Statistics of the epoch of reionization(EoR) 21-cm signal ... · Motivations Rajesh Mondal (IIT...

Statistics of the epoch of reionization(EoR) 21-cm signal:

power spectrum error covariance

Rajesh Mondal (IIT KGP) Non-Gaussianity 25 June 2015 1

Rajesh Mondal

A National Workshop: Cosmology with the HI 21-cm Line

Raman Research Institute, Bangalore, India

Department of Physics (CTS), Indian Institute of Technology Kharagpur

Introduction


𝐓𝒃 (𝐦𝐊)

Mpc

Mpc

Motivations


Motivations How accurately can we estimate the power spectrum from a

given EoR data?

It is commonly assumed that the EoR 21-cm signal is purely

Gaussian random variable in all sensitivity estimates studies(e.g. Morales 2005, McQuinn et al. 2006, Beardsley et al. 2013, Jensen et al. 2013,

Pober et al. 2014 etc.)

How good is this assumption?

Ionized bubbles introduce non-Gaussianity and 21-cm signal

is expected to become highly non-Gaussian as the reionization

proceeds.Mondal el al. 2015

Introduction


𝐓𝒃 (𝐦𝐊)

Mpc

Mpc

Introduction


𝐓𝒃 (𝐦𝐊)

Mpc

Mpc

Introduction


𝐓𝒃 (𝐦𝐊)

Mpc

Mpc

Introduction


𝐓𝒃 (𝐦𝐊)

Mpc

Mpc

Introduction


𝐓𝒃 (𝐦𝐊)

Mpc

Mpc

Introduction


𝐓𝒃 (𝐦𝐊)

Mpc

Mpc

Introduction


𝐓𝒃 (𝐦𝐊)

Mpc

Mpc

Introduction


𝐓𝒃 (𝐦𝐊)

Mpc

Mpc

Introduction


𝐓𝒃 (𝐦𝐊)

Mpc

Mpc

Introduction


𝐓𝒃 (𝐦𝐊)

Mpc

Mpc

Introduction


The error covariance Binned power spectrum estimator 𝑖𝑡ℎ bin

averaged over

Bin averaged power spectrum

The error covariance

𝚫𝒌

𝒌

Error covariance


The error covariance Using the definition of trispectrum (four-point statistics)

where

the square of the power spectrum averaged over the i th bin

the average trispectrum where 𝑘𝑎 and 𝑘𝑏 are summed over

the i th and the j th bins respectively

Error covariance


The error covariance

Error covariance For Gaussian random field


The error covariance for Gaussian random field

= 0

Error covariance For Gaussian random field


First: Diagonal, error in different bins are uncorrelated

Second:

The error covariance for Gaussian random field

Error covariance For 21-cm power spectrum


The error covariance for 21-cm power spectrum

The covariance matrix retains the 1/𝑉 dependence

First: no longer diagonal;

Second: diagonal terms deviate from behavior

The error variance saturates as the bin-width

approaches

Results SNR


signal to noise ratio 𝑆𝑁𝑅 = 𝑃𝑏 (𝑘)/𝛿𝑃𝑏(𝑘) = 𝑁𝑘

Mondal el al. 2015

Simulating the 21-cm maps


Simulating the 21-cm maps We have used semi-numerical simulations to study how non-

Gaussianity affects the error covariance predictions for the

EoR 21-cm power spectrum.

N-body Simulation: particle-mesh parallelized code,

coving volumes V1 = 150 Mpc 3 and 𝑉2 = 215 Mpc 3,

Mass resolution (𝑀𝑝𝑎𝑟𝑡) = 7.304 × 107ℎ−1𝑀⊙

Identifying Halos: Friends-of-Friends (FoF) algorithm,

linking length 0.2 times the mean inter-particle separation,

require a halo to have at least 10 particles

Generating the ionization map: homogeneous

recombination scheme (Choudhury et al. 2009), HI distribution

was mapped to redshift space (Majumdar et al. 2013)

Reference ensembles


Signal Ensemble (SE) An ensemble of 50 statistically independent realizations of

simulated EoR 21-cm signal and used this to estimate 𝐶𝑖𝑗.

“How do we interpret the estimated 𝐶𝑖𝑗?”

For a Gaussian random field: off-diagonal terms to be zero.

Reference ensembles Randomized Signal Ensemble


Any deviation from this as arising from non-Gaussianity

Then use these deviations to quantify the contribution from

the trispectrum.

Straight forward in concept complications in practice

from non-Gaussianity



The Randomized Signal Ensemble (RSE)

First complication: interpretation of the diagonal terms

It is not possible to use SE to independently determine the

We have over come this problem by constructing the RSE




k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

[SE]

Set 1

Set 2

Set …

Set 50

Each realization of RSE contains modes drawn from all 50

realizations in SE

We have ordered all the Fourier modes as 𝑘1, 𝑘2, 𝑘3 and so

on… And divided them into sets




k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_4 k_54 k_104 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

[SE]_1 [SE]_2 [SE]_50

For the firstRealization in

RSE




k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_4 k_54 k_104 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

[SE]_1 [SE]_2 [SE]_50


RSE




k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_4 k_54 k_104 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

[SE]_1 [SE]_2 [SE]_50


RSE




k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_4 k_54 k_104 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

[SE]_1 [SE]_2 [SE]_50


RSE




k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_4 k_54 k_104 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

[SE]_1 [SE]_2 [SE]_3

For the secondRealization in

RSE




k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_4 k_54 k_104 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

k_1 k_51 k_101 …

k_2 k_52 k_102 …

… … … …

k_50 k_100 k_150 …

[SE]_1 [SE]_2 [SE]_3

For the secondRealization in

RSE




Modes from [SE]1 is not correlated with those from [SE]2 The average trispectrum is at least 50 times

smaller for RSE as compared to SE. Assumed

We expect to have exactly the same value

in both SE and RSE.

values of 𝐶𝑖𝑖 are expected to have if the EoR signal were a

Gaussian random field with

It thus becomes possible to interpret any deviations from

this as arising from trispectrum.



The Gaussian Random Ensemble (GRE)

Second complication: SE has a finite number of realizations

The GRE contains 50 realizations of the 21-cm signal, the

signal in each realization is a Gaussian random field.

The signal at any mode k in the i th bin is calculated using

where a(k) and b(k) random variables, 𝑃𝑏(𝑘𝑖)power spectrum



One realization of SE, GRE and RSE

This three panels each show a single realization drawn from the

three different ensemble

All correspond to the same 𝑥𝐻𝐼 = 0.5 and same 𝑃𝑏(𝑘𝑖)



Ensemble of Gaussian Random Ensembles (EGRE)

The power spectrum estimated from any single realization

in GRE or from GRE will be different from 𝑃𝑏(𝑘𝑖)because of the limited number of realizations.

The off-diagonal terms of the error-covariance 𝐶𝑖𝑗 𝐺

estimated from GRE will not be zero

Compare the 𝐶𝑖𝑗 against the random fluctuation of 𝐶𝑖𝑗 𝐺in

order to determine whether 𝐶𝑖𝑗 is statistically significant or

Used 50 independent GREs to construct an EGRE which

we have used to estimate the variance of covariance 𝐶𝑖𝑗 𝐺.

Results Dimensionless covariance


Dimensionless form For convenience, use the dimensionless covariance matrix

Results Dimensionless trispectrum


The dimensionless bin-averaged trispectrum

;

Results Correlation coefficient


The correlation

coefficient



The correlation

coefficient



The correlation

coefficient



The correlation

coefficient



The correlation

coefficient



Summary Discussion


Summary and discussion The non-Gaussian components are correlated, which is

quantified through non-zero off-diagonal terms i.e.

trispectrum in error covariance matrix.

It is not possible to use the SE to independently determine the

value of 𝑃𝑏2(𝑘𝑖)

We have overcome this problem by constructing the RSE

The difference 𝐶𝑖𝑖 − 𝐶𝑖𝑖 𝑅𝑆𝐸 gives an estimate of trispectrum

The 𝑡𝑖𝑖~1 for 𝑘~0.1 Mpc−1, and it increases quite rapidly

with 𝑡𝑖𝑖 = 10 and ~103 at 𝑘~1 Mpc−1 and ~5 Mpc−1

respectively, dominating the error-covariance

Summary Discussion


Summary and discussion The off-diagonal terms

The 5 largest k bins (𝑘 > 0.5 Mpc−1) are correlated and

a correlation between 3 smallest (𝑘 < 0.3 Mpc−1) and 3

largest k bins (𝑘 > 1 Mpc−1). 2 smallest k bins (𝑘 <0.1 Mpc−1) are anti-correlated with the intermediate bins.

GRE: to appreciate the that the SE has a finite number of

realization

Statistical significance of the 𝑟𝑖𝑗 is determined by

comparing against the 𝛿𝑟𝑖𝑗 𝐺estimated from an EGRE.

Summary Implications


Implications The present analysis emphasises the fact that the non-

Gaussian effects play an important role in error prediction

for EoR 21-cm power spectrum

Generic results: not limited only to the EoR 21-cm signal

but can be applied to the galaxy redshift surveys

(Feldman & Peacock, 1994; Neyrinck, 2011; Carron, Wolk & Szapudi, 2014)

We plan to consider various EoR modelling and

implications for different EoR experiments in our future

work.


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Statistics of the epoch of reionization(EoR) 21-cm signal ... · Motivations Rajesh Mondal (IIT...

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