.xlsx’,’Template Fitting’);

data=xlsread(’SN data.xlsx’,’MLCS’);

redshift=data(:,1);

mu_0=data(:,2);

sigma_0=data(:,3);

%pre-allocation

chi_square_values=zeros(length(redshift),n,n);

omega_m_counter=0;

omega_lambda_counter=0;

%Running through values of omega_m and omega_lambda

for omega_m=linspace(0,2.5,n)

omega_m_counter=omega_m_counter+1;

for omega_lambda=linspace(-1,3,n)

omega_lambda_counter=mod(omega_lambda_counter+1,n);

if omega_lambda_counter==0

omega_lambda_counter=n;

end

omega_k=1-omega_lambda-omega_m;

%solving for distance modulus mu, stored in a 3d array f

if omega_k<0

fun=@(x)((1+x).^2.*(1+omega_m.*x)-x.*(2+x).*omega_lambda).^(-1/2);

sample_z=linspace(0,1,50);

v=arrayfun(@(z)integral(@(z)fun(z),0,z),sample_z);

integral_part=interp1(sample_z,v,redshift);

47

D_L=c*H_0^(-1).*(1+redshift).*(abs(omega_k))^(-1/2).*sin((abs(omega_k))^(1/2).*...

integral_part);

theo_mu=5.*log10(D_L)+25;

elseif omega_k==0





D_L=c*H_0^(-1).*(1+redshift).*integral_part;


else





D_L=c*H_0^(-1).*(1+redshift).*(abs(omega_k))^(-1/2).*sinh((abs(omega_k))^(1/2).*...

integral_part);


end

sigma_nu=(delta_v*5)./(c*redshift*log10(10));

chi_square_values(omega_lambda_counter,omega_m_counter,:)=(theo_mu-mu_0).^2./...

(sigma_0.^2+sigma_nu.^2);

end

end

chi_square=squeeze(sum(chi_square_values,3));

chi_square(imag(chi_square)~=0)=NaN;

%disp(chi_square)

[min_val,idx]=min(chi_square(:));

[I_row, I_col] = ind2sub(size(chi_square),idx);

disp(min_val)

lambda=linspace(-1,3,n);

m=linspace(0,2.5,n);

48

disp(m(I_col))

disp(lambda(I_row))

[M,L]=meshgrid(m,lambda);

zlim([min_val-1 min_val+20])

v=[min_val+2.3,min_val+6.18,min_val+11.83];

contour(M,L,real(chi_square),v,’black’)

colormap(cool)

xlabel(’omega m’)

ylabel(’omega lambda’)

end

B.2 MLE Calculation (ΛCDM Model)

function mle_lcdm

%Number of steps between omega_m and omega_lambda

n=20;

%sweeping through 2 parameters Omega_m and Omega_Lambda

parfor omega_m_counter=linspace(1,21,n+1)

for omega_lambda=linspace(0,1,n+1)

omega_m=(omega_m_counter-1)/n;

min_val=SN_MCMC_sampler(omega_lambda,omega_m)

stat=[num2str(min_val),’,’,num2str(omega_m),’,’,num2str(omega_lambda)];

disp(stat)

end

end

end

B.3 MCMC Sampler Code (ΛCDM Model)

function min_val=SN_MCMC_sampler(omega_m,omega_lambda)

49

global c H_0 N

c=299792.458;

%Hubble Parameter

H_0=70;

%Number of SNe1a sample data

N=740;

%Loading data from JLA dataset

exp_data=dlmread(’JLA.txt’);

%z

redshift=exp_data(:,1);

%Apparent magnitude m^*_b

apparent_m=exp_data(:,2);

%Stretch factor x_1

X=exp_data(:,3);

%Colour factor c

C=exp_data(:,4);

%Loading 740x740 covariance matrices

C_bias=fitsread(’C_bias.fits’);

C_cal=fitsread(’C_cal.fits’);

C_dust=fitsread(’C_dust.fits’);

C_model=fitsread(’C_model.fits’);

C_nonia=fitsread(’C_nonia.fits’);

C_pecvel=fitsread(’C_pecvel.fits’);

C_stat=fitsread(’C_stat.fits’);

sigma_lens=dlmread(’sigmalens_table.txt’);

sigma_z=dlmread(’sigmaz_table.txt’);

C_host=fitsread(’C_host.fits’);

%Total covariance matrix sigma_d

cov_mat=C_stat+C_bias+C_cal+C_dust+C_model+C_nonia+C_pecvel+sigma_z+sigma_lens;

%Calculating distance modulus mu based on the model

50

theo_mu=mu_calculator(redshift,omega_m,omega_lambda);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%MCMC CODE%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Number of MCMC samples

Nmcmc=2000;

%Number of burn-in samples

Nburnin=3000;

%Function to calculate log likelihood

likelihood=@logL_chisquare;

%Function to calculate mu based on the model

model=@mu_calculator;

%JLA data

data1=redshift;

data2=apparent_m;

data3=X;

data4=C;

data5=cov_mat;

%Specifying the parameters to sample, and its prior distributions

%Assume uniform (flat) priors with ranges specified from 2nd to 3rd input

prior=’a’,’uniform’,0,0.2,’fixed’;

’b’,’uniform’,3,5,’fixed’;

’M_0’,’uniform’,-20,-18,’fixed’;

’c_0’,’uniform’,-0.025,-0.01,’fixed’;

’x_0’,’uniform’,0.03,0.06,’fixed’;

’sigma_m_0’,’uniform’,0,1,’fixed’;

’sigma_c_0’,’uniform’,0,1,’fixed’;

’sigma_x_0’,’uniform’,0,1,’fixed’;

%Entra parameters

extraparams=’theo_mu’,theo_mu;

%Calling MCMC_sampler function with 30 ensemble walkers

[post_samples, logP] = mcmc_sampler(data, likelihood, model, prior, extraparams,...

51

’Nmcmc’, Nmcmc, ’Nburnin’, Nburnin,’NensembleStretch’, 30);

%Extracting points sampled for each parameter

a=post_samples(:,1);

b=post_samples(:,2);

M_0=post_samples(:,3);

c_0=post_samples(:,4);

x_0=post_samples(:,5);

sigma_m_0=post_samples(:,6);

sigma_c_0=post_samples(:,7);

sigma_x_0=post_samples(:,8);

%Calculating chi square value from log likelihood of points sampled

chisquare=-2.*logP;

%Finding minimum chi square value

[min_val,~]=min(chisquare(:));

end

B.4 Log Likelihood Calculation

function logL=logL_chisquare(data,model,parnames,parvals)

global N

%Extracting data arrays from input

redshift=data1;

apparent_m=data2;

X=data3;

C=data4;

cov_mat=data5;

%Extracting values of each parameter

for i=1:length(parnames)

switch parnamesi

52

case ’a’

a=parvalsi;

case ’b’

b=parvalsi;

case ’M_0’

M_0=parvalsi;

case ’c_0’

c_0=parvalsi;

case ’x_0’

x_0=parvalsi;

case ’sigma_m_0’

sigma_m_0=parvalsi;

case ’sigma_c_0’

sigma_c_0=parvalsi;

case ’sigma_x_0’

sigma_x_0=parvalsi;

case ’theo_mu’

theo_mu=parvalsi;

end

end

%Matrix A

A=[1 0 0; -a 1 0; b 0 1];

Ar=repmat(A, 1, N);

Ac=mat2cell(Ar, size(A,1), repmat(size(A,2),1,N));

block_A=blkdiag(Ac:);

%Vector Y_0

Y_0=repmat([M_0 x_0 c_0],1,N);

%Matrix Sigma_l

sigma_l=[sigma_m_0 0 0; 0 sigma_x_0 0; 0 0 sigma_c_0];

B=repmat(sigma_l, 1, N);

Bc=mat2cell(B, size(sigma_l,1), repmat(size(sigma_l,2),1,N));

block_sigma_l=blkdiag(Bc:);

53

%Calculating log determinant(first factor of L)

ATCOVlA=transpose(block_A)*block_sigma_l*block_A;

det_part=cov_mat+ATCOVlA;

chol_part=chol(det_part);

log_of_det=2*sum(log(diag(chol_part)))+3*N*log(2*pi);

%Calculating exponential factor of L (second factor)

abs_mag=apparent_m-theo_mu;

Z_hat=reshape(transpose([abs_mag X C]),[1,3*N]);

exp_part=(Z_hat-Y_0*block_A)*(det_part\transpose(Z_hat-Y_0*block_A));

%Sum of log determinant and exponential factor=-2logL=chi^2

chi_square=log_of_det+exp_part;

%Finding log likelihood from chi^2

logL=-chi_square/2;

end

B.5 Distance Modulus Calculation (ΛCDM Model)

function theo_mu=mu_calculator(redshift,omega_m,omega_lambda)

global c H_0

d_H=c/H_0;

omega_k=1-omega_m-omega_lambda;

%Calculation of d_L depends on value of Omega_k

if omega_k<0

%Hubble Function H(z)

fun=@(x)(omega_m*(1+x).^3+omega_lambda+omega_k.*(1+x).^2).^(-1/2);

%Interpolation of z to speed up integration

sample_z=linspace(0,1.33,25);



54

%Calculating luminosity distance d_L

D_L=d_H.*(1+redshift).*(abs(omega_k))^(-1/2).*sin((abs(omega_k))^(1/2).*integral_part);

%Calculating mu


elseif omega_k==0





D_L=d_H.*(1+redshift).*integral_part;


else





D_L=d_H.*(abs(omega_k))^(-1/2).*(1+redshift).*sinh((abs(omega_k))^(1/2).*integral_part);


end

end

B.6 MLE Calculation (wCDM Model Parametrization A)

function mle_wcdmA


n=10;



for omega_lambda=linspace(0,1,n+1)

for w_0=linspace(-1,0,n+1)


55

min_val=wSN_MCMC_sampler(omega_lambda,omega_m,w_0)

stat=[num2str(min_val),’,’,num2str(omega_m),’,’,num2str(omega_lambda),...

’,’,num2str(w_0)];

disp(stat)

end

end

end

end

B.7 MCMC Sampler Code (wCDM Model Parametrization A)

function wSN_MCMC_sampler(omega_lambda,omega_m,w)

global c H_0 N

c=299792.458;

%Hubble Parameter

H_0=70;


N=740;



%z




%Stretch factor x_1

X=exp_data(:,3);

%Colour factor c

C=exp_data(:,4);




56












theo_mu=wmu_calculator(redshift,omega_m,omega_lambda,w);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%MCMC CODE%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Nmcmc=2000;


Nburnin=3000;




model=@wmu_calculator;

%JLA data

data1=redshift;

data2=apparent_m;

data3=X;

data4=C;

data5=cov_mat;




57



’c_0’,’uniform’,-0.025,-0.01,’fixed’;

’x_0’,’uniform’,0.03,0.06,’fixed’;




%Entra parameters















chisquare=-2.*logP;



end

B.8 Distance Modulus Calculation (wCDM Model Parametrization A)

function theo_mu=wmu_calculator(redshift,omega_m,omega_lambda,w)

58

global c H_0

d_H=c/H_0;

omega_k=1-omega_m-omega_lambda;

%Calculation of d_L depends on value of Omega_k

if omega_k==0

%Hubble Function H(z)

fun=@(x)((omega_m.*(1+x).^3+omega_k.*(1+x).^2+omega_lambda.*(1+x).^(3+3*w)).^(-1/2));

%Interpolation of z to speed up integration




%Calculating luminosity distance d_L


%Calculating mu


elseif omega_k<0





D_L=d_H.*(1+redshift).*(abs(omega_k))^(-1/2).*sin((abs(omega_k))^(1/2).*integral_part);


else





D_L=d_H.*(abs(omega_k))^(-1/2).*(1+redshift).*sinh((abs(omega_k))^(1/2).*integral_part);


end

end

59

B.9 MLE Calculation (wCDM Model Parametrization B)

function mle_wcdmB


n=10;






omega_lambda=1-omega_m;

min_val=wbSN_MCMC_sampler(omega_lambda,omega_m,w_0,w_1)


’,’,num2str(w_0),’,’,num2str(w_1)];

disp(stat)

end

end

end

end

B.10 MCMC Sampler Code (wCDM Model Parametrization B)

unction min_val=wbSN_MCMC_sampler(omega_lambda,omega_m,w_0,w_1)

global c H_0 N

c=299792.458;

%Hubble Parameter

H_0=70;


N=740;

60



%z




%Stretch factor x_1

X=exp_data(:,3);

%Colour factor c

C=exp_data(:,4);















theo_mu=wbmu_calculator(redshift,omega_m,omega_lambda,w_0,w_1);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%MCMC CODE%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Nmcmc=2000;


Nburnin=3000;

61




model=@wbmu_calculator;

%JLA data

data1=redshift;

data2=apparent_m;

data3=X;

data4=C;

data5=cov_mat;






’c_0’,’uniform’,-0.025,-0.01,’fixed’;

’x_0’,’uniform’,0.03,0.06,’fixed’;




%Entra parameters










62






chisquare=-2.*logP;



end

B.11 Distance Modulus Calculation (wCDM Model Parametrization B)

function theo_mu=wbmu_calculator(redshift,omega_m,omega_lambda,w_0,w_1)

global c H_0

d_H=c/H_0;

%Omega_k=0 since flatness assumed

fun=@(x)((omega_m.*(1+x).^3+omega_k.*(1+x).^2+omega_lambda.*exp(3*w_1.*x).*(1+x).^...

(3+3*w_0-3*w_1)).^(-1/2));






end

B.12 MLE Calculation (wCDM Model Parametrization C)

function mle_wcdmC


n=10;


63





omega_lambda=1-omega_m;

min_val=wcSN_MCMC_sampler(omega_lambda,omega_m,w_0,w_1)


’,’,num2str(w_0),’,’,num2str(w_1)];

disp(stat)

end

end

end

end

B.13 MCMC Sampler Code (wCDM Model Parametrization C)

function min_val=wcSN_MCMC_sampler(omega_lambda,omega_m,w_0,w_1)

global c H_0 N

c=299792.458;

%Hubble Parameter

H_0=70;


N=740;



%z




%Stretch factor x_1

64

X=exp_data(:,3);

%Colour factor c

C=exp_data(:,4);















theo_mu=wcmu_calculator(redshift,omega_m,omega_lambda,w_0,w_1);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%MCMC CODE%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Nmcmc=2000;


Nburnin=3000;




model=@wcmu_calculator;

%JLA data

data1=redshift;

data2=apparent_m;

65

data3=X;

data4=C;

data5=cov_mat;






’c_0’,’uniform’,-0.025,-0.01,’fixed’;

’x_0’,’uniform’,0.03,0.06,’fixed’;




%Entra parameters















chisquare=-2.*logP;

66



end

B.14 Distance Modulus Calculation (wCDM Model Parametrization C)

function theo_mu=wcmu_calculator(redshift,omega_m,omega_lambda,w_0,w_1)

global c H_0

d_H=c/H_0;

%Omega_k=0 since flatness assumed

fun=@(x)((omega_m.*(1+x).^3+omega_k.*(1+x).^2+omega_lambda.*exp(3*w_1.*(1./(1+x)-1)).*(1+x).^...

(3+3*w_0+3*w_1)).^(-1/2));






end

67

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