effects of helium recombination in the early...

EFFECTS OF HELIUM RECOMBINATION IN THEEARLY UNIVERSE

By

ANDREW L. HILL

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2011

c© 2011 Andrew L. Hill

2

This is dedicated to everyone who helped me get to where I am today–my friends, myfamily, my brother, my mother, and myself. Also to my father, who would have loved the

chance to see this. Thanks everyone!

3

ACKNOWLEDGMENTS

Thanks to Dr. Fry for giving me the help and assistance to get this far. I couldn’t

have finished this without the many helpful conversations on physics and the universe,

or all of the useful suggestions and advice. I’d also like to thank Edmund Bertschinger

for creating COSMICS and Volker Springel for creating GADGET-2. Both of these programs

were invaluable tools, and are immensely useful.

I’d also like to thank all my friends and family, my grandparents, and especially my

mom and dad, who gave me their support and encouragement throughout the years. I

couldn’t have done it without all of you. Thanks a lot!

4

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER

1 INTRODUCTION TO RECOMBINATION AND THE EARLY UNIVERSE . . . . 11

2 BASICS OF RECOMBINATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1 Using the Saha Equation to Calculate the Recombination Time . . . . . . 192.2 Evolution of the Fractional Ionization of Hydrogen . . . . . . . . . . . . . . 22

3 ROUGH CALCULATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1 The Scale Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Rough Analytical Calculation of Recombination . . . . . . . . . . . . . . . 353.3 Rough Calculation of Recombination Starting From COSMICS Code . . . . 41

4 COSMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1 Modelling the Evolution of Density Perturbations . . . . . . . . . . . . . . 524.2 Addition of Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 INTRODUCTION TO GADGET-2 . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.1 Setting Up A Run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2 Running GADGET-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.3 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6 ANALYSIS OF HELIUM RECOMBINATION USING GADGET-2 . . . . . . . . . 70

6.1 Creating an Appropriate Set of Initial Conditions . . . . . . . . . . . . . . 706.2 Evolution Starting at a Redshift of 1250 . . . . . . . . . . . . . . . . . . . . 736.3 Evolution Starting at a Redshift of 1000 . . . . . . . . . . . . . . . . . . . . 896.4 Evolution Starting at a Redshift of 1250 With a Different Helium Mass . . . 97

7 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.1 COSMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.2 GADGET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

APPENDIX: SETTING INITIAL CONDITIONS FOR GADGET-2 . . . . . . . . . . . . 113

5

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6

LIST OF TABLES

Table page

6-1 Data conveying the hydrogen and helium composition of ”Clump 1.” . . . . . . 78









7

LIST OF FIGURES

Figure page

2-1 The fractional ionization of hydrogen (Saha equation). . . . . . . . . . . . . . . 21

2-2 The fractional ionization of hydrogen (case I). . . . . . . . . . . . . . . . . . . . 24

2-3 The fractional ionization of hydrogen (case II). . . . . . . . . . . . . . . . . . . . 25

2-4 The fractional ionization of hydrogen (case III). . . . . . . . . . . . . . . . . . . 27

2-5 The fractional ionization of hydrogen (case IV). . . . . . . . . . . . . . . . . . . 29

3-1 The line shows the evolution of the scale factor with time. . . . . . . . . . . . . 31

3-2 The line shows the evolution of the scale factor with time on a log-log plot. . . . 32

3-3 Evolution of the Hubble parameter with scale factor on a log-log plot. . . . . . . 34

3-4 The evolution of cold dark matter perturbations (case I). . . . . . . . . . . . . . 39

3-5 The evolution of cold dark matter perturbations (case II). . . . . . . . . . . . . . 40

3-6 The cold dark matter perturbations at k = 0.01. . . . . . . . . . . . . . . . . . . 44

3-7 The change in cold dark matter perturbations at k = 0.01. . . . . . . . . . . . . 45

3-8 The cold dark matter perturbations at k = 0.1. . . . . . . . . . . . . . . . . . . . 46

3-9 The change in cold dark matter perturbations at k = 0.1. . . . . . . . . . . . . . 47

3-10 The cold dark matter perturbations at k = 1. . . . . . . . . . . . . . . . . . . . . 48

3-11 The change in cold dark matter perturbations at k = 1. . . . . . . . . . . . . . . 49

3-12 The cold dark matter perturbations at k = 10. . . . . . . . . . . . . . . . . . . . 50

3-13 The change in cold dark matter perturbations at k = 10. . . . . . . . . . . . . . 51

4-1 The density perturbations of baryons and cold dark matter. . . . . . . . . . . . 55

4-2 The density perturbations during recombination. . . . . . . . . . . . . . . . . . 56

4-3 The density perturbations of helium. . . . . . . . . . . . . . . . . . . . . . . . . 58

5-1 The initial conditions for a run of GADGET. . . . . . . . . . . . . . . . . . . . . . . 66

5-2 The distribution of matter at z = 0. . . . . . . . . . . . . . . . . . . . . . . . . . 69

6-1 A view of the universe at z = 10 (run I). . . . . . . . . . . . . . . . . . . . . . . . 75

6-2 A view of the universe at z = 91 (run I). . . . . . . . . . . . . . . . . . . . . . . . 76

8

6-3 The motion of a cluster in three dimensions (run I). . . . . . . . . . . . . . . . . 83

6-4 A three dimensional plot of cluster motion (run I). . . . . . . . . . . . . . . . . . 84

6-5 The separation of the centroids (run I). . . . . . . . . . . . . . . . . . . . . . . . 85

6-6 The standard deviations of the clusters (run I). . . . . . . . . . . . . . . . . . . 86

6-7 The ratios of helium to hydrogen (run I). . . . . . . . . . . . . . . . . . . . . . . 88

6-8 A view of the universe at z = 50 (run II). . . . . . . . . . . . . . . . . . . . . . . 90

6-9 A view of the universe at z = 166 (run II). . . . . . . . . . . . . . . . . . . . . . . 91

6-10 The motion of a cluster in three dimensions (run II). . . . . . . . . . . . . . . . 95

6-11 The separation of the centroids (run II). . . . . . . . . . . . . . . . . . . . . . . 96

6-12 The standard deviations of the clusters (run II). . . . . . . . . . . . . . . . . . . 97

6-13 The ratios of helium to hydrogen (run II). . . . . . . . . . . . . . . . . . . . . . . 98

6-14 A view of the universe at z = 24 (run III). . . . . . . . . . . . . . . . . . . . . . . 100

6-15 A view of the universe at z = 101 (run III). . . . . . . . . . . . . . . . . . . . . . 101

6-16 The motion of a cluster in three dimensions (run III). . . . . . . . . . . . . . . . 105

6-17 The separation of the centroids (run III). . . . . . . . . . . . . . . . . . . . . . . 106

6-18 The standard deviations of the clusters (run III). . . . . . . . . . . . . . . . . . . 107

6-19 The ratios of helium to hydrogen (run III). . . . . . . . . . . . . . . . . . . . . . 109

9

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

EFFECTS OF HELIUM RECOMBINATION IN THEEARLY UNIVERSE

By

Andrew L. Hill

May 2011

Chair: James N. FryMajor: Physics

This study analyzes the recombination of hydrogen and helium in the early universe.

Specifically, I examined whether looking at hydrogen and helium as two different entities

causes any noticeable changes in the formation of structures in the early universe.

This distinction has been for the most part neglected, and yet it is an interesting area

of study. By tracking these two types of baryons independently, it is possible to see

some differences in the resulting structure formation. Since helium recombines before

hydrogen, it speeds up faster than hydrogen. This causes it to lead hydrogen, as well as

cluster together sooner than hydrogen. These are the effects I study in this paper.

10

CHAPTER 1INTRODUCTION TO RECOMBINATION AND THE EARLY UNIVERSE

Recombination is an important era of study in the evolution of the universe, and

accordingly, a large body of work is dedicated to it. From treatments in textbooks to

papers going into detail on specific aspects of the recombination process, there is a

lot of information to sift and sort through to find whatever one might need to know.

Much of the physics that govern evolution of the universe during this epoch is known,

but the more complicated one makes the model being studied, the more intractable,

difficult, and time consuming the problem becomes. Because many of the intricacies

and interactions during the recombination era can be ignored or neglected while still

giving useful results, these complications are often left out of recombination models.

One of the earliest studies of recombination was presented by Peebles (Peebles,

1968). This treatment was an excellent beginning to the subject, and today recombination

is part of all cosmological models. These basics are included in many textbooks on the

subject, and the theory is known by everyone studying cosmology (Liddle & Lyth,

2000; Peebles, 1993; Bertschinger, 2001; Carroll & Ostlie, 1996; Dodelson, 2003).

There are many refinements to the most basic model, some of which are more well

understood than others. Current calculations including all of the most relevant aspects

of recombination give good results, but more work still needs to be done on how the

different recombination times of hydrogen and helium affect structure formation and the

evolution of the universe (Wong et al., 2008). Work has already been done on aspects of

this subject, but there is still more to do (Switzer & Hirata, 2008a; Hirata & Switzer, 2008;

Switzer & Hirata, 2008b). This will be the subject of my study–the evolution of structure

formation when the earlier recombination time of helium is taken into account.

To really understand recombination, and helium’s role in the process, it must be

placed into the larger context of the Big Bang. The goal of this study is not to give an

exhaustive account of the creation of the universe, as entire books could be written

11

on this subject–and they have been. For an in-depth treatment of the Big Bang, or of

any one part of it, the reader may refer to the excellent material that already exists

(Hawking & Ellis, 1975; Kolb & Turner, 1990; Padmanabhan, 1993; Liddle & Lyth,

2000; Carroll & Ostlie, 1996). The beginnings of structure formation have their seeds

in the very early universe, during the inflationary epoch. Inflation occurred a very

short time after the universe began–from around 10−42 seconds to somewhere around

10−32 seconds. Introduced in 1981 (Guth, 1981), the model has gone through many

revisions to get to its current state. It is currently the best way to explain where the

density perturbations that give rise to structure come from. These arise from vacuum

fluctuations generated naturally from the inflation process, and are Gaussian in nature

(Guth, 1997; Mukhanov, 2005; Liddle & Lyth, 2000; Durrer, 2005). Without them, the

universe would be completely featureless, homogeneous and isotropic (in other words,

it would look exactly the same in every direction, with no variation of any kind). Of

course, today, on large enough scales, the universe is homogeneous and isotropic, but it

definitely has features. Inflation models tell us that these features start out as very small

inhomogeneities in the background of the universe, created from quantum fluctuations of

the inflaton field around its vacuum state–as mentioned, these fluctuations are Gaussian

and adiabatic, which fits the models for necessary conditions for large scale structure

formation perfectly (Peebles, 1980; Peacock, 1999; Liddle & Lyth, 2000).

After this initial inflationary period, the universe will undergo a period of reheating,

when the nonrelativistic particles left over after inflation will decay into relativistic

particles and, in the process, create a radiation-dominated universe. During this era, the

scale factor will depend on time in the following way:

a ∝ t1/2 , (1–1)

and the density will depend on the scale factor as:

ρR ∝ a−4 . (1–2)

12

These parameters will determine the evolution of the universe–and the density

perturbations which are of interest to us–until around 104 years, when the radiation and

matter start to equalize. During radiation domination, nucleosynthesis occurs–another

important step on the road to structure formation. It is during this process–which occurs

when the universe is only about 100 seconds old–that all the baryonic matter which will

one day create the structures that we are familiar with is first created. The hydrogen,

helium-4, and other light elements created during this process are nothing more than

ions at this point–because of the high binding energy of nuclei, these atoms can form

even at the very hot (0.1 MeV) temperatures of this primordial soup, but electrons would

be instantly stripped away (Liddle & Lyth, 2000; Peacock, 1999; Dodelson, 2003).

As the time passes, the universe expands and cools. It is this process that ends the

radiation-dominated era and allows the transition into the matter-dominated era, during

which the dependence of the scale factor on time changes to:

a ∝ t2/3 , (1–3)

and the density’s dependence on scale factor changes to:

ρM ∝ a−3 . (1–4)

It is during this era, when the temperature cools to around 3000 K and around 105

years have passed since the universe began, that recombination occurs. With the

average temperature of the universe cooled to this degree, neutral atoms can finally

start forming. At this point, the photons no longer have enough energy to keep the

electrons from binding to the previously ionized elements, and slowly at first, then faster

as the universe cools more and more, the electrons all get snatched up by atoms that

can finally bind electrons to themselves. Until this point, it is only the cold dark matter

perturbations that can grow; since they only interact through gravitational forces, the

radiation background does not affect them. On the other hand, while the radiation is

13

dominating the universe, the baryonic matter is not free to move–it is not until after

the matter decouples from the photons and forms electrically neutral atoms that it

is able to move freely under the influence of gravity. And it is not until the baryonic

matter is able to move solely as gravity dictates that baryonic matter perturbations

are able to grow (Liddle & Lyth, 2000; Peacock, 1999; Peebles, 1980; Kolb & Turner,

1990; Padmanabhan, 1993). It is the goal of this study to examine in more detail the

differences that arise between the two models when, rather than assuming that all

the baryonic matter is hydrogen, we instead treat the more accurate case of around

twenty-four percent of the baryonic matter forming into helium. Since helium-4 has a

higher binding energy than hydrogen, it is able to recombine and become electrically

neutral at an earlier time. This means that its density perturbations will start growing

before hydrogen’s, leading to possibly interesting effects in the universe during and

after recombination (Peebles, 1968; Seager et al., 2000; Switzer & Hirata, 2008a;

Dubrovich & Grachev, 2005; Wong et al., 2006).

The universe continues to cool as the perturbations continue to grow. Finally,

structures can start forming, creating the first objects, and eventually leading to the

universe we see today (Kolb & Turner, 1990; Peebles, 1980; Yoshida et al., 2003;

Gao et al., 2007). On large scales, we see that the universe is still homogeneous and

isotropic, but on small scales, it is most definitely inhomogeneous. The reason for this

structure is the once tiny density perturbations. As time passes, gravity acts on them,

pulling them closer and closer, and making them more and more tightly bound. As the

resulting structures become more and more massive, they will pull in more matter, and

eventually stars and galaxies are born from those once humble perturbations. This is

why we must understand helium recombination–we need to know what effect it has on

the evolving universe, and the growing structure that it contains.

An understanding of the large scale structure of the universe is therefore quite

important. The distribution of stars, galaxies, voids, matter, energy, the cosmic

14

microwave radiation background, and the like, are all important, and to understand

them in greater depth, statistics and techniques pertaining to the field, such as different

distribution functions, as well as correlation functions, fractals, and measure theory,

are important to one degree or another (Hartle, 2003). To start with, let us introduce an

equation for the density of objects:

ng(r) = n(1 + ξ(r)) . (1–5)

Here, n is the mean density of a species such as baryons or radiation (whatever is

represented by ”g”), and ξ(r) is a perturbation based on position. This equation will

be very useful when dealing with recombination and structure formation. It is only the

deviations from the background homogeneous density that give rise to structure–both

voids and clusters of matter–and these are represented by the ξ(r) term.1 From

large-scale structure, correlation functions, and scaling equations, we move on to

relativity, cosmology, and related areas. We can start with the Robertson-Walker metric

(Hartle, 2003; Liddle & Lyth, 2000):

ds2 = −dt2 + a2(t)dx2 . (1–6)

Using this metric, it is possible to obtain the one-particle Vaslov Equation by using the

Liouville Equation and Boltzman Distribution (Hartle, 2003; Liddle & Lyth, 2000; Peebles,

1993):

∂f

∂t+

∂f

∂xi

dxi

dt+

∂f

∂pi

dpi

dt= 0 . (1–7)

1 Among other sources, my conversations with Dr. Fry were very helpful in this area.

15

Now, if we let:

dxi

dt=

pi

ma2(1–8)

dpi

dt= −m

∂φ

∂xi

, (1–9)

we can get:

∂f

∂t+

pi

ma2

∂f

∂xi

−m∂φ

∂xi

∂f

∂pi

= 0 . (1–10)

If we integrate this equation with d3p pi and take the second moment, we can get the

Euler Equation (Liddle & Lyth, 2000; Peebles, 1993):

1

a

∂

∂t(avi) +

1

a(vk ∂

∂xk)vi +

1

a

1

δ + 1

∂

∂xk((1 + δ)Πij) +

1

a

∂φ

∂xi= 0 , (1–11)

where the third term contains the pressure force, and the fourth term is the gravitational

force. The continuity equation–a very important equation for studying evolution of the

universe and recombination–is (Liddle & Lyth, 2000; Peebles, 1993):

∂δ

∂t+

1

a

∂

∂xk((1 + δ)vk) = 0 . (1–12)

Poisson’s equation for comoving coordinates, another very important equation for

modeling recombination, can be written as (Liddle & Lyth, 2000; Peebles, 1993):

1

a2∇2φ = 4πGρδ . (1–13)

Using these three equations and making a few assumptions; namely, that Π = 0 and

that the perturbations, signified by δ, are very small, δ ≪ 1 (so that all the equations are

linear), it is possible to combine them. With these assumptions, the continuity equation

can be rewritten as (for a more detailed treatment, unnecessary for our purposes but

interesting nonetheless, see (Lima et al., 1997)):

∂δ

∂t+

1

a∇ · v = 0 . (1–14)

16

If this equation is multiplied by a2 and the partial derivative with respect to time is taken,

it can be turned into:

∂

∂t(a2∂δ

∂t) = − ∂

∂t(a∇ · v) . (1–15)

With the above assumptions, the Euler Equation becomes:

∂

∂t(av) = −∇φ . (1–16)

If we take the divergence of both sides (∇·), we can turn this equation into:

∇ · ( ∂

∂tav) = −∇2φ (1–17)

− ∂

∂t(a~∇ · ~v) = ∇2φ . (1–18)

This makes the right side of the continuity equation, Equation (1–15), equal to the left

side of the Euler Equation, so:

1

a2

∂

∂t(a2∂δ

∂t)− 1

a2∇2φ = 0 . (1–19)

And, combining this with the last equation we haven’t used yet, Equation (1–13), we

obtain an equation for the growth of linearized fluctuations:

∂2δ

∂t2+

2a

a

∂δ

∂t− 4πGρδ = 0 . (1–20)

The growth of linearized fluctuations is something which I want to keep track of with

respect to helium recombination, and so it makes sense to use this equation in some

simple cases. It can be directly solved in a couple of cases, including one with no

curvature and no dark energy, the case where there is just matter, and no pressure, and

the case where there can be some curvature, but there is no radiation. These exercises

helped familiarize me with the idea of linearized fluctuations, and how they could be

17

found with the help of the Friedmann Equation (Tamvakis, 2005):

(a

a)2 =

8π

3Gρ− k

a2+

Λ

3. (1–21)

18

CHAPTER 2BASICS OF RECOMBINATION

2.1 Using the Saha Equation to Calculate the Recombination Time

In order to understand how it was able to obtain density perturbations, it is important

to get better acquainted with the physics embodied in COSMICS. This means doing some

review of recombination era physics. To start with, we can do a rough calculation of

when recombination occurred. Using (Liddle & Lyth, 2000; Dodelson, 2003):

(ncnd

nanb

)eq = exp∆E/T , (2–1)

where the n’s are the number densities of each type of particle at equilibrium, ∆E is the

change in energy (taken to be the ionization energy below, in Equation (2–3)), and T is

the temperature. For the reaction:

p+ + e− ←→ H + γ , (2–2)

we see that Equation (2–1) gives:

nHnγ

npne

= expI0/T . (2–3)

Keeping in mind that:

nb

nγ= η = 6× 10−10 , (2–4)

where ”b” stands for baryons and ”γ” stands for photons during the early universe, and

that we can say:

np

nb

=ne

nb

= X , (2–5)

(so that the fraction of protons is the same as the fraction of electrons), we can show

that:

1− X

ηX2= expI0/T . (2–6)

19

With this equation, it is possible to create a graph showing how the Saha Equation,

without any other modifications or refinements, predicts the fractional ionization will

evolve. First, as recombination is the focus here, I0 can be replaced with ǫ0, the

ionization energy for hydrogen. This means that the ionization fraction, X, is actually

the ionization fraction of hydrogen, XH. Further, as I would like to see how the ionization

fraction of hydrogen depends on ǫ0/T, I will substitute x for ǫ0/T, so the equation I am

graphing is:

1− XH

ηX2H

= expx . (2–7)

Solving for XH using the quadratic formula gives:

XH =−1±

√1 + 4η expx

2η expx, (2–8)

and, since the fractional ionization can never be negative, the only physical solution is:

XH =

√1 + 4η expx − 1

2η expx. (2–9)

Graphing this function gives Figure 2-1, which shows how the fractional ionization of

hydrogen, XH, evolves as a function of ǫ0/T. Comparing this graph with the graphs

created with more complicated models, such as the one used to get Figure 2-2, it

seems that Figure 2-1 has the same general shape at early times, although it does

start dropping a little sooner. The other major difference is that it never levels out–the

graph just keeps getting smaller. Of course, that is why more complicated models are

desirable–by taking into account more variables, behavior closer to what should actually

occur can be seen.

At the crossover regime, where I0 and T are nearing each other, Equation (2–6) can

be used to find out approximately how much time has elapsed since the beginning of the

universe. If we say that X = 0.5, so that about half of the hydrogen has recombined, we

20

Figure 2-1. The line shows the evolution of the fractional ionization of hydrogen as afunction of x. The fraction starts dropping a bit before x = 20, whichcorresponds to a temperature of about 0.68 eV, and keeps dropping, gettingsmaller and smaller.

21

find that (since I0 = 13.6 eV for hydrogen) this temperature is around:

1− 0.5

η0.52= expI0/T (2–10)

T = 0.620 eV . (2–11)

This translates to a temperature of about 7000 K. If I then use Equation (18–42) from

Hartle’s book (Hartle, 2003), I can find the time at which this temperature occurred:

t = 1.3(1 MeV

T)2 s (2–12)

t0.5 = 3.4× 1012 s (2–13)

= 107000 years . (2–14)

This is a good approximation, but much more detailed calculations are possible. I will do

a more in-depth calculation using computer modelling in the next section; other studies

of ionization are also useful to consult (Jones & Wyse, 1985; Dubrovich & Stolyarov,

1997).

2.2 Evolution of the Fractional Ionization of Hydrogen

Another important aspect of the recombination era involves the amount of hydrogen

that has formed at any point in time. Modelling this fractional ionization is also a

nice introduction to the use of computer programming as a tool for solving otherwise

intractable problems in cosmology, astrophysics, and beyond. There is a problem

(exercise 8 on page 82) in Dodelson’s book (Dodelson, 2003) that provides us with an

equation to describe basic recombination. Being a rather complicated and ugly equation,

it would be nigh impossible to integrate using traditional means:

dXe

dx= Ax ln x(B

exp−x

x1.5−Xe(C

Xe

x3+ B

exp−x

x1.5))

× ((Dx1.5)/(1− Xe) + 8.227

(Dx1.5)/(1− Xe) + 8.227 + E(ln x/x) exp−x/4) , (2–15)

22

where A = 0.0451 cm3, B = 6.10 × 1020 cm−3, C = 3.25 × 107 cm−3, D = 4.714 × 10−5,

E = 5.618× 108, and:

x =ǫ0

T, (2–16)

and Xe is the fractional ionization of the hydrogen.

Although too intractable to solve analytically, computers give the ability to integrate

such an unpleasant equation. All that is necessary is to use a suitable integration routine

to solve this for the fractional ionization of hydrogen, which is what I am interested in.

Programs for integrating differential equations by the Runge-Kutta method of order 4 are

rather well-known and can be found on the internet or in basic programming textbooks,

and it was this routine that I modified to use with Equation (2–15) (Press et al., 1992).

After ironing out the kinks, I was able to obtain many graphs of the evolution of the

fractional ionization of hydrogen as a function of x (which, of course, relied inversely

on temperature, since the x value in Equation (2–15) is given in Equation (2–16), as

was mentioned earlier, and so it is in effect a measure of time). Figure 2-2 shows

a representative graph. In addition to my values, I included the values that Peebles

calculated for the ionization history (Peebles, 1968).

After I had this much working, I went back and altered it a bit. Dr. Fry suggested

that I replace my ln x with ln x + 0.86, as it would give slightly better results. When I reran

the program numerous times, I got new graphs, one of which can be seen in Figure

2-3. This figure has the values calculated by Peebles for the ionization history displayed

for comparison as well. The results are slightly better, but it is hard to see much of a

difference. It seems that the hydrogen fraction might start dropping a little sooner, but

my estimate on the other graph was so rough that the difference is not really detectable.

The same also holds true for when it starts levelling out–it might happen a bit sooner

in this version, but not enough to really make a note of. Although these differences

are small, they do seem to be for the better–they certainly do not make things any

worse–and so I left this minor change in all further revisions of the code.

23

Figure 2-2. The line shows the evolution of the fractional ionization of hydrogen as afunction of x. The fraction starts dropping around x = 30, which correspondsto a temperature of about 0.45 eV, and starts to level out at about x = 150,which is a temperature of about 0.09 eV. The squares show the values thatPeebles calculated for the ionization history.

24

Figure 2-3. The line shows the evolution of the fractional ionization of hydrogen as afunction of x, this time with the ln x + 0.86 term. The fraction starts droppingaround x = 30, which corresponds to a temperature of about 0.45 eV, andstarts to level out at about x = 150, which is a temperature of about 0.09 eV.Once again, the squares show the values for ionization history as calculatedby Peebles.

25

The next thing I did with this program was compare its output to that of Peebles

in his 1968 paper on recombination (Peebles, 1968); of course, this comparison can

also be seen in Figures 2-2 and 2-3, where his data is plotted on my graphs. This

comparison was important to make sure the program’s output was fairly close to

other simple models of recombination, which would imply that my model was at least

reasonably successful. Peebles had a table of the temperature variation of the matter on

page 10, which showed how hot the matter was at different fractional ionizations of the

hydrogen for three different present mean mass densities of the universe. To compare

my results with his, I had to make a few alterations to my equation so that it would match

up with each of his present mean mass densities. The equation I used assumed a value

of Ωm = 1, but Peebles did not do this. This meant I had to find Ωm for each of the

ρ0 values he used, and then plug them into my program. I know that (Peebles, 1993;

Liddle & Lyth, 2000):

ρm

ρcr

= Ωma−3 , (2–17)

and that a = 1 at the present time, so this means that:

ρ0

ρcr

= Ωm . (2–18)

Since ρ0 is the value Peeble gives in his table, and ρcr = 1.879h2 × 10−29 g cm−3

(with h = 0.5), it is possible to figure out Ωm for each of his given present mean mass

densities. Once I obtained these values, I used them in my program and compared my

results with his. The graph for ρ0 = 1.8 × 10−29 g cm−3, the first case he uses, can be

seen in Figure 2-4. Unfortunately, my values for the fraction of hydrogen were around

one and a half times what Peebles got. Presumably this is because he worked the

problem out forty years ago, and his code was probably quite a bit different from mine.

At this point I went back to my original equation and did some even longer runs

with it, in an effort to gain more information. To further this end, I added the equilibrium

26

Figure 2-4. The line shows the evolution of the fractional ionization of hydrogen as afunction of x. There is not much change from the last two graphs; thefraction starts dropping around x = 30, which corresponds to a temperatureof about 0.45 eV, and starts to level out at about x = 150, which is atemperature of about 0.09 eV.

27

values to the graph, so that I could see how recombination actually differed from the

equilibrium process. To do this, I looked for the values of Xe such that the differential

on the left hand side is always equal to 0. In other words, I solved Equation (2–15) for

dXe/dx = 0:

0 = (Aexp−x

x1.5− Xe(B

Xe

x3+ A

exp−x

x1.5)) , (2–19)

where A = 6.10 × 1020 cm−3 and B = 3.25 × 107 cm−3. When this is solved for Xe, the

resulting equilibrium equation is:

Xe = 9.38× 1012x1.5(− exp−x +

√

exp−2x +2.13× 10−13 exp−x

x1.5) . (2–20)

By creating a simple program to give numerical values for this equilibrium equation, I

was able to put a graph of it on the same graph as the recombination equation which

I had been working with; this graph can be seen in Figure 2-5. It is clear that the final

values of Xe determined from these equations are quite different, as Xe quickly goes to

zero in the equilibrium case, whereas it levels out at a fractional ionization of about 10−5

in actual recombination.

28

Figure 2-5. The solid line shows the evolution of the fractional ionization of hydrogen asa function of x. The dotted line, which drops quickly at about x = 50, orT = 0.272 eV, is the equilibrium equation.

29

CHAPTER 3ROUGH CALCULATIONS

3.1 The Scale Factor

Once I had gotten all that I could out of my basic recombination equation, I was

ready to move on. The next step was to do some rough calculations of the evolution of

the density perturbations, using just the basic analytical equations and without worrying

about the perturbations of the radiation. Instead of worrying about all the myriad ways

that different kinds of matter and radiation could interact, I started with some simple

coupling and rate equations, and figured out the evolution of the density perturbations of

dark matter, hydrogen, and helium under these conditions.

There was a chance that I would want or need to utilize time as my independent

variable, but all the equations I would be using for these calculations were dependent

on the scale factor, a. So, to prepare for this eventuality, and simultaneously check to

make sure the integration routine I was using was working properly, I started with a

simple graph of a vs. t. This turned out to be a good idea, as there were a number of

bugs in my code, and it took some time to track them all down and exterminate them. To

determine a, I used one of Friedmann’s equations (Liddle & Lyth, 2000; Peebles, 1993;

Dodelson, 2003):

H2 = (a

a)2 =

8πG

3

∑

ρi −k

a2+

Λ

3(3–1)

(a

a)2 =

8πG

3ρcritical,0(

Ωm

a3+

Ωk

a2+ ΩΛ) . (3–2)

The constants in front of the right hand side of Equation (3–2) can be turned into one

constant, the commonly used Hubble Constant, (H0)2 = (8πG/3)ρcrit,0 (Liddle & Lyth,

2000). So my equation for a is:

a = H0a

√

Ωm

a3+

Ωk

a2+ ΩΛ . (3–3)

30

Figure 3-1. The line shows the evolution of the scale factor with time.

Using this equation, I was able to make some graphs using my program. By running

it from a time of 10−12 seconds to a time of about 3 × 1018 seconds, I was able to create

a nice looking plot that showed me that everything was working properly. This plot, with

Ωm = 0.267, Ωk = 0, and ΩΛ = 0.733 and a Hubble constant of h = 0.704 can be seen

in Figure 3-1. This graph shows that a0 = 1 at around the current time of 4.32 × 1017

31

Figure 3-2. The line shows the evolution of the scale factor with time on a log-log plot.

seconds, or about 13.7 billion years. This is as it should be, so I checked the behavior

of a at early and late times to see what it looked like there. By plotting my data on a

log-log plot, which can be seen in Figure 3-2, I was able to examine the evolution of

the universe from a very early time to a time a little later than the current date. The

slope at early times on this log-log plot, from about one second to a time of around 1013

32

seconds, was 1/2. This makes sense, as a ∝ t1/2 for early times, when the universe is

radiation-dominated. After this point, until around 2017 seconds (which is getting very

close to the present day), the slope was 2/3, which meant that a ∝ t2/3–as it should be

when the universe is matter-dominated. All that was left was to make sure the behavior

at late times was correct, but that was slightly more complicated. I know that the graph

should grow faster and faster as a increases past one, and that behaviour can be seen

in my graph. To gain a little more information on what happens to a as it grows larger

than one, I solved for a as a function of t at later times:

a = H0a

√

Ωm

a3+ ΩΛ . (3–4)

Of course, the first term disappears when a gets large, so this equation becomes:

∂a

∂t= H0a

√

ΩΛ (3–5)∫

∂a

a=

∫

H0

√

ΩΛ∂t (3–6)

log a = H0t√

ΩΛ . (3–7)

This means that if the data is graphed on a log-linear scale, there should be a constant

slope of H0

√ΩΛ = 1.95 × 10−18 s−1. Once I made this graph, I saw that the slope did

appear to be constant, which was heartening. Upon making a rough estimate of the

slope, I determined it to be around 1.5 × 10−18 s−1, which was close enough to convince

me that my equation and program were both working.

There was one more graph I wanted to make at this point. A graph of the Hubble

parameter, H, as a function of the scale factor, a, seemed to be a good idea, since I

already had the necessary equations. To make this graph, I started with Equation (3–3)

and rearranged it a bit, then added a term for the radiation density parameter, to end up

with Equation (3–26). The Ωk drops out, as I assumed the universe was flat. Using this

equation, I created Figure 3-3, which shows a log-log plot of H as a function of a. To

create this graph, I used values of Ωr = 0.00002, Ωm = 0.2792, ΩΛ = 0.721, and H0 = 70.4

33

Figure 3-3. Evolution of the Hubble parameter with scale factor on a log-log plot.

34

(km/s)/Mpc. Since this is a log-log graph, it appears to have one slope at early times,

when the universe is radiation-dominated (of −2, since the radiation term dominates the

others), another at middle times when the universe is matter-dominated (of −3/2, since

the matter term dominates), and the Hubble parameter seems to level out, decreasing

slowly, around the present time. As can be seen on the graph, the crossover from

radiation to matter domination occurs around a = 0.0001. Once I was convinced that my

integration routine was working, I was able to move on and utilize it on something more

interesting.

3.2 Rough Analytical Calculation of Recombination

Before getting into long, detailed calculations of recombination that take into

account everything that can possibly happen, it is important to understand and be

able to produce simple results from a rough calculation of recombination, taking into

account only the most important factors. The shapes of these graphs are well known,

and can be found in several places (Liddle & Lyth, 2000; Peebles, 1968; Seager et al.,

2000; Ma & Bertschinger, 1995). These graphs should start with very small density

perturbations at very early times, and as the scale factor increases, the perturbations

should grow linearly. The cold dark matter starts increasing very early, while the

hydrogen and helium perturbations can’t start to grow until they recombine, leading

to different starting times for both. The first step towards doing this is calculating the

density perturbations, δ, of cold dark matter alone.

Assuming that δ ≪ 1 so that the equations are linear, it is a fairly simple matter to

find the equations that need to be numerically integrated. Starting with the continuity

equation, Equation (1–14):

δ +1

a∇ · v = 0 , (3–8)

35

and with ∇ · v = θ, we have:

δ = −1

aθ . (3–9)

This equation describes the evolution of δ with time; now I need a similar one for θ. To

get it, I can start with F = ma, (Liddle & Lyth, 2000; Peebles, 1993):

1

a

∂

∂t(av) = −1

a∇φ (3–10)

av + a∂v

∂t= −∇φ . (3–11)

If I take the divergence of both sides, I find:

a(∇ · v) + a∂

∂t(∇ · v) = −∇2φ (3–12)

aθ + a∂

∂tθ = −∇2φ . (3–13)

Poisson’s Equation, Equation (1–13), tells us that ∇2φ = 4πGa2ρδ, so:

θ = − a

aθ − 4πGaρδ , (3–14)

and the equation for the evolution of θ with time is:

θ = −Hθ − 4πGaρδ . (3–15)

Now I have equations for ∂δ/∂t and ∂θ/∂t, but it is more convenient to use a as the

independent variable, so I need equations for ∂δ/∂a and ∂θ/∂a. This should be easy

enough; all I need to do is use the chain rule:

∂f

∂t=

∂f

∂a

∂a

∂t. (3–16)

I know that H = a/a, so:

f =∂f

∂a(Ha) (3–17)

f ′ =f

Ha, (3–18)

36

where the prime denotes a derivative with respect to a. With this, I finally have the two

equations I need to numerically integrate δ:

∂δ

∂a= − θ

Ha2(3–19)

∂θ

∂a= −Hθ + 4πGaρδ

Ha. (3–20)

Using these equations and my integration program from Section 3.1, I did a number

of runs of recombination. To start with, I only used dark matter. Before adding in the

more complicated substances, I wanted to make sure the basic program was running

correctly. To do this, I assumed that the δ and θ values for everything else–hydrogen,

helium, radiation, and anything else I might want to add, like neutrinos–were 0, so

that I could replace ρ with just ρc. With these assumptions, I ran my program multiple

times, but kept getting inaccurate results. I spent a lot of time tracking down bugs and

annihilating them, each time making my results a bit more accurate, but never quite

achieving the correct graphs I was looking for. In an effort to improve my results, I

decided to add Ωradiation to the Hubble factor, hoping the added accuracy would fix some

of the problems I was having (until this point, I was only treating the density parameters,

Ω, of matter and dark energy in my Hubble factor). I know that (Liddle & Lyth, 2000):

ρr = gπ2

30

(kT)3

(~c)3

kT

c2. (3–21)

Since I am using a as an independent variable, rather than T, I needed to change this

equation slightly. I can do this because I know that at a = 1, a = T0/T, where T0 is the

current background temperature of the cosmic microwave background. So:

ρr = gπ2

30

(kT0)3

(~ca)3

kT0

ac2. (3–22)

37

Finally, I can use this equation to figure out what Ωrad will be. I know the critical density

at today’s time, and I can figure out what ρr is at the present time by setting a = 1, so:

Ωr =ρr,0

ρcr,0

(3–23)

Ωr = (gπ2

30

(kT0)3

(~ca)3

kT0

ac2)/(

3H20

8πG) (3–24)

Ωr =8π3Gg(kT0)

4

90H20~

3c5, (3–25)

where g = 3.3626439, since it includes both photons and neutrinos in its calculation. With

this new Ω, my new Hubble factor is:

H = H0

√

Ωr

a4+

Ωm

a3+ ΩΛ . (3–26)

Of course, as so often happens with these things, the lack of Ωr was not the

problem. It made things a bit more accurate, and was certainly a good thing to add to

my equations, but it did not fix the problems I was having. So I went over my code again,

even more carefully, and eventually figured out the problem–which was in the coding.

Of course, it had nothing at all to do with the physics, and was purely a programming

error. After treating this problem, I got beautiful graphs that looked just like I wanted

them to, which was quite satisfying. A representative graph can be seen in Figure 3-4.

This graph has ΩH = 0.0334, ΩHe = 0.01056, Ωc = 0.223, and ΩΛ = 0.733, and has an

initial δc = 1 and an initial θc = 0, as well as H0 = 70.4 km/Mpc/s and T0 = 2.725 K.

To be sure that my program was giving me correct values, I really had to see the slope

of the graph my program produced. To that end, I made another graph of the slope of

the perturbations of cold dark matter as a function of the log of the scale factor, which

can be seen in Figure 3-5. As expected, the slope starts off at zero, then rapidly rises

until it gets to almost one (about 0.9 in my graph) during the matter-dominated phase

of the universe’s evolution, and then finally drops rapidly as it approaches the present

time, due to the increasing domination of dark energy. It appears that this undertaking

38

Figure 3-4. This graph shows the evolution of cold dark matter density perturbations asthe scale factor increases.

39

Figure 3-5. This graph shows the slope of the evolution of the cold dark matter densityperturbations graph as the scale factor increases.

40

was a success; the next step was to add matter to my equations, and to do that, I figured

the best place to start would be with the code from COSMICS. It was my hope that I could

extract the equations pertaining to the essential elements of recombination while leaving

behind the parts of the code that would lead to small changes.

3.3 Rough Calculation of Recombination Starting From COSMICS Code

The COSMICS code gives me equations that are a bit more accurate than the ones I

was using in the last section. The creators of the COSMICS code started from scratch (i.e.,

the Einstein Equations) and worked out exactly what the resulting perturbation evolution

equations would be (Bertschinger, 1995; Ma & Bertschinger, 1995; Bertschinger, 2001).

The equations they found for the evolution of the density perturbations for cold dark

matter are:

δ = −θ + 3φ (3–27)

θ = − a

aθ + k2φ . (3–28)

Of course, since my program uses a as the independent variable, rather than conformal

time τ , as their program uses, I will have to once again divide both these equations by

(a/a)a to get the proper results.

Clearly, these equations are a bit more complicated than the ones we were working

with. What are the differences? How are these equations calculated? At the risk of

going into too much detail, I will describe exactly how they work. First of all, there is the

constant, grhom, which has a value of 3.3379× 10−11H20 Mpc−2 (assuming that the units

of H20 have already been added to the end of the number; this is where the units come

from). This comes from:

grhom =8πGρ0

c2(3–29)

=8πG

c2

3H20

8πGΩ(3–30)

=3H2

0

c2. (3–31)

41

If we take the units out of the H20 and the speed of light, we get:

grhom =3H2

0

c2

km2 s2

Mpc2 s2 m2. (3–32)

By cancelling the seconds and adding a conversion factor for the meters, we can leave

just the units we want, so:

grhom =3H2

010002

c2Mpc−2 (3–33)

= 3.3379× 10−11H20 Mpc−2 . (3–34)

This constant is the one used when finding the Hubble variable in conformal time:

grho =(grhom)(Ωc + ΩH + ΩHe)

a+ (grhom)ΩΛa2 , (3–35)

and so,

a

a=

√

grho

3. (3–36)

This is not enough to find φ, however, since it depends on the density perturbations and

how they are changing. So to get the rest of the variables we will need, we have to have

values for (these are variables from the code):

dgrho = (grhom)Ωcδc

a(3–37)

dgtheta = (grhom)Ωcθc

a. (3–38)

With these values, we can find φ, which is (with the wave number ”k” in units of Mpc−1):

φ = − 1

k2(dgrho +

3(a/a)(dgtheta)

k2) . (3–39)

Finally, we have φ, which is:

φ = − a

aφ +

dgtheta

2k2. (3–40)

42

These equations all seem right, as far as they go (they could definitely be more

accurate, since I am currently only including Ωc in dgrho and dgtheta), and so I made

graphs with them, at k = 0.01, 0.1, 1, and 10 Mpc−1. As can be easily ascertained,

something is off, at least with the smallest values of k. This is particularly frustrating

as the smallest values of k should be the easiest thing to compute. Although the graph

looks okay for later times, the early times–when perturbations increase by a factor of

1026–are clearly wrong. Figures 3-6 to 3-13 show each graph, followed by its slope. They

all have ΩH = 0.0334, ΩHe = 0.01056, Ωc = 0.223, and ΩΛ = 0.733, with an initial δc = 1

and an initial θc = 0, as well as H0 = 70.4 km/Mpc/s and T0 = 2.725 K.

As can be seen, the perturbation graphs for k = 10, 1, and even 0.1 are pretty close

to what would be expected, but the graph for k = 0.01, which can be seen in Figure

3-6, is way off from what it should be. I spent a lot of time trying to figure out where my

equations went wrong, or where some small, stupid error in my code was amplified into

a horrendous error causing the rapid growth of perturbations at small times for k = 0.01,

but despite a lot of testing and rewriting, I could not make the problem go away. I did

find several small errors that caused almost imperceptible problems, but they did not

help much with my k = 0.01 graph. In the end, I gave up on trying to solve this problem.

There are much better codes out there–COSMICS, for instance–which I know work, and

which model the growth of density perturbations, which is what I was interested in

studying. Spending more time on this code, which at the very least has some serious

problems, did not seem like a productive use of my time. Still, it was worth the effort to

try–it helped me understand recombination era physics better, and allowed me to get

familiarized with how the COSMICS code worked in more detail.

43

Figure 3-6. This graph shows the evolution of the cold dark matter density perturbationsas the scale factor increases, with k = 0.01.

44

Figure 3-7. This graph shows the slope of the evolution of the cold dark matter densityperturbations as the scale factor increases, with k = 0.01.

45

Figure 3-8. This graph shows the evolution of the cold dark matter density perturbationsas the scale factor increases, with k = 0.1.

46

Figure 3-9. This graph shows the slope of the evolution of the cold dark matter densityperturbations as the scale factor increases, with k = 0.1.

47

Figure 3-10. This graph shows the evolution of the cold dark matter densityperturbations as the scale factor increases, with k = 1.

48

Figure 3-11. This graph shows the slope of the evolution of the cold dark matter densityperturbations as the scale factor increases, with k = 1.

49

Figure 3-12. This graph shows the evolution of the cold dark matter densityperturbations as the scale factor increases, with k = 10.

50

Figure 3-13. This graph shows the slope of the evolution of the cold dark matter densityperturbations as the scale factor increases, with k = 10.

51

CHAPTER 4COSMICS

4.1 Modelling the Evolution of Density Perturbations

Once I had a solid understanding of the physics underlying the recombination

era, I obtained a copy of COSMICS, which was written by Ma and Bertschinger to

model the growth of density perturbations in the early universe (Bertschinger, 1995;

Ma & Bertschinger, 1995; Bertschinger, 2001). It was a very long, complicated code

written in FORTRAN 77, and it took into account many different factors, such as massless

and massive neutrinos, tightly coupled and uncoupled baryon and photon motion, the

temperature of the baryons, ionization fractions, and the density and pressure of the

neutrinos. It did all this, and was written in an old language that I was unfamiliar with.

Nevertheless, I was eager to find out what kind of results it would give me, so I read

through the information that came with it and started examining the code.

I knew that before decoupling, the electrons scatter off of the photons, due to the

intense heat and density, and therefore the electrons are unable to go too far before

interacting with something (Peebles, 1993; Liddle & Lyth, 2000; Dodelson, 2003). At

the point of decoupling, it is likely that there was about 74 percent hydrogen ions and

24 percent helium ions–this is an appreciable amount of helium. This enrichment of

helium will likely effect the decoupling and deionization process, since helium ionizes at

a higher energy than hydrogen, which allows the helium ions to recombine sooner than

the hydrogen. It was this problem that I was hoping COSMICS might eventually shed some

light on for me. Using the equation for density perturbations, I would be able to find out

what effect neutral helium would have on the early universe (Liddle & Lyth, 2000):

∂δj

∂t+ 2

a

aδj − 4πG

∑

i

ρiδi = 0 , (4–1)

52

where∑

i ρi comes from the densities of all the different kinds of matter and energy in

the universe:

ρ = ρDM + ρH + ρHe + ρrad + ρΛ . (4–2)

This physics was all included in COSMICS, so I started plugging in values and examining

my outputs.

The first problem I noticed was that COSMICS took the initial values and evolved

the universe to a time chosen by the user starting from those initial conditions. The

output it gave was the current structure of the universe at the chosen a value. That was

fine, but I was not interested in any one particular time–I wanted to know what was

happening during the whole recombination era. I knew that information had to be in

there somewhere, I just had to go into the inner workings of the program and figure out

how to make it do what I needed it to. Of course, I could just run the program repeatedly

with slightly different ending times, but that would be incredibly inefficient and would

take much, much longer to get usable data from. Instead, I chose to undertake the more

difficult, but ultimately more rewarding task, of making the program give me usable data.

After some trial and error, I understood how COSMICS worked well enough to change

it to my specifications. By inserting my own code and altering the existing program

judiciously, I was able to make the program print out as many time steps as I required,

between any starting and any ending time. Needless to say, this pleased me a good

deal, and I experimented with some short, simple runs to iron out the remaining kinks.

No sooner had I got this modified program running, though, than I realized I had

another problem. The output I was getting could not be easily graphed, because it

showed the evolution of many different wavenumbers, all jumbled together. I was able

to graph the first few easy test runs because they only had five or ten points, so I just

cut and pasted the wavenumbers I needed into a separate file, but I knew this would

never work for really long runs. Lacking any other options, I went back into the code and

53

added more changes in an effort to fix this newest problem. In the end, I succeeded,

and persuaded the program to do what I needed it to. My modified version of COSMICS

now picked out whatever wavenumber I chose and followed the evolution of the universe

at that wavenumber from any starting time I chose to any ending time I chose. After

some more testing, things seemed to be working fine, and so I moved on to some real

runs.

I started off with large time periods and only a few time steps, until I was sure

that I understood what I was getting and that it was accurate. The baryons were all

evaluated together, so they just gave me one line, but I was able to find the evolution of

the dark matter density perturbations in addition to the baryonic density perturbations.

As I made bigger and bigger runs, with more and more data points, various problems

arose. However, they were all programming problems with FORTRAN 77, not problems

with the physics model in COSMICS, so I was able to iron out all these difficulties without

major changes to the program itself. One of the later runs of the program gave me

the following graphs, which are quite interesting. Figure 4-1 shows the evolution of

both particle species over a long period of time. Figure 4-2 takes a look at a smaller

period of time in the interest of taking more data points in the interesting areas. In

both figures, the densities of baryonic matter are 0.044, cold dark matter is 0.223, and

dark energy is 0.733. The Hubble constant is 70.4 (km/s)/Mpc, the temperature of the

cosmic microwave background (CMB) today is 2.725 K, the initial fraction of helium is

0.24, and the wavenumber for these plots is 0.1 Mpc−1. In both graphs, we see that the

baryonic matter oscillates around the recombination time–the solid lines and dots are

the actual values, whereas the open points and dotted lines are actually negative values

of density perturbations which had their signs flipped in order to graph them on a log-log

plot like this. These representative graphs show the main features of the evolution of

density perturbations of different species with time. Once I got COSMICS working properly,

I moved on to the next stage. Seeing how baryons evolve in time is quite interesting, but

54

0.0001 0.001 0.01 0.1

10

100

1000

a

Figure 4-1. The smooth, solid line shows the evolution of the cold dark matter while thepoints show the evolution of the baryonic matter. The density perturbationsof each species are plotted versus a.

55

0.0001 0.001

10

100

a

Figure 4-2. The smooth, solid line shows the evolution of the cold dark matter while thepoints show the evolution of the baryonic matter. The density perturbationsof each species are plotted versus a.

56

my focus is on helium recombination. Somehow, I had to be able to track it separately

from hydrogen.

4.2 Addition of Helium

Delving into the depths of COSMICS, I found certain things that made me think I might

be able to split the baryons into hydrogen and helium, giving me more data on what

was happening during recombination. The way COSMICS was initially set up treated all

baryons the same. These baryons were affected by the physics of recombination and

treated as one substance. There were a couple parts of the code that made it clear that

COSMICS should be able to distinguish between hydrogen and helium, and treat each

differently during recombination, but the way COSMICS was coded, it did not allow the

user to do anything specific with the different baryons, or keep track of them separately

(Siegel & Fry, 2006).

While running, COSMICS takes everything that needs to be tracked and puts it into

one very large array, y. The first thing I tried to do was add another array to keep track

of the helium results, yHel, and I got a good way through the changes I wanted to make

before I realized that it would ultimately be futile. The things I was trying to keep track of

in yHel (such as the perturbation density and velocity at which the perturbation density

was changing at each point at each time, as well as the other substances, like cold dark

matter and neutrinos) depended on the elements of another array, and the only way to

take a time step and iterate my arrays was to send my array to another program, dverk,

which evaluated the derivatives. Since dverk could only accept one array at a time, this

method was a dead end.

The next thing I tried was to add two elements to the array that contained all the

other values the program solved for, y. This method was a lot more complicated, since

I had to change every part of the program that relied on this array (and basically the

whole program relied on this array), but after a good deal of work, I had made all the

changes I thought were necessary. After all of my changes, the program now had

57

0.0001 0.001 0.01 0.1 11

10

100

1000

This is lg(a)

Figure 4-3. The dotted line shows the evolution of the cold dark matter while the squarepoints and solid line show the evolution of the hydrogen and the x points andtheir dotted line show the evolution of the helium. The log of the densityperturbations of each species are plotted versus log(a).

two more things it kept track of: δ and θ for helium. One long run I made with this

new addition to the program can be seen in Figure 4-3. In this figure, the density of

hydrogen is 0.03344, helium is 0.01056, cold dark matter is 0.223, and dark energy is

0.733. The Hubble constant is 70.4 (km/s)/Mpc, the temperature of the CMB today is

58

2.725 K, the initial fraction of helium is 0.24, and the wavenumber for this plot is 0.1

Mpc−1. Once more, we see that the baryonic matter oscillates around the recombination

time, and although the helium perturbations seem to start out lower than the hydrogen

perturbations, they cross over and become larger after the oscillations end, eventually

growing faster than the dark matter and hydrogen density perturbations. This doesn’t

seem right, as helium should recombine before hydrogen, which would lead to it having

density perturbations that would grow faster than hydrogen. There’s the added problem

that all three types of matter should end up at the same value, since they are all matter.

The fact that somehow the helium perturbations increase above the hydrogen and cold

dark matter perturbations is a clear indication that in trying to expand the capabilities

of COSMICS, I instead managed to break it. I was dissatisfied with these results, and

this was one of the better graphs I got. I came to the conclusion that this was another

dead end, and lacking any other options, decided to move on to another program which

promised me new, better results–GADGET-2.

59

CHAPTER 5INTRODUCTION TO GADGET-2

GADGET-2 (which stands for GAlaxies with Dark matter and Gas intEracT) is an

extremely powerful piece of software written by Volker Springel, and designed to

simulate particle interactions on cosmological scales (Springel, 2005a). Many different

programs have been created to perform these types of simulations, and they all have

their strong and weak points (Heitmann et al., 2008; Tasker et al., 2008; O’Shea et al.,

2004). Dr. Fry recommended I try GADGET, and so I spent some time learning it. It

seemed to do everything I needed it to, and in the end, I decided GADGET would work

quite well for my purposes. GADGET can model the universe using Newtonian dynamics

or cosmological integrations with comoving coordinates, with or without periodic

boundary conditions. By changing the many available options, the user can set up a

variety of universes, with their own parameters governing particle motion. The settings

I personally found most useful were cosmological comoving coordinates in a periodic

box using the adaptive tree particle mesh grid. This allowed me to simulate structure

formation on a cosmological scale, which is exactly what I was trying to do.

There is no question that GADGET is a complicated program, but it is not hard to

describe what it does. General parameters (like the periodic boundary conditions which

I employed) are set in the makefile, after which the program is compiled. (This means

that it must be recompiled whenever you want to change the parameters.) Once the

program for the galaxy that is required is created, it needs two things to run properly:

• A parameter file, which prepares the program to accept the initial conditions.• An initial conditions file, which distinguishes one run from another.

The parameter file tells GADGET where to read the input file from, where to write the

output to, when to start the simulation, how often to take snapshots (which are the only

way of recording the progress of the program), the cosmological parameters (like the

Hubble Constant and the density values, such as ΩCDM), and various other options

that control how the program runs. The initial conditions file is in a binary format and

60

is rich with information about the state of the galaxy at the time the simulation starts.

It includes things such as the types of particles involved, their masses, positions, and

velocities, and other factors that might come in handy when tracking the evolution of

many particles.

5.1 Setting Up A Run

In order to make a run of GADGET , as was pointed out above, three things need

to be changed. The program itself is never altered–by putting all the changes in

separate files, usability and portability are greatly increased. GADGET is made up of

many files–around thirty–and it would be very difficult to make the proper alterations in

multiple files. The first file that needs to be changed is the makefile. There are many

features that can be changed in here, but most are for special cases and are not useful

for my purposes. A few are worth setting, though. This is where the periodic boundary

conditions are turned on, which I always used, as well as the zoom and high resolution

settings, which I did not need to turn on for my purposes.

The makefile is also the place where you turn on the TreePM option, which was

used for cosmological integrations. This is a hybrid force calculation that uses two

different methods to determine the strength of the force acting on an individual particle,

depending on whether said particle is at short range or long range. Short range forces

are found using a tree method. Finding the forces between every pair of particles gets

prohibitively computationally expensive rather quickly–luckily, it is also unnecessary.

Only very small errors are introduced by treating clusters of particles that are (relatively)

far away as one large particle–the tree routine is how these distances and replacement

particles are chosen. When trying to find the force on a particular particle, the program

walks up the tree, finding the nearest neighbors and treating them individually for the

closest particles, and as increasingly larger clusters of particles the further away from

the target particle the program goes in the tree (Hernquist, 1987; Hockney & Eastwood,

1988; Kravtsov et al., 1997).

61

For longer range forces, GADGET relies more heavily upon an adaptive particle mesh

routine. This is easier and faster than walking the tree, as it is less computationally

expensive. Space is broken up into a grid, and the particles contribute density to the

vertices nearest themselves. Once this density grid is created, the gravitational potential

can be found at any point from this density distribution using the Poisson Equation:

∇2Φ = 4πGρ . (5–1)

By going to the frequency domain by using a Fourier transform, Poisson’s Equation

becomes quite easy to solve:

Φ = 4πGρ

k2. (5–2)

In this notation, the hats mean that the Fourier transform has been taken of the

gravitational potential Φ and the particle mesh density ρ. The k represents the comoving

wavenumber. To find the gravitational field from Equation (5–2), one only needs to

multiply by k and then take the inverse Fourier transform. This process is much quicker

than other alternatives, like walking the tree or calculating the force between each

particle pair directly (Hockney & Eastwood, 1988; Villumsen, 1989).

These are the most important options available in the makefile. As can be seen,

essentially, it contains the settings that tell the program what kind of universe to use.

The parameter file is utilized to tell the program specific information about the run that

is taking place in the universe established by the makefile. It contains many options,

but most of them do not need to be fiddled with, as the defaults handle a regular

cosmological universe quite well. To model an expanding universe with cosmological

parameters, comoving integration has to be turned on. This tells the program to use

comoving coordinates, and rather than regular time (which is what the program uses

for plain Newtonian runs), the expansion factor a is used. Specific values must also

be chosen for things like the start time (or, in my case, the expansion factor a, since

62

comoving integration was used for my runs), the expansion factor corresponding to

the time at which the simulation ends, and the amount of time that passes between

each snapshot file that the program writes. Since these snapshots are the output of the

program that the user examines to see how the universe is evolving, it is important to

make sure enough snapshots are being output to give an accurate view of the simulation

over time. Other parameters set here include the size of the box the simulation takes

place in (in GADGET’s units, an internal length unit of 1.0 equals 1.0 kiloparsecs/h, and I

set my box to 50000, or 50 megaparsecs/h), the Hubble Constant (which I took to be 70.4

(km/s)/Mpc), the matter density Ω0, including all matter (baryonic and cold dark matter),

and which I set to 0.3, and the vacuum energy density, ΩΛ, which I set to 0.7.

The final piece of this puzzle involves the initial conditions file. This file is written in

a specific binary format, and if all the data it contains is not in a very specific order, the

program will have no idea what the universe is supposed to look like at the beginning

of the simulation. In fact, if the initial conditions file is wrong, GADGET will probably just

crash, and it can be very difficult to find the problem when it is buried deep inside a

binary file that can not be opened and examined for problems in Notepad or VI. On the

bright side, the initial conditions file that GADGET uses is rich with information, and keeps

track of pretty much everything the user could want to know about. The corresponding

drawback to this wealth of variables is that it makes the initial conditions file rather

unintuitive and learning how it is structured and where it places all the information

needed to conduct an analysis, or, even worse, how to change it to create new initial

conditions can be a daunting task indeed. Fortunately, GADGET’s author anticipated this

problem, and tried to explain the structure of the initial conditions file to the user as best

he could.

This explanation was done in two major ways. First, there is a brief chapter in the

user’s guide for GADGET that goes over the initial conditions file, what it contains, how it

is ordered, and how to access specific values of interest (Springel, 2005b). This is fine

63

as far as it goes, but frankly it is a bit TOO brief, and could stand to be extended several

more pages, allowing room for a more in-depth explanation of how the initial conditions

files work. The author of the program realized that if he was to explain everything that

needed to be known about the initial conditions file in words, it would be hard to follow

and be an inefficient use of space. Rather than write a very long, detailed description

in the user’s guide, he included a C program designed to read the initial conditions file.

By studying this program, it is possible to figure out all that is needed to access any part

of the initial conditions file one might need to–and once you can access something, it

is usually possible to figure out a way to write over any part you may need to, or even

create your own set of initial conditions, assuming that is the way you want to go. With

these two files working in tandem, it is possible to figure out how the initial conditions

files work, which is certainly necessary to use GADGET effectively.

The author of GADGET does supply one sample initial conditions file for the case

in which I am interested in–a cosmological, expanding universe. This file became my

template for constructing future initial conditions files, so I spent a lot of time getting to

know and understand it. Once you get used to it, the structure of these files does kind of

make sense. The data is broken up into blocks of similar information, so once you find

the block you are looking for, the whole thing can be read into whatever program you

are using to analyze the data and worked with in whatever way seems best. The first

block is the header information, containing information that is true for all the particles

in the simulation. This block tells the user things like how many particles of each type

(baryons, cold dark matter, etc.–it can keep track of up to 6 different types of particles)

are in the simulation, the time that the snapshot belonging to that header block was

taken, and constants like the Hubble Constant, the box size, or the matter and energy

densities. The sample initial conditions file included 32768 dark matter particles and

32768 gas particles, used a box size of 50 megaparsecs/h, and an initial time of z = 10.

64

The next block of information in the initial conditions file lists the positions of all the

particles. This and the future blocks of information split the particles into groups, one

for each particle type being used, so that the order of, say, the gas particles can change

in the position block, but all the gas particles stay together. Of course, since this is a

three dimensional simulation, the x, y, and z coordinates of each particle are recorded

in the position block together. The next block contains the velocity information for each

particle, once again in three dimensions, and once again all particles of the same type

stay together. When using cosmological integration, these are calculated to be in units

of km/sec, and to get peculiar velocities from these, they just need to be multiplied by√

a. However, as has been pointed out, these particles are free to move around in their

data block in order to allow parallel processing by the computer and make for quicker

file input and output. This means that the only way to track a particle is to determine its

unique particle identification number–which all particles have–and use this identification

number to see where a specific particle is in any given simulation. The identification

numbers for all the particles are given in the next block of data. These are the most

important blocks, and they must be included in every initial conditions file. There are

several more optional blocks, however, which may be useful depending on the type of

simulation being run. For instance, the next block contains the masses of each particle,

and allows the user to input different masses for each particle if it becomes necessary.

Figure 5-1 shows an x-y overhead view of the positions of the particles in this

sample initial conditions file. The red particles are the gas particles and the black

particles are the cold dark matter particles. As can be seen, the initial conditions are

based on a grid, with the clumps of gas at the centers of the squares outlined by the

cold dark matter clumps. Clearly, the particles are not all clustered exactly on top of the

grid points, but are rather spread out around each grid point. These grid points are taken

more as approximate centers, around which the accompanying particles are perturbed.

65

00

Figure 5-1. Initial conditions supplied with GADGET. 50 Mpc/h by 50 Mpc/h grid with blackparticles dark matter and red particles gas.

66

Once all of these things (the makefile, the parameter file, and the initial conditions file)

are properly adjusted and ready to go, GADGET can finally be run.

5.2 Running GADGET-2

Because GADGET is such a complex and complicated program, it is difficult to mess

about with it successfully. Unless you have a lot of experience with it already, trying to

alter the code itself is a losing proposition. GADGET is a massively parallel code, written

using the Message Passing Interface (MPI), so to make alterations to the code, one

must also have a sound knowledge of MPI. And although the use of MPI does make the

code more complicated, the trade-off in complexity is definitely worth it, as the ability

to run GADGET in parallel can save the user a good deal of time when making especially

long runs.

I have already mentioned the large number of source files from which GADGET is

compiled. It would probably be a paper in and of itself to try to explain exactly how

GADGET takes the initial conditions that the user provides and uses this data to simulate

the entire universe, from an early time to a later time. The general terms mentioned

so far–that is, that GADGET uses a TreePM algorithm to model gravitational forces on the

different types of particles in an expanding cosmological universe–should be sufficient to

explain the basics of the physics that GADGET uses to make its simulations. As I did not

alter anything in the source files, if more information is desired, I recommend looking up

the code paper on GADGET-2 and the user guide that explains how to operate it (Springel,

2005a,b). More information can be found in the original code paper for GADGET-1, the

original incarnation of GADGET (Springel et al., 2001). These should more than satisfy

any curiosity one could have about GADGET-2 and its usage.

5.3 Output

Instead, I would like to jump ahead to the output of GADGET-2. I mentioned briefly

that GADGET outputs snapshot files containing information about the universe at times

specified by the user in the parameter file. These files are how the user examines the

67

universe he has created from nothing. The author of GADGET supplies a short program to

demonstrate an example of how to take a snapshot file and display it graphically, which

can definitely help with analysis. Using this program is not strictly necessary, however,

for the simple reason that the snapshot files are formatted in the same way that the

initial conditions files are structured. This means that if one has spent time learning

the initial conditions files–which is fairly likely, since to make your own runs of GADGET,

you have to be able to create new initial conditions–then everything that is needed to

manipulate the information in the snapshot files is already known. If the user wants to

use a different graphing program than the one in which the example program is written,

it should not be a problem with knowledge of how the snapshot files are structured.

Similarly, the user is not limited to the type of graph the supplied example program can

produce as long as they know where to find the information they want to analyze in the

snapshot files.

Still, despite the wealth of information available in the snapshot files, one of the

most useful things is perhaps the x, y, and z coordinates of each particle at each time

a snapshot file is taken. In Figure 5-2, I created another graph with the z dimension

flattened into a plane. This figure shows the universe at the present time (with a redshift

of 0, or an expansion factor of a = 1), and shows some important features. There is

a great deal of difference between this figure and the figure demonstrating the initial

conditions distribution of particles at a redshift of 10. Rather than a fairly homogeneous

distribution, large clumps of gas particles are scattered here and there about the

grid, surrounded by large voids empty of matter. This is what would be expected, as

observations of the universe today obviously show galaxies separated by large voids, so

I can be fairly confident that GADGET does what it is advertised to do–and now that I know

how it works, I can use it to run simulations of my own, with initial conditions of interest

to me.

68

Figure 5-2. Distribution of gas particles at a redshift of 0 in an x-y plane. Once again, theunits are Mpc/h.

69

CHAPTER 6ANALYSIS OF HELIUM RECOMBINATION USING GADGET-2

6.1 Creating an Appropriate Set of Initial Conditions

To examine the effects of helium recombination, I obviously needed to create initial

conditions that had helium in them. As mentioned in Chapter 5, the author supplied a set

of initial conditions with the GADGET-2 code, and so it seemed the easiest way to create

my own initial conditions would be to modify this existing file.

The first change that had to be made was changing the starting time of my

initial conditions. I needed to be able to choose different starting times; even if I had

decided to make all my runs from the same starting time, the supplied initial conditions

corresponded to a starting redshift of z = 10, which was far too late in the structure

formation process to be of any use for my purposes. I was able to make this alteration

to the initial conditions by scaling them with the expansion factor. A more detailed

treatment is given in the appendix, but to summarize, the fluctuations depend in a

relatively simple way on the expansion factor as:

δ ∝ a (6–1)

v ∝√

a . (6–2)

With these equations, it is possible to take the given set of initial conditions and scale

them back to whatever expansion factor one is interested in.

The next step is to separate out the helium atoms from the hydrogen atoms,

rather than treat all the baryonic matter as one category. Of course, one of the main

differences between hydrogen and helium during the recombination era is their

relative velocities. Since helium will start to recombine before hydrogen, at a redshift

of about z = 2500, helium atoms will be moving faster than hydrogen atoms during the

recombination era, which do not start recombining until closer to z = 1300 (Seager et al.,

2000; Switzer & Hirata, 2008a; Rubino-Martın et al., 2008; Dubrovich & Grachev, 2005;

70

Wong & Scott, 2007). Before each element starts recombining, its fluctuations are

stationary–only after each element starts to recombine can these fluctuations start to

change. So when it comes to changing the initial conditions file, the velocities of the

cold dark matter particles can be left alone, as they are free to change from a much

earlier time than the baryon fluctuations, and the hydrogen and helium velocities can be

scaled as fractions of the cold dark matter particle velocities. Starting from the continuity

equation (Liddle & Lyth, 2000)1 :

dρ

dt= −3Hρ (6–3)

H =1

3a∇ · v (6–4)

δ = −31

3a∇ · v (6–5)

δ +1

a∇ · v = 0 , (6–6)

and making the substitution:

Θ =1

a∇ · v , (6–7)

I am left with:

δ +a

a(1

a∇ · v) = 0 (6–8)

δ +a

aΘ = 0 (6–9)

Θ = − δa

a. (6–10)

In order to get Θ as a function of the expansion factor, I need to find out how δ depends

on the expansion factor. Fortunately, Equation (6–1) tells me that the density fluctuations

are proportional to the expansion factor, so if I take the derivative of this equation with

1 Conversations with Dr. Fry were also very helpful in making these calculations.

71

respect to time, I find:

δ ∝ a . (6–11)

Combining Equations (6–10) and (6–11), I get:

Θ ∝ − aa

a(6–12)

Θ ∝ −a . (6–13)

This means that I have three equations for Θ, one for each species I am keeping track

of. As has already been mentioned, hydrogen and helium atoms do not start moving

until after they start to recombine, so their velocities do not start to grow according to

this equation until after their decoupling times. Keeping this in mind, here are the three

equations for Θ that I have been looking for:

ΘCDM = −Ca (6–14)

ΘHe = 0, a < aHeDec (6–15)

ΘHe = −C(a− aHeDec), a ≥ aHeDec (6–16)

ΘH = 0, a < aHDec (6–17)

ΘH = −C(a− aHDec), a ≥ aHDec . (6–18)

By taking the ratios I am interested in, I can eliminate the constant C and find how

the velocities of hydrogen and helium relate to the velocities of the cold dark matter

particles. Taking the decoupling redshifts to be z = 2500 for helium and z = 1300 for

hydrogen gives values for the expansion factor of aHeDec = 0.00040 and aHDec = 0.00077,

so:

ΘHe

ΘCDM= 1− 0.00040

a(6–19)

ΘH

ΘCDM= 1− 0.00077

a. (6–20)

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By solving these two equations for whatever expansion factor I am interested in using as

a starting point, I can change the speed of the helium and hydrogen particles in order to

distinguish them from one another.

Obviously, it is important to keep track of which particles are helium and which

are hydrogen. Because of the way GADGET works, the specific order of the particles in

the initial conditions files gets mixed up as the program runs, so I am not able to just

call the first quarter of the gas particles helium and then treat the first quarter of the

gas particles as helium in the output files. What I did instead was randomly change

twenty-four percent of the baryonic matter to helium and leave the rest hydrogen. I kept

track of which particles I made helium through the use of a large array, which I wrote to

a file when I was done assigning helium particles. Since the ID numbers of the particles

in GADGET never change, this file uniquely identified which particles were helium. By

utilizing this file along with the output snapshots from GADGET, I was able to keep track of

which particles were hydrogen and which were helium when analyzing my data. So, by

manipulating the initial conditions file for GADGET in all of these ways, I was able to make

runs of GADGET that kept track of helium separately from hydrogen and then interpret my

results.

6.2 Evolution Starting at a Redshift of 1250

For this run, I used the supplied initial conditions file with the alterations outlined

above. The matter density parameter, Ω0, was set to 0.3, the density parameter for the

cosmological constant was set to ΩΛ = 0.7, and the dimensionless Hubble parameter

was h = 0.704. I set the start time to z = 1250, made half of the particles cold dark

matter and the other half baryonic matter, and of the baryonic matter, I made twenty-four

percent helium and left the rest as hydrogen. To make this differentiation, I followed the

procedure outlined above, and using Equations (6–19) and (6–20), I got velocity ratios

73

for z = 1250, or a = 0.0007994, of:

ΘHe

ΘCDM= 0.50 (6–21)

ΘH

ΘCDM= 0.037 . (6–22)

Using these initial conditions, I obtained several output snapshots at various

times. At a redshift of 10, the universe can be seen in Figure 6-1. Clusters of matter

can clearly be seen by this time. They are evident at earlier times as well–Figure 6-2

shows just the hydrogen and helium–which are of primary interest to me–at a redshift

of 91. Looking at different snapshots of my results is not the best way to analyze this

data, though. From these graphics, it is clear that the matter is clustering properly,

as it should in the real world, but the differences between the hydrogen and helium

are not particularly noticeable. To determine the effects that the earlier recombination

time of helium will have on the resulting structures, I have to examine the clustering

more mathematically. The snapshot files tell me where everything is at any point in

time, so by tracking a clump’s composition and structure as the redshift gets smaller

(or, equivalently, as the expansion factor increases), I can see how the helium evolves

relative to the hydrogen. To collect this statistical and mathematical data, I had to

write a new program which determined all the quantities I wanted to find. To use this

program, I first selected a clump of interest. I fed my program the approximate x, y, and

z coordinates of my clump, as well as the approximate radius of said clump, and then let

it loose.

The program that I designed to fulfill this function was fairly complicated. It took

the initial coordinates I supplied it with and found the centroid of the sphere of mass

dictated by the radius I also supplied. Of course, GADGET used a periodic box to model

the universe, so if I took a clump too close to an edge, the program had to correctly

account for the radius extending to the other side of the box without creating any errors

or giving wildly inaccurate centroid values. Since the centroids are proportional to the

74

00

Y

Figure 6-1. A y-z view of the universe at z = 10. The box has sides of 50 Mpc/h, and theblack particles are dark matter, the red particles are hydrogen, and the blueparticles are helium.

75

00

Y

Figure 6-2. A y-z view of the universe at z = 91. The box has sides of 50 Mpc/h, the redparticles are hydrogen, and the blue particles are helium.

76

summation of all the position values, if you have some very large values and some very

small values, your ”centroid” will be somewhere in the middle of the box, rather than at

the actual centroid. It is these kinds of unfortunate mistakes that took the most effort to

correct for and prevent in my program.

Once the program found the centroid using the initial values I supplied it with,

it checked to see how far away from the estimated position the calculated centroid

was located. If it was too far away (and ”too far away” is dictated in my program as a

certain fraction of the radius of the clump that I supplied it with), the program picked a

point fairly close to the calculated centroid as the new ”initial position” and calculated

the centroid again. It repeated this process until the calculated centroid was close

enough to the ”initial position,” at which point it had a clump to evaluate. My program

also found the standard deviations of the hydrogen and helium particles in each clump

that I was tracking, which was not too hard once I had the overhead in place to find the

centroids–the hardest part of finding these various values for each clump was clearly

delineating the clump, regardless of such things as whether it was split across the box

or not. After I knew what particles belonged to the clump, it was not too difficult to get

the centroid of the hydrogen and helium particles, as well as their standard deviations

and the number of each type of particle in the clump. Furthermore, since I knew how

massive each particle type was, once I knew how many particles of each type was in

the selected clump, I could easily figure out the total masses for all the particles of a

particular species in a clump.

In order to get a good overview of my data, I tracked the evolution of four different

clumps. This way, I could make sure no anomalous data or statistical outliers interfered

with my results. The data for each clump is collected in Tables 6-1, 6-2, 6-3, and 6-4.

There is a lot of data contained in these tables, and several different ways to analyze

it. First of all, the motion of the centroids is obvious by just looking at the locations

for each cluster as time passes. Obviously, they all move around, some substantially.

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Table 6-1. Data conveying the hydrogen and helium composition of ”Clump 1.”Particle Num. H Centroid He Centroid Stand. Dev.

z H He x (kpc) y (kpc) z (kpc) x (kpc) y (kpc) z (kpc) H (kpc) He (kpc)380 190 48 24762 20994 10819 24940 21115 10994 3232 3156300 185 46 24858 20316 10837 24666 20250 10792 2692 2592236 301 89 25096 19161 10882 25132 18872 10997 3043 3128186 186 50 25161 19734 11039 25156 19707 10987 1343 1407147 262 74 25624 18942 11101 25725 18855 11220 1233 1294116 356 101 26230 18132 11126 26197 18176 11138 935 89691 398 112 26858 18032 11439 26885 17994 11499 1043 104772 476 143 27538 18055 11684 27530 18007 11762 986 103756 560 178 28342 18234 11795 28310 18133 11893 1015 108444 612 189 29340 18383 11778 29374 18351 11869 1000 103035 875 262 30438 18721 11792 30447 18651 11814 914 95427 943 273 31360 18913 12222 31343 18864 12249 1153 118221 1026 303 31727 19125 12991 31752 19080 12990 1076 110817 1081 310 33019 19875 13681 33005 19850 13705 950 94513 1104 326 34253 20518 14658 34252 20543 14647 880 90510 975 281 36661 22577 15315 36699 22577 15330 697 692

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79



80



81

Since this motion is in three dimensions, a graph of their paths is a little complicated

to produce. The best way I found of doing this was to give three graphs, one for each

dimension. I also decided to just graph one cluster; when I graphed the entire 50 Mpc

box, the individual clusters did not move too much, and the individual motions of the

hydrogen and helium were almost impossible to see. However, because Clusters Two

and Four do end up joining at very late times (around z = 13 or so), I did manage to get

the tail end of Cluster Four’s motion in these graphs. The main focus of Figure 6-3 is

the motion of Cluster Two, however. The starting points of the centroids can be found by

looking at Table 6-2; for the x-y graph, they start in the middle, towards the bottom. In

the y-z graph, they start in the bottom left, and in the z-x graph, they start at the left. The

solid lines marked by plus signs indicate the motion of the helium, and the dotted lines

marked by the asterisks indicate the motion of the hydrogen. When Cluster Four enters

the graph at late times, the line types are the same, but the hydrogen is distinguished by

diamonds and the helium is distinguished by squares.

From these graphs, one thing is clear–as time passes, the two centroids do get

closer and closer together, as would be expected. Stronger conclusions are harder

to draw. Looking at my graphs, as well as the data for my other clusters, it appears

that for most times–excluding the very early times, where it is hard to see exactly what

is happening, and late times, where the centroids have basically joined–the helium

centroid may lead the hydrogen centroid a bit. I wish I could be a bit more certain, but

it is hard to say what is going on for sure. I would say that the helium lines are a bit

smoother and that the hydrogen motion is following their motion to a small degree, but

there do appear to be times where the hydrogen centroid seems to be on the inside of

the curves that the motion is making, which seems to imply that the helium is following

the hydrogen at these times. The problem may lie in the fact that it is just too hard to

put together these three graphs into a reasonable approximation of three dimensional

motion. I did try to make a three dimensional graph of the motion of Cluster One, and it

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Figure 6-3. Motion of Cluster Two in three dimensions. The dotted lines with asterisks indicate the hydrogen and the solidlines with plus signs indicate the helium.

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appears that the helium is on the inside of the curve their motion makes for most of their

evolution time, which would imply that the hydrogen is following the helium. This graph

can be seen in Figure 6-4, but it is kind of hard to tell exactly what is happening.

x axis

y axis

9000 10000 11000 12000 13000 14000 15000 16000

z axis

H ClusterHe Cluster

22000 24000 26000 28000 30000 32000 34000 36000 38000

17500 18000 18500 19000 19500 20000 20500 21000 21500 22000 22500 23000

z axis

Figure 6-4. This is a three dimensional plot of the motion of the centroids of Cluster One.The solid line is hydrogen and the dashed line is helium. The motion startsin the lower left, and moves to the right as time passes.

Something I can be a bit more definitive on is the separation between the two

centroids. There is no question that as time passes, the distance between the hydrogen

and helium centroids gets smaller. This makes sense–as time passes, the velocities of

all the baryonic matter tend to go to about the same amount. Since the velocities of the

clusters themselves will also tend to slow as they get larger, the centroids of both the

helium and hydrogen will go to the center of these clusters and then slow themselves.

In Figure 6-5, the separation between the helium and hydrogen centroids is shown for

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each cluster. For some the trend is more dramatic than for others, but the overall motion

Figure 6-5. This graph shows the separation of the helium and hydrogen centroids, inkiloparsecs, as time passes. The solid line with diamonds is Cluster One, thedotted line with circles is Cluster Two, the dashed line with asterisks isCluster Three, and the dash dotted line with plus signs is Cluster Four.

clearly shows the two centroids getting closer together as time passes. Despite starting

out at a variety of different displacements, all the clusters eventually decrease their

separations to around fifty kiloparsecs, which definitely shows that as time gets closer

to the present, the hydrogen and helium particles will intermix and have overlapping

centers of mass.

Another important statistical tool for analyzing the motion of these clusters is the

variance, and its square root, the standard deviation. The standard deviation can be

thought of as a measure of how closely clumped together the matter particles are, as it

85

is basically a measure of how fast the density falls off. The standard deviations for each

of my clusters is plotted in Figure 6-6. This graph is kind of busy, but the later times,

Figure 6-6. This graph shows the standard deviations of the hydrogen and heliumclusters as time passes in kiloparsecs. Each cluster is represented by adifferent line type–Cluster One is the solid line, Cluster Two is the dotted line,Cluster Three is the dashed line, and Cluster Four is the dash dotted line.The hydrogen is further demarcated by plus signs, while the helium isrepresented by asterisks.

where they all tend to be pretty close together (which makes them hard to tell apart),

are not as important. Clearly, the standard deviations all get much smaller at those late

times, which makes sense. As the time approaches the present, the clusters should be

well formed, and these certainly appear to be that. By the time the redshift approaches

ten, all of the standard deviations are around the 100 kpc range. In fact, they stay fairly

86

constant around that range from around a redshift of thirty or forty onwards. It is the

earlier behavior with which I am concerned. At early times, the standard deviations tend

to be fairly large, but start to trend downward rather sharply. As the helium particles

begin to move before the hydrogen particles, I would expect them to start to form

clusters sooner, which would therefore start to collapse sooner as well. For Clusters

Two and Four, this can be seen in a pronounced manner–for a large amount of the

time, the helium is consistently more tightly bound together than the hydrogen cluster

that is gathering around it. However, there is little difference between the standard

deviations for the hydrogen and helium for Cluster One, and the hydrogen standard

deviation is actually smaller than the helium standard deviation for much of Cluster

Three’s evolution. Since clusters are hard to clearly delineate, a certain amount of error

is introduced when trying to do this type of analysis on them–this may account for these

discrepancies. I do find it encouraging that two of the clusters I was tracking seem to

give the expected results.

A final measure of the evolution of these clusters lies in the ratio of helium to

hydrogen as a function of the cluster’s total mass. In the tables above, I listed the

number of each type of particle, hydrogen and helium, as time passed. Each particle

has a mass of 4.235 × 1010 M⊙/h, so both the ratio of helium to hydrogen and the total

mass for each cluster is easy to calculate:

mtotal = (nHe + nH)× 4.235 . (6–23)

The units are, as stated above, 1010 M⊙/h. This graph, for each of the clusters I tracked,

appears in Figure 6-7. It is known that about twenty-four percent of the baryonic matter

formed in the Big Bang is helium, with the rest being hydrogen (Peebles, 1993). If the

particles were all distributed completely homogeneously throughout the universe, they

would give a ratio of:0.24

0.76= 0.32 . (6–24)

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Figure 6-7. This graph shows the ratio of helium particles to hydrogen particles as afunction of the cluster’s mass in units of 1010 M⊙/h. The solid line withdiamonds is Cluster One, the dotted line with circles is Cluster Two, thedashed line with asterisks is Cluster Three, and the dash dotted line withplus signs is Cluster Four.

Significant deviations from this mass ratio are interesting, and are likely the result of

the earlier recombination time of helium. Figure 6-7 clearly shows large deviations from

this mean value. Because this graph is a function of total mass rather than redshift, it

is a bit harder to see how these clusters evolve in time. However, the data shows that

clusters get bigger as time passes (which, as described above, is what was expected),

so the right ends of the lines are the latest times, while the left ends of the lines are early

times. Looking at these graphs, it is clear that as time passes, the mass ratios for all the

clusters head towards the expected value of 0.32. In fact, the bigger clusters reach these

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values at late times and remain relatively constant, whereas the smaller clusters are not

able to reach these completely homogeneous ratios. Clusters Two and Three are highly

enriched in helium, whereas Clusters One and Four are deficient in helium.

6.3 Evolution Starting at a Redshift of 1000

The only difference between this run and the previous run was the initial starting

time. The matter density parameter, Ω0, was still set to 0.3, the density parameter for the

cosmological constant remained at ΩΛ = 0.7, and the dimensionless Hubble parameter

was h = 0.704. However, I did change the start time to z = 1000. I left half of the

particles as cold dark matter and the other half as baryonic matter, and of the baryonic

matter, I made twenty-four percent helium and left the rest as hydrogen. To make this

differentiation, I followed the procedure outlined above, and using Equations (6–19) and

(6–20), I got velocity ratios for z = 1000, or a = 0.0009990, of:

ΘHe

ΘCDM= 0.60 (6–25)

ΘH

ΘCDM= 0.23 . (6–26)

Using these initial conditions, I obtained several output snapshots at various times.

At a redshift of 50, the universe can be seen in Figure 6-8. I did not make this run last

as long, because as I mentioned a few times above, the late times are not as interesting

to me. It is mainly the early and intermediate times during which effects due to the

earlier recombination time of helium will be most noticeable. An earlier view of just the

hydrogen and helium for this run can be seen in Figure 6-9, which is taken at a redshift

of 166. Once again, I can see that the matter is clustering properly, but the differences

between the hydrogen and helium are not particularly noticeable. To get a better idea

of how the structures are evolving, I must once again turn to the program I designed for

that purpose.

In this case, I decided to track two clusters, since I was not expecting any really

drastic changes from the z = 1250 runs. The data for these clusters is collected in

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00

X

Figure 6-8. An x-y view of the universe at z = 50. The box has sides of 50 Mpc/h, andthe black particles are dark matter, the red particles are hydrogen, and theblue particles are helium.

90

00

X

Figure 6-9. An x-y view of the universe at z = 166. The box has sides of 50 Mpc/h, thered particles are hydrogen, and the blue particles are helium.

91

Tables 6-5 and 6-6. Using the data contained in these tables, I can follow the same

analysis that I described above. First, I will check the centroid motions. Once again, I

am just going to look at the motion of one cluster–Cluster Two, this time. This motion

can be seen in Figure 6-10. The starting points of the centroids can be found by looking

at Table 6-6; in all three cases, the motion starts in the upper right of the graphs. The

solid lines marked by plus signs indicate the motion of the helium, and the dashed lines

marked by the asterisks indicate the motion of the hydrogen.

In the z-x graph, it appears that the two centroids remain fairly close together

throughout the range of motion–not a lot can be gleaned from this graph. However,

looking at the other two graphs, it does appear that the helium leads the hydrogen a

little bit. In the x-y graph, especially towards the end, it appears that the helium does

not change too much, and the hydrogen centroid is slowing approaching the helium

centroid. This trend seems a bit more clear in the y-z graph–the features in the helium

graph seem to lead the same features in the hydrogen graph. One prominent feature

is the spike to the lower right at y = 17000 kpc–the hydrogen spike is clearly lagging

by about 500 kpc, and a similar trend of helium leading can be seen later in the graph,

although it is nowhere near this pronounced.

This takes me to the centroid separation. Once again, there is no question that the

two centroids are getting closer together with time, as would be expected. This graph

can be seen in Figure 6-11. Since I only carried out the analysis to about a redshift of

50, the clusters end up around 100 kiloparsecs apart, but they are definitely trending

downward, and would presumably get even closer as time approaches the present. It

is the early and intermediate times that are of interest for me, however, so this late time

behavior does not particularly concern me.

Once again, looking at the standard deviations should provide some interesting

data. The standard deviations for both of these clusters appears in Figure 6-12. My

results for these two clusters seem to agree with my expectations. For most of the

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z H He x (kpc) y (kpc) z (kpc) x (kpc) y (kpc) z (kpc) H (kpc) He (kpc)811 241 62 10998 20341 33906 11057 19984 33545 5003 4946666 301 79 1038 15860 30326 1382 15581 30319 5018 4918546 202 65 4338 13708 27587 4485 13494 27854 4148 4237448 246 76 3961 15069 28150 3772 14327 28489 4304 4332367 154 45 4300 14532 28372 4083 14155 28688 3373 3369301 167 53 2896 14588 28241 2669 14281 28679 3321 3318247 124 39 5922 14247 27482 6539 14184 27624 2677 2658203 198 57 4910 15002 27351 4974 14906 27513 3053 3014166 147 48 4189 15100 27555 4083 15121 27722 2293 2444136 173 52 2677 15605 27094 2643 15584 27218 2109 2048111 212 62 1927 16497 27070 1842 16366 27043 1570 152391 247 73 2135 16913 26725 2189 16980 26668 1179 109375 346 101 2631 17191 26176 2572 17254 26110 1195 111561 363 101 2534 17441 25783 2469 17513 25743 865 76450 420 120 2158 17512 25022 2124 17623 25022 1006 911

93


z H He x (kpc) y (kpc) z (kpc) x (kpc) y (kpc) z (kpc) H (kpc) He (kpc)811 347 102 35075 21188 19004 34826 20793 19200 5707 5731666 449 129 26506 19712 10844 26760 20042 11680 5607 5829546 426 115 24535 18177 9204 24220 17686 9298 5382 5642448 303 79 24961 17286 9246 25104 16778 9208 4546 4659367 223 56 24792 18212 8489 24918 17518 8496 3835 4010301 190 46 24792 17504 9198 25142 16933 9312 3334 3322247 125 37 24995 16109 9190 25024 15589 9194 2502 2473203 172 47 25103 16292 9739 25201 15681 9796 2630 2419166 126 39 25565 15614 10093 25428 15551 10125 1807 1712136 166 43 25635 16725 10154 25586 16308 10225 1678 1487111 318 89 25838 18188 10994 25823 18075 10923 1369 146491 371 104 26277 18116 11156 26220 18174 11115 984 90275 375 99 26746 18051 11410 26611 18139 11389 1001 96961 426 131 27376 18040 11723 27301 18188 11738 913 93350 493 155 28060 18090 11735 27974 18174 11739 935 946

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Figure 6-10. Motion of Cluster Two in three dimensions. The dashed lines with asterisks indicate the hydrogen and thesolid lines with plus signs indicate the helium.

95

Figure 6-11. This graph shows the separation of the helium and hydrogen centroids, inkiloparsecs, as time passes. The solid line with asterisks is Cluster Oneand the dotted line with plus signs is Cluster Two.

intermediate times, the helium in both clusters is more tightly bound than the hydrogen

in those same clusters. In fact, even at the later times, the helium still seems to be more

tightly bound, although the standard deviations for the two elements are getting closer

together by z = 50.

The last thing to check is the ratio of helium to hydrogen as a function of the

cluster’s total mass. In the tables above, I listed the number of each type of particle,

hydrogen and helium, as time passed. Once again, each particle has a mass of 4.235×

1010 M⊙/h, so both the ratio of helium to hydrogen and the total mass for each cluster is

easy to calculate in the same manner outlined in Section 6.2. This graph, for both of my

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Figure 6-12. This graph shows the standard deviations of the hydrogen and heliumclusters as time passes in kiloparsecs. Each cluster is represented by adifferent line type–Cluster One is the solid line and Cluster Two is thedotted line. The hydrogen is further demarcated by plus signs, while thehelium is represented by asterisks.

clusters, appears in Figure 6-13. This graph shows that both of the selected clusters

seem to be rather deficient in helium. In fact, they are rarely above the mean expected

value of 0.32 throughout their evolution. By looking at the data, it is clear that the late

times for both clusters are in the upper right, so it does seem that both clusters increase

in size throughout their lifetimes, but it would appear that they are quite rich in hydrogen.

6.4 Evolution Starting at a Redshift of 1250 With a Different Helium Mass

For this run, I used the same set of initial condition I used in Section 6.2. The

matter density parameter, Ω0, was set to 0.3, the density parameter for the cosmological

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Figure 6-13. This graph shows the ratio of helium particles to hydrogen particles as afunction of the cluster’s mass in units of 1010 M⊙/h. The solid line withasterisks is Cluster One and the dotted line with plus signs is Cluster Two.

constant was set to ΩΛ = 0.7, and the dimensionless Hubble parameter was h = 0.704.

I set the start time to z = 1250, made half of the particles cold dark matter and the other

half baryonic matter, and of the baryonic matter, I made twenty-four percent helium and

left the rest as hydrogen. To make this differentiation, I followed the procedure outlined

above, and using Equations (6–19) and (6–20), I got velocity ratios for z = 1250, or

a = 0.0007994, of:

ΘHe

ΘCDM= 0.50 (6–27)

ΘH

ΘCDM= 0.037 . (6–28)

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The only change I made was to the mass of the helium particles. GADGET allows for each

particle’s mass to be changed separately, so with a bit of tinkering around with my initial

conditions file, I was able to figure out how to change the masses for the particles I was

designating helium. I made them four times as massive as the hydrogen particles, and

then ran my simulation again.

Using these initial conditions, I obtained several output snapshots at various times.

At a redshift of 24, the universe can be seen in Figure 6-14. A graph showing only

hydrogen and helium at a redshift of 101 can be seen in Figure 6-15. Looking at these

two graphs, there is plenty of clustering, as there should be. However, at a glance,

it appears that the helium is actually clustering more tightly than the hydrogen, and

forming the seeds around which the hydrogen clusters form. This is what I would expect,

but this effect was not visible by just looking at the graphs in my first two runs. It would

already appear that increasing the mass of the helium particles is having a definite effect

on the evolution of these clusters. I need to see if this is more than an illusion, though,

and so the next step is to perform the same analysis on this data that I did on the last

two runs.

For this run, I tracked the evolution of three different clusters, in order to get a nice

sampling. The data for each clump is collected in Tables 6-7, 6-8, and 6-9. To start

with, I have included a graph of centroid motion in Figure 6-16. This time the cluster I

decided to show is Cluster One. By looking at its data, I can see that in all three graphs,

the centroids started in the lower left and evolved from there. The dashed lines with the

asterisks show the motion of the hydrogen, and the solid lines with plus signs show the

motion of the helium. As expected, the centroids are further apart and move faster at

earlier times, slowing down and getting close together at later times. In both the x-y and

y-z graphs, the helium seems to be leading the hydrogen–it seems to almost always

be on the inside of the curves, and the sharp bends and kinks in the hydrogen curves

seem to be lagging the same features in the helium curves. The hydrogen and helium

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00

Z

Figure 6-14. A z-x view of the universe at z = 24. The box has sides of 50 Mpc/h, and theblack particles are dark matter, the red particles are hydrogen, and the blueparticles are helium.

100

00

Z

Figure 6-15. A z-x view of the universe at z = 101. The box has sides of 50 Mpc/h, thered particles are hydrogen, and the blue particles are helium.

101


z H He x (kpc) y (kpc) z (kpc) x (kpc) y (kpc) z (kpc) H (kpc) He (kpc)1249 134 40 33141 12065 28174 33389 12379 27998 3887 38261068 113 33 33453 12483 29188 33730 12715 29042 3602 3468913 122 38 33933 12146 29851 34460 12114 30468 3679 3526781 115 38 33976 12307 30017 34478 12150 30481 3574 3496667 103 38 34437 12147 31448 34605 12314 31938 3365 3233571 98 36 34614 12206 32373 34500 12688 32473 3190 2948488 94 34 34946 11945 32625 34656 12510 32766 2990 2780417 128 43 35244 11323 35359 34991 11425 34929 2879 2870356 207 66 35974 12033 35339 35618 11941 35155 3479 3504305 78 29 35731 12293 33931 35491 12364 33948 2081 1817260 104 35 35961 12439 34363 35757 12400 34209 2119 1828222 167 54 35475 12297 37138 35227 12330 36678 2250 2268190 188 56 37907 12938 39955 38079 13339 39797 2498 2593162 163 51 35988 13024 38573 35848 13037 38612 1863 1994139 136 39 36405 13737 39405 36163 13596 39500 1492 1501118 433 149 36975 14881 38219 36994 14820 38244 1741 1730101 468 156 36844 15544 38241 36898 15513 38207 1382 107286 463 167 36709 16307 38180 36760 16369 38221 1306 92174 649 239 36146 18225 38212 36261 18035 38315 1868 159263 1163 445 35985 19319 38705 36016 19277 38708 1494 94154 825 460 36246 19373 38567 36169 19405 38736 1441 121146 936 499 36339 19614 38454 36392 19508 38552 1471 97139 733 504 36586 19763 38267 36578 19659 38319 1311 89033 1182 590 36724 19834 37716 36751 19788 37841 1709 108228 996 600 36913 19913 37620 36975 19992 37544 1383 89924 881 603 37144 20112 37358 37204 20200 37422 1201 745

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103



104

Figure 6-16. Motion of Cluster One in three dimensions. The dashed lines with asterisks indicate the hydrogen and thesolid lines with plus signs indicate the helium.

105

curves seem to cross a bit more often in the z-x graph, but this could just be a matter of

perspective.

The centroid separation can be seen in Figure 6-17. Clearly, as in the other runs,

Figure 6-17. This graph shows the separation of the helium and hydrogen centroids, inkiloparsecs, as time passes. The solid line with circles is Cluster One, thedotted line with asterisks is Cluster Two, and the dashed line with plussigns is Cluster Three.

the centroid separation gets smaller as time passes. In this case, it levels out around a

redshift of 75 or so at about 100 kpc.

The next thing that needs to be examined is the standard deviation. This will tell me

if the helium is clustering more tightly than the hydrogen, as it should be. The standard

deviations for this data are graphed in Figure 6-18. Putting all three clusters on one

graph makes for a figure that conveys a lot of information, all on top of each other, but

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Figure 6-18. This graph shows the standard deviations of the hydrogen and heliumclusters as time passes in kiloparsecs. Each cluster is represented by adifferent line type–Cluster One is the solid line, Cluster Two is the dottedline, and Cluster Three is the dashed line. The hydrogen is furtherdemarcated by plus signs, while the helium is represented by asterisks.

I think the three clusters are different enough to see everything that needs to be seen.

This graph shows a much larger separation between hydrogen and helium than either of

the graphs for my first two runs. At early times, the standard deviations for the hydrogen

and helium remain pretty close for Cluster Two, but in Clusters One and Three, there

is a clear and noticeable separation between the standard deviations for hydrogen and

helium. Sometimes they get close to one another, but at almost all times, the helium is

more tightly bound than the hydrogen. Especially at late times, all three clusters show

a large difference in the standard deviations for hydrogen and helium, which seems

107

to remain fairly constant after a redshift of around 75 or so onwards. These are pretty

interesting results that seem to show that the helium is definitely forming clusters at

earlier times than the hydrogen, around which the hydrogen clusters form.

Finally, this brings me to the ratios of helium to hydrogen as a function of each

cluster’s total mass. This time, as I mentioned, my helium was four times as massive as

my hydrogen, giving it a mass of 16.94× 1010 M⊙/h. This changes the equation I need to

find the total mass of a cluster to:

mtotal = (4nHe + nH)× 4.235 . (6–29)

The units are still 1010 M⊙/h. This graph, for each of the clusters I tracked, appears in

Figure 6-19. This is a pretty interesting figure. The ratios for all three clusters seem to

have a fairly similar shape, so the behaviour that is being displayed is probably common

to most clusters. Of course, time starts on the left side of the figure, when the clusters

are all small and therefore contain less mass. It appears that at early times, the clusters

are highly enriched in helium, as would be expected. As time passes, this ratio falls

towards the mean value of 0.32, and at later times–around a redshift of 100 or so for

Clusters One and Three and 50 or so for Cluster Two–the ratio even falls below this

value. None of the clusters stay below this value for long, though, as they all increase

back over this value shortly afterwards–in some cases by quite a bit. Presumably this is

a function of the clusters I was tracking merging with other clusters, also highly enriched

in helium. It is not the later times that are of interest to me, though–it is the early and

intermediate times. At these redshifts, the ratios for all three clusters show substantial

helium enrichment, which makes sense. The helium should start forming clusters before

the hydrogen, and as it does, the hydrogen will gradually fall in towards these clusters.

Since most of the helium has already formed into clusters, as time passes, the ratios

will all fall (as shown here) as the hydrogen starts forming around the helium. The main

way of changing the amount of helium in these clusters after they form will be merging

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Figure 6-19. This graph shows the ratio of helium particles to hydrogen particles as afunction of the cluster’s mass in units of 1010 M⊙/h. The solid line withcircles is Cluster One, the dotted line with asterisks is Cluster Two, and thedashed line with plus signs is Cluster Three.

with another cluster, which will not happen until after the clusters have already had some

time to develop.

109

CHAPTER 7CONCLUDING REMARKS

7.1 COSMICS

Understanding the creation and growth of perturbations in the early universe is

essential to being able to model large scale structure formation in the universe. Before

the creation of these perturbations, the universe was completely featureless and

homogeneous. Once they start growing, they form the seeds for structure formation

on long time scales. In addition to my own relatively simple calculations and programs

modeling the evolution of the early universe, I worked with COSMICS, a code developed

by Bertschinger and explained in detail in his paper on its use (Bertschinger, 1995;

Ma & Bertschinger, 1995).

COSMICS was originally designed to treat all baryonic matter as hydrogen, but with

some alterations, I was able to track the hydrogen perturbations separately from the

helium perturbations. Unfortunately, COSMICS is a complicated program, and breaking

the baryonic matter into two different groups like this gave rise to some problems in the

code which I was unable to resolve. At this point, I am unsure whether the problem was

a programming issue on my behalf, or whether COSMICS is set up in such a way as to

make separating the helium from the hydrogen impossible without majorly reworking

the code. This is an issue that I may come back to in the future, as having a program

like COSMICS that evolves the different kinds of baryonic matter perturbations separately

without neglecting any of the relevant physics would be a very useful tool that might give

interesting insights into the recombination era, the evolution of these perturbations, and

the growth of large scale structures.

7.2 GADGET

Using GADGET, I was able to model the formation of structures when helium is

allowed to start recombining before hydrogen. Throughout the three runs I performed,

several things seem certain. The centroids of the helium particles and hydrogen

110

particles do begin separate from each other, implying that the helium must be moving

to form structures before the hydrogen. If it was not, the centroids would be in the

same place, since the hydrogen and helium is initially distributed randomly and

homogeneously. Therefore, this separation in centroids must arise from the fact that

they are different particles, with different properties. Of course, as time passes, the

velocities of the hydrogen and helium particles tend to approach the same value, and

the centroids of the two different types of particles approach each other as everything

reaches the same velocity and gravity pulls everything into towards each cluster’s center

of mass.

The standard deviations of each type of particle are also interesting to look at. For

the first two runs, it appears that for the majority of time, and for most of the cluster’s

tracked, the standard deviations of the helium particles are smaller than the hydrogen

particles, as would be expected. For the last run, which had an increased mass for

the helium particles in addition to the fact that it started moving at an earlier time, this

effect was even more pronounced. From these results, I would draw the conclusion that

the helium particles do tend to be more tightly clustered than the hydrogen particles.

Since the helium recombines first, and can therefore start forming structures earlier, this

certainly makes sense.

Finally, the ratios of helium particles to hydrogen particles as a function of each

cluster’s total mass gives a useful measure of whether each structure formed is richer

in helium than would be expected from the average value. Of course, there is much

more hydrogen than helium in the universe–about three times as much. This would

lead me to expect that structures formed at early times, when helium first starts to

recombine, but before hydrogen is able, should be enriched in helium, which will form

the seeds for these structures once hydrogen is able to start forming. Conversely, since

much of the helium will be locked up in these early structures, any structures that form

at intermediate times, after hydrogen really starts recombining, would be deficient in

111

helium. Of course, much of the hydrogen would be attracted to the gravitational pull from

the clusters already formed from the helium, but with so much hydrogen in the universe,

it is entirely possible that hydrogen rich structures might start forming in the areas

relatively far away from the helium structures. This may explain why a couple of the

clusters I tracked in runs one and two are deficient in helium, rather than enriched, which

is what I would have expected. Of course, maybe this effect is weaker than I expect–all

three of the clusters I tracked in Run Three were greatly enriched in helium, so perhaps

by neglecting the increased mass of the helium particles, I was over simplifying the

problem. I do have more confidence in my results from Run Three, as they include more

of the relevant physics than Runs One and Two by including the helium mass.

Everything I did in this study leads me to believe that including the earlier recombination

time of hydrogen in the modeling of large scale structure formation is worth doing in any

detailed calculations. Although it seems to introduce only small effects that probably will

not cause any problems in most calculations, the effects it does produce are noticeable,

and should be studied more. This study provides a good starting point, but more work

in this area is certainly justified to figure out to just what extent these effects should be

included in models of recombination, large scale structure formation, and the evolution

of the universe.

112

APPENDIX: SETTING INITIAL CONDITIONS FOR GADGET-2

Creating a proper initial conditions file for the helium recombination runs that I wish

to do is an important part of my project. Using the initial conditions provided for the

Lambda-Cold Dark Matter (or ΛCDM) model supplied with GADGET is a good starting

point, but these initial conditions were created for a specific time–as mentioned in

the paper, this time is z = 10. I know, as mentioned in Section 6.1, that the density

fluctations scale as functions of the expansion factor during the era of matter domination

(which is the time during which I will use GADGET), and so it is easy to scale the given

initial conditions file to whatever time is required.

There are two things that need to be changed in the initial conditions file to change

the corresponding time. Each particle has a position and velocity associated with it; the

positions of the particles are determined by an equation like:

x = na + δ . (A–1)

The na determines the points of the grid. The n variable tells which grid point to base

that particular particle on (n can range from 1 to the number of points on one side of

the grid), and the a variable determines the distance between grid points. Its value is

the length of the box divided by the number of grid points. The δ is the variable that

determines the fluctuation sizes, and it is only this part of the position variable for each

particle that I need to change. So, to scale these positions with time, I need to figure

out how the density fluctuations, represented in the δ term, scale with the expansion

factor. The velocity terms are a bit easier–they do not have a constant term, and are

just functions of the velocities of the fluctuations, so it is a good deal easier to scale the

velocities once I know how they depend on the expansion factor.

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To find how the density fluctuations depend on the expansion factor, I will start with

Poisson’s Equation (Liddle & Lyth, 2000; Peebles, 1993)1 :

∇2Φ = 4πGδρ . (A–2)

This calculation is easier in Fourier space, so I will make that switch with the relation:

∇2 = −(k

a)2 , (A–3)

where k is the wavenumber of a particular density fluctuation (all of which evolve

independently of each other), a is the expansion factor, and k/a is the physical

wavenumber in comoving coordinates. Φk, the gravitational potential in Fourier space,

is constant in time, as well. Technically, all of the density fluctuations should carry a k

subscript as well, since these are density fluctuations in Fourier space, but I will take

them to be understood to save some time. With these relations, I find:

− (k

a)2Φk = 4πGδρ . (A–4)

It is the density contrast that is most often used in cosmological equations, and it is the

density contrast used in the initial conditions equation described in (A–1), so that is the

next substitution I will make. The density contrast is defined by:

δ =δρ

ρ, (A–5)

which gives:

− (k

a)2Φk = 4πGρδ . (A–6)

During matter-domination, I know that:

ρ ∝ a−3 . (A–7)

1 Conversations with Dr. Fry were also very edifying here.

114

So, making this substitution and simplifying, I find:

−(k

a)2Φk ∝

4πGδ

a3(A–8)

4πGδ ∝ −k2aΦk (A–9)

δ ∝ −k2aΦk

4πG(A–10)

δ ∝ −2

3

k2

a2H2Φk (A–11)

δ ∝ a . (A–12)

And so I find that the density fluctuations are proportional to the expansion factor.

This leaves the velocities of the particles to be determined. Once again, I will just

look at the matter-domination era, where ρ ∝ a−3 and using the Friedmann Equation with

no curvature and a cosmological constant of 0, H2 = ((8πG)/3)ρ (Liddle & Lyth, 2000).

First, I want to find how time depends on H:

a

a= H (A–13)

da

dt= aH (A–14)

∫ t

0

dt =

∫ a

0

da

aH(A–15)

t =

∫ a

0

a3/2da

a√

(8πG)/3(A–16)

=1√

8πG3

∫ a

0

a1/2da (A–17)

=2

3

1√8πG3

a3/2 (A–18)

=2

3

1

H(A–19)

t =2

3

1

H. (A–20)

Once I have this equation, I can combine it with another equation from (Liddle & Lyth,

2000), given by:

~v = −t~∇Φ . (A–21)

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Converting this to Fourier space using the relation ~∇ = i~k/a, I find:

~v = −2

3

1

Hi~k

aΦk . (A–22)

I want to get rid of the vectors, so I will take the dot product of both sides with their

complex conjugates and then take their square roots to get:

v =2

3

k

aHΦk , (A–23)

and, with (A–11), this becomes:

v = −aH

kδ . (A–24)

Since H ∝ √ρ, H ∝ a−3/2, and so:

v ∝ −a

ka−3/2δ (A–25)

v ∝ δ√a

. (A–26)

And, along with Equation (A–12), this allows me to state that:

v ∝√

a . (A–27)

So the velocities of the particles are proportional to the square root of the expansion

factor.

116

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BIOGRAPHICAL SKETCH

Andrew L. Hill was born in California in the year 1980. He spent most of his life

there, eventually settling down in Northern California. He attended Del Oro High

School in Loomis, graduating in 1999. Interested in space from an early age, he

decided to attend the California Polytechnic State University in San Luis Obispo,

mostly for the aerospace engineering program, but partly for the proximity to the ocean.

When he discovered he was more interested in theory than the practical discipline of

engineering, he switched to physics and graduated with a bachelor’s degree. Unsure of

the usefulness of a bachelor’s degree in physics on the job market, Andrew then decided

he had little choice but to go to graduate school and get a doctorate. Still interested

in space, he decided astrophysics was the field for him, and started his research into

cosmology and the large scale structure of the universe. Future plans: take a nice,

long vacation, and then hopefully find a job with at least some small connection to

astrophysics.

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effects of helium recombination in the early...

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