Title: A Linear Perturbation Theory of Inhomogeneous Reionization
1A 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
3Introduction
- Outline
- Observations
- Existing Theoretical Methods
- Our Method
4Introduction
Reionization
Credit Djorgovski et al, Caltech.
5Introduction
- Current Constraints on Reionization
High redshift quasar spectra
WMAP
Credit WMAP team SDSS team
6Introduction
- Future Constraints on Reionization
Credit Chicago/MSFC S-Z group
21cm emission from neutral hydrogen
SZ effect from CMB
CMB polarizations
7Introduction
- A Key Thing to Understand
- The distribution of HI/HII (neutral/ionized
hydrogen)
- Existing Theoretical Methods
- Analytic models
- Numerical simulations
8Introduction
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.)
9Introduction
- 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.
10Introduction
- Problems in existing analytic models
- Assuming ionization topology
- Not good for general source spectra.
11Introduction
- Numerical Simulations on Large Scales
- Kohler et al. (2005)
- Iliev et al. (2005 2006)
- Zahn et al. (2006)
- Mellema et al. (2006)
12Introduction
- Problems For Numerical Simulations
- Hard to cover large dynamical range
- Expensive to explore a large parameter space
- Not enough for understanding physics.
13Introduction
- 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
14Our Method
- Outline
- Equations
- How to Solve the Equations
- Some Caveats
15Our Method
Ionized Hydrogen
Total Hydrogen
Photon
Source Emissivity
16Our Method
Peculiar Velocity
1. Ionization Balance
Number density of HII
Recombination
Ionization
2. Radiative Transfer
Diffusion
Radiation Intensity
Redshift
17Our Method
1. Ionization Balance
18Our Method
1. Ionization Balance
Peculiar Velocity
Photo-Ionization
Recombination
19Our Method
2. Radiative Transfer
20Our Method
Diffusion
2. Radiative Transfer
Redshift
Source
Photo-Ionization
21Our 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
22Our 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
23Our 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.
24Our 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
25Our Method
- 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
26Our Method
- 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.
27Our Method
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
)
28Our Method
C?H(1) C?H(2) 1 for all cases
CHII
10 if ß - 3 or - 2 1 if ß - 1
.
29Our Method
- Neglect the High Order Terms
- Neglect Helium Reionization
- Neglect the Temperature Dependence of the
Recombination Rate
30Results
fHII lt n HII gt / lt n H gt
- The evolution of the ionized fraction fHII as a
function of redshift z.
31Results
- The bias factors of the HII regions with respect
to the dark matter at k0.01Mpc-1 .
32Results
z 13 z 10 z 9
- The HII bias as a function of scale.
33Results
f HI 1 - f HII
- The redshift dependence of the HI bias
(multiplied by the mean neutral fraction) at
k0.01Mpc-1.
34Results
z 9.8 z 13.6 z 18.7
- The spectrum of the radiation background.
35Results
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.
36Results
Surprise No.1 Why does the bias of the soft
photons shots up quickly?
37Results
Surprise No.2 Why does the bias of the soft
photons oscillates after the reionization is
complete?
38Discussion
- Outline
- Ionization Topology
- Some Prospects
39Discussion
- The Clumping Factors
- Set by hand
- Only appear in the 0th order equations
- Change quantitative details, but not
qualitative behaviors
40Discussion
- 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.
41Discussion
- The Ionization Topology
- Inside-out even for hard source spectrum
- Inside-out even if the source bias is lowered
42Discussion
- 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 .
43Discussion
- 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.
44Summary 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
45Prospects
- 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
46Prospects
- To understand the high order terms
100 Mpc
47Prospects
- To understand the high order terms
100 Mpc