Cosmic 21-cm Fluctuations from Dark-Age Gas Kris Sigurdson Institute for Advanced Study Cosmo 2006 September 25, 2006 Kris Sigurdson Institute for Advanced

Cosmic 21-cm Fluctuations from Dark-Age GasCosmic 21-cm Fluctuations from Dark-Age Gas

Kris Sigurdson

Institute for Advanced Study

Cosmo 2006

September 25, 2006

Kris Sigurdson


Cosmo 2006

September 25, 2006

What are the properties of neutral hydrogen atoms 20 to 100 million years after the big bang?

How do we calculate their observational signatures?

What are the properties of neutral hydrogen atoms 20 to 100 million years after the big bang?

How do we calculate their observational signatures?

Kris Sigurdson


Cosmo 2006

September 25, 2006

Kris Sigurdson


Cosmo 2006

September 25, 2006

C. Hirata and KS (astro-ph/0605071)

Cosmic 21-cm Fluctuations: Why?

Cosmic 21-cm Fluctuations: Why?

• The Epoch of Reionization (e.g. Furlanetto et. al 2004). (Covered by Steve, Peng, and Miguel.)

• Measure the Primordial Power Spectrum at high redshift! 3D instead of a 2D CMB. (e.g. Loeb and Zaldarriaga 2004)

• Another probe of Inflation; exotic particle physics effects on the Matter Power Spectrum. (e.g. KS and Cooray 2005; Profumo, KS, Ullio and Kamionkowski 2004)

• If measured they will leave us with an embarrassment of riches the likes of which the world has never seen!

• The Epoch of Reionization (e.g. Furlanetto et. al 2004). (Covered by Steve, Peng, and Miguel.)

• Measure the Primordial Power Spectrum at high redshift! 3D instead of a 2D CMB. (e.g. Loeb and Zaldarriaga 2004)

• Another probe of Inflation; exotic particle physics effects on the Matter Power Spectrum. (e.g. KS and Cooray 2005; Profumo, KS, Ullio and Kamionkowski 2004)

• If measured they will leave us with an embarrassment of riches the likes of which the world has never seen!

What I am not talking about.What I am not talking about.

• 21-cm fluctuations from the epoch of reionization (EOR).

(Steve, Peng, and Miguel will cover or already covered that!)

• 21-cm fluctuations from the epoch of reionization (EOR).

(Steve, Peng, and Miguel will cover or already covered that!)

What I am talking about.What I am talking about.

• 21-cm fluctuations before reionization physics becomes important. Bewtween recombination and reionization.

• Smooth, slightly lumpy Universe.

• Main Players: Neutral Gas and the CMB

• Roughly Speaking 20 < z < 100

• 21-cm fluctuations before reionization physics becomes important. Bewtween recombination and reionization.

• Smooth, slightly lumpy Universe.

• Main Players: Neutral Gas and the CMB

• Roughly Speaking 20 < z < 100

21-cm Hyperfine Transition21-cm Hyperfine Transition

Calculate: Atomic Distribution Function

Calculate: Atomic Distribution Function

• Determines the 21-cm line profile.

• The integrated line profile determines the total 21-cm emissivity.

• The 21-cm emissivity (and fluctuations in the emissivity) are needed when calculating the power spectrum of 21-cm fluctuations.

• Determines the 21-cm line profile.

• The integrated line profile determines the total 21-cm emissivity.

• The 21-cm emissivity (and fluctuations in the emissivity) are needed when calculating the power spectrum of 21-cm fluctuations.

The PlanThe Plan

First: Calculate the spin-resolved distribution function of atomic hydrogen.

Then: Calculate the 21-cm Line Profile, the 21-cm Emissivity, and the 21-cm

Power Spectrum.

First: Calculate the spin-resolved distribution function of atomic hydrogen.


Power Spectrum.

The Atomic H Distribution FunctionThe Atomic H Distribution Function

Statatistical Mechanics Basics:Statatistical Mechanics Basics:

Maxwell-Boltzmann

Number Density

H atom distribution function

The Spin Temperature*The Spin Temperature*

Radiative interactions with the CMB vs. Atomic Collisions:Radiative interactions with the CMB vs. Atomic Collisions:

* Before Ly- photons and the Wouthuysen-Field Effect turns on

Collision Threshold

Thermal Spin-Change Cross Section

Einstein A Coefficient

(Dalgarno 1961; Allison & Dalgarno 1969)

Atomic Spin-Change CollisionsAtomic Spin-Change Collisions

Schrödinger

Phase Shifts

Spin-Change Cross Section (Dalgarno 1961; Allison and Dalgarno 1969)

Spin-Change Cross SectionSpin-Change Cross Section

Thermal Cross SectionThermal Cross Section

Spin-Temperature EvolutionSpin-Temperature EvolutionAbsorption Against the CMB

(Loeb & Zaldarriaga, PRL 2004)

What’s Wrong?What’s Wrong?

Some Clues:Some Clues:

Thermal Spin-Change Cross Section (Velocity Independent)

(A Velocity Independent Function of T)

Thermal Cross SectionThermal Cross Section

(A Velocity Independent Function of T)

Spin-Change Cross SectionSpin-Change Cross Section

(A Velocity dependent Function of E)

What’s wrong?What’s wrong?

• Distribution does not factor!

• Collision time comparable to the radiative time

• Spin degrees of freedom are correlated with the kinetic degrees of freedom!

• Distribution does not factor!

• Collision time comparable to the radiative time

• Spin degrees of freedom are correlated with the kinetic degrees of freedom!

Quantum AstrophysicsQuantum Astrophysics

Solve the Boltzmann equation:Solve the Boltzmann equation:

Dominant Terms No Ly Early Mostly Neutral


Steady State Solution:Steady State Solution:

Radiative Term

Blackbody Formula


Collision Term:Collision Term:Product of Cross Section and Relative Velocity

Scattering out of v

Scattering in to v Probability of F


Equations are nonlinear and nontrivial to solve

However as:

May solve in a perturbation series in

about the thermal equilibrium solution:

Equations are nonlinear and nontrivial to solve

However as:

May solve in a perturbation series in

about the thermal equilibrium solution:

Perturbation

Spins thermalized at Tk


Expand in orthogonal modes:Expand in orthogonal modes:

Smooth

Hermite

The SolutionThe Solution

The steady state solution is

where

The steady state solution is

where

The Answer!!!!

Ts(v)Ts(v)

The spin-resolved distribution functions are:

For comparison define:

The spin-resolved distribution functions are:

For comparison define:

Velocity-Dependent Spin Temperature

Ts(v) Ts(v)

The Observable:The Brightness Temperature


A function of redshift, density, and velocity(and direction on the sky)



Linear

Power Spectrum

Direction cosine between wavevector and line of sight

Fourier Space



Power SpectraPower Spectra(Naoz and Barkana, astro-ph/0503196)

Power Spectra ChangePower Spectra Change

Power Spectra ChangePower Spectra Change

21-cm Line Profile21-cm Line Profile

Line Profile WidthLine Profile Width

Fourier Transform of ProfileFourier Transform of Profile

The EndThe End

• The spin and velocity degrees of atomic hydrogen in primordial gas are correlated and the spin-resolved distribution function of atomic hydrogen is nonthermal.

• The 21-cm line profile is not Gaussian. Total emissivity altered.

• Redshift and projection dependent effect of up to 5% on the large scale power spectrum, and an order unity effect on the small scale power spectrum of 21-cm fluctuations.

• Details: (See C. Hirata and KS, astro-ph/0605071)

• The spin and velocity degrees of atomic hydrogen in primordial gas are correlated and the spin-resolved distribution function of atomic hydrogen is nonthermal.

• The 21-cm line profile is not Gaussian. Total emissivity altered.

• Redshift and projection dependent effect of up to 5% on the large scale power spectrum, and an order unity effect on the small scale power spectrum of 21-cm fluctuations.

• Details: (See C. Hirata and KS, astro-ph/0605071)

The EndThe End

Ts(v) Ts(v)



The PlanThe Plan

First: Calculate the distribution function of atomic hydrogen.


Power Spectrum.

First: Calculate the distribution function of atomic hydrogen.


Power Spectrum.

21-cm Emissivity21-cm Emissivity

Photon Phase Space Density

Gaussian


Solve the EquationSolve the Equation

Matrix Structure:Matrix Structure:

Radiative

H-H

H-He

Rotate BasisRotate Basis

The key to the solution:The key to the solution:SumDifference Helium

A SimplificationA Simplification

In the new basis:

Note that both and have no source termand do not depend on

In the new basis:

Note that both and have no source termand do not depend on

It can be shown

A SimplificationA Simplification

We thus have:

with the solution:

We thus have:

with the solution:

Kinematic Distributions of H and He Relax to Thermal Equilibrium


Most Generally:

Simplifies If:

A) Spin and velocity relaxation times are fast compared to the expansion, rotation, shearing, diffusion or free-streaming times. Steady State. Homogenous.

B) Isotropic radiation field with smooth frequency dependence (such as the CMB). Radiative Rates Independent of Direction.

C) Collisional transitions dominated by simple spin exchange mechanisms.No Atomic Polarization


Quantum Numbers

Density Matrix

How do we characterize neutral H atoms in the electronic ground state?


Spin-Resolved Distribution Function


Radiative

HH Collision Matrix

H-H Atomic Collision TermH-H Atomic Collision Term

The end result of all this formalism:The end result of all this formalism:

He-H Atomic Collision TermHe-H Atomic Collision Term

Should account for Helium as:

Introduce:

Helium collision term:

Should account for Helium as:

Introduce:

Helium collision term:

No F changing collisions as He is spin singlet

Solve the EquationSolve the Equation

Boltzmann Equation in Matrix formBoltzmann Equation in Matrix formRelaxation Matrix Source Vector


Fourier Transform of ProfileFourier Transform of Profile

21-cm brightness temperature fluctuations

must be convolved with the 21-cm line

profile in the radial direction or in terms of

power spectra multiplied by

21-cm brightness temperature fluctuations

must be convolved with the 21-cm line

profile in the radial direction or in terms of

power spectra multiplied by

Fourier Transform


Photon Phase Space Density

Not Gaussian

Documents

Cosmic 21-cm Fluctuations from Dark-Age Gas Kris Sigurdson Institute for Advanced Study Cosmo 2006 September 25, 2006 Kris Sigurdson Institute for Advanced