Galaxies as Clocks and the Universal Expansion

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IN

DEGREE PROJECT ENGINEERING PHYSICS, SECOND CYCLE, 30 CREDITS

,

STOCKHOLM SWEDEN 2021

Galaxies as Clocks and the

Universal Expansion

ANDERS AHLSTRÖM KJERRGREN

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

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Master’s Thesis

Galaxies as Clocks and the

Universal Expansion

Anders Ahlstr¨

om Kjerrgren

Supervisor:

Prof. Edvard M¨

ortsell

KTH Royal Institute of Technology School of Engineering Sciences

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Examensarbete f¨or avl¨aggande av masterexamen i Teknisk fysik, inom ¨ amnes-omr˚adet Teoretisk fysik.

Master’s thesis for the degree of Master of Science, in the subject area of Theoretical Physics.

TRITA-SCI-GRU 2021:016 c

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Abstract

The Hubble parameter H(z) is a measure of the expansion rate of the universe at redshift z. One method to determine it relies on inferring the slope of the redshift with respect to cosmic time, where galaxy ages can be used as a proxy for the latter. This method is used by Simon et al. in [1], where they present 8 determinations of the Hubble parameter. The results are surprisingly precise given the precision of their data set. Therefore, we reanalyze their data using three methods: χ2 _{minimization, Monte Carlo sampling, and Gaussian processes.}

The first two methods show that obtaining 8 independent values of the Hubble parameter yields significantly larger uncertainties than those presented by Simon et al. The last method yields a continuous inference of H(z) with lower uncertainties. However, this is obtained at the cost of having strong correlations, meaning that inferences at a wide range of redshifts provide essentially the same information. Furthermore, we demonstrate that obtaining 8 independent values for the Hubble parameter with the same precision as in [1] requires either significantly increasing the size of the data set, or significantly decreasing the uncertainty in the data. We conclude that their resulting Hubble parameter values can not be derived from the employed data.

Sammanfattning

Hubbleparametern H(z) är ett m˚att p˚a universums expansionshastighet vid rödskift z. En metod som bestämmer parametern bygger p˚a att hitta lutningen av sambandet mellan rödskift och kosmisk tid, där det sistnämnda g˚ar att ersätta med galax˚aldrar. Denna metod används av Simon et al. i [1], där de presenterar 8 värden av Hubbleparametern. Resultaten är förv˚anansvärt precisa, med tanke p˚a precisionen i deras data. Vi omanalyserar därför deras data med tre metoder: χ2_{-miniminering, Monte Carlo-sampling och Gaussiska processer. De tv˚}_{a f¨}_orsta

metoderna visar att när 8 oberoende värden av Hubbleparametern bestäms f˚as mycket större osäkerheter än de som presenteras av Simon et al. Den sistnämnda metoden ger en kontinuerlig funktion H(z) med lägre osäkerheter. Priset för detta ¨

ar dock starka korrelationer, det vill säga att resultat vid m˚anga olika rödskift inneh˚aller i princip samma information. Utöver detta visar vi att det krävs antingen en mycket större mängd data eller mycket mindre osäkerheter i datan för att kunna bestämma 8 oberoende värden av Hubbleparametern med samma precision som i [1]. Vi drar slutsatsen att deras värden av Hubbleparametern inte kan f˚as med den data som använts.

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Preface

The master’s thesis that lies before you is my final work at the Royal Institute of Technology in Stockholm, Sweden. With it, I will have completed the degree programme in Engineering Physics, which awards the Swedish title civilingenj¨or. It is also the last part of the master’s programme in Engineering Physics, where I have specialized in Theoretical Physics.

During my studies I have had the opportunity to explore many fields within physics, but none have piqued my interest as much as general relativity and cos-mology. I just had to do my master’s thesis within one of these two fields. This made me contact Edvard M¨ortsell, Professor in Observational Cosmology at Stock-holm University. He responded quickly with an idea for a project, which one could crudely summarize as “disprove this paper with hundreds of citations, written by cosmologists with thousands of citations” - a quite intimidating task for a master’s student. As evident from the existence of this thesis, I decided to face the challenge with Prof. M¨ortsell as my supervisor.

I am certainly glad that I did. Not only did we manage to disprove said article, but in doing so I have learned a lot: how to effectively research new topics, broader understanding of cosmology in general, many tools for data analysis, proficiency in Python, and much more.

All data analysis in this thesis has been done using Python, with much help of SciPy, NumPy, and Pandas. The figures have been made using Matplotlib.

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Acknowledgements

I would like to extend my deepest gratitude to my supervisor Prof. Edvard M¨ortsell for his excellent guidance during this project. Many unexpected problems have been resolved after a short discussion with him. Furthermore, he has always been quick to reply to any questions or set up meetings to discuss the project - a quality I have appreciated immensely throughout the whole project.

As you may have figured out, this thesis project has been completely conducted during the COVID-19 pandemic. Thus, working on this project would have been a lonely time, were it not for a few select friends who I would like to thank for keeping me company during this time: Anna, Martin, and Ariel.

Many people have impacted my time at KTH in a positive way, and all deserve my thanks for making it such a memorable experience. Nevertheless, I would like to directly thank a few who I have spent a significant amount of time studying with these past years: Thomas, Max, and Erik. I have had many enlightening discussions with all of you which have contributed to much of the knowledge I have gained during my studies, and I hope that many more discussions are ahead of us. Of course, I would also like to thank my beloved mother and father for their never-ending support. You helped me stay motivated during the more trying times, and for that I am very grateful.

Last of all, I want to thank Assoc. Prof. Mattias Blennow for being the formal supervisor at KTH during this project, as well as for helping me find an examiner: Prof. Sandhya Choubey, who I thank for taking on this task.

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Introduction

Back in 2005 the paper titled “Constraints on the redshift dependence of the dark energy potential” was published in Physical Review D [1], and it was written by Joan Simon, Licia Verde, and Raul Jimenez - all prominent scientists within the field of cosmology. As of today, the paper has over 900 citations and the number is still growing. However, the many citations are not due to the constraints mentioned in the title. Instead, they are mostly due to the results in their first figure in which they obtain 8 independent values for the Hubble parameter at a large range of redshifts to a relatively high degree of precision. These values are being used to test other aspects of cosmology, see for instance [2–6]. Some papers use the results, but cite compilations which include the results from [1] instead; examples include [7] who cited [8], and [9] who cited [10]. The Hubble parameter values obtained by Simon et al. clearly permeate much of cosmological research, and in recent years it seems to only permeate more of it - with over 300 of the 900 citations being from the last 3 years.

The purpose of this work is to investigate whether the resulting values for the Hubble parameter can actually be correct. The method used in [1] seems simple at the surface of it, but if one digs deeper it is clear that it is not as trivial as one might think from a first glance. First of all, it requires determining the ages of galaxies accurately and precisely. The same type of galaxies’ ages will then have a dependence on the redshift, t(z). The second step is to infer the derivative dt/ dz of this relation using the galaxy age data, a much more difficult task than simply inferring the function values in the sense that it, in general, leads to larger uncertainties. Lastly, one must obtain the reciprocal of the derivative distribution, a simple task if the uncertainties are small but requires extra care if the uncertainties are large. Even assuming that the first step is done flawlessly, we show that the results in [1] can not be correct due to the difficulty in making precise inferences of derivatives. They obtain 8 values for the Hubble parameter, and therefore compute 8 derivatives, with uncertainties between 10 and 20% using 32 galaxies with 12% uncertainties in the ages. It seems too good to be true - because it is.

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1.1 Executive Summary

The question: Can the resulting Hubble parameter values obtained by Simon et al. in [1] be correct? Their method relies on determining the redshifts and ages of galaxies, where the latter is a difficult task. The Hubble parameter is then derived by determining the derivative of the relation between age and redshift. Determining derivatives from data is notoriously known for introducing large uncertainties, yet the uncertainties for the Hubble parameter values in [1] are relatively small.

Importance: Due to the seemingly precise values for the Hubble parameter the paper has been cited over 900 times. The values are used to infer other cosmological parameters. Thus, if the values are incorrect many subsequent cosmological studies are in turn rendered incorrect and the values should cease to be used in future works.

Results: Using multiple statistical tools we analyze the data used in [1] and conclude that the Hubble parameter values they claim to obtain can not be correct. We follow the instructions given by Simon et al. in an attempt to reproduce their results. We find that it is not even possible to obtain the same effective redshifts, i.e. the binning of the data becomes different from what they must have used. Moreover, when working backwards from the effective redshifts to find a possible binning used in [1], we find that multiple data points belong to two bins; implying that the Hubble parameter values they obtain are not independent.

Due to the failure of reproducing the results, we perform our own analysis using the same galaxy age and redshift data as in [1] using two methods: χ2_minimization

and Monte Carlo sampling. Both methods show that it is not possible to obtain as many independent Hubble parameter values with as small uncertainties as is claimed by Simon et al.

We also consider a method which requires no binning: Gaussian processes. This method yields a continuous inference for H(z), but the different Hubble parameter values are correlated. This results in much lower uncertainties, but with strong correlations. However, even with the strong correlations the uncertainties are not as small as in [1] for the whole redshift range.

Lastly, we show that it is necessary to either include hundreds of galaxies in the data set or determine the galaxy ages with uncertainties between 1 and 3% in order to obtain as small uncertainties for the Hubble parameter as Simon et al. claim to obtain - many more than 32 galaxies or significantly lower uncertainties than the 12% in their data set.

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1.2. Structure of the Thesis 3

Recommendations

Given that:

(i) following the instructions written by Simon et al. in [1] fails to repro-duce their results,

(ii) analyzing the same data as Simon et al. using χ2 _{minimization and}

Monte Carlo sampling it is not possible to obtain as many independent Hubble parameter values with as small uncertainties as they do, (iii) using Gaussian processes, it is not even possible to obtain as small

uncertainties when the results are highly correlated,

(iv) much more data or significantly better age determination is necessary to obtain as precise results as in [1],

we urge researchers to not include the Hubble parameter values from [1] in their analysis. Furthermore, researchers who have performed analysis using the Hubble parameter values from [1] should consider redoing the analysis without these values.

1.2 Structure of the Thesis

In chapter 2 we cover the theoretical background necessary to understand this work. More specifically, section 2.1 gives a relatively detailed overview of cosmology and general relativity for readers unfamiliar with these fields. Those who have studied general relativity but not cosmology may skip section 2.1.1. In section 2.2 we describe the theory behind the methods used in this work: χ2minimization, Monte Carlo sampling, and Gaussian processes. Chapter 3 describes how one can use galaxies to infer the Hubble parameter. Section 3.1 breaks down the idea behind the method used by Simon et al. and includes both their data and their results. In section 3.2 we briefly describe how galaxy redshifts and ages can be determined. Chapter 4 contains our analysis of the data in Simon et al., as well as an attempt to reproduce the results using their description. We conclude in chapter 5 with a summary of the thesis as well as a discussion of the results.

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Chapter 2

Theoretical Background

To understand this work one must first get a grasp of cosmology, particularly the Hubble parameter, and the methods we use in later chapters. Therefore, section 2.1 contains a relatively detailed description of cosmology for readers with little or no background within the field, whereas section 2.2 presents the theory behind the three methods we use in our analysis of the data in Simon et al.: χ2 _{minimization,}

Monte Carlo sampling, and Gaussian processes.

2.1 Cosmology and General Relativity

Although this section contains a relatively high degree of detail, starting at the Einstein field equations and building from there, we can not cover all aspects of cosmology and general relativity. If one desires more details we encourage the reader to turn to the cited sources. A recommended start is to look in [11], which much of this section relies on and is the source unless explicitly stated.

2.1.1 The Einstein Field Equations

The Einstein field equations (EFE) are of central importance in the field of general relativity (GR), and hence essential in the study of cosmology. They describe how the curvature of spacetime, what we perceive as gravity, responds to energy and momentum. There are several ways to arrive at the EFE, see for instance [11, 12]. The derivation is lengthy and technical, and will not be presented here, but instead the result will simply be stated. It is a tensor equation which in component form looks deceptively simple,

Gµν = 8πGNTµν , (2.1)

where Gµν is the Einstein tensor, GN is Newton’s constant of gravitation, and Tµν

is the energy-momentum tensor. The notation is extremely compact, and we will soon unveil what is underneath the hood. Instead of going top down, that is ripping

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equation (2.1) into smaller and smaller constituents, we will take the opposite route and start with its smallest constituents and build all the way up to the EFE.

In GR spacetime is described by a manifold, crudely defined as a space which globally can have a complicated geometry, but locally looks like Rn_{. The geometry}

of the manifold is encoded in the symmetric metric tensor with components gµν,

the components of which can conveniently be displayed1 through the line element ds2 as

ds2= gµνdxµdxν , (2.2)

where xµ are the coordinates, and the Einstein summation convention is implied. Furthermore, one often defines the metric to be non-degenerate, such that the de-terminant g = |gµν| is non-zero. One can then define the inverse metric components

gµν _via

gµνg_νσ= g_λσgλµ = δ_σµ , (2.3)

where δµ

σ is the Kronecker delta, i.e. one when σ = µ and zero otherwise. All

indices can take on four values, such that we have the coordinates (x0_{, x}1_{, x}2_{, x}3_),

since spacetime has one temporal component, and three spatial components. The metric can be combined in many ways to form new objects, where the Christoffel symbols Γσ

µν are of notable importance. These are also known as

connection coefficients of a covariant derivative ∇. Acting on a vector V the co-variant derivative is defined as

∇µVµ= ∂µVµ+ ΓνµλV

λ_, _(2.4)

which is a coordinate-independent correction to the regular partial derivative to account for a curved space. We now impose that the connection coefficients are torsion-free, meaning that

Γλµν = 1 2 Γλµν+ Γλνµ , (2.5)

i.e. the Christoffel symbols are symmetric in their lower indices, and impose metric compatibility, meaning that

∇ρgµν = 0 . (2.6)

With these two restrictions one obtains a unique formula for the connection coeffi-cients, which are the Christoffel symbols,

Γσ_µν= 1 2g

σρ _∂

µgνρ+ ∂νgµρ− ∂ρgµν . (2.7)

One can understand the Christoffel symbols further by considering the geodesic equation, with λ parameterizing the world line xµ(λ),

d2xµ dλ2 + Γ µ ρσ dxρ dλ dxσ dλ = 0 , (2.8)

1_{For details on how to rigorously define the metric tensor, see for instance [13, 14]. In short,}

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2.1. Cosmology and General Relativity 7 which describes straight lines in a curved space. It is clear that if all Γµ

ρσ = 0 one

obtains the usual notion of a straight line in a flat space.

Moving one step closer to the EFE we define the Riemann tensor Rρσµν= ∂µΓρνσ− ∂νΓρµσ+ Γ

ρ µλΓ

λ

νσ− ΓρνσΓλµσ . (2.9)

This tensor, although not obvious from the above expression, gives an even better description of curvature than the metric. It can be hard to tell if the space is flat or not by inspecting the metric tensor, as even in a flat space the components may have complicated expressions due to the choice of coordinates. The Riemann tensor however, has an easier interpretation. If it vanishes, the space is flat; if it does not vanish, the space is curved.

We have arrived at the final step toward the left hand side of the EFE. We form the Ricci tensor through a contraction of the Riemann tensor,

Rµν = Rλµλν . (2.10)

The trace of this tensor is the Ricci scalar,

R = Rµµ= gµνRµν . (2.11)

We finally define the Einstein tensor as Gµν = Rµν−

1

2Rgµν . (2.12)

The right hand side of the EFE is proportional to the energy and momentum tensor Tµν. The component Tρσ is the flux of the ρ:th component of the

four-momentum across a surface of constant xσ. We have now uncovered all of the components that make up the EFE, which are often written in the slightly less compact form

Rµν−

1

2Rgµν = 8πGNTµν , (2.13)

and we have seen that the left hand side contains all information about curvature, whereas the right hand side contains the information about energy and momentum. The EFE may seem quite daunting after unpacking them. However, an equiv-alent way of expressing the information contained in the EFE which is easier to decode takes the form [15]

¨ V V _t=0 = −1 2 ρ + px+ py+ pz , (2.14)

where V (t) is the volume of a small ball of freely falling particles initially at rest with respect to each other, ρ is the energy density, and piis the pressure in the i:th

direction. From this equation one easily identifies the gravitational effects of energy density and pressure: Both positive energy density and pressure curve space-time

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in such a way as to shrink the volume of the ball. Recalling Einstein’s formula E = mc2 _{and that we are working in units where c = 1 we see that mass and}

energy are on equal footing. Thus, the gravitational effect of massive objects is attractive. One should think carefully about the fact that positive pressure attracts gravitationally, in contrast to what one might expect from our everyday experience of pressure, which tends to push things apart. The two effect are indeed separate; the gravitational effect cares only about the value of the pressure, whereas the other effect relies on pressure differences. [15]

2.1.2 Cosmology: Basic Principles

This project is in essence about cosmology and not general relativity. Thus, the EFE are not the central equations, but will be used to obtain the ones that are. However, before we can do that we need to state the assumptions in the theory of cosmology which will lead us to the desired equations.

First, we state the Copernican principle: The assumption that the universe looks the same everywhere. This assumption leads to two other, namely that the universe is isotropic and homogeneous in space. Note that these are only ap-proximations which hold at large scales (greater than about 100 MPc [16]), and become better the larger the scale one considers.

The most compelling evidence that isotropy is a reasonable assumption is the cosmic microwave background (CMB). Its black-body radiation temperature TCMB= 2.7255 ± 0.0006 K [17]. Evidently, the temperature is severely constrained

with very small variations no matter in which direction we look.

Homogeneity is a tougher nut to crack. What one can do is propose alternative inhomogeneous models and see how well they fit the observed data. One simple model of this kind is the large spherically symmetric void described by the Lemaˆıtre-Tolman-Bondi metric. This model was considered in [18], in which they concluded that an observer must be within ∼ 1% of the void radius from the center of the void in order to fit data from both type Ia supernovae, and the cosmic microwave background dipole anisotropy. This requires a lot of fine-tuning and one would then have to ask how likely it is that we occupy such a special place in the universe -which also contradicts another statement of the Copernican principle; that we do not occupy a special place in the universe.

Another assumption made in cosmology is that one can model matter and energy in the form of a perfect fluid, such that the energy-momentum tensor takes the form

Tµν = (ρ + p)UµUν+ pgµν . (2.15)

This assumption is argued for in [13] in the following way: On very large scales galaxies can be considered as “particles”. These “particles” are components of a “gas” which permeates the universe. To simplify things one ignores that these “particles” cluster and have internal structure (such as stars). Simplifying even

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2.1. Cosmology and General Relativity 9 further, one ignores the particulate nature of the gas by treating it in the perfect fluid approximation2_.

2.1.3 Robertson-Walker Metrics and the Friedmann

Equations

Using the isotropy and homogeneity one can obtain the Robertson-Walker met-ric, ds2= − dt2+ a2(t) " dr2 1 − κr2+ r 2_dΩ2 # , (2.16)

where a(t) is known as the dimensionless scale factor, t is the time coordinate, r is the radial coordinate, κ is a curvature parameter, and dΩ2_{= dθ}2_{+ sin}2_{θ dφ}2_{is the}

metric on the two-sphere. The sign of κ determines whether the spatial part of the metric, and hence the universe, is open (κ < 0), flat (κ = 0), or closed (κ > 0). The time coordinate is called the cosmic time and the spatial coordinates are known as the comoving coordinates [20]. Any observer who stays on constant spatial coordinates are called “comoving”.

The Friedmann equations are just a special case of the EFE (2.13) using the Robertson-Walker metric and the perfect fluid approximation in equation (2.15). The perfect fluid is isotropic in its rest frame, and the Robertson-Walker metric is isotropic; for both of these statements to remain true, the two frames must coincide. Thus, the four velocity in equation (2.15) must be Uµ _{= (1, 0, 0, 0). Now, with the}

Ricci scalar and tensor obtained in appendix A we have everything we need. The µν = 00 component of the EFE gives

¨ a a = −

4πGN

3 (ρ + 3p) , (2.17)

and the µν = ij components (with Latin indices corresponding to spatial compo-nents) give ¨ a a+ 2 ˙a a 2 + 2κ a2 = 4πGN(ρ − p) . (2.18)

Combining the two equations yields ˙a a 2 = 8πGN 3 ρ − κ a2 , (2.19)

often known as the Friedmann equation, while equation (2.17) is often referred to as the second Friedmann equation or the acceleration equation.

2_{It is also possible to argue what the components of the energy-momentum tensor must be in}

the rest frame of the fluid using only isotropy and homogeneity. The generalization to any frame is then straightforward, see [19].

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The most important parameter in this work is the Hubble parameter. Here we will only define it, and save a more detailed description for a later section,

H ≡ ˙a

a , (2.20)

such that the Friedmann equation (2.19) can be written

H2= 8πGN

3 ρ −

κ

a2 . (2.21)

We see from the definition of the Hubble parameter that it is a measure of the expansion rate of the universe.

2.1.4 Cosmological Parameters

The Hubble parameter is central in cosmology, and even more so in this work. Nev-ertheless, there are several other parameters of importance which will be described here.

The perfect fluid model used in cosmology usually follows the simple equation of state for each energy component

pi= wiρi , (2.22)

where wi is a time-independent constant in the current concordance model for

cosmology, and i stands for the energy component: matter (M), radiation (R), or vacuum (Λ). The conservation of energy in general relativity is given by

0 = ∇µT µ

0 = −∂0ρ − 3

˙a

a(ρ + p) . (2.23)

Combining this with the equation of state yields ˙ ρi ρi = −3(1 + wi) ˙a a , (2.24)

which, with a constant wi, yields

ρi ∝ a−3(1+wi). (2.25)

With the above equation one can find how the energy density should develop in dif-ferent universes dominated by difdif-ferent energy components; matter-dominated, radiation-dominated, and vacuum-dominated. Table 2.1 summarizes the dif-ferent behaviors.

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2.1. Cosmology and General Relativity 11 Table 2.1: Energy components’ dependence on the scale factor.

Energy component w ρ vs. a

Matter (M) 0 ρM∝ a−3

Radiation (R) 1₃ ρR∝ a−4

Vacuum (Λ) −1 ρΛ= const.

The dimensionless density parameter is defined as Ωi = 8πGN 3H2 ρi= ρi ρcrit (2.26) for each energy component, with

ρcrit=

3H2

8πGN

. (2.27)

being the so called critical density. Its name comes from the fact that it determines whether the universe we live in is flat, open, or closed. One can see this by rewriting the Friedmann equation in terms of the total density parameter,

Ω − 1 = κ

H2_a2 . (2.28)

Thus, the sign of κ is determined by whether Ω is greater than, equal to, or less than 1, which in turn is determined by whether the total ρ is greater than, equal to, or less than ρcrit.

The final parameter introduced in this section is the redshift z. This can be related to the dimensionless scale factor and we will show how. Consider a light ray emitted from r = remat time temtraveling radially towards an observer located

at r = 0 that observes the light ray at time tobs. The radial coordinate therefore

decreases as time increases. In general, light rays follow null-geodesics and therefore obey ds2_{= 0. Combined with equation (2.16) one obtains [19]}

dt = −a(t)√ dr 1 − κr2 =⇒ Z tobs tem dt a(t) = − Z 0 rem dr √ 1 − κr2 . (2.29)

There is no time dependence on the right hand side of the second equality, and it therefore holds for a light ray emitted slightly after the first as well, such that

Z tobs tem dt a(t) = Z tobs+δtobs tem+δtem dt a(t) ⇐⇒ Z tem+δtem tem dt a(t) = Z tobs+δtobs tobs dt a(t) . (2.30) For short times δtem/obs the scale factor is essentially constant, yielding

δtem

a(tem)

= δtobs a(tobs)

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Considering the signals as subsequent wave crests, the emitted frequency and ob-served frequencies are fem= 1/δtem and fobs= 1/δtobs, respectively. Then,

a(tobs) a(tem) = fem fobs =λobs λem . (2.32)

Combining this with the definition of redshift z = λobs− λem λem , (2.33) we obtain a(t) = 1 1 + z(t) , (2.34)

where we dropped the subscript “em” and considered an observation taking place today, where by convention we set a(tobs) = 1 (an arbitrary choice). The redshift

is therefore directly connected to the scale factor, and thus the expansion of the universe3_.

2.1.5 The Hubble Parameter

Thus far we have only stated the definition of the Hubble parameter in equation (2.20). The present day value of this parameter is called the Hubble constant, denoted H0. It is often parameterized, using a dimensionless quantity h, as

H0= 100h km/sec/Mpc . (2.35)

As hinted by the unit in the above equation, the Hubble constant measures a velocity which is larger at larger distances. This is made clearer in the Hub-ble–Lemaˆıtre law

v = H0d , (2.36)

where v is the recession velocity of an object at a distance d away from us due to the expansion of the universe. Care must be taken when using this law though. In a curved spacetime one can not strictly speak of relative velocities unless the two objects are located at the same point in spacetime (i.e. same place at the same time). The notion of distance is not well defined either, but for sufficiently short distances one can use the instantaneous physical distance, which is the distance between the two objects along the current spatial hypersurface. This approximation works well for sufficiently small distances.

3_{Some would object that this is merely a consequence of the choice of coordinates. In other}

coordinates the redshift might just as well be interpreted as a Doppler shift (see [21] for more details). While a valid objection, the comoving coordinates which give rise to the interpretation of an expanding universe also yield a simple mathematical description and are derived by using relatively intuitive symmetry arguments, as done in section 2.1.3 and is thus an excellent choice of coordinates. For this reason we will consistently use these coordinates and write about physics as if they are the “true” coordinates of the universe.

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2.1. Cosmology and General Relativity 13 By taking the derivative of equation (2.34) with respect to time one obtains the following formula for the Hubble parameter as a function of redshift,

H(z) = − 1 1 + z

dz

dt . (2.37)

Therefore, if one can determine the redshift and its derivative with respect to the cosmic time, one can determine the Hubble parameter. This is the central equation in this work and the equation used in [1] to obtain the Hubble parameter from galaxy ages, see section 3.1.

The Hubble constant is relevant when determining the lookback time, which is the time difference between the age t0of the universe today and the time t∗when

a photon at redshift z was emitted, that is t0− t∗= Z t∗ t0 dt = Z 1 a∗ da aH(a) = H −1 0 Z z∗ 0 dz0 (1 + z0_)E(z0₎ . (2.38)

The function E(z) takes different forms depending on which kind of universe one considers. In general it takes the form

E(z) =   X i(c) Ωi0(1 + z)−3(1+wi)  , (2.39)

where the wi can be seen in table 2.1, and the (c) means that we also sum over the

fictitious energy density ρc= − 3κ 8πGNa2 ⇐⇒ Ωc= − κ H2_a2 , (2.40)

corresponding to spatial curvature. This is introduced only to make notation more compact, e.g. the Friedmann equation (2.19) takes the compact form

H2= 8πGN 3 X i(c) ρi ⇐⇒ 1 = X i(c) Ωi . (2.41)

Therefore, with observed values of all the energy density parameters and the Hubble constant, one can calculate the total age of the universe by letting t∗ → 0, or

equivalently letting z∗→ ∞ in equation (2.38).

The Hubble parameter is of importance in other aspects of cosmology as well, one example is the theory of inflation covered in appendix A.

2.1.6 The ΛCDM Concordance Model of Cosmology

There is one cosmological model which stands out among all others, and it is the flat ΛCDM model - often refered to as the concordance model of cosmology. Λ

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refers to the cosmological constant, whereas CDM stands for Cold Dark Matter. Chapter 1 of [22] introduces this model and some observational data which fits the model very well, showing that ΛCDM is indeed a very good model for the universe. Due to the success of the ΛCDM model we would like to compare the results we obtain later with the predictions of this model. First of all, the Hubble parameter is obtained by first rewriting equation (2.41), such that

H2= H₀2ΩΛ0+ ΩM 0a−3+ ΩR0a−4

, (2.42)

where we replaced the energy densities with the density parameters and introduced the scale factor dependence. The curvature term is excluded due to it being zero in a flat universe. Now, simply take the square root and replace the scale factor a by using equation (2.34) to obtain

H(z) = H0

q

ΩΛ0+ ΩM 0(1 + z)3+ ΩR0(1 + z)4 . (2.43)

2.1.7 Cosmological Parameters: Current Values

At any point in this report where cosmological parameter values are necessary we refer to the Particle Data Group, which has a review of particle physics and much of cosmology including up-to-date lists of physical constants [23]. The values relevant for us are listed in table 2.2 and their table 2.1. We should note that there is currently disagreement for what the Hubble constant is between different observational methods and the value in the table is obtained by considering the CMB. The disagreement is referred to as the Hubble tension, see chapter 25.3.1 of [23] for a short discussion of different methods to obtain the Hubble constant. Table 2.2: A few cosmological constants from [23] used in this report.

Parameter Symbol Value

Hubble constant H0 67.4 km/s/Mpc

CMB radiation density ΩR 5.38 × 10−5

Dark energy density ΩΛ 0.685

Matter density ΩM 0.315

2.2 Methods

Here we cover the necessary theoretical background for the three methods used in this work: χ2 _{minimization, Monte Carlo sampling, and Gaussian processes.}

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2.2. Methods 15

2.2.1 χ

2

_Minimization

χ2 minimization is a method for obtaining the optimal parameters given a model and a set of data. One prerequisite for using this method is that the data is normally distributed [24].

Definition

Given a set of measured data D, and a model M with some parameters θ one wants to maximize the likelihood function

L(D; M, θ) = P (D|M, θ) , (2.44)

which is the probability to obtain the observed data, given a certain model with some given parameters [24]. By assuming that all measured data points are inde-pendently normally distributed one obtains [25]

ln L = −1 2χ

2

+ C , (2.45)

where C is some constant and χ2=X i yi− f (xi; θ) 2 σ2 yi , (2.46)

where (xi, yi) are the measured data, σyiare the corresponding standard deviations for the normally distributed y-data, and f is the function for the model at hand. The x-data is assumed to be known with infinite accuracy. We see that maximizing the likelihood is equivalent to minimizing the χ2in the case of normally distributed data.

Linear Fit

Using any polynomial model simplifies matters, since the minimization can be done analytically with matrix multiplications, as described in [26]. Consider fitting data to a linear function f (x) = mx + b in equation (2.46), such that the parameters to use for optimization are m and b. Given N pairs of independent data points (xi, yi), with corresponding uncertainties σyi in y, we then construct the matrices

Y =       y1 y2 .. . yN       , A =       1 x1 1 x2 .. . ... 1 xN       , C =       σ2 y1 0 . . . 0 0 σ2 y2 0 .. . . .. ... 0 0 . . . σ2yN       . (2.47)

When the equation

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can not be solved due to being over-constrained, the best fit parameters for a straight line is b m = X =hATC−1Ai −1h ATC−1Yi, (2.49)

which are also the parameters which minimize the χ2_{. Furthermore, the parameter}

covariance matrix is given by

Σ =hATC−1Ai

−1

, (2.50)

such that the standard deviation of the intercept b is σb =

√

Σ11, the standard

deviation of the slope m is given by σm=

√

Σ22, and the covariance is σmb= Σ12.

For a linear fit the resulting parameter distributions are exactly Gaussian, since the likelihood is then a bi-variate normal distribution.

2.2.2 Monte Carlo Sampling

Monte Carlo sampling methods are quite popular within cosmology; as pointed out by [27] most cosmological parameters are estimated with Monte Carlo methods in some way due to the efficiency when the parameter space is large, as is often the case in cosmology. If the reader is familiar with cosmology, the mind might wander toward Markov Chain Monte Carlo methods, which is probably one of the most common Monte Carlo methods within cosmology. For instance, the Data Analysis Recipes series by David W. Hogg focuses solely on this particular implementa-tion of Monte Carlo [28]. However, for our purposes the Markov Chain Monte Carlo method is not necessary, and a much simpler random sampling algorithm is presented below. It might be more fitting to simply call it a random sampling algorithm, but as described in [29], all Monte Carlo methods follow a pattern. Our algorithm follows this pattern as well, namely

1. Model the problem in terms of a series of probability density functions. 2. Draw samples from the probability density functions multiple times. 3. Statistically analyze the resulting samples.

The algorithm we use suffices to solve a specific problem we faced with the method of χ2_{minimization, namely that regular uncertainty propagation fails for too large}

relative uncertainties of the normal distribution, see the results in section 4.2.2 and Appendix B for details.

2.2.3 Gaussian Processes

The above two methods can only give a finite number of independent inferences. The method of Gaussian processes (GPs) instead generates a continuous inference where every inferred point uses all of the data to some extent. The trade-off is

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2.2. Methods 17 that the inferences at different points are correlated. This method has been used extensively in cosmology for instance to infer properties of dark energy [30–34], the Hubble constant4 _{[6, 35–37], and other applications [29, 38–40]. Thus, we consider}

this method to infer the Hubble parameter.

For an extensive overview of Gaussian processes we refer to [41], from which most of the author’s knowledge of Gaussian processes comes and is the source for this chapter unless explicitly stated otherwise.

The Gaussian process is often referred to as a generalization of the Gaussian (normal) probability distribution; instead of a distribution over real numbers, it is a distribution over functions. A sample from the Gaussian process is thus a function, and not a number. Combing this with the fact that the method is non-parametric5

gives rise to a problem. A general continuous function is defined at an infinite number of points. How can we possibly handle this numerically? Luckily, the process fulfills the so called marginalization property, which in simple terms states that it is enough to infer the function for a finite number of points and for these points the result would have been the same if one considered all infinite number of points.

General

The formal definition of a GP as stated in [41] is:

A Gaussian process is a collection of random variables, any finite number of which have a joint Gaussian distribution.

Describing this in more mathematical terms we first point out that the real Gaussian process for a function f (z) is completely determined by the mean function m(z) and the covariance function k(z, ˜z) defined as6

m(z) = E[f (z)] ,

k(z, ˜z) = E[(f (z) − m(z))(f (˜z) − m(˜z))] . (2.51) We will cover how we choose both the mean function and the covariance function in a later chapter, but for now we will take them as given. The Gaussian process for f (z) is then written as

f (z) ∼ GP m(z), k(z, ˜z) . (2.52)

For each point z∗the function “value” f (z∗) can then be seen as a univariate normal

distribution with mean µ = m(z∗) and variance σ2= k(z∗, z∗), and the correlations

4_{Note that many papers use the resulting Hubble parameter values from [1] in their inference}

of H0.

5_{I.e. the process attempts to fit a function with no specific form to the data. This contrasts a}

parametric method where one might fit data to, for instance, a straight line y = mx + b, with the

parameters m and b. χ2_{minimization is an example of a parametric method.}

6_{Our application requires only one-dimensional inputs in the form of redshift, so we use z as}

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to other points z is described by k(z∗, z). A sample vector of length N is then

obtained from the multivariate Gaussian distribution as f (z1), f (z2), . . . , f (zN)

T

∼ N m(z1), m(z2), . . . , m(zN) T

, Σ, (2.53)

where Σ is the covariance matrix constructed using the covariance function, i.e.

Σ =        k(z1, z1) k(z1, z2) . . . k(z1, zN) k(z2, z1) k(z2, z2) ... .. . . .. ... k(zN, z1) . . . k(zN, zN)        . (2.54)

Equation (2.53) is written more compactly as

f ∼ N (µ, Σ) . (2.55)

We are now in a position to clarify the marginalization property. Say that the GP specifies (f1, f2)T ∼ N (µ1, µ2)T, Σ11 Σ12 Σ21 Σ22 ! , (2.56)

then it must also specify f1∼ N (µ1, Σ11), and similarly for f2. What this means

is that considering a larger set of output points does not change the distribution for the smaller subset of output points.

Inferring Function Values

The above only shows how to draw random functions from a GP, but how are inferences made? First of all, we need N measurements {(zi, ti), i = 1, 2 . . . , N }

which we place in vectors t and z. The data is assumed to be noisy and thus perturbed from the actual function as ti = f (zi) + i, where i is Gaussian noise

with variance σ2

i [31]. We form the correlation matrix C, which for independent

data is just diag σ2

1, σ22, . . . , σ2N. Thus, the distribution for the observed values is

t ∼ N (µ, K(z, z) + C) , (2.57)

where K(·, ·) is the covariance matrix formed using the covariance function k(·, ·), i.e. [K(x, y)]ij = k(xi, yj).

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2.2. Methods 19 We now want to make inferences at z∗ = (z∗1, z∗2, . . . , z∗M)T which we place

in the vector f∗. The joint probability distribution for the data vector t and the

inference vector f∗ is then

" t f∗ # ∼ N µ,K(z, z) + C K(z, z∗) K(z∗, z) K(z∗, z∗) ! , (2.58)

Now, we are only interested in the distribution of the target values f∗given the data,

and the desired target points, i.e. f∗|z, z∗, t. The appendix of [31] demonstrates that

this results in the distribution

f∗|z, z∗, t ∼ N (f∗, cov(f∗)) , (2.59) where f∗= µ∗+ K(z∗, z)K(z, z) + C −1 (t − µ) , cov(f∗) = K(z∗, z∗) − K(z∗, z)K(z, z) + C −1 K(z, z∗) . (2.60)

Inferring the Derivative

Another step has to be taken at this point, since we also want to infer the derivative of a function and not just the function itself. Luckily, the derivative of a Gaussian process is also a Gaussian process. The only requirement is that the mean and covariance functions are differentiable - twice in the case of the covariance function, since cov(fi, fj) = k(zi, zj) =⇒ cov fi, ∂fj ∂zj ! = ∂k(zi, zj) ∂zj =⇒ cov ∂fi ∂zi ,∂fj ∂zj ! = ∂ 2_k(z i, zj) ∂zi∂zj . (2.61)

The Gaussian process for the derivative is then

f0(z) ∼ GP m0(z),∂

2_{k(z, ˜}_z)

∂z∂ ˜z !

. (2.62)

The distribution for the derivative given the data and the target points is then obtained in the same way as for the function, resulting in

f_∗0|z, z∗, t ∼ N (f∗0, cov(f∗0)) (2.63) with f0 ∗= µ0∗+ K0(z∗, z)K(z, z) + C −1 (t − µ) , cov(f_∗0) = K00(z∗, z∗) − K0(z∗, z)K(z, z) + C −1 K0(z, z∗) . (2.64)

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Training the Hyperparameters

Whatever covariance function one chooses, there will also be hyperparameters7 to choose. For instance, the squared exponential covariance function,

k(z, ˜z) = σ2_fexp −(z − ˜z)

2

2l2

!

, (2.65)

has the hyperparameters σf, which characterizes the change of the function in the

t-direction, and l, which characterizes the covariance length scale or rather the separation one needs for a significant difference in function value. Changing these hyperparameters can change the behavior of the function dramatically, and they should therefore not be chosen by manual inspection of the data. Instead, one should use a method which trains the hyperparameters. We will use the marginal likelihood8, since we have not come across any other method for training the hy-perparameters in the cosmology literature.

The marginal likelihood is defined as the integral over the likelihood times the prior,

p(t|z, σf, l) =

Z

p(t|f , z)p(f |z, σf, l) df , (2.66)

where the marginalization is over the function values f . The prior is Gaussian, with

f |z, σf, l ∼ N (µ, K(z, z)) , (2.67)

and the likelihood distribution is

t|f ∼ N (f , C) . (2.68)

All together, the log marginal likelihood becomes ln p(t|z, σf, l) = − 1 2(t − µ) T_{[K(z, z) + C]}−1_{(t − µ)} −1 2ln K(z, z) + C− n 2log 2π , (2.69)

where the dependence on the hyperparameters is hidden in the covariance matrix K(z, z). The first term is the only one containing the data, and is thus called the data-fit term. The second term is a complexity penalty term, although not obvious. The last term is just a normalization constant. Now, most cosmologists just maxi-mize the log marginal likelihood9_{with respect to the hyperparameters and use the}

7_{Hyper since they do not specify the form of a function, but instead only specifies its}

charac-teristics.

8_{There are many other methods, for instance [41] also covers Leave-One-Out Cross-Validation,}

and [42] considers a few other methods. However, comparing all different methods is beyond the scope of this work.

9_{Since the logarithm is monotonically increasing this is equivalent to maximizing the marginal}

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2.2. Methods 21 hyperparameters which correspond to the maximum. However, this is merely an approximation which works well when the marginal likelihood is sharply peaked. As noted by [6] this approximation essentially assumes that the distributions for the hyperparameters are given by Dirac deltas at the location for which the marginal likelihood is maximal. If the distributions are not sharply peaked one should sample from the hyperparameter distributions using Markov Chain Monte Carlo methods. This would propagate the uncertainties in the hyperparameters, and is done in [6]. We will not consider the MCMC, but will instead simply optimize the log marginal likelihood. A further discussion of the hyperparameters is left in appendix C.

We should add that one can use another hyperparameter modelling the intrinsic noise of the function, as is done in [38]. To do this one replaces K(z, z) + C with K(z, z) + C + σ2

II (where I is the identity matrix) in equation (2.57) and the ones

that follow. However, in our analysis the marginal likelihood forced this parameter to be zero, and we therefore chose not to include it as the difference is negligible.

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Chapter 3

The Hubble Parameter from

Galaxy Ages

3.1 Method and Results from Simon et al.

The idea behind the method in [1] is rather ingenious, although it turns out to be quite difficult to implement in practice. Nevertheless, in this section we want to focus on the idea; how do they propose to use galaxy ages to determine the Hubble parameter?

The equation of importance was derived in the previous chapter, namely equa-tion (2.37), which we restate here:

H(z) = − 1 1 + z

dz

dt . (3.1)

Thus, one needs the redshift and its derivative with respect to the cosmic time to determine the Hubble parameter at said redshift. It turns out that galaxies are good candidates for determining both of these quantities. One reason is that the redshift of a galaxy can be determined with essentially negligible uncertainty.

The derivative need not be measured directly. Instead, a relation between z and t can first be obtained, and the derivative is then obtained from this relation. The derivative simplifies matters a bit, as any constant systematic error in the relation between z and t is eliminated by it.

Not any galaxies can be compared to one another however. To cover a large range of redshifts one must consider galaxies that formed at high redshifts and have evolved in the same way since their formation. Different types of galaxies will have a different offset from the cosmic time and the derivative would then not eliminate the offsets, resulting in incorrect proxies for the cosmic time.

The relation between z and t does not need to be derived explicitly to obtain the derivative. One can instead do as Simon et al., who obtain the redshifts and

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ages for 32 galaxies, presented in table 3.1 and figure 3.1. These are then placed in bins with galaxies which are close in redshift in the same bin and the derivative is approximated as the slope obtained from a linear fit to the data in the corresponding bin. From the original redshifts of the galaxies in a bin, the effective redshift can be computed by the arithmetic mean of these. Using this method at most n/2 Hubble parameter values can be obtained from n galaxies, since at least two data points are needed for a linear fit. They obtain 8 values presented in table 3.2 and figure 3.1.

To summarize, the idea behind the method is the following:

1. Identify galaxies of the same type (preferably at a large range of redshifts). 2. Determine the redshifts and the ages of the galaxies.

3. Place the galaxies in bins, with those close in redshift in the same bin. 4. Compute the effective redshifts.

5. Perform a linear fit for the data in each bin.

6. Identify the slope as the derivative, and compute its uncertainty.

7. With the obtained derivatives and effective redshifts use equation (2.37) to obtain the Hubble parameter at these effective redshifts. Propagate the un-certainties in the slopes.

With these steps several Hubble parameter values can be obtained, with the total amount depending on the binning.

Figure 3.1: Left: The galaxy data used by Simon et al. in [1]. Right: The values for the Hubble parameter obtained by Simon et al. in [1].

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3.1. Method and Results from Simon et al. 25

Table 3.1: Ages and redshifts of galaxies used in [1] and available in table form in [43]. Note that all 1-σ uncertainties are 12% of the ages as was given to [43] by R. Jimenez in a private communication.

Redshift Age Uncertainty Redshift Age Uncertainty

z t (Gyr) σt(Gyr) z t (Gyr) σt(Gyr)

0.1171 10.2 1.224 1.2260 3.5 0.420 0.1174 10.0 1.200 1.3400 3.4 0.408 0.2220 9.0 1.080 1.3800 3.5 0.420 0.2311 9.0 1.080 1.3830 3.5 0.420 0.3559 7.6 0.912 1.3960 3.6 0.432 0.4520 6.8 0.816 1.4300 3.2 0.384 0.5750 7.0 0.840 1.4500 3.2 0.384 0.6440 6.0 0.720 1.4880 3.0 0.360 0.6760 6.0 0.720 1.4900 3.6 0.432 0.8330 6.0 0.720 1.4930 3.2 0.384 0.8360 5.8 0.696 1.5100 2.8 0.336 0.9220 5.5 0.660 1.5500 3.0 0.360 1.1790 4.6 0.552 1.5760 2.5 0.300 1.2220 3.5 0.420 1.6420 3.0 0.360 1.2240 4.3 0.516 1.7250 2.6 0.312 1.2250 3.5 0.420 1.8450 2.5 0.300

Table 3.2: The resulting Hubble parameter values in [1] extracted from the right hand side of their figure 1. We also compute the relative uncertainties.

Redshift Hubble parameter Uncertainty Relative uncertainty

z H km/s/Mpc σH km/s/Mpc σH/H 0.17 83.0 8.0 9.6% 0.27 70.0 14.0 20.0% 0.40 87.0 17.0 19.5% 0.88 117.0 23.0 19.7% 1.30 168.0 17.0 10.1% 1.43 177.0 18.0 10.2% 1.53 140.0 14.0 10.0% 1.75 202.0 40.0 19.8%

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3.2 Obtaining the Data

We focus now on step 2 in the above description of the idea in Simon et al. Obtaining reliable results in science is inherently dependent on obtaining accurate and precise measurements or observations of physical quantities. There are two such quantities of interest in this work: The redshifts and the ages of galaxies.

3.2.1 Redshift

We have already mentioned that redshift can be determined both accurately and precisely. That is not to say it is an easy task. There are many difficulties involved and we will touch upon some of them here.

20 of the 32 galaxies considered in [1] come from the Gemini Deep Deep Survey (GDDS) paper [44] and we will therefore consider how GDDS determined redshifts. For an exhaustive understanding of how all redshifts were obtained we urge the reader to explore the cited data in [1]. The process for determining redshift in the GDDS is described in section 3 of [45]1_{. The first step is of course to capture}

the light of the galaxy with a telescope and CCD. This must be done under good weather conditions to have minimal noise, but techniques which reduce the noise further are still necessary. GDDS uses a technique called Nod & Shuffle to subtract the night sky emissions, which almost completely removes the night sky noise. The technique is described in detail in [46]. Furthermore, special software was created to extract the spectra of the galaxies, from which the redshift can be determined by identifying absorption and emission lines. A system which assigns a confidence class to the redshift of each galaxy is used which reflects the consensus probability among at least five of their team members that the assigned redshift is correct. A high confidence means that the redshift is correct with at least 95% certainty. To obtain the confidence they consider many factors of the spectrum: Signal-to-noise ratio, number of emission and absorption features, local continuum shape near prominent lines, and global continuum shape.

One possible culprit to redshift determination is not mentioned in the GDDS paper, namely peculiar velocities. These are velocities within the comoving coor-dinates which contribute to the total redshift in the form of Doppler shift, but we are only interested in the cosmological redshift in this context. The effect of pecu-liar velocities on cosmological parameter inferences using supernovas was studied by [47], in which they find the effects to be small. Nevertheless, for the moment we are only interested in typical uncertainties in redshift. They mention several phenomena which affect the redshift. First, they note that the maximum change in redshift due to our own motion is σCMB

z = 0.00124 when considering sources

directly aligned with the Cosmic Microwave Background dipole2_{. This effect can}

1_{As to not make this section longer than necessary, many difficulties GDDS faced with}

ob-taining good spectra are omitted. See [45] for a full description of every step involved.

2_{A clear dipole anisotropy has been measured in the CMB, corresponding to a peculiar velocity}

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3.2. Obtaining the Data 27 easily be corrected for. Furthermore, random peculiar velocities of observed galax-ies also affect the observations. A commonly used value for this is σrand

v ' 300

km/s, corresponding to σrand

z ' 0.001 using v = cz. Lastly, they note that

ob-servational uncertainty for the redshift of supernova host-galaxies is approximately σobs

z = 0.0005 for most surveys. The total uncertainty, ignoring the one due to the

CMB dipole anisotropy, is then σtot_z = p(σrand

z )2+ (σzobs)2 ≈ 0.0011, yielding a

maximum relative uncertainty of just under 1% for our sample in table 3.1.

3.2.2 Galaxy Age

The second quantity of utmost important in the analysis done in this work and in [1] are the ages of galaxies. Determining this is a much more difficult process than de-termining the galaxy redshift. Several methods have been developed to estimate a galaxy’s age, but in this section we will only give a brief explanation of the method used by [1], namely the SPEED model developed in [48]. The age determination starts at the, relatively, small scale of modeling individual stars. This process is too complicated to summarize here in a meaningful way, but it involves: Adaptive meshing of the star, advection and convection modeling, composition equations for certain elements, opacity models, nuclear reaction and neutrino loss rates, mini-mizing the free energy to obtain the equation of state and adding Coulomb and quantum corrections, different mass-loss functions for different categories of stars, and more - see [48] for details. With the model for a single star they compute many stellar tracks and isochrones3for varying masses and metallicities4, and find a varying degree of success when comparing to data. Nevertheless, overall fits are relatively good.

The next step is to build a stellar population from single star evolution. They define a single stellar population (SSP) as a stellar population formed in a uniform chemical environment in an instantaneous burst, i.e. the duration for star formation is modeled by a Dirac delta function. The only missing ingredient to obtain the luminosity of an SSP is the initial mass function (IMF), a distribution over the masses of stars in a stellar population. Then, with the age, metallicity, and IMF the luminosity function is obtained through a series of integrals. They present a few example spectra obtained in their figure 7. The Salpeter IMF is used throughout the paper.

Now, the task is to work backwards; obtain the age and metallicity from an observed spectrum. By adding noise to a simulated spectrum they find that they can specify the age and metallicity using a χ2 _{minimization fit given that the}

spectrum covers a sufficiently large range of wavelengths.

3_{An isochrone is a curve in the luminosity-temperature plane (the Hertzsprung–Russell}

dia-gram) for which all stars on the curve have the same age.

4_{In astronomy, metallicity is a measure of the amount of elements heavier than hydrogen and}

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Although very impressive work, some things are worth pointing out. As they mention themselves, no galaxy is truly an SSP since star formation is never instan-taneous and there is never a single metallicity describing the whole galaxy. Some galaxies are however better approximated by SSPs, namely most elliptical galaxies. For this reason, they only use elliptical galaxies in [1] and they claim to discard galaxies “for which the spectral fit indicates an extended star formation”, but it is not entirely clear what indicates this except for a reference to the “decaying rate” of the “declining exponential”, which they do not define.

Lastly, as mentioned in [49], to use galaxy ages to obtain the Hubble parameter in the way done in [1] one must correctly identify populations of galaxies at lower redshifts which are just older versions of galaxies at higher redshift, or make the necessary corrections due to events such as mergers. We borrow their concluding remark about this method in general:

These papers provide extensive discussions of systematic uncertainties and argue that they can be well controlled. Nonetheless, the reliance on population synthesis and galaxy evolution models means that this method will face a stiff burden of proof if it finds a discrepancy with simple dark energy models or other observational analyses, particularly if the differences are at the sub-percent level that is the target of future dark energy experiments.

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Chapter 4

Analysis

4.1 Attempt at Reproducing the Results

We have already spoiled our results in the introduction of this work. Neverthe-less, we have of course attempted to reproduce the results in [1] by following their instructions and we present our attempt here.

4.1.1 Reproducing the Binning

One might think that their description for obtaining the Hubble parameter from the galaxy ages is straight forward, after all it fits in just a few paragraphs. However, as will be demonstrated here, following what they write we do not manage to reproduce the results.

We begin by attempting to reproduce the binning and we separate their descrip-tion into three steps, which are visualized in figure 4.1 (step zero is the unprocessed redshift data for the galaxies in their sample), where the color coding is described in the figure caption.

The first step is contained in the following quote from [1]:

First we group together all galaxies that are within ∆z = 0.03 of each other.

Which should be an easy step, simply place galaxies that are within ∆z = 0.03 in the same bin. However, already at this first simple step, something appears to go wrong. They say that this interval is

...large enough for our sparse sample to have more than one galaxy in most of the bins.

We do not obtain this. Instead, nineteen bins are obtained, where only eight bins contain two or more galaxies. Ignoring the above discrepancy, we move on. Next, they write:

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Then for each bin we discard those galaxies that are more than 2σ away from the oldest galaxy in that bin.

This requires some interpretation. Which galaxy’s σ is one supposed to use? We tried the two extremes: The oldest, and the youngest galaxy in the bin. Neither approach resulted in any change from the previous step. Therefore, we call this step 1.5 and do not display it in figure 4.1.

Step two also requires some interpretation:

We then compute age differences only for those bins in redshift that are separated more than ∆z = 0.1 but no more than ∆z = 0.15.

Further interpretation is required. Where does a bin start and where does it end? At the redshift corresponding to the galaxy in the bin with lowest redshift, or the highest, or the middle, or the mean? There are many possibilities. We tried all four possible combinations of lowest and highest redshift, as well as the middle, and the mean. All result in similar results, but we choose to compare the middle of the bins1, i.e. the lowest galaxy redshift plus the highest galaxy redshift in the bin divided by two. Then the bin separation ∆z is defined as the difference between two bins’ middle redshifts. Now, instead of computing the differential ages2_{, we}

merge bins which fulfill ∆z ∈ (0.1, 0.15]. This is effectively done when computing the differential ages between bins. Note in step two in figure 4.1 that this results in some data points being used twice, for two different differential age computations, implying that some resulting Hubble parameter values are not independent.

The final step is to compare the effective redshifts for our attempt at reproducing their results with the effective redshift they have displayed in the right hand side of figure 1 in [1]. This is step three in figure 4.1, where the effective redshift for a bin is defined as the arithmetic mean of the redshifts in that bin3_{. We obtain}

a quite different set of effective redshifts and for this reason do not compute the Hubble parameter at our obtained effective redshifts. The difference is already large enough to conclude that it is indeed difficult to reproduce their results using their instructions.

1_{This choice was made only to make the figure less cluttered. All could be shown, but the}

differences were so small we deemed it unnecessary.

2_{On that note, it is not clear how they compute the differential ages. One can only assume}

that they use a linear χ2 _{minimization fit and identify the slope as the derivative, since that is}

what they do in their feasibility article [50]

3_{The effective redshift is another quantity which they do not define, in fact it is not even}

mentioned (in the feasibility article [50] it is mentioned, but not defined). Even when choosing another sensible definition for the effective redshift, maximum redshift plus minimum redshift in a bin divided by two, the results we obtain differ similarly from the results in Simon et al. as when using the arithmetic mean.

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4.1. Attempt at Reproducing the Results 31

Figure 4.1: Visualization of our attempt of reproducing the binning in [1]. Step 0: Unprocessed redshift data for the 32 galaxies in their sample. Step 1: Galaxies within ∆z = 0.03 have been grouped together. Different colors correspond to different groups. Step 2: Groups from the previous step which are separated by at least 0.1 but no more than 0.15 have been placed in the same bin (see text for details). Data points with the same shape and color correspond to the same bin. Note that some data points are in two different bins. Step 3: The effective redshifts from [1] (black and marked with stars), and the effective redshifts computed for every bin from the previous step (colored as the bins in the previous step and marked with crosses).

4.1.2 Matching the Effective Redshift

Since attempting to reproduce their effective redshifts by following their description proved to be difficult, we will in this section disregard it and instead attempt to find the bins which yield the effective redshifts they report.

To achieve this we first construct all subsets of the 32 data points which have a size ranging from two to ten data points, and for which the maximum redshift minus the minimum redshift is lower than 0.2. This limit is chosen since it is slightly more generous than than their limit of ∆z = 0.15 combined with the grouping criteria of ∆z = 0.03, discussed above4. After this filtering, the effective redshifts are computed to two decimals for all remaining subsets, defined as the arithmetic mean of the redshifts in each subset. As a last step we compare the obtained effective redshifts with the ones in [1] and choose the bins with matching effective redshift5_{. The resulting bins are shown in the middle panel of figure 4.2.}

All redshifts can be matched, except for the last where we instead choose the bin producing the closest effective redshift.

4_{The reason we do not set it to 0.18 is because we did not want to be too restrictive, considering}

that there is something which we can not understand in their binning process, as demonstrated in the previous section.

5_{In the case of multiple matches, we choose the bin which produces the closest effective redshift,}

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Figure 4.2: Top: Unprocessed redshift data. Middle: The bins which produce (almost) the same effective redshifts as in [1]. Data corresponding to the same bin have the same color. Note that some data points are used twice, i.e are in two bins. Bottom: The effective redshifts of the above bins, with the same color coding. Only the last redshift was not managed to be matched.

Using these bins we would like to perform the same analysis as in [1] to com-pute the derivative dt/ dz. Unfortunately, they do not clearly communicate which method they use. Therefore, we choose to assume a standard χ2 _{minimization fit,}

as described in section 2.2.1, since it is what they use in their feasibility article [50]. The result is shown in figure 4.3, which demonstrates uncertainties around 100% (some even larger), much larger than the 10 to 20% reported by Simon et al. Thus, even using bins which are credible to be close to the ones they used, the uncertain-ties in dt/ dz are too large to produce Hubble parameter values with the precision they reported.

4.1.3 Remark about the Data

All the analysis we perform later in this chapter will use the same age and redshift data as Simon et al. Nevertheless, we should note that there are things we find suspicious about the data set. It is a combination of three smaller sets; the first set being 10 galaxies from their feasibility article [50] which they obtain by discarding galaxies which indicate “an extended star formation”, a process which is not entirely clear (as pointed out in the previous chapter). Furthermore, it is not clear which of the 32 galaxies in figure 3.1 and table 3.1 are the 10 from [50].

Our biggest concern is with the GDDS data set. 20 of the 32 galaxies in Simon et al. come from this survey and the paper they cite is [44], which contains precisely 20 galaxies for which the redshifts and ages have been determined (among other quantities, see their table 1). Simon et al. write

We have reanalyzed the GDDS old sample using SPEED models and obtained ages within 0.1 Gyr of the GDDS collaboration estimate.

It is unclear what would prompt such a statement. We compare the higher redshift data in Simon et al. with the data in table 1 of the GDDS paper [44], see figure 4.4.

Galaxies as Clocks and the Universal Expansion

Galaxies as Clocks and the

Universal Expansion

ANDERS AHLSTRÖM KJERRGREN

Master’s Thesis

Galaxies as Clocks and the

Universal Expansion

Anders Ahlstr¨

om Kjerrgren

Supervisor:

Prof. Edvard M¨

ortsell

Abstract

Sammanfattning

Preface

Acknowledgements

Contents

Chapter 1

Introduction

1.1

Executive Summary

1.2

Structure of the Thesis

Chapter 2

Theoretical Background

2.1

Cosmology and General Relativity

2.1.1

The Einstein Field Equations

2.1.2

Cosmology: Basic Principles

2.1.3

Robertson-Walker Metrics and the Friedmann

Equations

2.1.4

Cosmological Parameters

2.1.5

The Hubble Parameter

2.1.6

The ΛCDM Concordance Model of Cosmology

2.1.7

Cosmological Parameters: Current Values

2.2

Methods

2.2.1

χ

Minimization

2.2.2

Monte Carlo Sampling

2.2.3

Gaussian Processes

Chapter 3

The Hubble Parameter from

Galaxy Ages

3.1

Method and Results from Simon et al.

3.2

Obtaining the Data

3.2.1

Redshift

3.2.2

Galaxy Age

Chapter 4

Analysis

4.1

Attempt at Reproducing the Results

4.1.1

Reproducing the Binning

4.1.2

Matching the Effective Redshift

4.1.3

Remark about the Data

_Minimization