Radiative feedback during inhomogeneous reionization

(1)

Introduction

Reionization is a milestone in the history of the universe which is still poorly understood. We know that after recombination there was a time in which all the matter was neutral and the universe was opaque to light because of the atomic absorbtion lines; then the growth of the perturbations in the dis-tribution of matter made matter itself begin to collapse and form structures. These structures emitted light, which is believed to be responsable of reion-ization of the universe and so of the end of the dark ages: after reionreion-ization the universe became transparent to light again.

The aim of this work is to analyze some of the physical mechanisms which reg-ulate reionization: in particular we are focusing on the fact that the growth of structures is different in an ionized region than in a neutral one (this effect was first pointed out by Shapiro, Giroux and Babul [1]). There are two different mechanisms causing the baryons not to collapse efficiently onto halos in ionized regions, where the temperature (typically close to 104 _{K) is}

much higher than the temperature in neutral regions. So the collapse of the baryons onto halos can be impeded in ionized regions because the pressure opposes the gravitational attraction and because the cooling processes that allow the baryons to collapse into stars are less efficient. The final effect is that halos growing in an ionized (and so hotter) region of the universe have a smaller baryonic mass than halos with the same total mass collapsing in a neutral region. This effect is expected to affect the history of reionization: it is a feedback mechanism in the sense that the more the universe is ionized, the more the accretion of baryonic matter on halos is suppressed; since the source of the ionizing radiation is the collapsing baryonic matter, further reionization is slowed down and we have a negative feedback mechanism. In this work we are going to analyze quantitavely the impact of the radia-tive feedback on the reionization history. Modelling the impact of radiaradia-tive feedback during reionization presents great difficulties: it is necessary to fol-low the evolution of the density and of the velocity fields to the non-linear regime relevant in modelling the reionization process and the spectrum of the light emitted by the first structures (responsable of the heating of the

(3)

intergalactic medium) is highly uncertain. In order to capture both the ef-fect of radiative feedback on the single halos and the global history of the reionization process, we would have to resolve the structure of single halos in performing cosmological numerical simulations with a box side ∼ 100 Mpc. This is clearly impossible for our computational possibilities, so we decide to adopt a hybrid approach:

• We focus on single halos and we look for an analytic expression of the critical mass scale at which a halo evolving in presence of a UV background have lost half of its baryons with respect to the cosmic mean.

• To implement the effect of radiative feedback, we keep track of the redshift when every single pixel of our box enters an ionized region; then we only regard halos above the critical mass scale we have found as capable of emitting ionizing photons.

When we focus on single halos we adopt a 1D spherically-symmetric grav-ity/hydrodynamic code in order to explore a wide range of the parameters space relevant to our problem, as necessary to get robust conclusions. After we have found the dependence of this mass scale on the collapse redshift, on the redshift when the ionizing UV background is turned on and on the intensity of the background, we perform a cosmological numerical simulation of the reionization process. In this way we can analyse properly the impact of radiative feedback on the global evolution of the neutral fraction and on the topology of the reionization process.

The work is divided as follows: in the first chapter we describe the theoreti-cal framework in which every cosmologitheoreti-cal study has to be set. We present the standard cosmological model of the homogeneous, isotropic universe, in-cluding the Friedmann-Robertson-Walker (FRW) metric and the Friedmann equations which describe the dynamics of expansion. Then we briefly out-line briefly the main events in the termal history of the universe, focusing in particular on recombination.

In the second chapter we try to quantify the suppression of the accretion of baryons onto halos in ionized regions and we look for a new formula that ex-presses the minimum mass of halos emitting ionizing photons. As a first step we abandon the theory of homogeneous, isotropic universe and we describe the linear growth of structures: in particular we focus on the Jeans mass effect, which makes baryonic matter not to follow the collapse of the dark matter in sufficiently small structures because of the pressure which opposes the gravitational forces. Using the linear perturbation theory we find an ex-pression for the mass scale we are looking at making the hypotheses that the

(4)

temperature of the gas is null in neutral regions and equal to 104 K in ionized ones. Then we try to extend the Jeans mass scale to the non-linear regime we are interested in: we present the theory for the non-linear collapse of spheri-cal fluctuations and we get an estimate of the mass sspheri-cale below which the gas inside halos can not efficiently cool through atomic processes. With the help of numerical simulations performed with a code by Thoul and Weinberg [2], we eventually find a formula which quantifies the suppression of the growth of baryons onto halos bigger than the threshold for efficient atomic cooling: this formula for the minimum mass of halos capable of emitting ionizing pho-tons depends on the redshift when the halo enters a ionized region and on the redshift it collapses: the basic feature of our formula is that the effect of the feedback is not instantaneous, in the sense that the suppression does not happen as soon as a halo enters a ionized region. So we propose a new model in which we use our formula to quantify the effect of radiative feedback and we also show that we are likely to overestimate the importance of radiative feedback with our model; this happens because we choose a high value of the intensity of the UV background and we neglect the effect of the self-shielding of the halos, which causes the inner regions of the halos to be reached by a lower amount of radiation.

In the third chapter, we briefly describe reionization and in particular the experimental constraints we have on it. Then, using a code by Mesinger and Furlanetto [3], we investigate which are the effects of the suppression of the collapsed baryonic mass on the history of reionization. In particular we want to compare our model with a model in which the effect of feedback is instan-taneous (i.e. the minimum mass for halos to emit ionizing photons suddenly increases when regions become ionized, extensively used in the literature). We analyze both the global evolution of the neutral fraction and the topology of the ionized bubbles to put a reliable upper limit to the importance of the feedback mechanisms on the reionization process.

(5)

Chapter 1 The standard cosmological

model

In the most popular cosmological model, the universe is regarded as homo-geneous (on a scale of 100 Mpc) and isotropic. If we look at sufficiently early times, it makes sense to use these assumptions or at most the linear pertur-bation theory: in fact the fluctuations in the cosmic microwave background radiation (CMB) are of order 10−4 [4], so we think isotropy to be a good assumption; then the galaxy surveys roughly indicate the homogeneity scale given above [5]. Instead, we obviously have to abandon these hypotheses if we are interested in studying the growth of structures which happens at later times. This introductory chapter is devoted to the study of the homogeneous, isotropic universe: as a reference, see Weinberg’s book [6].

1.1 FRW metric and redshift

With the assumptions of homogeneity ad isotropy, it is possible to show [7] that, in the reference frame of an observer at rest relative to the CMB, the universe has to be described by a Friedmann-Robertson-Walker metric

ds2 = dt2− a2_(t) dr2 1 − Kr2 + r 2_dΩ2 (1.1) where K is a term describing curvature: given our assumptions on homo-geneity and isotropy, it is possible to show that its allowed values are 1, 0 or −1. The time-dependent parameter a(t) which appears in the FRW metric describes the expansion of the universe. As usual we distinguish between the comoving and the proper coordinate system: we will call r the comoving coordinate, while the proper distance is defined putting dt = 0 in equation

(6)

(1.1).

Let us consider how the expansion of the universe is responsable of the ob-served redshift of light coming from distant sources. Photons move on null geodesics (the ones with ds2 = 0), so their motion is described by the equa-tion dt a(t) = ± dr √ 1 − Kr2 (1.2)

Let us consider a wavepacket which is emitted at time temand whose duration

is ∆tem; the wavepacket will be absorbed at time tobs and will have a duration

∆tobs. Using the equation of motion of the photons, if the emitter and the

observer are at fixed comoving coordinates, we deduce that ∆tem

a(tem)

= ∆tobs a(tobs)

(1.3) So if light with a frequency νem is emitted at time tem, now we observe a

lower frequency ν0 = νem a(tem) a(t0) ≡ νem (1 + z) (1.4)

where t0 is the present epoch; the redshift z of the source is defined by

a(t0) = a(tem)(1 + z). For low redshift sources, it is possible to derive a

simple relation between redshift and distance: this is the Hubble’s law. In this case 1 + z = a(t0) a(tem) ' a(t0) a(t0) + H0(tem− t0) ' 1 + H0 t0− tem a(t0) (1.5) where H0 = ˙a(t0)/a(t0) is the Hubble constant. Using the equation of motion

for the photons we conclude

z = H0d (1.6)

where d is the distance between the emitter and the observer (in this case there is no distinction between proper and comoving distance because we have set the normalization a(t0) = 1). Present day measurements on low

redshift sources, using equations (1.4) and (1.6), give H0 = 100h km s−1

Mpc−1, with h = 0.73 ± 0.03 [4].

1.2 Dynamics of the expansion

Now we apply the Einstein’s equation of general relativity (Rµν − 1₂Rgµν =

8πGTµν, where Rµν is the Ricci tensor and R ≡ Rµµ) to the FRW metric

(7)

it is necessary to identify the stress-energy tensor Tµν: if the fluid which

constitutes the universe has density ρ and pressure p, it is possible to derive the Friedmann equation

˙a2+ K = 8πGρa

2

3 (1.7)

and the conservation law ˙

ρ = −3 ˙a

a (ρ + p) (1.8)

If we define the Hubble constant as H(t) = ˙a(t)/a(t) (and now we see that it is not really a constant, and that what we measure is only H0 ≡ H(t0)),

from equation (1.7) we get

H2+K a2 =

8πGρ

3 (1.9)

From the equation above it is apparent that there is a critical density ρc=

3H2

8πG (1.10)

if ρ > ρc the universe is close and has a curvature K = −1, if ρ = ρc the

universe is flat and has K = 0 and if ρ < ρc the universe is open and has

K = +1. Present day measurements of H0and ρ seem to indicate that K = 0

[4]; so we are going to assume that the universe is flat, and the FRW metric (1.1) becomes

ds2 = dt2− a2_{(t) dr}2_{+ r}2_dΩ2

(1.11) In the standard cosmological model there are three types of fluid: matter (baryonic, the ordinary type of matter, and dark, which doesn’t interact with light), radiation and dark energy: matter has negligeble pressure, so from equation (1.8) we deduce that ρM(z) = ρM(0)(1 + z)3; radiation has

p = ρ/3, so ρR(z) = ρR(0)(1 + z)4; dark energy was introduced to reproduce

the experimental evidence of an accelerated expansion of the universe [8][9]: it has p = −ρ (and we have really no idea of what it is!), so ρΛ(z) = ρΛ(0)

and the density does not depend from the redshift. It is interesting to notice that, since ρΛ does not depend on the redshift and ρM ∝ (1 + z)3, the

importance of the dark energy component rapidly decreases with respect to that of the matter component: so if we are interested in high redshifts, we can neglect the presence of dark energy. Since the universe is flat we have ρM(0) + ρR(0) + ρΛ(0) = ρc; usually people write ρM(0) = ΩMρc, ρR(0) =

ΩRρc and ρΛ(0) = ΩΛρc, with ΩM + ΩR + ΩΛ = 1. The measurements

(8)

ΩΛ = 0.73 ± 0.02 and ΩR = (7.5 ± 0.4) · 10−5 [4]. The evolution with the

redshift of the three different types of fluid in the universe give

ρ(z) =ΩΛ+ ΩM(1 + z)3+ ΩR(1 + z)4 ρc (1.12)

Now it is possible to derive the value of the Hubble constant at every redshift H2(z) = H₀2ΩΛ+ ΩM(1 + z)3+ ΩR(1 + z)4

(1.13) From the equation above, using H = ˙a/a, we obtain the age of the universe at every redshift t(z) = 1 H0 Z +∞ z dz0 (1 + z0_{) [Ω} Λ+ ΩM(1 + z0)3+ ΩR(1 + z0)4] 1/2 (1.14)

With the current measurements of all the parameters, the previous formula gives a present age of the universe of about (13.7±0.2) Gyr, in agreement with the age of the most ancient structures we observe. If the matter component dominates both on the radiation and on the dark energy, it is possible to make an analytic integration of equation (1.14)

t(z) = 1 H0 Z +∞ z dz0 Ω1/2_M (1 + z0₎5/2 = 3 2H0Ω 1/2 M (1 + z) 3/2 (1.15)

Using this equation we also see that, if the universe is matter-dominated, we have t ∝ (1 + z)−3/2 and so a(t) ∝ t2/3_.

1.3 Brief thermal history of the universe

With the standard cosmological model in mind, we now outline briefly the main events in the thermal history of the universe. At the beginning, the matter in the universe was very dense and hot, and all the species of particles were in thermal equilibrium (even also with their antiparticles at sufficiently high redshifts); photons were also in thermal equilibrium with matter and their temperature was proportional to a−1. We can see why by the following reasoning: let us consider the number of photons between frequency ν and ν + dν in a volume dV , with a black-body spectrum at temperature T

nT(ν)dνdV =

8πν2

c3

1

ehν/kT _{− 1}dνdV (1.16)

As time passed, the frequency of photons decreased as a−1 because of cos-mological redshift and the volume increased as a3 _{because of the expansion}

(9)

of the universe: so the spectrum kept always the same form, with a temper-ature decreasing as a−1. At very early times nuclei were not stable. Then the temperature dropped and nuclei began to form when it reached almost 109 K: this process is called cosmological nucleosynthesis, and it produced hydrogen (with a relative abundance of 76%) and helium (with a relative abundance of 24%); we will not describe it further here. Istead, we are fo-cusing on recombination: this is the time when the matter in the universe became neutral because the protons recombined with electrons; after recom-bination the matter remained neutral untile the age of reionization, when the light emitted from the first stars ionized the universe again. The process of reionization will be the subject of the next chapters.

1.3.1 The recombination epoch

After the cosmological nucleosynthesis the matter in the universe was all ionized for a long time, and it was in thermal equilibrium with the photons due to Thompson scattering between the photons and the free electrons. In this era the temperature dropped as a−1. We can try to use the Saha equation to describe the ionization state of matter. If the reaction p + e H1s is in

equilibrium (1s is the ground state of hydrogen), the Saha equation predicts n1s npne = mekT 2π~2 −3/2 eB1/kT _(1.17)

where B1 = 13.6 eV is the bynding energy of the ground state of hydrogen,

meis the electron mass and the number densities npand neare equal because

of the charge neutrality of matter; as we have said, the theory of cosmological nucleosynthesis predicts that np+ n1s = 0.76nB, where nB is the total baryon

number density. Combining all together we get an equation for the fractional hydrogen ionization X ≡ np/(np+ n1s): X " 1 + 0.76nB mekT 2π~2 −3/2 eB1/kT_X # = 1 (1.18)

From the equation above it is possible to see that the fractional ionization of hydrogen decreases rapidly from X = 0.97 at T = 4200 K to X = 0.01 at T = 3000 K.

After the protons recombined with the electrons, the distribution of the pho-tons continued to be planckian with a temperature dropping as a−1. We expect to observe these photons now, and this is what has been discovered with the observation of the CMB: we see an almost isotropic background

(10)

radiation with a black-body spectrum at T0 = (2.725 ± 0.002) K. Since the

temperature of the radiation decreased as T (z) = T0(1 + z), we deduce that

recombination happened around z = 1100. The Saha equation predicts suc-cesfully when recombination happened, but fails in predicting the evolution of the fractional hydrogen ionization at later times: in fact in our derivation we have supposed that the reaction p + e H1s was in thermal equilibrium,

and this is no longer true when X drops to very small values. In particular this derivation fails in predicting a residual fractional ionization which is able to mantain the matter at the same temperature of radiation roughly down to z = 150 [10].

(11)

Chapter 2 The growth of structures

In this chapter we abandon the theory of the homogeneous, isotropic universe and we begin to study the development of cosmological perturbation: the growth of these perturbations, which are visible even in the CMB, determines the birth of proto-galaxies, the first structurs capable of emitting light: since this light is responsable of reionization, it is necessary to study the growth of these structures. This chapter is divided as follows: in the first section we present the linear theory of cosmological perturbations; using this linear theory we show that the growth of baryonic perturbations is suppressed with respect to the growth of dark matter perturbations (Jeans mass effect). When we study the reionization process, it is important to understand in detail the behaviour of the baryonic matter component because it is the only type of matter capable of emitting light.

The linear regime is not suitable to describe the growth of the first halos, in which the density contrast with respect to the background can be very high. So in the second part of the chapter we consider the non-linear growth of structures and we try to derive a new formula for the mass scale at which halos have lost half of their baryons with respect to the cosmic mean. We use numerical simulations performed with a code by Thoul and Weinberg [2]: to model the entrance of a halo in an ionized region we turn on a UV background radiation heating the intergalactic medium to temperatures close to 104 _K.

In the next chapter we are going to use our new formula to implement the effect of the radiative feedback on the reionization process.

2.1 The linear growth of structures

In this section we consider the linear evolution of perturbations, both of dark and baryonic matter in a matter dominated universe; as a reference,

(12)

see Weinberg’s and Padnamabahn’s books [6, 11]. This linear theory is a useful background to understand the non-linear behaviour we are going to analyze later. Using the linear theory we see that the density contrast of dark matter, in which we can neglect pressure, always grows as the scale factor a(t) (in the linear regime). Instead the perturbations in the baryonic matter component can be suppressed because pressure competes with gravity during the collapse: we find a mass scale (the Jeans mass) below which halos are almost free of baryonic matter and above which, in contrast, they are able to retain all their baryonic matter.

2.1.1 The case with no pressure

We begin to consider a non-relativistic fluid without pressure. If we consider a region much smaller than the Hubble radius c/H, we can use the equa-tions of fluid dynamics without taking into account the general relativistic corrections. The density ρ and the velocity v of any fluid element obey the continuity equation (2.1), the Euler equation (2.2) and the Poisson equation (2.3) ∂ρ ∂t + ∇ · (ρv) = 0 (2.1) ∂v ∂t + (v · ∇)v = −∇φext (2.2) ∇2φext= 4πGρ (2.3)

where φext is the external potential per unit mass acting on a fluid element.

These equations have an unperturbed solution of the form

ρ = ρ0(a0/a)3 v = HX φext= 2πGρa2X2/3 (2.4)

where H ≡ ˙a/a and a(t) satisfies the equation

˙a2+ K = 8πGρa2/3 (2.5)

with ρ0, a0 and K constants. The last equation is analogous to the Friedmann

equation (1.7), but we will not try to push this analogy too far. If now we add small perturbations δρ, δv and δφ to the unperturbed solution and we expand the equations (2.1)-(2.3) to the first order in the perturbations we find the equations describing the evolution of perturbations in a fluid without pressure

∂δρ

∂t + 3Hδρ + HX · ∇δρ + ¯ρ∇ · δv = 0 (2.6) ∂δv

(13)

∇2δφ = 4πGδρ (2.8) where ¯ρ is the density in the unperturbed solution (2.4). It is convenient to use the comoving coordinate x = X/a and to write the perturbations as Fourier transforms:

δρ(X; t) = Z

d3q eiq·X/a(t)δρq(t) (2.9)

The equations (2.6)-(2.8) become dδρq dt + 3Hδρq+ ia −1 ¯ ρq · δvq= 0 (2.10) dδvq dt + Hδvq = −ia −1_qδφ q (2.11) q2δφq = −4πGa2δρq (2.12)

There are two types of solution for equations (2.10)-(2.12): there are vector modes in which all the scalars δρq, δφq and q · δvq vanish. In this case

equations (2.10) and (2.12) are automatically satisfied, while equation (2.11) becomes

dδvq

dt + Hδvq= 0 (2.13)

The solution is δvq ∝ a−1and this mode decays: so this solution is suppressed

and it is not interesting in studying the growth of cosmological perturbations. There are also scalar modes in which the velocity can be expressed as the gradient of a scalar function u; so for the Fourier transform we have δvq =

iqδuq. Using equation (2.12) to eliminate δφq, equations (2.10) and (2.11)

give dδρq dt + 3Hδρq− a −1 ¯ ρq2δuq = 0 (2.14) dδuq dt + Hδuq = 4πGa q2 δρq (2.15)

Using equation (1.9) we see that if K = 0 we have 4πGρ = 3H2_{/2 and the}

unperturbed density is proportional to a−3; so we can eliminate δuq from

the two equations above. We get a second order differential equation for the growth of the density perturbation:

d dt t4/3 d dt δρq ρ − 2 3t −2/3 δρq ρ = 0 (2.16)

The solution of this equation is a combination of a decaying mode δρq/ρ ∝ t−1

(14)

2.1.2 The case with pressure: the Jeans mass

Now we add the contribution of pressure in equation (2.1)-(2.2) in order to understand the role of baryons in collapsing structures. If we also consider pressure the continuity equation (2.1) is unchanged, while the Euler equation (2.2) must take into account also the pressure forces:

∂v

∂t + (v · ∇)v = − ∇p

ρ − ∇φext (2.17)

We now consider a two-component fluid to reproduce the stuff of which the universe is made: it is actually composed of dark matter (a component with-out pressure) and ordinary baryonic matter.

If δρD and δρB are the perturbations in the density and δuD and δuB are the

perturbations in the velocity potential, we can write two continuity equation of the form (2.14): dδρD dt + 3HδρD − a −1 ¯ ρDq2δuD = 0 (2.18) dδρB dt + 3HδρB− a −1 ¯ ρBq2δuB = 0 (2.19)

The gravitational potential in the Euler equation must take into account of both dark and baryonic matter. We must also include pressure forces in the equation for baryonic matter; since ∂pB/∂ρB = c2s, where cs is the velocity

of sound, we can write two equation of the type (2.15): dδuD dt + HδuD = 4πGa q2 [δρD + δρB] (2.20) dδuB dt + HδuB = 4πGa q2 [δρD+ δρB] − c2_s a ¯ρB δρB (2.21)

It is convenient to introduce the density contrast δ = δρ/ρ; using the fact that the unperturbed density ¯ρ ∝ a−3 ∝ t−2 _{and eliminating the velocity}

potentials these equations become ¨ δD+ 4 3t˙δD = 2 3t2βδB+ (1 − β)δD (2.22) ¨ δB+ 4 3t˙δB= − 2α 3t2δB+ 2 3t2βδB+ (1 − β)δD (2.23) where the parameters α and β are defined by

α ≡ 3q 2_c2 st2 2a2 β ≡ ¯ ρB ¯ ρM = ΩB ΩM ' 0.17 (2.24)

(15)

In the era when baryonic matter was in thermal equilibrium with radiation (roughly for z > 150, as we have said in section (1.3.1)), α did not depend on the time because T ∝ a−1 ∝ t−2/3_{. In this case we can find a solution of}

these equations setting

δD ∝ tν δB = ξδD (2.25)

where ν and ξ are time-independent quantities (in principle they can depend on q, as ξ will actually do). We can write simple solutions which make us to gain physical insight in the problem in the limit of a dark matter-dominated universe (β 1, while in the real universe we have β = 0.17). In this limit we find three decaying modes (ν < 0) and a growing mode (ν > 0): this growing mode has

ν = 2/3 ξ = 1

1 + α (2.26)

As in the case without pressure the fractional overdensity of dark matter grows as the scale factor a, but now we see that the growth of the baryonic fractional overdensity can be suppressed. To get the mass-scale at which baryonic perturbations are suppressed, we compute the lenght scale q_J set-ting α = 1 and we calculate the corresponding mass scale

MJ = ¯ρM 2πa qJ !3 = π G !3/2 c3 s ¯ ρ1/2_M (2.27)

In halos with a mass much greater than the Jeans mass baryons collapsed with dark matter (α 1, so ξ ' 1), while in halos with a mass much less than this ar nearly free of baryons (α 1, so ξ ' 0).

When matter was in thermal equilibrium with radiation, the Jeans mass (with the cosmological parameters we are using) is about 6 · 105_M

: this

mass scale is meaningful at very high redshifts (roughly z > 150 as we have said), much higher than those we are interested in. So we need to extend the Jeans mass theory to lower redshifts, when matter has already decoupled from radiation: a first attempt will be presented in the next section.

2.1.3 The filtering mass

In the previous section we used the fact that the parameter α did not depend on time because the baryons were in thermal equilibrium with the radiation and so their temperature dropped as T ∝ a−1. This is no longer true at z < 150, when the temperature of matter began to decreases as a−2 and the Jeans mass, defined as in (2.27) putting α = 1, is now time-dependent. So,

(16)

even in the linear regime, it is no longer clear if the Jeans mass gives a good estimate of the mass at which halos have lost half of their baryons. Gnedin and Hui [12] showed that it is possible to find a solutions of equations (2.22)-(2.23) in the case of a time-dependent α. They showed that, in the limit of small q2 _{(and so of a big mass of the structure collapsing), the growth of}

baryonic matter perturbations was related to that of dark matter by δB(t; q) δD(t; q) = 1 − q 2 q2 F (2.28) where the filtering scale q_F is defined by

1 q2 F = 1 a(t) Z t 0 dt0a2(t0)¨a(t 0_{) + 2H(t}0_{) ˙a(t}0₎ q2 J(t0) Z t t0 dt00 a2_(t00₎ (2.29)

There are two main conceptual problems if we want to apply this theory to the growth of structures during reionization: above all, all this derivation relies entirely on linear theory, which is no longer applicable at sufficiently low redshifts; morover, this filtering scale is stricly meaningful (because of the derivation Gnedin and Hui proposed) only when q qF: but at this

scales halos are likely to have retained almost all their baryons, and we are interested in halos losing a significant quantity of them. In any case, it is useful to analyze the predictions of this theory for the growth of structures during reionization.

If we consider the intergalactic medium entering a ionized region at redshift zIN, we can set its temperature to be T = 0 if z > zIN and T = T (J ) later

(J is the ionizing radiation heating the ionized intergalactic medium). The temperature before the moment of reionization is not precisely zero (we have seen in equation (2.27) that even at high redshift it gives rise to a positive Jeans mass), but it is certainly much lower than temperature reached in ionized regions (typically near 104 _{K and only dependent on the intensity of}

the ionizing background radiation; as a reference see Shu’s book [13]). With such a sudden increase in the gas temperature the time-dependent Jeans mass instantaneously increases, while computing the integral (2.29) in this case we get 1 q2 F = 3 10q2 J h 1 + 4 1 + z 1 + zIN 5/2 − 5 1 + z 1 + zIN 2i (2.30) where the dependence from the temperature is inside the Jeans lenght qJ.

(17)

filtering mass scale MF = 1.1 · 1010 T 104 _K !3/2 1 + z 10 !−3/2 · · " 1 + 4 1 + z 1 + zIN !5/2 − 5 1 + z 1 + zIN !2#3/2 M (2.31)

where the temperature of the gas T depends only on the ionizing background J . An important point about the filtering mass is that MF = 0 when z ≤ zIN:

this mass scale is a continuous function of the redshift and it does not increase instantaneously as the Jeans mass: we are going to imit this two points building a formula for the critical mass at which halos have lost half of their baryons with respect to the cosmic mean. Several simulations have been performed in order to see if the filtering mass (2.31) reproduced the critical mass scale: the first simulations were performed by Gnedin [14], who gave a positive answer. But more recently Okamoto, Gao and Theuns [15] showed that this model had to be changed in order to reproduce the results of their more accurate simulations. So we decided to perform new simulations in order to understand better the dependence of the critical mass on the intensity J of the ionizing background and on the redshifts z and zIN in the

non-linear regime we are interested in.

2.2 The non-linear growth of structures

In the previous section we have seen how to calulate the mass scale at which, if the linear theory is applicable, the halos have lost half of their baryons. In this section we present the non-linear theory of the collapse of structures:

• We assume spherical simmetry and we analytically calculate the dy-namics of the collapse: this turns out to be useful both in calculating an expression for the critical mass and in studying the reionization process, the subject of the next chapter.

• Considering only the efficiency of the cooling processes, we find a for-mula for the minimum mass of halos emitting ionizing photons.

• We look for a new formula of the critical mass for halos growing in ionized regions which is valid in the non-linear regime we are interested in.

(18)

2.2.1 Non-linear spherical collapse

Let us consider a spherically symmetric shell with radius r and mass ∆m containing a total mass M ; let us suppose also that r RH, where the

Hubble radius is defined by RH = c/H, so that we can neglect the general

relativistic corrections to the newtonian equations for gravitation. The total energy of the shell is

E = 1 2∆m ˙r

2₋ GM

r ∆m (2.32)

In the initial instant of its evolution the shell is expanding following the Hubble flow, so we can write its kinetic energy K

K = 1 2∆mH 2 ir 2 i (2.33)

where Hi and ri are the initial values of the Hubble constant and of the radius

of the shell. If we assume that there is a uniform initial density contrast δi

inside the sphere we can also write the potential energy U of the shell as U = −GM ri ∆m = −4πG 3ri r_i3ρ¯i(1 + δi) ∆m = − 1 2ΩiH 2 ir 2 i (1 + δi) ∆m (2.34)

where we have used ¯ρi = Ωiρcfor the average density of matter in the universe

and the formula (1.10) to calculate the critical density. From these equations we can find the maximum radius the shell reaches before collapsing:

rM AX =

Ωi(1 + δi)

Ωi(1 + δi) − 1

ri (2.35)

We see that if we neglect the contribution of dark energy we have Ωi = 1 and

all the perturbations with δi > 0 collapse, while if Ωi < 1 only the

pertur-bations with Ωi(1 + δi) > 1 collapse. We can now solve the equation (2.32)

with our initial conditions and making the assumptions Ωi ' 1 and δi 1

(small initial perturbations in a matter-dominated universe, as appropriate to describe the situation at high redshifts). We can parametrize the solution as r = ri 2δi (1 − cos θ) (2.36) t = δ −3/2 i 2Hi (θ − sin θ) (2.37)

From equation (2.36) we see that at θ = 2π the perturbation collapses at r = 0. Using equation (2.37) and equation (1.15) to express time in function

(19)

of redshift in a universe where ΩM = 1 we find an expression for the redshift

zcoll when the collapse happens:

1 + zcoll = δi(1 + zi)

4 3

2/3

· (2π)−2/3 ' 0.356δi(1 + zi) (2.38)

We can now compare this result of the non-linear evolution with the solution of the linearized equations described in section (2.1.1). The solution of the linearized equations has a growing mode ∝ a and a decaying mode ∝ a−2/3; so the solution of the linearized equations for a static perturbation ( ˙δ = 0 at z = zi) is δlin(z) = 3 5δi 1 + zi 1 + z + 2 5δi 1 + z 1 + zi 3/2 ' 3 5δi 1 + zi 1 + z (2.39)

So from equations (2.38) and (2.39) we see that at the redshift of collapse z = zcoll the linear theory predicts a critical density contrast

δc= 1.686 (2.40)

We are going to use this result later, identifying collapsed structures in re-gions where the average density contrast is bigger than 1.686.

In our treatment of the sherical collapse we have approximated the pertur-bations as spherical, so that their final state is a single point. However in the real world the perturbations are not exactly spherical, so they end up to form virialized structures. The virial theorem asserts that 2K + U = 0 or equivalently U = 2E, where E is the conserved total energy. If we evaluate E at the moment of maximum expansion (so that E ∝ 1/rM AX) and U at

the moment of virialization (so that U ∝ 1/rvir, with the same constant of

proportionality) we get an expression for the radius at which the structure virializes: rvir = rM AX/2. From equations (2.36)-(2.37) we see that this

hap-pens at θ = 3π/2 and this corresponds to a density inside the shell we are considering

ρvir = 18π2ρ¯ (2.41)

where ¯ρ is the average density in the universe at the moment of virialization. The relation (2.41) is useful because, using equation (2.41) and the virial theorem in the form GM2_/r

vir = 3N kTvir, we can find the temperature of

the virialized, ionized gas inside a halo at some redshift z. For a mixture made by 76% of hydrogen and by 24% of helium we get

Tvir= M 108_M 1 + z 10 3/2 104 K (2.42)

(20)

This equation is important because it gives us a first intuition of a physical process which, in the non-linear regime, is very important in determining the critical mass scale we are looking for: baryons have to cool to collapse onto halos and eventually form stars, and Thoul and Weinberg [18] showed that the gas inside halos can efficiently cool through atomic processes only if Tvir & 104 K. Molecular hydrogen cooling is not considered because H2

is very fragile and it can be easily be dissociated during reionizetion [19]; however the contribution of H2 cooling is still unclear: we are going to return

on this point in section (2.3.2). So if we consider only halos with Tvir> 104 K

as capable of emitting ionizing photons, from equation (2.42) we get a mass threshold Mcoolbelow which halos can not emit ionizing photons because the

baryons can not cool inside them

Mcool= 108

1 + z 10

−3/2

M (2.43)

We conclude that halos with mass M < Mcool can’t emit ionizing photons;

we are going to use this point in the next section when we look for a new formula for the critical mass. We also point out that now we can say nothing about halos with mass M > Mcool: we will see that they also can be free of

baryonic matter if they grow in an ionized region.

2.2.2 A new formula for the critical mass

Now we derive a new formula for the critical mass scale at which a halo growing in an ionized region has lost half of its baryons with respect to the cosmic mean. We are interested in the case when we can’t apply the linear theory and we have to take into account both the effect of pressure and of the cooling and heating processes: first of all we try to understand what should be the functional form for this critical mass and eventually we are going to adjust it to fit our numerical simulations. We think that the critical mass MC should depend on the redshift z at which we calculate MC, on the

redshift zIN at which the halo enters an ionized region and on the intensity

J of the ionizing radiation heating the halo gas (in principle this flux can depend on the redshift, but in our simulations we are making the simplifying assumption of treating it as time-independent). As we have said we model the entrance of a halo in a ionized region turning on a UV background that heats the gas. Since there are several complex mechanisms to be considered (first of all the cooling processes of the baryons), we do not try to build a complete theory but we first try to understand what should be the general form for MC: we begin with its asymptotic limit, and then we consider its

(21)

behavior when z approaches zIN. Eventually we adapt it to fit the numerical

simulations.

From our simulations and from the ones performed by Mesinger and Dijk-stra [16], it seems that the dependence of the critical mass from J factors out, so that MC(J ; z; zIN) = f (J )g(z; zIN). This fact is also evident from

the formula (2.31) for the filtering mass, in which the dependence from the temperature factors out. Now we try to understand what is the form of the function g, while for the function f from now on we are going to rely entirely on our numerical simulations.

Now we consider the limit zIN z, and we make a further assumption: the

system hasn’t too much memory, so the limit is approached independently on zIN. This seems reasonable because, if a lot of time has passed since the halo

has entered an ionized region, the baryonic mass of the halo is not sensitive to the precise value of zIN. So we can write

MC(J ; z; zIN) = f (J )g1(z)g2(z; zIN) (2.44)

with g2(z; zIN) = 1 if zIN z; the function g1 gives the asymptotic limit

we are looking for. For zIN z the filtering mass computed in the previous

section is proportional to (1 + z)−3/2, so it could be attracting to assume that in the limit considered we have also g1 ∝ (1+z)−3/2. However, the simulations

we performed suggest a different asymptotic behaviour: we assume that g1

always depends on the redshift as a power law, but with a different exponent to be determined to fit the numerical simulations. So finally we assume

MC(J ; z; zIN) = M0f (J )

1 + z 10

a

g2(z; zIN) (2.45)

where M0 is a mass scale to be chosen to fit the numerical simulations; we

will see that our fit give M0 = 2.78 · 109M for a reasonable value of J , not

so far from the asymptotic limit proposed by Gnedin: so, apart the precise dependence on the two redshifts of interests, the filtering mass and the Jeans mass give a reasonable estimate of the order of magnitude of the critical mass.

Now we try to undertand what is the form of the two functions f and g2. As

we have said, for f we look only to our simulations: it seems that f (J ) follows a power law, so we assume f (J ) ∝ Jb_{. Morover, we obviously want the critical}

mass to be a continuous function of the redshift. Since the temperature of the gas before the ionizing background is turned on is much lower that its temperature at later times, we can assume that the critical mass is null at z ≤ zIN; this is also the basic fact we pointed out when we presented the

(22)

z zIN; the specific formula to be used is very uncertain: we just try to

imit the filtering mass formula (2.31) and we assume

g2(z; zIN) = 1 − 1 + z 1 + zIN cd (2.46) So the final formula we are going to use to fit the data from our numerical simulations is MC = M0 1 + z 10 a_J J0 b 1 − 1 + z 1 + zIN cd (2.47) where M0, a, b, c and d are parameters to be determined by fitting. We are

going to apply this formula only to halos with M > Mcool; instead, following

the argument of the previous section, we will consider almost baryon-free halos with M < Mcool.

2.3 Numerical simulations

In this section we are going to show the results we got running the code by Thoul and Weinberg [2]:

• We briefly describe the code we used.

• We use our results to find the all the unknown parameters of our for-mula (2.47) and we find a satisfying forfor-mula for the critical mass at which the halos have lost half of their baryons with respect to the cosmic mean.

• We derive a simple expression which gives the fraction of baryonic mass in halos whose mass is different from the critical mass.

2.3.1 Brief description of the code

Now we describe the code we used [2]: we used a one-dimensional grav-ity/hydrodynamic code which evolves a mixture of gas and collisionless dark matter to study isolated, collapsing density perturbations. The code evolves a mixture of dark matter and baryon fluids by moving concentric spherical shells of fixed mass in the radial direction. As in Dijkstra et al. [17] we start with a gaussian density profile sampled by 6000 shells for the dark matter and by 1000 shells for the baryons; we also fix the total mass M collapsing at redshift z if the perturbation evolves without pressure. The gas responds

(23)

æ æ æ æ æ ææ ææ æ ææ æ æ æ æ_æ æ æ æ æ æ æ æ æ æ æ æ æ æ à à à à à àà àà à àà à à à à_à à à à à à à à à à à à à à ì ì ì ì ìì ìì ì ìì ì ì ì ì ì ìììì_{ì ìì} ì ì_ì ì ì ì ì ò ò ò ò òò òò ò òò ò ò ò òò ò ò ò ò ò òò ò ò ò ò ò òò ZIN = 18 ZIN = 11 ZIN = 9 0 12 1 10 20 30 t tcoll R 109R

Figure 2.1: Radius of the shell versus time (in units of tcoll, the Hubble time

(1.14) at z = zcoll) in a halo of mass 1.2 · 109M collapsing at z = 6. The blue

circles show the situation without the background; in the other situations the background is turned on at zIN = 9, zIN = 11 and zIN = 18.

æ æ ææ ææ ææ æ æ ææ æ æ æ æ æ æ æ æ æ æ_æ æ æ æ æ æ æ æ à à àà àà àà à à àà à à à à à à à à à à_à à à à à à à à ì ìì ìì ìì ì ì ìì ì ì ì ì ì ì ì ì ìì_{ì ìì}ì ì ì ì ì ì ò ò òò òò ò òò ò ò ò òò ò òò ò ò ò ò ò ò ò ZIN = 13 ZIN = 8 ZIN = 7 0 12 1 10 20 30 t tcoll R 109R

Figure 2.2: Radius of the shell versus time in a halo of mass 108Mcollapsing

at z = 6. The blue circles show the situation without the background; in the other situations the background is turned on at zIN = 7, zIN = 8 and

(24)

æ æ æ æ ææ ææ ææ ææ æ æ æ æ æ æ æææ_æ æ æ æ æ æ æ æ æ à à à à àà àà àà àà à à à à à à ààà_à à à à à à à à à ì ì ì ìì ìì ìì ì ì ì ìì ì ì ìììì_{ì ì ì}ìì ìì ìì ì ò ò ò òò òò òò ò ò òò ò ò òò òò òò ò ò ZIN = 7 ZIN = 9 ZIN = 13 0 12 1 10 20 30 t tcoll R 109R

Figure 2.3: Radius of the shell versus time in a halo of mass 3·108_M

collaps-ing at z = 6. The blue circles show the situation without the background; in the other situations the background is turned on at zIN = 7, zIN = 9 and

zIN = 13.

to gravity and pressure forces, it can be heated by adiabatic compression, by shocks and by the energy input from an ionizing background, and it can cool by a variety of atomic radiative processes. As we have said we treated the ionizing background as time-independent. The spectral shape of the ionizing background we used was

J (ν) = J21(ν/νH)−5· 10−21 erg s−1 Hz−1 cm−2 sr−1 (2.48)

suitable to describe the situation at high redshift (νH is the Lyman α

fre-quency) [16, 17, 18]. In our simulations we used J21 = 0.01, J21 = 0.1 and

J21= 1, as proposed by Dijkstra et al. [17], where the highest value

overesti-mates the intensity of the background at high redshifts: we expect that the intensity of the background decreases at higher redshifts and the estimates by Haardt and Madau give, for example, J21 ∼ 0.1 at z = 1 and J21 ∼ 0.06

at z = 5 [20]. The geometry in our calculations is idealized, but the ralative simplicity of the code allowed us to explore a wide range of the parameters J , z and zIN, as necessary to get robust conclusions.

The gaseous component is described by the fluid equations of a perfect gas: there are the continuity equation (2.1), the Euler equation (2.17), the energy

(25)

equation (2.49) and the equation of state (2.50) du dt = p ρ2 B dρB dt + Γ − Λ ρB (2.49) p = (γ − 1)ρBu (2.50)

where ρB, p and u are the baryonic matter density, the pressure and the

internal energy per unit mass; γ is the adiabatic index of the gas, Γ is the heating rate and Λ is the cooling rate. The dark matter is simply treated as a non-relativistic fluid with no pressure. A detailed description of the heating and cooling rates used can be found in two works by Thoul and Weinberg [2, 18].

The effect of the ionizing background on the motion of the collapsing shells is shown in figures (2.1)-(2.3). In the first figure we show the radius of the shell containing half of the baryonic mass of the halo versus time (in units of tcoll, the Hubble time at the collapse redshift) in a 1.2 · 109M halo collapsing

at z = 6. The blue circles show the situation without the background; in the other cases the background is turned on at zIN = 9, zIN = 11 and zIN = 18,

as indicated in the figure. The second figure shows the same information for a more external shell containing the total baryonic mass in a 108_M

halo.

In this case the ionizing background is turned on at zIN = 7, zIN = 8 and

zIN = 13, as indicated in the figure. In the third figure we show again a shell

containing the total baryonic mass in a 3·108_M

halo. In this case the ionizing

background is turned on at zIN = 7, zIN = 9 and zIN = 13. The values of

zIN presented here and then used in the next section are chosen in order to

explore a large range around the redshift of reionization zreion = 10.6 ± 1.2

(calculated assuming an instantaneous, homogeneous process from WMAP data [4]). If we simulate the collapse without the ionizing background the evolution of the two shells is quite similar (the shell shown in the figure (2.2) collapse before only because it is more internal): in all the cases we see that, after a period of expansion, the shells begin to collapse. The time between the beginning of the evolution and the turn-around point is exactly half of the time from the beginning to the final collapse to a single point: it is the same result one would get from the analytic solution of the evolution of a spherical-symmetric shell of pressureless gas so, in the mass range we are looking at, we identify the situations without the ionizing background and without pressure.

The situation is radically different if the background is turned on: for the biggest halo the collapse of the shell is simply delayed, while for smaller halos the collapse is stopped at all, and shells begin to expand again after the ionizing background is turned on. If we choose other shells the behaviour

(26)

is similar: fixing the mass of the halo, the collapse of the inner shells is generally only delayed, while the collapse of the more external ones can be stopped and these shells begin to expand again after the background is turned on. We conclude that the general behaviour of the collapsing shells is that, in presence of an UV background, the collapse of the more external shells in light halos is stopped and these shells begin to expand again, while the collapse of the inner shells in massive halos is simply delayed. In any case the fraction of the baryonic mass collapsing at a given redshift significantly decreases in small halos in the presence of a UV background.

2.3.2 Results of the simulations

The aim of this section is to present all the main results of the new simulations we performed. In order to identify the fraction of baryonic matter collapsing on halos we ran the code with and without the ionizing background: we looked at the trajectories of various shells and we found the number of shells collapsing at redshift z in the two cases. The ratio fb of the shells collapsing

in the two situations gives the fraction of baryonic matter collapsing on the halo with respect to the cosmic mean (in the code we used the mass between a shell and the next one is a constant); we eventually identified MC by the

condition fb = 1/2.

In the first part of this section we fit all the unknown parameters in our formula (2.47) for the critical mass. After we find the final expression for the critical mass MC we look for another formula giving the fraction fb of

baryonic matter collapsing in halos with mass different from MC. In the last

part of the section we conclude our work writing down the final expression for the minimum mass MM IN of halos emitting ionizing photons: we are going

to use this formula in the next chapter when we implement the effect of the radiative feedback on the reionization process.

Fitting the parameters of MC

Now we are going to find all the parameters of the formula for the critical mass (2.47). The dependence of the critical mass on the redshift is shown in the figures below: for the data shown in figures (2.4)-(2.9) the ionizing background was turned on at zIN = 99, zIN = 19, zIN = 16, zIN = 13,

zIN = 10, zIN = 9 respectively. In all the figures we show the fitting formula

of the form (2.47) we found

MC = 2.78 · 109 1 + z 10 −2.11 J₂₁0.17 " 1 − 1 + z 1 + zIN 2.02#2.45 M (2.51)

(27)

æ æ æ æ à à à à ì ì ì ì

Z

IN

= 99

6

7

8

9

10

11

12

10

8

10

9

Z

M

Figure 2.4: Critical mass versus redshift with zIN = 99. Dot-dashed : J21 =

0.01. Dashed : J21= 0.1. Dotted : J21 = 1. æ æ æ æ à à à à ì ì ì ì

Z

IN

= 19

6

7

8

9

10

11

12

10

8

10

9

Z

M

(28)

æ æ æ æ à à à à ì ì ì ì

Z

IN

= 16

6

7

8

9

10

11

12

10

8

10

9

Z

M

0.01. Dashed : J21= 0.1. Dotted : J21 = 1. æ æ æ æ æ à à à à à ì ì ì ì ì

Z

IN

= 13

6

7

8

9

10

8

10

9

Z

M

(29)

æ æ æ æ æ à à à à à ì ì ì ì ì

Z

IN

= 10

6

6.5

7

7.5

8

10

8

10

9

Z

M

0.01. Dashed : J21= 0.1. Dotted : J21 = 1. æ æ æ æ à à à à ì ì ì ì

Z

IN

= 9

6

6.5

7

7.5

10

8

10

9

Z

M

(30)

In all the figures we show three different values of the UV background inten-sity J21: with the dot-dashed line we show J21 = 0.01, with the dashed line

J21= 0.1 and with the dotted line J21 = 1.

A formula for fb(M )

Now we are going to find a formula that, given the critical mass MC at

some redshift, reproduces the fraction of baryonic mass fb which collapses

on a halo with a mass different from MC (normalized to the situation with

no ionizing background, which we found to be equal to the cosmic mean). Clearly we have to find fb(M ) = 1 if M MC (if the mass of the halo is

much bigger than the critical mass, the gravitational attraction dominates the pressure forces and all the baryonic matter collapses) and fb(M ) = 0 if

M MC (if the mass of the halo is much smaller than the critical mass,

the gravitational attraction is much weaker than the pressure forces and the baryons can not collapse); by definition we have fb(MC) = 1/2. In the figures

(2.10) and (2.11) we show the baryon collapse fraction we got turning on the ionizing background at zIN = 16 versus the mass of the halos in units of M,

with different choices of the collapse redshift and of the ionizing background (zcoll = 10 and J21= 0.01 in figure (2.10), zcoll = 10 and J21 = 0.01 in figure

(2.11), as indicated on the panels). We show the same things in the figures (2.12) and (2.13), but in this case the ionizing background was turned on at zIN = 19 (here zcoll = 12 and J21 = 0.01 in figure (2.12), zcoll = 10 and

J21 = 0.01 in figure (2.13), as indicated on the panels). In all the figures we

show the fitting formula we found for fb

fb(M ) = 2−MC/M (2.52)

In the form this formula is quite different from the one originally proposed by Gnedin to calculate fb [14], but plotting fb versus M with the two different

choices gives very similar results. Final remarks

To efficiently implement the effect of the radiative feedback in our cosmolog-ical simulation of the reionization process we have to estimate the minimum mass MM IN of halos that can emit ionizing photons. We can not

extrapo-late the formula for the critical mass (2.51) that we have got using Thoul and Weiberg’s code and then apply it for halos with mass smaller than the threshold Mcool (2.43) for efficient atomic cooling: in fact the code neglects

all cooling mechanisms involving molecular hydrogen and the survival possi-bilities of the H2 molecule inside halos during reionization are uncertain; so

(31)

æ æ æ æ æ æ æ ZIN = 16 Zcoll= 10 J21 = 0.01 108 109 1010 0 0.5 1 M M fb

Figure 2.10: Baryon collapse fraction versus halo mass; the ionizing back-ground is turned on at zIN = 16. The collapse redshift is zcoll = 10 and the

intensity of the ionizing background is J21= 0.01.

æ æ æ æ ææ æ æ ZIN = 16 Zcoll= 12 J21 = 0.1 108 109 1010 0 0.5 1 M M fb

(32)

æ æ æ ææ æ æ ZIN = 19 Zcoll= 12 J21 = 0.01 108 109 1010 0 0.5 1 M M fb

intensity of the ionizing background is J21= 0.01.

æ æ æææ æ æ æ ZIN = 19 Zcoll = 12 J21 = 0.1 108 109 1010 0 0.5 1 M M fb

(33)

we can only apply our formula for the critical mass to halos with M > Mcool.

Since we can not model the presence of molecular hydrogen during reioniza-tion self-consistently, we just assume that halos with M < Mcool does not

emit ionizing photons. So, since the cutoff in the formula (2.52) is very sharp, we are going to consider only the halos whose mass is bigger than the max-imum between Mcool (2.43) and MC (2.51) as capable of emitting ionizing

photons:

MM IN = max (Mcool, MC) (2.53)

2.4 Limitations of our method

Now we discuss briefly our assumptions. We have only considered a time-independent J21; this is certainly a rough assumption, but we don’t think

that using a time dependent background gives radically different results: in fact we have found a very weak dependence of the critical mass from the value of J21 (a variation of J21 of two orders of magnitude produces only a

multiplication of the critical mass by a factor ∼ 2.3). The value J21 = 1

is probably too big to describe the situation at high redshifts as discussed in section (2.3.1); since we have no precise idea of what is the precise value of J21 at these redshifts, we just explore a wide range of intensities. Also

about this point we have to remark that changing J21 does not alter our

results substantially. In any case in the continuation of the work we are always considering the case J21 = 1, so that we can put an upper limit on

the importance of the feedback effect.

There are two further assumptions to be discussed. The code by Thoul and Weinberg neglects colling mechanisms through molecular hydrogen, which are expected to be important in small halos (roughly the ones with a mass less than 108_M

): several works suggest that these halos are unimportant

to the process of reionization because molecular hydrogen is very fragile and is expected to be easily dissociated in the presence of a ionizing background [19]; however there is still not any work considering in detail the contribution of these small halos to the reionization process. There is another important point, which is the most radical assumptions we do: we neglect the self-shielding of the halos and we assume an isotropic background. The effect of self-shielding can decrease a lot the ionizing radiation reaching the inner parts of the structures, where the stars form, and so reduce the importance of the feedback mechanism we are studying. The reasons for ignoring self-shielding in our calculations are (i ) it speeds up the simulations: properly accounting for self-shielding requires one to compute the increasing hardness of the spectrum as one moves deeper into the cloud; (ii ) in more realistic 3D

(34)

æ æ æ æ à à à à ì ì ì ì ò ò ò ò ò ò 6 6.5 7 7.5 8 8.5 108 109 Z M M

Figure 2.14: Critical mass versus redshift calculated with different models. Green triangles: results of the model by Okamoto [15]. Dot-dashed : our model with J21 = 0.01. Dashed : our model with J21 = 0.1. Dotted : our

model with J21 = 1. We also show (long-dashed ) the threshold (2.43) above

which halos can cool through atomic processes.

models of collapsing gas clouds, ionizing photons may penetrate deeper into the cloud via paths of lower HI column densities than one expects from a 1D calculation. Hence, if one accounts for self-shielding using a 1D simulation, one may underestimate the impact of the UVB, while throughout this paper we have conservatively chosen to overestimate the impact of the UVB. For example, Dijkstra et al. found that ∼ 95% of the gas was able to collapse into dark matter halos with a mass of ∼ 108M that collapsed at z = 11

for zIN = 17 and J21 = 0.01 when self-shielding was included. This is a

significant rise compared to the ∼ 10% fraction that was found for a model in which self-shielding was ignored [17]. Since this is the typical mass scale we are interested in, we think that the effect of self-shielding can significantly decrease the impact of the radiative feedback.

It is also useful to compare our results with those found by Okamoto, Gao and Theuns [15], although their results are surprisingly low. In this work they calculated the dependence on the redshift of the critical mass assum-ing zIN = 9 and an UV background from Haardt and Madau [20]: in figure

(2.14) we show with the green triangles the values of the critical mass ver-sus redshift calculated by Okamoto and the results of our model (2.51) with different values of J21 (the dot-dashed line when J21 = 0.01, the dashed line

(35)

when J21 = 0.1 and the dotted line when J21 = 1), assuming zIN = 9. We

also show the threshold (2.43) above which halos can cool through atomic processes with the long-dashed line.

Our model clearly overestimates the importance of the suppression of the baryonic mass for high values of the critical mass, as expected from the pre-vious discussion, but it underestimates the value of the critical mass found by Okamoto at redshifts soon after zIN, when the critical mass is small.

The region where we begin to underestimate the value of the critical mass is the region where its value becomes smaller than the threshold for efficient atomic cooling: so we think that we underestimate the value of the critical mass when it is small because the code we used neglects the effect of molec-ular hydrogen cooling.

As we have said in section (2.3.2), when we model reionization we consider only the emission of photons from halos that can cool through atomic pro-cesses, so in the mass range we are studying we are overestimating the im-portance of this radiative feedback mechanism: in fact the entrance of a halo in a HII region is modelled turning on an UV background with J21= 1 (and

sometimes with J21= 0.01 for comparison). With the UV background (2.48)

we are using, this corresponds to set a temperature close to 104 K in ionized regions. The equilibrium temperature is evaluated putting Γ = Λ, where Γ and Λ are the heating and the cooling rates in equation (2.49) [18].

In figure (2.14) the dotted line (critical mass with our model when J21 = 1)

is always above Okamoto’s data when they are bigger than the long-dashed line (threshold for molecular cooling), so we are likely to overestimate the importance of the radiative feedback.

So there are two possible scenarios: if we find that the feedback mechanisms have a negligeble impact on the history of reionization also when J21 = 1,

our result will be quite safe; otherwise, if we conclude that they are playing an important role, further investigation will be needed. In particular the impact of self-shielding and a better model for the UV background intensity to model the situation in ionized bubbles at high redshift will be needed.

(36)

Chapter 3 The reionization process

In this chapter we are going to study the reionization process and to analyze the impact of the feedback mechanism we have just described. In the first part of the chapter we are giving an introduction to the reionization process: we are focusing on the experimental constraints we have on it (we will see that we can say when reionization finished and when its middle point was) and on the future observational perspectives; then we are describing an analytic model of reionization. In the last part of the work we are going to present the results of our new simulations, performed with a modified version of the code by Mesinger and Furlanetto [3], and to investigate in detail the impact of the feedback process we are interested in.

3.1 Experimental probes of reionization

As we have said, we can say when reionization finished and when its mid-dle point was. The first problem is solved by the Gunn-Peterson effect [21]: observing high redshift quasars we notice that spectra from quasars from red-shifts higher than a critical value exibit a strong absorbtion; at sufficiently high redshifts in the universe there was enough neutral hydrogen to absorb the light from the quasars through atomic lines.

The second problem is solved by WMAP observations of the optical depth to reionization [4]: after reionization there were free electrons in the universe, and so the CMB photons suffered Thompson scattering producing addic-tional anisotropies in the CMB; the measurement of the amplitude of these anisotropies gives an integral estimate of the quantity of neutral matter in the universe.

We are now going to describe in detail both these processes. Then in the last part of this section we are going to present the main possible future

(37)

developments of the experimental probes of the reionization epoch; the most interesting experiments in preparation are those aiming to measure the 21 cm radiation [22]: this is the hyperfine emission line of neutral hydrogen, and its measurements would provide a constraint on the quantity of neutral hydrogen in the universe at every redshift.

3.1.1 The Gunn-Peterson effect

Let us suppose that a source emits light at some time t1 with an

inten-sity I1. Along its journey to us the frequency of the light will redshift as

ν(t) = ν1a(t1)/a(t); if the intergalactic medium absorbs light at a rate Λ(ν; t),

taking into account both absorbtion and stimulated emission we can write an equation for the intensity I of the light

˙

I(t) = − 1 − e−hν1a(t1)/kT (t)a(t) Λ (ν

1a(t1)/a(t); t) I(t) (3.1)

The solution of the equation above gives an observed intensity

I(t0) = e−τI(t1) (3.2)

with an optical depth τ =

Z t1

t0

1 − e−hν1a(t1)/kT (t)a(t) Λ (ν

1a(t1)/a(t); t) dt (3.3)

The absorbtion rate is given by Λ(ν; t) = n(t)σ(ν) where n(t) is the number density of absorbing neutral atoms and σ(ν) is the absorbtion cross section. Often the absorbtion cross section is sharply peaked at a frequency νR, so the

absorbtion only take place at a time tR = ν1a(t1)/νR. Thus we can simplify

the expression (3.3) getting

τ ' n(tR) 1 − e−hνR/kT (tR)

Z t1

t0

σ (ν1a(t1)/a(t)) dt (3.4)

Changing now the variable of integration from the time to the frequency we eventually get τ ' n(tR) 1 − e−hνR/kT (tR) ˙a(tR) a(tR) 1 νR Z σ(ν)dν (3.5)

where the integral is taken over a small range of frequencies containing the absorbtion line. If we focus on the Lyman α line (νR ≡ νH) we see that

(38)

Figure 3.1: Observed intensity versus wavenlenght for four high-redshift quasars [23]. In the bottom panel the spectrum of the z = 6.28 quasar is shown: it exibits complete suppression in the range of frequencies discussed in the text. The vertical dashed lines indicate the redshifted wavenlenghts of various spectral lines.

(39)

the intergalactic medium; so the exponential factor in the optical depth is close to zero and

τ ' n(tR) ˙a(tR) a(tR) 1 νH Z σ(ν)dν (3.6)

For a source at redshift z, if the intergalactic medium is neutral, the observed spectrum is completely suppressed at νH/(1 + z) ≤ ν0 ≤ νH. From equation

(3.6) it is possible to see that if, for example, there were a fraction f of neutral hydrogen at z = 5, the corresponding optical depth for the Lyman α line is 3.8 · 105f : so even a very small fraction of neutral hydrogen gives a large suppression in the spectrum.

For many years the search for the Lyman α absorbtion was unsuccesful. In 2001 a quasar with redshift 6.28 was discovered by the Sloan Digital Sky Survey, and it was found to show a complete absorbtion in the range of wavenlenghts between 8845 ˚A (the redshifted Lyman α frequency) and 8450 ˚A, indicating a significant fraction of neutral hydrogen down to redshift 8450/1215 − 1 = 5.95. So a redshift of order 6 can be considered as the end of the dark ages [23]. The spectra of some quasars observed at redshifts around z ∼ 6 are shown in figure (3.1). In particular in the bottom panel the spectrum of the z = 6.28 quasar is shown, and it exibits complete suppression in the range of frequencies discussed above; the vertical dashed lines indicate the redshifted wavenlenghts of various spectral lines.

3.1.2 The optical depth to reionization

Let us suppose that reionization happens instantaneously at some redshift zR. We now calculate the corresponding optical depth for Thompson

scatter-ing, which is responsable of the birth of secondary anisotropies in the CMB (these anisotropies are called secondary to distinguish them from the primary anisotropies, generated at the epoch of recombination). The optical depth over a proper lenght dl is dτ = nσTdl, where σT is the Thompson scattering

cross section and n = n0(1 + z)3 the number density of free electrons; so if

we assume that reionization happens instantaneously at time tR, the total

optical depth will be τ = Z t0 tR σTn(t)cdt = σTn0c Z t0 tR (1 + z)3dt (3.7)

where n0 is the present number density of electrons. We can simplify this

expression using equations (1.13) and (1.14): using the fact that

dt = dz

Radiative feedback during inhomogeneous reionization

Contents

Introduction

Chapter 1

The standard cosmological

model

1.1

FRW metric and redshift

1.2

Dynamics of the expansion

1.3

Brief thermal history of the universe

1.3.1

The recombination epoch

Chapter 2

The growth of structures

2.1

The linear growth of structures

2.1.1

The case with no pressure

2.1.2

The case with pressure: the Jeans mass

2.1.3

The filtering mass

2.2

The non-linear growth of structures

2.2.1

Non-linear spherical collapse

2.2.2

A new formula for the critical mass

2.3

Numerical simulations

2.3.1

Brief description of the code

2.3.2

Results of the simulations

Z

= 99

6

7

8

9

10

11

12

10

10

Z

M

M

Z

= 19

6

7

8

9

10

11

12

10

10

Z

M

M

Z

= 16

6

7

8

9

10

11

12

10

10

Z

M

M

Z

= 13