Probing high-redshift galaxies with intensity mapping of the Ly-alpha line

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Universit`

a di Pisa

FACOLT `A DI SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di Laurea in Fisica Teorica

Probing high-redshift galaxies with

intensity mapping of the Ly-α line

Tesi di Laurea in Fisica

Relatore:

Chiar.mo Prof.

Andrea Ferrara

Presentata da:

Paolo Comaschi

Anno Accademico 2013-2014

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Abstract

We studied the 3D fluctuations in Ly-α emission with a particular focus on the epoch of reionization. Our goal was to assess both the feasibility of an intensity mapping survey and the physics that it could probe.

We assumed a reionization scenario dominated by UV galaxy emission and considered separately population II and population III stars. We developed an analytical model for the star formation rate, then we calculated the reionization history and the Ly-α emission accordingly. We computed the 3D power spectrum of the line fluctuations, finding that the dominant source is the IGM emission from scattering of redshifted Lyman photons. The fluctuations trace the star formation rate, therefore a Ly-α intensity mapping can probe both the large scale galaxy distribution and the star formation history.

Then we considered the prospects of an intensity mapper for measuring the fluctuations, dealing also with the foreground removal. We found that a small space instrument can probe the large scale structure up to z = 7. Therefore, Ly-α intensity mapping is an extremely powerful tool that can observe the large scale structure of the galaxy distribution in the late stages of the epoch of reionization.

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1 Introduction

We know from CMB experiments that, after the formation of neutral hydrogen and the decoupling of baryons from photons, the universe was a hot gas, almost homogeneous and with tiny density fluctuations (δρ/ρ < 10−5). This has no-thing in common with the local universe, where we can observe a large variety of gravitationally bound objects, from planets to cluster of galaxies.

One of the most fascinating research field in cosmology is the study of this transition: the formation of the first stars and galaxies and their evolution during the cosmic time. This phase in the history of the universe is usually called Epoch of Reionization (EoR), because the photons generated by the first stars ionized the neutral hydrogen in the diffuse intergalactic medium (IGM). This topic is even more interesting for theorists because it is at the edge of our observational capabilities and the next generation of instruments promises to probe it in detail. Therefore we need detailed predictions to test our theoretical model and to project new observations.

In this work we study a particular type of survey, intensity mapping. It consists in the observation of the 3D structure of a large patch of the sky, measuring the fluctuations in line emission (in this case the Ly-α line). The purpose of these observations is not to detect individual sources, but to study the large scale structure of the universe and the global evolution of its constituents. Therefore, they require a low spatial resolution, a relatively small telescope and a short exposure time.

The great advantage of this survey is that it is complementary to deep high redshift observations, such as the Hubble’s one or the scheduled JWST’s. In fact during reionization most of the stars are in very small galaxies that are not detectable by any of the instruments under construction. An intensity mapping survey will detect the intregrated emission from all the sources and therefore it will probe galaxies totally unreachable otherwise.

The scope of this thesis is to assess the observability of the 3D power spectrum of Ly-α emission. Therefore, this is only a preliminary study, that do not pretend to make a prediction to be tested by observations; on the other side our goal

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1 Introduction

is to understand the physics involved in Ly-α emission (and thereby what an observation could probe) and the difficulties that could emerge in an observation. We developed an analytical model for the star formation in galaxies; this al-lowed us to compute the emission of ionizing photons and a reionization history. Then, we considered all the physical processes that produce Ly-α photons and used our model to estimate both the mean intensity and the power spectrum of Ly-α fluctuations.

Finally, we considered the prospects of an intensity mapper for measuring Ly-α fluctuations. We studied the impact of both continuum sources of foreground and interloping lines, and some of the relative removal techniques.

The results from this work are encouraging, because Ly-α emission is a tracer of star formation and therefore its mapping could observe the large scale distribution of high-redshift galaxies. We found that the 3D power spectrum is observable on large scales up to z = 7, opening the possibility of probing the late stages of reionization.

The thesis is organised as follows:

• Chapter 2: it is an introduction to the cosmological background. It il-lustrates the ΛCDM cosmology and the basic tools needed for the study of structure formation.

• Chapter 3: here we consider the physics of the intergalactic medium and the current understanding of reionization. We will develop the theory that we will use in chapter 5 for the calculation of the reionization history.

• Chapter 4: it is an introduction to the intensity mapping technique. It will analyse it in general, with a examination of the most studied lines in literature. We will also present the difficulties that this type of survey presents, with a particular focus on foregrounds in the near infrared.

• Chapter 5: in this chapter we present our analytical model for Ly-α emis-sion. We will examine in detail the physics involved and present an estimate of the Ly-α 3D power spectrum.

• Chapter 6: we analyse the observational prospects of the previous esti-mates, with an analysis of both the instrumental uncertainties and of the foregrounds removal techniques.

• Chapter 7: we conclude this thesis with a brief summary of the work done and of the results obtained. We also consider the limitations of our work and the possible future improvements.

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2 Cosmological background

In this chapter we will introduce the cosmological background essential to the contextualization of this thesis. Obviously, we will not go into detail of many relevant topics and we will only try to make clear the fenomenology and the physics in play.

The first part is about the properties of the large scale universe, in a regime where it can be considered homogeneous and isotropic; the second one handles the collapse of the gravitational instabilities and the resulting formation of structure.

2.1 The expanding universe

After the work of Edwin Hubble, it was clear that objects outside of the Milky Way (and sufficiently far away) are receding away from us, with a velocity pro-portional to the distance. This discovery is easily explained by general relativity, that describes both the gravitational interaction of the content of the universe and its geometry.

In this section we will approximate the universe as homogeneous and we will address the physics of this expansion. This hypothesis is obviously false on small scale and after the gravitational collapse that led to our existence, but it is an excellent approximation on large scale or in the very early universe.

In §2.1.1 and §2.1.2 we will introduce the cinematic and the dynamics of the expansion, as well as the standard model of cosmology. §2.1.3 is about the thermal history of the baryons, but for brevity reasons we will concentrate ourselves on the formation of the hydrogen atom, because it is the most important topic for the contextualization of this work.

2.1.1 The cosmological principle and the cinematic of the

expansion

The cosmological principle is basically a generalisation of the Copernican prin-ciple and states that the place from which we observe the universe is identical,

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2 Cosmological background

on large scales, to every other place. This is equivalent to say that the universe is homogeneous and isotropic.

Even if this is a statement that cosmologists belive willingly for philosophical reasons and so far there is not a experimental certainty, it has been tested obser-vationally by CMB measurments (introduced in §2.1.3) and galaxy surveys; both support it on scales larger than 100 Mpc (1 pc ≈ 3.1 × 1018 cm).

In general relativity there are only three metrics that describe a universe ho-mogeneous and isotropic: the Robertson-Walker metrics.

ds2 = dt2− a2_(t) dR2 1 − kR2 + R 2_(dθ2_{+ sin}2_θdφ2₎ (2.1)

where a(t) is a scale factor (with dimensions of length) that describes the expansion of the spatial coordinates in time, k determines the geometry of the space and can have values +1, 0 or -1 for a closed, flat or open universe. R, θ and φ are called the comoving coordinates, because they are constant for an object with no proper motion, at rest relative to the expanding Hubble flow.

As we will see in the next subsection, observations suggest that our universe is flat, that it is expanding (therefore a(t) is increasing) and that the expansion is accelerating.

One of the consequences of the Hubble flow is that the radiation emitted from distant galaxies is redshifted. This leads to a simple relation between spectral features and emission coordinates: for example if we identify an emission line in a galaxy we can infer the distance and the time of emission. From the simple cinematic described by this equation we can already derive the relation between redshift, expansion and the Hubble law.

Lets consider an electromagnetic wave emetted at a time ti, received at a time

tf and with time separation between the fronts of ∆ti,f. We can calculate the

relation between ∆ti and ∆tf using that photons propagate through the gedesics

(ds2 _{= 0) and equating the distance traveled by the fronts.}

Z dR = Z tf ti cdt a(t) = Z tf+∆tf ti+∆ti cdt a(t) (2.2)

This gives ∆tf/∆ti = a(tf)/a(ti). The frequency of the radiation is inversely

proportional to the ∆ti,f and the redshift is defined by

z = ∆λ λi

= νi νf

− 1 (2.3)

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scale factor:

1 + z = a(tf) a(ti)

(2.4)

That tells us, for example, that in the z ≈ 1 universe the typical distance between galaxies was half that of the local universe.

Eq. (2.4) is a generalization of the Hubble law, that states that in the local universe the redshift of galaxies is proportional to their distance. Expanding in dt = tf − ti ≈ dR/c we find:

z ≈ ˙a adt =

H0

c dR (2.5)

where H0 is the Hubble constant. In this work we will use the 9-year WMAP

data release values: H0 = 0.697km/s/Mpc (Hinshaw et al. 2013).

2.1.2 Dynamics and ΛCDM cosmology

In the previous subsection we introduced the cinematics of the homogeneous universe. We found a interesting relation between redshift and scale factor only using the form of the metric; but, to fully describe the expansion of the universe, we need to link its dynamical properties with the matter content. In particular we want to find the dependence of the scale factor from time and also to understand the relation between the evolution of the universe and the different types of fluid that it contains.

In order to find the differential equations for a(t) it is necessary to use the equations of energy conservation and general relativity. Here we will skip the derivation; with the hypothesis of homogeneiy and isotropy the Einstein equations reduce to only two indipendent equations, the Friedmann equations:

H(t)2 = ˙a(t) a(t) 2 = 8πG 3 ρ − k a(t)2 (2.6) ¨ a(t) = −4π 3 Ga(t)(ρ + 3P ) (2.7) Eq. (2.6) and (2.7) do not contain all the necessary information, because we have not yet specified the content of the universe and the relative equation of state P = P (ρ). Nevertheless there are two important fact to be highlighted:

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• in eq. (2.6) there is a critical density ρc=

3H2

8πG (2.8)

for which the universe is flat. A denser universe is closed (k < 0) while a rarefied one is open (k > 0). As we will see, observations seem to favour a flat universe.

• For ordinary matter, with P (ρ) > 0, the expansion is always decelerating (¨a < 0 for eq. (2.7)). Moreover, if the pressure is greater the deceleration is higher. This seem to contradict the common sense, because we expect pressure to make matter drift apart. This argument is incorrect because there is no pressure gradient and therefore no force; on the other hand the expansion of a gas with pressure causes its internal energy to diminish, because it does work.

Using the conservation of energy and the homogeneity we can find the redshift evolution of the energy densities. There could not be any temperature gradient and heat transfer, therefore the conservation of energy becomes dU = −P dV . If we consider a volume V = V0a(t)3, U = ρV and a simple equation of state

P = ωρ, we find dρ ρ = −3(1 + ω) da a =⇒ ρ ∝ a −3(1+w) (2.9)

In 1998 two research groups found evidence that the expansion of the universe is accelerating. This led to the acceptance by the scientific community of the standard model of cosmology, the so called ΛCDM, which states that the density of the universe is composed by four components:

1. Ordinary matter: this category includes both baryons and leptons (without neutrinos), even if it is improperly called baryonic matter. At z = 0 it comprises a small part of the total density, the 4.64%; in the redshift range in which we are interested its equation of state is non-relativistic (i. e. P (ρ) ρ) and thus its pressure can be neglected.

Therefore ω = 0 for baryons and from eq. 2.9 the baryonic density is inversely proportional to the cube of the scale factor:

ρbar(z) = ρbar,0

a(t₀) a(t)

3

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This is very intuitive, because baryons (and in general non-relativistic mat-ter) are simply diluted by the expansion.

2. Radiation and relativistic matter: the known components of this group are photons and neutrinos. Although this is the dominant component by num-ber, its z = 0 density is negligible (≈ 10−5) and in the redshift range relevant here can be neglected. The density evolution with redshift is steeper than the baryonic one because radiation in addition to be diluted is redshifted and looses energy. Indeed for radiation P = ρ/3 → ω = 1/3 and this gives:

ρrad(z) = ρrad,0(1 + z)4 (2.11)

3. Dark Matter (DM or CDM): this is the first of the two types of fluid of unknown nature. It is basically a fluid with the same behaviour of baryonic matter, but with no electromagnetic interaction (because we can not see it). Its existence is required in order to explain observations of galactic dynamics (the reason for which it was first introduced), cluster of galaxies dynamics and cosmology. It is presupposed to be composed by weakly interacting particles and cosmological simulations suggest an heavy (> 1GeV) mass; up to date there is no direct detection although. Current measurments suggest that CDM is the 23.5% of the total, being more than the 83% of the total matter.

Clearly the density evolution with redshift is the same of the baryonic mat-ter:

ρCDM(z) = ρCDM,0(1 + z)3 (2.12)

4. Dark Energy (DE): the existence of this exotic fluid is required to explain the accelerating expansion of the universe. For eq. 2.7 we need PDE/ρDE <

−1/3 and it is clear that it is impossible for ordinary matter. We can obtain a reasonable model for it by introducing a cosmological constant in the GR equations or a constant vacuum energy (they are equivalent from a gravitational point of view).

This leads to the equation of state:

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Figure 2.1: Redshift evolution of Ωm, Ωr and ΩDE. Dark energy dominates the

energy content only at z < 1, then the universe is matter dominated until very high redshift.

Even though the exact constant of proportionality between P and ρ is matter of research. We will assume the constant energy density hypothesis, therefore:

ρDE(z) = ρDE,0 (2.14)

WMAP measurments assess DE to be more the 71.8% of the total.

We have know all the information necessary to connect the redshift of a source to its distance or time of emission. As it is common in literature, we will indicate the various components of the universe as fractions of the critical density:

ΩX,0 =

ρX,0

ρc

(2.15)

In figure 2.1 we plotted the redshift evolution of the Ωs assuming a flat universe (as it seems from the observational data). We have three phases in the history of the universe:

• At low redshift z < 1 the cosmological constant (i.e. dark energy) is the main component and therefore the Hubble expansion is accelerated.

• At 1 < z <∼ 3300 the universe is matter dominated. This is the regime more relevant for the scope of this thesis because it encopasses the epoch

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of structure and galaxy formation. If we assume that radiation and dark energy are negligible, eq. (2.6) has a simple analytical solution

a(t) ∝ t2/3 (2.16)

H(t) = 2 3t

• At very high redshift z > 3300 the universe becomes radiation dominated. As we will see, this has a very important effect on the power spectrum of large scale fluctuations.

Now that we have defined how the density of the various components of the universe evolve, we can write the equation of the Hubble parameter. Eq. (2.6) becomes

H(z) = H0Ωm,0(1 + z)3+ Ωrad,0(1 + z)4+ ΩDE,0+ (1 − Ω)(1 + z)2

1/2

(2.17)

Where Ω = Ωm,0 + Ωrad,0 + ΩDE,0. The plot of eq. (2.17) is in figure 2.2. The

Hubble parameter is very important to understand what physical processes are relevant in a cosmological context. In fact the inverse of H(z) is the timescale of the expansion at redshift z, therefore any astrophysical process with a longer timescale is usually negligible.

Separating the differentials in eq. (2.17) we can integrate numerically for the time elapsed since the big bang and for the comoving distance between the local universe and redshift z:

τ (z) = Z +∞ z dz0 (1 + z0_)H(z0₎ (2.18) R(z) = Z z 0 cdz0 H(z0₎ (2.19)

In this work we will use the nine year release of the WMAP data, with Ωbar,0 =

0.046, ΩCDM,0 = 0.23, ΩDE,0 = 0.72 and H0 = 70km/s/Mpc. These values are

consistent with a flat universe. In figure 2.3a and 2.3b it is possible to find the plot of the relations (2.18) and (2.19).

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Figure 2.2: Hubble constant as a function of redshift computed with the latest WMAP data (Hinshaw et al. 2013).

(a) Age of the universe (b) Comoving distance

Figure 2.3: Figure (a) shows the time elapsed since the Big Bang (eq. (2.18)). Figure (b) shows the redshift evolution of the comoving distance from us (eq. (2.19)).

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2.1.3 The very early universe

This thesis is focused on the relatively low redshift universe (z < 10 − 15), therefore we will not cover topics such as inflation, relic particles and nucleosyn-thesis. However it is worthwhile to investigate the recombination of protons and electrons and the successive decoupling of baryons and radiation, because it is essentially the inverse of reionization.

Our universe is pervaded by a background radiation, the cosmic microwave background (CMB), almost isotropic and with a blackbody spectrum. This radiation has cosmological origin and, along with relic neutrinos, is accounted by Ωr in the Friedmann equations. The z = 0 temperature of the CMB is

TCMB(z = 0) ≈ 2.73 K, this means that at sufficiently high redshift its

temper-ature was higher than the hydrogen ionization potential BH = 13.6 eV. In this

enviroment electrons and protons cannot bound, because the resulting hydrogen atom is immediately ionized.

But, as soon as the radiation temperature drops below 13.6 eV, we expect the formation of hydrogen atoms and the IGM to neutralize. This process is quite complex and requires a numerical solution in its general case. We will begin with a simplified analysis and then describe qualitatively the general case, reporting only the final results.

At the beginning of the recombination the rate of the reaction

e + p *) H + γ (2.20)

is higher than the Hubble time, therefore we can assume termal equilibrium. This implies the equality of the chemical potentials and permits to write the Saha equation for hydrogen density:

nH= nenp meT 2π −3/2 exp BH T (2.21)

We can write it in terms of nb = np+ nHand of the ionized fraction χe= ne/nb

1 − χe χ2 e = nb meT 2π −3/2 exp BH T (2.22)

Using that T = Tr(z) = TCMB(0)(1 + z) we can invert numerically this equation

and find z as a function of χe. It is usual to define the recombination epoch when

χe= 0.1, corresponding to a redshift

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This equation is quite surprising because the recombination temperature is Trec ≈ 0.3eV BH. This happens because the photon number density is

sev-eral orders of magnitude higher than the baryons one, therefore the high energy photons in the Wien tail are sufficient to keep the IGM ionized to such a low temperature.

This equation for zrec is not accurate because the rate of the reaction in eq.

2.20 scales as χ2

e; therefore in the late stages of recombination it is higher than

the Hubble time and there is not thermal equilibrium.

The non-equilibrium analysis of the problem is quite complex because when an electron and a proton bound usually generate a ionizing photon that ionize immediately another hydrogen atom. Moreover even if they bound to an excited state of H there is a high probability to emit a Ly-α photon. These photons resonate with hydrogen atoms and are scattered multiple times before redshifting; therefore Ly-α photons keep H to a excited state that is easily ionized by the blackbody radiation or other Ly-α photons.

Thus there are only two processes that permit the recombination of e and p:

• recombinations to an excited state that cascade to the 2S level. This level is metastable and decays with a two photon emission. These photons have energies lower than the Ly-α, therefore do not interact with neutral hydro-gen.

• the redshifting of Ly-α photons.

From a detailed analysis emerges that the first is the dominant one.

It is interesting and very important for the following thermal hystory of the IGM that not all the electrons recombine but a relic ionized fraction still exists until reionization and is almost constat for z < 700. From a detailed solution emerges χe ≈ 1.2 × 10−5 p Ωm,0 Ωb,0h ! (2.24) at z ≈ 200.

Apart from resonant lines, hydrogen atoms are neutral and do not interact with photons; therefore we expect radiation to decouple from baryons and propagate freely. This is indeed the effect that generates the CMB, detected for the first time by Penzias and Wilson in 1964.

Photons are scattered by electrons with a rate ΓT = neσTc, where σT is the

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2 Cosmological background σT = 8π 3 e2 mec2 2 ≈ 6.65 × 10−25cm2 (2.25) collisions with protons are negligible because of the mass dependence σT ∝

m−4.

Clearly we expect photons to decouple from baryons when ΓT < H, with the

usual cosmological parameters and the equilibrium hypothesis this appens at zdec ≈ 1100, Tdec ≈ 0.26 eV. As expected photons decouple shortly after the IGM

neutralize.

Without the equilibrium hypothesis, a good fit to the Thomson optical depth is τ (z) = Z z 0 neσT dt dzdz ≈ 0.37 z 1000 14.25 (2.26)

This let us calculate the probability of scattering between z and z + dz: P (z) = e−τ dτ_dz; this distribution has a sharp peak at zdec ≈ 1067 and a width ∆z ≈ 80.

Even if photons decouple from baryons, the relic ionized fraction of eq. 2.24 is sufficient to thermally couple the baryons to the CMB. This asymmetry happens because of the relatively higher density of photons compared to the baryons that results in a shorter mean free path for electrons

le

lγ

= nγ ne

≈ 2.72 × 10−8(χeΩb,0h2) (2.27)

If the two fluid where decoupled we would expect a different temperature evolu-tion, with TCMB ∝ (1 + z) and Tbar ∝ (1 + z)2. But, the number of interactions

between free electrons and CMB photons is sufficient to thermally couple them through Compton scattering. The entropy of photons is dominant, therefore the baryons’ temperature follow the CMB one until redshifts as low as 150.

2.2 Structure formation

Our local universe is very dishomogeneous, as it is obvious considering the difference between the typical densities on Earth ( > 10−3g/cm3) and in the IGM (≈ 10−30g/cm3). But we know from CMB experiments that at very high redshift the universe was very homogeneous, with density relative fluctuations smaller than 10−5. These were the seeds of structure formation, because the gravitational instability made overdense regions grow at the expense of the underdese ones.

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by few parameters to the one that we are used to. In §2.2.1 we will study the linear evolution of the primordial perturbations and in §2.2.2 we will examine their statistical properties. Afterwards we will consider the nonlinear collapse, in §2.2.3 we will see an approximation that grasps the qualitative features of the gravitational collapse, while in §2.2.4 we will solve exactly the problem in spherical symmetry. Finally in §2.2.5 we will develop the mass function, an important instrument to model the galaxy population.

2.2.1 Growth of linear perturbations

We will study here the evolution of this small perturbations in the linear regime and the resulting power spectrum.

The fluctuations field is described by

δ(x, t) = ρ(x, t)

ρ(t) − 1 1 (2.28) or

∆(x, t) = δρ(x, t)

ρ(t) (2.29)

If we consider fluctuations smaller than the Hubble radius and with nonrela-tivistic proper velocities, we can use the Newton law for gravity and the classical equations for fluid dynamics.

∂ρ ∂t r + ∇r· (ρv) = 0 ∂v ∂t r + (v · ∇r)v = − ∇rP ρ − ∇φ (2.30) ∇2_rφ = 4πGρ

It is simpler to use comoving coordinates x = r/a(t), v = ˙ax + u. The differential operators in the equations change as

∂f ∂t r = ∂f ∂t − ˙a ax · ∇f (2.31) ∇ = a∇r

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an equation of state for the fluid. For the moment we are more interested in the high redshift universe, therefore we will neglect the cosmological constant (that suppress the growth of perturbations, as we will see in §2.2.5):

φ = δφ + 2 3πGρa 2_x2 _(2.32) ∇P = dP dρ∇ρ = c 2 sρ∇δ

where cs is the sound speed in the fluid.

Combining the previous equations we obtain:

∂2_δ ∂t2 + 2 ˙a a ∂δ ∂t = 4πGρδ + c2 s a2∇ 2_δ _(2.33)

If we neglect for a moment the pressure term we can analyse the physics of dark matter or of cold baryons on large scale. In a matter dominated universe, a(t) ∝ t2/3_{, therefore the equation simplifies to:}

∂2δ ∂t2 + 4 3t ∂δ ∂t = 2 3t2δ (2.34)

This equation has two independent solution, one vanishing with δ ∝ t−1 and the other

δ ∝ t2/3 ∝ a(t) (2.35) Therefore we can see that in an Einstein-deSitter universe the perturbations grow proportionally to the growth factor.

In a radiation dominated universe we have a(t) ∝ t1/2_{, with the analogous}

calculation it easy to show that the nonvanishing solution for δ is

δ ∝ t ∝ a(t)2 (2.36)

Actually this result is misleading because during radiation domination the Hub-ble time is comparaHub-ble to the free fall time. Moreover baryons are tightly coupled with radiation until the recombination and its pressure and free streaming block any gravitational collapse. Therefore baryons fluctuations do not grow at all in this regime and dark matter fluctuations grow logarithmically.

Now we can include pressure in order to study baryons on small scales. This is actually an oversimplified analysis because baryons are only ∼ 15% of the matter and we should write the coupled equations for the interaction with dark matter.

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Nevertheless the final results are similar and therefore we will ignore this coupling. As it is usual when dealing with linear problems, we write eq. (2.33) in Fourier transform: ∂2_δ k ∂t2 + 2 ˙a a ∂δk ∂t = 4πGρ − k 2_c2 s a2 δk (2.37)

We can see from the RHS that there are two different regimes depending on k being smaller or greater than kJ

λJ = 2πa kJ = πc 2 s Gρ 1/2 (2.38)

if k > kJ the RHS is negative and eq. (2.37) is the equation for a dumped wave.

Therefore for fluctuations smaller than λJ (which is called the Jeans length) and

with mass smaller than

MJ = 4π 3 ρλ 3 J = 4π 3 πc2 s Gρ1/3 3/2 (2.39)

there could not be any gravitational collapse. On the other side if M > MJ

(k < kJ) the fluctuations can collapse and form a nonlinear object.

As we will see in §2.2.5 it is important also to consider fluctuations on scales larger than the Hubble radius. This requires to include general relativity and we will not present a detailed analysis here. We will report the final results in table 2.1 (from Padmanabhan 1993).

Epoch radiation DM baryons t < tenter< teq ∝ a2 ∝ a2 ∝ a2

tenter< t < teq osc ∝ ln(a) osc

teq < t < tdec osc ∝ a osc

tdec < t osc ∝ a ∝ a

Table 2.1: Evolution of linear fluctuations in different regimes. tenter is the time

at which a fluctuation enter the Hubble orizon, teqis the transition time between a

radiation and a matter dominated universe, tdecis the time of decoupling between

radiation and baryons.

The main feature to notice is that during radiation domination perturbations grow as long as they enter the Hubble radius. Therefore we expect that the evolution of small scale perturbations is suppressed during this phase compared with the large scale ones.

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2.2.2 Statistical properties of the density fluctuations

In this section we will develop statistical tools to describe the overdensity field δ(x) = ρ(x)/ρ−1. There is a subtlety here, because we have only one universe and therefore wondering about the probability of its density field seems a misplaced question.

We can bypass the problem exploiting the cosmological principle and dividing the universe in boxs of volume V . If these regions are sufficiently large, they are perfectly equivalent on average and with equal statistical properties. Therefore it is legitimate to ask what is the probability functional P (δ(x)) of having a overdensity field δ(x).

Thus we will use the same approach as in statistical mechanics and we will not try to describe the actual realisation of the density field, but we will see it as the product of a random process and we will analize its statistical properties.

The formalism is more simple working in Fourier transform

δ(x) = 1 V X n ρnexp(ikn· x) (2.40) ˜ δ(kn) = Z V δ(x) exp(−ikn· x)d3x

because the probability functional P [δ(x), z] (that contains complete informa-tion about the statistical properties of δ(x)) decomposes

P [˜δ(k), z] =Y

n

gkn(˜δ(kn), z) (2.41)

where kn = 2πn/L, n is a vector of integers and V = L3. This property is

conserved during the evolution as long as perturbations remain linear.

We need to determine the form of the function gk(˜δ(k), z) (we will take the

continuum limit V → ∞ henceforth) at an early time; then we know how different modes evolve in the linear regime (see §2.2.1) and we can compute gk(˜δ(k), z) at

any redshift. One of the simplest and more resonable prescription is a gaussian: there are several reasons for this choice, for example it is predicted by inflationary theories, but for this thesis it is sufficient to note that experimental data from the CMB are in agreement with it.

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2 Cosmological background Px(δ1, δ2, ..., δn) = exp1 2 Pj iδi(M −1₎ ijδj [(2π)n_{det(M )]}1/2 (2.42) Mij = hδiδji

is that it is completely specified by its second moment, the covariance matrix M . In an isotropic universe this means that the correlation function

ξ(r) = hδ(x)δ(x + r)i (2.43)

contains all the information about our density field.

We can write ˜δ(k) = ak + ibk = rkexp(iφk) (with a−k = ak, b−k = −bk,

r−k = rk and φ−k = −φk ). ak and bk are a linear combination of gaussian

variables, therefore they have a gaussian distribution:

gk(ak, bk, z)dakdbk = 1 πσ2 k exp −a 2 k+ b2k σ2 k dakdbk (2.44) gk(rk, φk, z)drkdφk= 2rkdrk σ2 k exp −r 2 k σ2 k dφk 2π (2.45)

It is easy to prove that if we have homogeneity and isotropy σk is

h˜δkδ˜∗pi = (2π) 3_σ2

kδ 3

D(k − p) (2.46)

where we took the continuos limit V → +∞.

Thus the Gaussian probability distribution is described only by σk (where we

used σk = σk for the cosmological principle). Usually P (k) = σ2k is called the

power spectrum of the fluctuations. P (k) is tightly related to the correlation function, one being the Fourier transform of the other

ξ(r) = 1 2π2

Z

dkk2P (k)sin(kr)

kr (2.47)

Inflation theory predicts a power law for the primordial power spectrum

Pi(k) ∝ kn (2.48)

with n = 0.97 from Hinshaw et al. 2013. As we saw in §2.2.1, before the matter domination different modes have a different evolution because they enter the horizon at a different time. Therefore the actual power spectrum that seeds

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structure formation is not the primordial one, but has a different shape. It is usual to include this multiplying Pi for the transfer function T (k)2

P (k) ∝ Pi(k)T (k)2 (2.49)

In this work we used the transfer function from Eisenstein and Hu 1998 (figure 2.4a). In figure 2.4b there is the processed power spectrum used in this work. The effect of the transfer function is a suppression of the small scale fluctuations for λ < 100 Mpc. This is a conseguence of the analysis of the growth of linear perturbations in §2.2.1.

(a) Transfer function (b) Power spectrum

Figure 2.4: (a): transfer function from Eisenstein and Hu 1998. (b): a comparison between the primordial power spectrum and the processed one. There is a peak around k ≈ 10−2 while smaller scales are suppressed.

We will not address the problem of the normalization of the power spectrum here; indeed we will see in §2.2.5 how it can be done analysing the distribution of galaxies.

2.2.3 The Zeldovich approximation

The Zeldovich approximation is a simple extension of the linear theory that grasps the qualitative fenomenology of the nonlinear collapse.

During the epoch of reionization about 90% of the matter in the universe is outside galaxies, in the intergalactic medium (IGM). Therefore it is of primary importance to understand its matter distribution, both to explain observations and to understand its relation with galaxies. Fortunately the physics of the IGM is way more simple than that of galaxies and its description is at reach of the numerical simulations.

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Thus it is well understood that the IGM is organized in a structure (the ”cosmic web”, see figure 2.5) composed of sheets, filaments and large voids in between. Filaments are in the intersections of sheets, while large DM halos are in the intersection of filaments.

Figure 2.5: Slice through the Millennium simulation, a massive computer sim-ulation of cosmological structure formation. Large voids are separated by dense filaments (which can be also slices through sheets of matter) while galaxies and clusters of galaxies form in their intersections. Image from Loeb and Furlanetto 2013.

The Zeldovich approximation consists in the assumption that the initial pertur-bation in the gravitational potential remain constant, while collisionsless matter continues to collapse. We can see the time evolution as a change of comoving coordinates

r(t) = r(t0) − b(t)∇Φ (2.50)

where b(t) is a growing function of time (for example b(t) = t2_{/2 for free fall)}

and Φ is the gravitational potential. We can now consider the conservation of energy: an infinitesimal volume d3r(t0) contains an energy ρd3r(t0) and it is

transformed in an infinitesimal volume d3r(t), therefore

ρ(r, t)d3r(t) = ρd3r(t0) (2.51)

Thus we can write the density as a function of the Jacobian of the transforma-tion ρ(r, t) = ρ det ∂ri(t) ∂rj(t0) = ρ det δi,j − b(t) ∂ 2_Φ ∂ri∂rj (2.52)

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The matrix in the determinant is real and symmetric, therefore it has three eigenvectors:

ρ(r, t) = ρ

[1 − b(t)λ1][1 − b(t)λ2][1 − b(t)λ3]

(2.53)

This equation contains all the information that we need to understand the formation of the cosmic web. Indeed there are four cases, each one for each combination of the signes of the eigenvectors of ∂i∂jΦ:

1. three negatives: the density goes to zero and we form a void in that point.

2. 1 positive and 2 negatives: only one direction collapse, therefore the point belongs to a sheet but not to a filament or a halo.

3. 2 positives and 1 negative: the same as in the previous point, but with two directions of collapse and the formation of a filament

4. 3 positives: the point belongs to a big halo.

Note that for the cases 3 and 4, in general the magnitude of the positive eigenvec-tors is different, therefore we have an ordered collapse that procedes first along one axis (forming a sheet), then along a second one (forming a filament) and eventually along the third one (collapsing in a halo).

2.2.4 Spherical collapse

The dynamics of the gravitational collapse in the nonlinear regime is very complex and requires numerical simulations in the general case. However we can have an insight of the process by studying a simplified model, such as the collapse of a structure with spherical symmetry.

In the nonlinear regime it is no longer convenient to work in the Fourier space, because different modes become coupled. Therefore we will express the density in the x-space:

ρ(x, t) = ρb(t) + δρ(x, t) = ρb(t)[1 + δ(r, t)] (2.54)

where we used the spherical symmetry hypothesis in the last equation. r is the proper coordinate: r = a(t)|x|.

If we consider perturbations with a size much smaller than the Hubble radius, we can use the Newtonian approximation. Moreover in a spherically simmetric system the gravitational force at r is determined only by the matter with r0 < r.

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We can write the equation for the conservation of energy for a point particle at r: 1 2 dr dt 2 − GM r = E (2.55) with M = ρm 4π₃ ri3 (1 + δi) = 1/2ΩHi2ri3(1 + δi).

We have a simple criterion to determine if an overdensity is going to collapse: if E ≥ 0 the kinetic term must be always positive and therefore the system can not stop its expansion and collapse, on the other end if E < 0 the gravitational potential will become always smaller, forcing the kinetic therm to be null and the system to contract and collapse.

In the following we will assume to be in a flat, matter dominated universe; therefore Ω = 1, Hiti ≈ 2/3 and δi ∝ t

2/3 i .

The equations of motion of the collapsing system have a simple parametric solution

r = A(1 − cos θ) (2.56) t = B(θ − sin θ)

Expanding r and t at the lowest orders in θ and sobstituting in the energy equation we find

A3 = GM B2 (2.57)

A = GM −2E

We need an expression at the first order in δi for vi = dri/dt in order to relate

A, B and E to the initial conditions δi, ri and ti. We can obtain this using the

mass conservation dM/dt = 0: vi = Hiri 1 − 1 3Hiti δi 1 + δi d ln δi d ln ti ≈ Hiri 1 − δi 3 (2.58)

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2 Cosmological background A = 1 2 ri 5δi/3 B = 3 4 ti [5δi/3]3/2 (2.59) E = −5 6H 2 ir 2 iδi

Therefore we can see that in a flat universe all the overdense regions will col-lapse. However we must keep in mind that the collapse time depends on the initial overdensity and diverges for δi → 0; plus we did not consider the

cosmo-logical constant that become the dominant component of the universe at z ≈ 1. Therefore it is not true that all the overdense regions are collapsed at low redshift. Now that we have solved the dynamics of the problem, we can concentrate in the understanding of the physics: at t ≈ tithe perturbation is expaning with with

the Hubble flow, but, as it is obvious from eq. 2.56, the expansion progressively slows until it stops at θ = π. This is the radius of maximum expansion and the turn around point:

rta = 2A (2.60) tta = πB = 3π 4 ti [5δi/3]3/2

Apparently, the overdensity reaches full collapse at θ = 2π, tcoll = 2tta. This is

a non-physical result and arises because we assumed that different shells do not cross. This hypothesis ceases to be valid in the late phases of the collapse, when the shells cross many times and the gravitational interactions make the structure virialize and reach a static equilibrium. Nevertheless from more refined analysis it emerges that by tcolthe structure is virialised and therefore we will anyway use

it as the collapse time.

It is useful for §2.2.5 to compare the prediction of the linear density evolution at collapse; we saw in §2.2.1 that δl(t) = δia(t)/a(ti) ≈ δi(t/ti)2/3, therefore

δl(tta) = δi tta ti 2/3 ≈ 1.06 (2.61) δl(tcoll) = 22/3δl(tta) = 1.686

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2.2.5 The halo mass function

The halo mass function is the mass distribution of the density of DM halos

dn(M, z)

dM dV dM = mean number of halos with M ∈ [M, M + dM ] in dV

Press and Schechter 1974 derived a formula based on simple considerations about the spherical collapse (§2.2.4); in this section we will derive the same theory with a different approach: the excursion set formalism.

To compute the main goal of this section, the mass function, we will need the statistics of the excess of mass contained around a point x:

δM (x) = Z d3yδρ(y)W (x − y) = Z d3_k (2π)3δ˜kW (k)e˜ ik·(x−y) _(2.62)

where W (x) is a window function and VW =R d3xW (x) is the equivalent of the

volume of this window. For the central limit theorem, δM/M has a gaussian distribution and has a standard deviation

σ_M2 (R) = h(δM/M )2i = 1 2π2_V2 W Z ∞ 0 k2dkσ_k2W˜_k2 (2.63) therefore σ2

M(R) can be computed from the power spectrum that we introduced

in §2.2.2

Two examples of window functions are:

WR(x) =

(

1 for x < R

0 otherwise Top hat (2.64)

WF(x, R) = 4πR3

sin kR − kR cos kR

(kR)3 Top hat in k−space

These examples are particularly important: for example the first is used for the normalization of the power spectrum (a problem left open from §2.2.2). We can survey the large scale galaxy distribution and then analize its overdensity on a scale R (conventionally R = 8h−1Mpc, a scale on the edge between linear and nonlinear fluctuations). We will see next that δgal = bδm, where b is the bias

parameter, therefore we can impose

σm(8h−1Mpc) =

σgal(8h−1Mpc)

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and fix observationally the normalization of the primordial power spectrum. As usual, in this work we will use σm(8h−1Mpc) = σ8 = 0.820 from Hinshaw

et al. 2013.

The second is a low-pass filter in k-space and we will use it for the formal derivation of the mass function.

(a) σM (b) Growth fractor

Figure 2.6: These figures show on the left σM(z = 0) and on the right the growth

factor from Carroll et al. 1992.

Figure 2.6a shows σM at z = 0. It is a decreasing function of mass, because

σ2

M ∝R d3kP (k) ∝ kn+3∝ M

−(n+3)/3_{. The processed power spectrum has always}

n > −2 and therefore σM is larger for smaller mass. We can write σM2 as

σ_M2 ≈ M M∗

−(n+3)/3

(2.66)

where M∗ is the mass-scale at which the fluctuations become nonlinear. This can

lead to a simple argument that proves that in our universe the growth of structure is bottom up, that is smaller structures form first.

We saw in §2.2.1 that in a matter dominated universe all the perturbation grow with the scale factor δ ∝ D(z) = 1/(1 + z). This is a little more complicated by the presence of the cosmological constant that slow the gravitational collapse at low redshifts. In this work we used the fitting formula of Carroll et al. 1992 for the growth factor:

D(z) = g(z) g(0)(1 + z) (2.67) where g(z) = 5 2ωm(z) ωm(z)4/7− ΩDE(z) + 1 + ωm(z)/2 1 + ΩDE(z)/70 −1 (2.68)

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D(z) is plotted in figure 2.6b along with the ΩDE = 0 Einstein-deSitter case.

The growth factor is normalized to unity at z = 0, therefore D(z) > DEdS(z)

means that the growth of structure is slowed down at low redshifts.

σ_M2 (z) is a quadratic function of the fluctuations, therefore σ_M2 (z) = D(z)2σ_M2 (0). This translates in a redshift evolution for the parameter M∗ in eq. (2.66).

Be-cause D(z) is a decreasing function of redshift, M∗(z) should also be a decreasing

function of redshift. This proves that the higher the redshift the smaller the mass-scales that are nonlinear, therefore smaller nonlinear structures form first.

Now we have developed all the formalism necessary for the mass function. The basic idea starts from the spherical collapse theory (§2.2.4): we saw that when an object virializes, linear theory predicts an overdensity δc = 1.686. Press and

Schechter exploited this fact arguing that it is reasonable to assume the inverse implication: if linear theory predicts a mean overdensity δM (x, R)/M = δR(x) >

δc than we can consider that region collapsed.

Therefore the fraction of mass collapsed in halos with mass greater than M (F (M )) is the probability that, taken a point x, exists a radius R > (3M/4πρ)1/3

such that δR(x) > δc. From F (M ) it is easy to compute the mass function

dn/dM = (ρ_m/M )dF/dM .

To compute F (M ) we will use the excursion set formalism, a technique inspired by the random walk problem and based on the diffusion equation. We will use the window function WF(x, R) (eq. (2.64)) and divide the range R ∈ [0, +∞] in

finite steps ∆R, with R0 → +∞ and Ri+1 = Ri − ∆R. Clearly δR0(x) → 0 and

we want to find the maximum R such that δR(x) > δc.

Decreasing progressively R, δR(x) random walks with steps

l2_i = h(δRi+1− δRi) 2_{i = ... = σ}2 i+1− σ 2 i = ∆σ 2 i (2.69)

where σi = σM(Ri). Using the window function WF different steps are not

correlated (h(δi+1− δi)(δj+1− δj)i = 0 if i 6= j); therefore we can write a simple

equation for the probability density P (δ, Ri)dδ: the probability of having an

overdensity δi ∈ [δ, δ + dδ] at the step i.

P (δ, Ri+1) =

1

2P (δ + li, Ri) + 1

2P (δ − li, Ri) − P (δ, Ri) (2.70) That, in the continuum limit, becomes the diffusion equation in σ units (σM is

a decreasing function of M , therefore we can use it as independent variable)

∂P (δ, σ2₎ ∂σ2 = 1 2 ∂2_{P (δ, σ}2₎ ∂δ2 (2.71)

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The general solution startin from δ = δ0 and σ = σ0 is:

P (δ, σ2) = 1 p2π(σ2 _{− σ}2 0) exp − (δ − δ0) 2 2(σ2_{− σ}2 0) (2.72)

Firstly we will compute the unconditional mass function: δ0 = 0 and σ0 = 0.

F (M ) is the fraction of random paths that cross δc(z) at least once at M0 > M .

There are two kind of such paths: those with δ(M ) > δc(z) and the others with

δ(M ) < δc(z) but δ(M0) > δc(z) at M0 > M . These two groups are in bijection,

because we can pass from one to another simply reflecting around δc(z) at the

first crossing. Therefore

F (M, z) = 2 × Z +∞

δc

P (δ, σ2_M)dδ = erfc (νc) (2.73)

where νc= δc/σ(M, z). And the mass function:

dn(M, z) dM = − ρ M ∂F (M, z) ∂M = −2 ρ M 1 2π νc ∂ log σ(M, z) ∂M e −ν2 c _(2.74)

This is the Press-Schechter mass function; it agrees well with numerical simu-lations, but underestimates very massive halos and overestimates low mass halos. Therefore in our thesis we will use a fit from Tinker et al. 2008:

dn(M, z) dM = −A " σM(z) b −a + 1 # e c σM (z) 2 ρ_m,0 1 σM(z) 1 M ∂σM(z) ∂M (2.75) where A = 0.18(1 + z)−0.14, a = 1.43(1 + z)−0.06, b = 2.70(1 + z)−0.77, c = 1.2. The Tinker mass function along with the Press-Schecheter one are plotted in figure 2.7a. The Press and Schecheter overestimated the number density for low mass halos at any redshift. As it is obvious from the shape of σM, low mass

halos are less abundant than the high mass ones. Moreover it seems that the number density of low mass halos saturates at high redshift, evolving very little from z = 12 to z = 0. On the other side there is a strong redshift evolution in the massive end of the mass function.

The excursion set formalism permit us to study also the large scale clustering properties of halos. We want to study the mass function in a large scale region with an initial overdensity δb; this can be done modifying the initial conditions

in eq. (2.72) to δ0 = δb0 = δb/D(z) and σ0 ≈ 0 (if the mass of the region is

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A background overdensity modifies the mass function in two ways:

• the halo mass function is in comoving units, that is in units of mass rather than volume. An overdense region will occupy a smaller comoving volume, thus dn dM _com = dn dM(1 + δb) (2.76) This is a simple unbiased modification to the mass function: the number of galaxies in a volume is proportional to the mass in it.

• if we are in the linear regime (as it is expected for large scale overdensities) we can expand the mass function in νc:

dn dM ps = dn dM + d2_n dνcdM dνc dδ_b0δ 0 b (2.77)

This is the biased correction to the mass function and, as we will see, as important cosmological implications.

The linear bias is defined as

dn dM δb = dn dM(1 + bδb) (2.78)

and b is the sum of the two previous corrections

bP S = 1 + d2_n dνcdM dνc dδ0 b dn dM −1 (2.79)

And for the Tinker mass function (Tinker et al. 2010):

bT(M, z) = 1 + γν2 c − 1 + 2η δc + 2φ/δc 1 + (βνc)2φ (2.80) where β = 0.589(1 + z)0.2, φ = −0.729(1 + z)−0.08, η = −0.243(1 + z)0.27 and γ = 0.864(1 + z)−0.01.

If the halo density were proportional to the mass density, we would have b = 1, but the overhead in the collapse deriving by the large scale overdensity increases further the collapsed mass fraction, leading to high biases b > 1. This effect is more important for massive halos, because they are exponentially suppressed in the mass function and this enhance the effect of the initial overdensity (see figure 2.7b). In figure 2.7c we plotted the average bias weighted with the star formation rate in the halos (see §5.1.2); as it is expected it is a growing fuction of the redshift.

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(a) Mass functions

(b) Linear bias _{(c) Mean linear bias}

Figure 2.7: The figure on the top shows the Tinker mass function and the Press-Schechter one at z = 0, z = 6 and z = 12. On the left there is the Tinker linear bias as function of mass at z = 0, z = 6 and z = 12. On the right the mean linear bias weighted with the star formation rate from §5.

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3 The cosmic dawn

Reionization is the last phase transition of the IGM. At sufficiently high re-dishfit, the universe was so dense and hot that hydrogen atoms could not form, because there were too many ionizing photons in the black body spectrum. But with the adiabatic expansion and subsequent cooling, as we explained in §2.1.3, the recombination of protons and electrons becomes possible and the IGM neu-tralises almost completely (z ≈ 1100).

This state is stable and persists as long as the first ionizing sources form (z ≈ 30?) and reionize the IGM again.

Reionization is a phenomenon at the edge of our observational capabilities and its exploration is one of the main objectives for cosmologists and astronomers. It has an obvious scientific interest and charm, because it involves the formation of the first stars and galaxies. The fact that we cannot directly observe it so far, but we expect to have new decisive observations in the next years makes it a challenging topic for theorists, because we need predictions to better understand the data and to project new experiments.

Because of its mainly speculative nature, we don’t have absolute certainties on reionization; therefore we will adopt a model that is the most accepted by the scientific community, but with the awareness that the limited scope of this work imposes us to assume untested hypothesis.

In the first part of the chapter, §3.1, we will introduce the basic physics involved, while in §3.2 we will present our current understanding of reionization.

3.1 Physics of the IGM

In this section we will examine three physical processes that are essential in the understanding of reionization and in the development of this work. In §3.1.1 we will introduce the cosmological equation for radiative transfer, that is the law describing the diffusion of radiation and its interaction with matter. In §3.1.2 we will apply the radiative transfer equation to the study of the absorption of photons by the Ly-α line while in §3.1.3 we will study the physics of ionizations

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3 The cosmic dawn

and recombinations.

3.1.1 Radiative transfer equation

The radiative transfer equation determines the evolution of the specific intensity of radiation Iν. Iν is defined in the following way: if we consider an infinitesimal

area dA and a solid angle dΩ around its perpendicular, the energy per unit time passing through dA and dΩ with photons of frequencies between ν and ν + dν is

dEν

dt = IνdAdΩdν (3.1) It is simple to derive the equation of cosmological radiative transfer from basic considerations. We can consider a volume ∆V with N = nν∆V ∆ν photons with

frequencies between ν and ν + ∆ν. This volume expands with the scale factor as ∆V ∝ a(t)3 _{and photons frequecies redshift ∆ν ∝ a(t)}−1_{; without absorption or}

emission of photons the total number N is constant. Thus we can easily compute the time derivative of nν and of the specific intensity Iν = (c/4π)hPνnν:

dIν

dt = −3H(t)Iν (3.2) This is the evolution of Iν without interactions and has a simple solution Iν ∝

(1 + z)3_.

Introducing absorption and emission and expanding the temporal derivative we have the final equation:

∂Iν ∂t − νH ∂Iν ∂ν = −3HIν − cκνIν + c 4πν,P (3.3) where ν,P is the proper emissivity. Hereinafter we will use the comoving

emis-sivity ν = ν,P/(1 + z)3.

We can change the independent variable from t to z and integrate the previous equation: Iν(z) = c 4π(1 + z) 3 Z ∞ z dz0 1 H(z0_{)(1 + z}0₎ν0(z 0_)e−τef f(ν0,z0) _(3.4)

where ν0 = ν(1 + z0)/(1 + z) and τef f is the effective optical depth

e−τef f ₌

Z

d∆P (∆)e−τ (∆) (3.5)

P (∆) is the probability of a fluctuation ∆ in the IGM and τ (∆) is the optical depth in that fluctuation (see §3.1.2).

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3 The cosmic dawn

We can also consider a different problem (useful in the study of the 21cm radiation, §4.2.2): a background radiation passes through a cloud. We want to find the intensity after the absorption and the emission in the cloud. In this problem the Hubble expansion is not relevant and we can approximate H(z) ≈ 0, finding the usual equation for radiative transfer:

dIν

dτ = −Iν + Sν (3.6) where dτ = kνds is the differential of the optical depth and Sν = ν/kν is the

source term. If we consider the source term to be constant the solution of the previous equation is

I_ν0 = Iνe−τν + (1 − e−τν)Sν (3.7)

where I0 is the intensity exiting the cloud, I is the background intensity and τν

is the total optical depth of the cloud.

3.1.2 Gunn-Peterson optical depth

Most of the information that we have on the IGM cames from its absorption properties on sources in the background. In particular the SDSS found QSOs at redshifts as high as 7 and the analysis of their spectrum can shed light on the properties of the cosmic web introduced in §2.2.3.

If we consider a photon emitted with a wavelength λ < λα = 0.1216µm, it

will propagate through the IGM and redshift. At a particular redshift it will enter in the resonance with the Ly-α line of HI and it has a certain probability of absorption, depending on the ionization state of the IGM. Therefore we can observe the spectrum of a distant source and map the ionization state of the IGM as a function of redshift along that line of sight.

Figure 3.1 shows the spectrum of a QSO from Songaila and Cowie 2010. There are clear discontinuities in the spectrum at the Ly-α and Lyman limit rest wavelengths. Between the Ly-α and the Ly-limit there is a complex absorp-tion structure, the Lyman forest, that reveals that the IGM is not homogeneous but presents fluctuations in density and ionized fraction.

In the following we will find an expression for the optical depth (and hence the flux suppression) as a function of the neutral fraction and the local density fluctuations.

The cross section of an hydrogen atom is:

σα(ν) = 3λ2 αΛ2α 8π (ν/να)4 4π2_{(ν − ν} α)2+ Λ2α/4(ν/να)6 (3.8)

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3 The cosmic dawn

Figure 3.1: Spectrum from a z = 5 QSO, from Songaila and Cowie 2010

where Λα = 8π2_e2_f α 3mecλ2α ≈ 6.25 × 108s−1 (3.9) is the decay rate and fα = 0.4162 is the oscillator strength.

The line shape is altered by both the termal motion of the atoms and the proper velocity of the IGM perturabations. This shifts the resonance for each single atom and broadens the cumulative line through the doppler redshift. Nevertheless these doppler shifts are negligible if compared with the cosmological redshift; therefore we can approximate the line with a delta function:

σα(ν) =

3Λαλ2α

8π δ(ν − να) (3.10) This simplifies the integral for the optical depth:

τα = Z drσα(r)nHI(r) = 3Λαλ3α 8π xHInH(z) H(z) ≈ 1.6 × 10 5_x HI(1 + δ) 1 + z 4 3/2 (3.11) This expression for the Gunn-Peterson optical depth gives us a lot of informa-tions on the ionization state of the IGM, indeed the huge factor multiplying the neutral fraction ensures that even a higly ionised medium with xHI ≈ 10−3 can

be completely opaque to Ly-α photons. We will analize the implications when we will discuss the current observations on the state of the IGM, in §3.2.6.

We computed the Gunn-Peterson optical depth for the Ly-α line but we did not use any specific characteristic of it (a part from the numerical values describing the line). Therefore this calculation is valid for every line and we will see an application in §4.2.2 also for the 21cm radiation.

3.1.3 Hydrogen ionization

The ionization state of a gas is the result of the balance of two opposite pro-cesses: the ionization of the atoms and the recombination of electrons with ions.

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3 The cosmic dawn

In this subsection we will present the physics involved and we will write the differential equations for the evolution of the ionized fraction of a gas.

An hydrogen atom can be ionized in two ways: photoionization and collisional ionization. The latter is relevant only in particularly hot enviroments, like in galaxy clusters, and therefore is negligible in this context.

Photoionization happens in presence of photons more energetic than the ion-ization potential of HI (13.6eV = hPνT). The ionization rate of an atom in a UV

background with specific energy uν is

ΓHI= c Z ∞ νT dν uν hPν σ_νHI (3.12) where σ_νZ = A0 Z2 ν_T ν 4 e4−(4 tan −1_)/ 1 − e−2π/ (3.13)

is the photoionization cross section for an hydrogenic atom with atomic number Z (A0 ≈ 6.30 × 10−18cm2 and = (ν/νT − 1)1/2) .

The opposite process, recombination, presents some subtleties. We will call “case A” all the recombinations, while “case B” the recombinations to an excited state. The ionization cross section at the resonance with the Rydberg energy is so big that photons are immediatly absorbed by neutral hydrogen. This has the effect of neutralizing recombinations directly to the ground state in the IGM, and therefore these recombinations should be ignored in the equations.

On the other hand, if an atom recombines in a thick cloud with a relatively high neutral fraction, the ionizing photon generated will ionize another atom in the cloud, an so on until a recombination to a excited state happens. Therefore in these tipe of recombinations (very important at low redshift for the strong X-ray background generated by QSOs) we should consider case A recombinations.

In this work we will ignore this issue and use only case B recombinations, because at high redshift it is the best physically motivated choice and at low redshift we will use a neutral fraction derived from observations.

The case B recombination coefficient is temperature dependent:

αB(T ) = 2.6 × 10−13T4−0.76cm 3_s−1

(3.14)

and the recombination rate is simply neαB(T ). T4 is the temperature in units

of 104_{K. α}

B ≈ 0.62αA, therefore about one recombination out of three is to the

ground state.

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3 The cosmic dawn

and ionized fraction:

dxHI dt = −xHIΓHI+ xHIIneαB(T ) (3.15) dxHII dt = − dxHI dt

The equilibrium solution is:

xeq_HII = neαB(T ) ΓHI+ neαB(T )

(3.16)

The most important limiting case in this equation is

xHII ≈

neαB(T )

ΓHI

1 (3.17)

that describes the residual neutral fraction in the ionized regions.

3.2 Current understanding of reionization

In this section we will introduce the reionization scenario used in this work. Even if it is partially based on theoretical works, it is sustained by all the available data (see §3.2.6) and it is the most resonable choice so far.

We will be mainly qualitative, because in chapter 5 we will implement a simple model for reionization and the ionizing sources; therefore we will go more into detail there.

Reionization is an interaction between ionizing sources and the IGM, we will discuss the former in §3.2.1 and in §3.2.2, while the latter in §3.2.3, §3.2.4 and §3.2.5. Finally in §3.2.6 we will discuss the observational constraints that we have so far and the promising techniques for the future.

3.2.1 Ionizing sources

There is a wide range of possible sources that produce ionizing photons: massive stars, QSOs, intermediate mass black holes (miniquasars) and decaying exotic particles. Current observations seems to favour a reionization powered by stars, ruling out the other possible hypothesis:

• QSOs become extremely rare at z > 3.5 and their ionizing radiation is not sufficient to reionize the IGM already at redshifts where we know it is ionized. Therefore the accretion on very massive black holes is known to

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3 The cosmic dawn

have a marginal role in hydrogen ionization. On the other hand they are the main source for the HeII ionization, because of their harder spectrum.

• Miniquasars are another possible ionizing source; they could form as the remnants of the first massive stars and produce ionizing photons accreting material. But, for their hard spectrum, they would produce hard X-ray pho-tons and a resulting X-ray background; the current constaint on it seems to exclude black holes as a relevant source. Indeed if they where the dominant source of ionizing photons their X-ray background would be comparable to the current unresolved X-ray background; therefore they are likely to violate this constraint.

On the other side they could contribute partially to reionization and this is likely given that each low redshift galaxy has a central black hole with a mass of ≈ 10−4 of the total.

• Decaying exotic particles could be an efficient reionization mechanism. In-deed only 10−8 of the total DM rest energy is enough to reionize the IGM. On the other side they are unlikely relevant because they are supposed to produce gamma photons and a relative unobserved background.

On the other hand, the observation of high redshift galaxies (z as high as 11) supports the hypothesis that simple stars are the main source of ionizing photons. However, as we will see in chapter 5, a conclusive theoretical model of this sources is full of unconstrained parameters and it is beyond our capabilities; therefore this is the most plausible and accepted hypothesis, but a definitive proof is still lacking.

First of all, we do not know what type of stars where formed during reionization. The quantity of ionizing photons generated by stars is strongly dependent on the initial mass function (IMF) and on the metallicity; indeed only massive stars are sufficiently hot to emit in the UV and also there is an inverse relation between surface temperature and metallicity. Moreover metals are the dominant mean in the cooling of giant molecoular clouds (the site of star formation in galaxies), therefore metallicity has a great impact on the Jeans mass and on the IMF. The precise connection between metals and star formation is not quantitatively well understood, plus we do not even know observationally or theoretically the metallicity of the stars responsible for reionization.

The second uncertainty factor in the estimate of relevance of early galaxies is that we have strong uncertainties in the formation rate of new stars (the star formation rate SFR); therefore we do not know both the charactestics and the