A Linear Perturbation Theory of Inhomogeneous Reionization

1
A Linear Perturbation Theory of Inhomogeneous
Reionization
Jun Zhang (UC Berkeley) Lam Hui, Zoltan Haiman
(Columbia Univ.)
Journal Reference MNRAS, 375, 324 (2007)
2

• Introduction
• Our Method of Studying Reionization
• Results
• Summary Prospects

3
Introduction

• Outline
• Observations
• Existing Theoretical Methods
• Our Method

4
Introduction
Reionization
Credit Djorgovski et al, Caltech.
5
Introduction

• Current Constraints on Reionization

High redshift quasar spectra
WMAP
Credit WMAP team SDSS team
6
Introduction

• Future Constraints on Reionization

Credit Chicago/MSFC S-Z group
21cm emission from neutral hydrogen
SZ effect from CMB
CMB polarizations
7
Introduction

• A Key Thing to Understand
• The distribution of HI/HII (neutral/ionized
  hydrogen)

• Existing Theoretical Methods

• Analytic models
• Numerical simulations

8
Introduction

• Existing methods

• Analytic models

Inside - Out
Outside - In
neutral
ionized
Soft Source Spectrum (eg. Haiman Loeb 1997)
Hard Source Spectrum (eg. Miralda-Escudé et al.
2000, Oh 2001, Venkatesan et al. 2001.)
9
Introduction

• Existing methods

• More recent analytic models
• Furlanetto, Zaldarriaga Hernquist (2004)
  includes source clusterings.
• Furlanetto Oh (2005)
• Includes recombinations.
• Cohn Chang (2006)
• Considers the role of major mergers in modifying
  the photon production rate.

10
Introduction

• Existing methods

• Problems in existing analytic models
• Assuming ionization topology
• Not good for general source spectra.

11
Introduction

• Existing methods

• Numerical Simulations on Large Scales
• Kohler et al. (2005)
• Iliev et al. (2005 2006)
• Zahn et al. (2006)
• Mellema et al. (2006)

12
Introduction

• Existing methods

• Problems For Numerical Simulations
• Hard to cover large dynamical range
• Expensive to explore a large parameter space
• Not enough for understanding physics.

13
Introduction

• Our Method A Linear Perturbation Theory

• Advantages
• From first principles ( the ionization topology
  is not assumed from the outset)
• Good for sources of general spectral shapes and
  clustering properties
• Good for studying HI/HII properties on large
  scales (when combined with small scale
  simulations)
• Taking into account most of the relevant
  physical processes

14
Our Method

• Outline
• Equations
• How to Solve the Equations
• Some Caveats

15
Our Method

• Definitions

Ionized Hydrogen
Total Hydrogen
Photon
Source Emissivity
16
Our Method

• Equations

Peculiar Velocity
1. Ionization Balance
Number density of HII
Recombination
Ionization
2. Radiative Transfer
Diffusion
Radiation Intensity
Redshift
17
Our Method

• Equations

1. Ionization Balance
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Our Method

• Equations

1. Ionization Balance
Peculiar Velocity
Photo-Ionization
Recombination
19
Our Method

• Equations

2. Radiative Transfer
20
Our Method

• Equations

Diffusion
2. Radiative Transfer
Redshift
Source
Photo-Ionization
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Our Method

• Linear Perturbation Theory (example)

n (x,t) lt n (t) gt dn (x, t)
0th Order
n1 (x,t) n2 (x,t) lt n1 (t) gt lt n2 (t)
gt lt dn1 (x, t) dn2 (x, t) gt dn1 (x, t) lt
n2 (t) gt dn2 (x, t) lt n1 (t) gt dn1 (x, t)
dn2 (x, t) - lt dn1 (x, t) dn2 (x, t) gt
1st Order
2nd Order
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Our Method

• Linear Perturbation Theory (continued)

f (n1, n2, ) 0
f (n1, n2, ) lt f (n1, n2, ) gt S ?if
dni
lt f (n1, n2, ) gt 0 S ?if dni 0
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Our Method

• How do we solve the equations ?

• Fourier transform
• Neglect the high order terms on large scales
• Neglect the multiple moments of the source
  emissivity on large scales.

24
Our Method

• How do we solve the equations ?

Goal to solve for lt nHII gt , lt n ? gt , d nHII
, d n ? .
Using Radiative Transfer Equation
Step 1 ( 0th Order)
lt nHII gt
lt n ? gt
Using Ionization Balance Equation
Using Radiative Transfer Equation
Step 2 (1st Order)
d nHII
d n ?
Using Ionization Balance Equation
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Our Method

• Input No.1

• Model Source Clustering

• Minimum halo mass Tvir 104 K
• Constant photon/baryon ratio in collapsed
  objects
• Extended Press-Schechter model to relate the
  collapsed fraction with the mass overdensity

26
Our Method

• Input No.2

• Source Spectrum

• The spectral shape E? ? ? ß
• Consider three cases ß - 1, - 2, - 3
• The total emissivity Chosen to fix the mean
  optical depth at 0.088, suggested by WMAP 3-year
  data.

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Our Method

• Input No.3

• The Clumping Factors

lt n1 n2 gt
lt dn1 d n2 gt
C12
1
lt n1 gt lt n2 gt
lt n1 gt lt n2 gt
In Our Calculations
CHII lt nHII2 gt / lt nHII gt 2 C?H(1) lt
nHI n ? ? gt / ( lt nHI gt lt n ? gt lt ? gt
) C?H(2) lt nHI n ? gt / ( lt nHI gt lt n ? gt
)
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Our Method

• Input No.3

• The Clumping Factors

C?H(1) C?H(2) 1 for all cases
CHII
10 if ß - 3 or - 2 1 if ß - 1
.
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Our Method

• Some Caveats

• Neglect the High Order Terms
• Neglect Helium Reionization
• Neglect the Temperature Dependence of the
  Recombination Rate

30
Results
fHII lt n HII gt / lt n H gt

• The evolution of the ionized fraction fHII as a
  function of redshift z.

31
Results

• The bias factors of the HII regions with respect
  to the dark matter at k0.01Mpc-1 .

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Results
z 13 z 10 z 9

• The HII bias as a function of scale.

33
Results
f HI 1 - f HII

• The redshift dependence of the HI bias
  (multiplied by the mean neutral fraction) at
  k0.01Mpc-1.

34
Results
z 9.8 z 13.6 z 18.7

• The spectrum of the radiation background.

35
Results
E 25KeV E 580eV E 170eV
Left k0.01Mpc-1 Right k0.1Mpc-1

• The bias in the ionizing background photon
  density as a function of redshift.

36
Results
Surprise No.1 Why does the bias of the soft
photons shots up quickly?
37
Results
Surprise No.2 Why does the bias of the soft
photons oscillates after the reionization is
complete?
38
Discussion

• Outline
• Ionization Topology
• Some Prospects

39
Discussion

• The Clumping Factors
• Set by hand
• Only appear in the 0th order equations
• Change quantitative details, but not
  qualitative behaviors

40
Discussion

• The Ionization Topology
• It is determined by the sign of lt d dXgt , where
  dX is the fluctuation in the ionized fraction.
• Since lt d dX gt lt d ( dHII - d ) gt , the sign
  of lt d dX gt is determined by whether the HII bias
  is larger or smaller than one.
• It is a scale - dependent statement.

41
Discussion

• The Ionization Topology
• Inside-out even for hard source spectrum
• Inside-out even if the source bias is lowered

42
Discussion

• The Ionization Topology for hard source spectrum
  

• The HII bias as a function of mean ionized
  fraction. The solid, dotted, and dashed curves
  correspond to energy cut-offs at 170eV, 270eV,
  and 450eV respectively in the ß - 1 source
  spectrum .

43
Discussion

• The Ionization Topology for a low source bias

• The HII bias as a function of redshift in the
  model with ß - 3 source spectrum, but assuming
  an unbiased source population.

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Summary Prospects

• Developed a perturbation theory of cosmic
  reionization
• From first principles
• Good for general source properties
• Using the Press-Schechter model for the source
  clustering, and three power-law forms for the
  source spectral shape, we found
• Soft source spectrum the HII bias remain high
  for long
• Hard source spectrum the HII bias drops
  continuously

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Prospects

• To understand the high order terms
• To include helium ionization
• To include stochastic source bias
• To include feedback effects
• To study further about putting constraints on
  source properties and cosmology

46
Prospects

• To understand the high order terms

100 Mpc
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Prospects

• To understand the high order terms

100 Mpc