Morphology of Cosmological Fields during the Epoch of Reionization - - PowerPoint PPT Presentation

Morphology of Cosmological Fields during the Epoch of Reionization

Akanksha Kapahtia

Indian Institute of Astrophysics (Joint Astronomy Program, IISc Bengaluru)

With Pravabati Chingangbam

(Indian Institute of Astrophysics, Bengaluru)

Stephen Appleby

(Korea Institue of Advanced Studies, Seoul)

AK, P. Chingangbam et al JCAP 2018; AK, P. Chingangbam et al arXiv:1904.06840

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Epoch of Reionization

21cm Cosmology for Epoch of Reionization

Cosmic inflation would have amplified minute quantum fluctuations (pre-inflation) into slight density ripples of overdensity and underdensity (post-inflation) It is these fluctuations that are the seeds of structure formation in the universe.

Image Credit: Roen Kelly- Discover Magazine 2 / 36

Epoch of Reionization

Observational evidence of EoR

Ly-alpha

Spectra of distant quasars show an absorption trough. xHI ≃ 10−4Ω1/2

m h(1 + z)3/2τα

Universe is highly ionized atleast till z ≃ 6.(Fan et.al.

AnnRev.AA 2006)

CMB

The scattering of CMB photons induces polarizations and temperature anisotropies. Optical depth to last scattering, τls ∼ 0.054. (Planck 2018)

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Epoch of Reionization

The 21cm spin flip transition

Transition between 11s1/2 & 10s1/2

Image Credit: nrao.edu

The relative populations of hydro- gen atoms in the two spin states de- fines the spin temperature Ts, (T∗ = 68mK,ν0 = 1420 MHz): n1 n0 = 3 exp −T∗ Ts

• E= 5.87 × 10−6 eV

, A21 = 2.88 × 10−15 sec ∼ 11 (Myr)−1

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Epoch of Reionization

Brightness temperature

The transfer of radiation through thermally emitting matter can be described in terms of the specific intensity: dIν dτν = −Iν + Bν(T) The temperature of a black body having the same specific intensity as Iν is the Brightness Temperature. In RJ regime Iν = 2ν2

c2 kT and so the above equation can be written in

terms of the brightness temperature, with solution: T ′

b(ν) = TS(1−e−τν)+T ′ R(ν)e−τν

The background radiation is usually CMB, so T ′

R(ν) = Tγ(z)

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Epoch of Reionization

21cm Brightness Temperature

Image Credit:lunar.colorado.edu/dare/science δTb(ν) = TS − Tγ 1 + z (1 − e−τν0 ) ≈

27xHI(1 + δnl)

• H

dvr/dr + H 1 − Tγ TS 1 + z 10 0.15 ΩMh2 1/2 Ωbh2 0.023

• mK

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Epoch of Reionization

Spin Temperature

δTb(ν) ∝ xHI (1 + δnl)

• 1 − Tγ

TS

• δTb < 0 if Ts < Tγ: absorp-

tion δTb > 0 if Ts > Tγ : emission δTb = 0 if Ts = Tγ: No signal Three competing processes determine Ts:

• 1. Absorption of CMB photons(and stimulated emission by CMB

photons)

• 2. Collisions with other hydrogen atoms, free electrons, and protons
• 3. Scattering of Lyman alpha photons (Wouthuysen-Field Effect)

T−1

s

= T−1

γ

+ (xc + xα)T−1

k

1 + xc + xα

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Epoch of Reionization

Evolution of Ts and δTb:Models

6 8 10 12 14 16 18 20

z

101 102 103

T (K)

Tγ Tk TS

ζ ∼ 17 (Fid) ζX ∼ 1056

6 8 10 12 14 16 18 20

z

101 102 103

T (K)

Tγ Tk TS

ζ ∼ 10.9 ζX ∼ 1056

6 8 10 12 14 16 18 20

z

101 102 103 104

T (K)

Tγ Tk TS

(Fid) ζX ∼ 1057

−120 −80 −40 δTb(mK)

Fiducial ζ = 10.9 ζ = 23.3

6 10 14 18 z 6 12 σTb −120 −80 −40 δTb(mK)

Fiducial ζX = 1 × 1057

6 10 14 18 z 6 12 σTb

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Epoch of Reionization Credit:Pritchard and Loeb (2012) 9 / 36

Epoch of Reionization

Observations of the 21cm brightness temperature

There are two ways the signal is detected:

Global Signal:EDGES,LEDA,DARE Fluctuations:GMRT,LOFAR,SKA,PAPER Observational Challenge: Foregrounds are 5 orders of magnitude greater than the signal Power spectrum → advantageous for observations. Fields are highly non-Gaussian → methods to include higher n-point statistics. Analyzing the morphology in real space → disadvantage as large sky volume is required for analysis. We use simulations to develop the method and make predictions for physical models.

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Epoch of Reionization

Morphology of cosmological fields

Cosmological fields are random fluctuation fields in 2 or 3 dimensional space. Excursion set: all spatial points with field values higher than or equal to a chosen threshold. In 2D, boundaries form closed contours enclosing connected regions

• r holes.

Betti Numbers: Topological quantities nc = no of connected regions nh = no of holes

200MPc

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Epoch of Reionization

Tensor Minkowski Functionals in 2D space

Alesker 1997, Hug 2008, Beisbart et al 2002, Schroeder-Turk et al 2011

n r

W m =

• r m da

W m,n

1

=

• C
• r m ⊗ ˆ

nn dℓ W m,n

2

=

• C
• r m ⊗ ˆ

nn κ dℓ κ ≡ local curvature ( x ⊗ y )ij ≡ 1 2 ( xiyj + xjyi ) m + n ≤ 2

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Epoch of Reionization

Scalar Minkowski Functionals :

m = 0, n = 0 W0 =

• da

− → area W1 =

• C

dℓ − → contour length W2 =

• C

κ dℓ − → genus

Cosmological application: Gott 1990

Tensor Minkowski Functionals: W 1,1

1

,W 0,2

1

,W 1,1

2

,W 0,2

2

W 1,1

2

=

• C
• r ⊗ ˆ

n κ dℓ =

• C

ˆ T ⊗ ˆ T dℓ

κ = |d ˆ T/dℓ| (Schroeder-Turk et al 2011, Chingangbam,KP Yogendran et al 2017) Translation invariant Gives the size and shape information of the curve : Trace(W 1,1

2

) =

• C

dℓ

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Epoch of Reionization

Shape and Alignment measure using W 1,1

2

Single Curve: W 1,1

2

− → λ1, λ2, λ1 < λ2 β ≡ λ1 λ2 0 ≤ β ≤ 1 Many curves: : ¯ β ≡ λ1 λ2

• Average over all curves

− →

• W 1,1

2

• −

→ Λ1, Λ2 α ≡ Λ1 Λ2 , 0 ≤ α ≤ 1

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Epoch of Reionization

Shape and Alignment measure using W 1,1

2

β:intrinsic shape of each curve α:relative alignment of many curves

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Epoch of Reionization

Simulating EoR

Messinger et. al., 2010

The brightness temperature field was generated using the publicly available code 21cmFAST. Uses a combination of the excursion set and perturbation theory to generate full 3D realizations of:

◮ Evolved density Field ◮ Ionization field - ζ and Tvir ◮ Spin temperature field - ζX and Tvir ◮ Brightness temperature field

Simulation Simulated δTb, Ts, δnl and xHI. Box size = 200 Mpc, resolution = 5123 grid. Combinations of ζ and Tvir to correspond to reionization ending at z ≈ 6, τre = 0.054 (PLANCK 2018).

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Epoch of Reionization

Morphology of fields during EoR

Physical questions How is the shape of structures related to the underlying physics of EoR? To study the morphology of δnl, Ts and xHI, and see how the morphology is reflected in δTb. To discriminate models of reionization To trace ionization and heating history of the IGM

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Epoch of Reionization

Quantities of interest: Single curve

λ1, λ2 − → Eigenvalues of W1 βch ≡ λ1/λ2 rch ≡ (contour length)/(2π) nc(ν), nh(ν) − → Betti Numbers at a given ν Average quantities at each threshold, ν λi,x(ν) ≡ nx(ν)

j=1 λi,x(j)

nx(ν) rx(ν) ≡ nx(ν)

j=1 rx(j)

nx(ν) βx(ν) ≡ nx(ν)

j=1 βx(j)

nx(ν)

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Epoch of Reionization

Isotropic Gaussian random field

The analytical forms for scalar Minkowski functionals (Tomita 1986,

Schmalzing:1998) and α (Chingangbam, Yogendran et al.:2017)for Gaussian

random fields is known . Their variation with threshold is same for a gaussian random field, irrespective of it’s power spectrum. However, their amplitude depends upon σ0 (standard deviation) and σ1 (standard deviation of the field derivative). The variation of Betti numbers is sensitive to the power spectrum (Park et al. 2013, Pranav 2018) and their analytical forms is not known. The analytical form for variation of β is not known but it is sensitive to power spectrum. rch gives a measure of perimeter of individual curves. It is sensitive to power spectrum of the field.

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Epoch of Reionization

Isotropic Gaussian random field

On varying ν from top to bottom: Isolated small connected regions around the highest peaks of the field and their number gradually increases Some of these small connected regions merge thereby decreasing their number. Connected regions all merge to form a single connected region with holes puncturing it which shrink in size and disappear as we go lower in threshold.

(Image Credit:Feldbrugge and Engelen,University of Groningen (2012)) 20 40 60 80

ncon, hole

con hole

200 400

rcon, hole(Mpc)

• 4
• 2

2 4

ν 0.45 0.65 0.85

βcon, hole

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Epoch of Reionization

Redshift Evolution of average quantities

Condense the ν dependence to get a single quantity at each redshift

Nx(z) ≡ νhigh

νlow

dν nx(ν, z) λch

i,x(z)

≡ νhigh

νlow dν nx(ν, z)¯

λi,x(ν) Nx(z) rch

x (z)

≡ νhigh

νlow dν nx(ν, z)¯

rx(ν) Nx(z) βch

x (z)

≡ νhigh

νlow dν nx(ν, z)¯

βx(ν) Nx(z) νhigh and νlow can be suitably chosen based on physical interpretation.

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Epoch of Reionization

Morphology of δnl

200Mpc

0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20

Tb[mK]

z=15.73

200Mpc

0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20

Tb[mK]

z=16.41

200Mpc

0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20

Tb[mK]

z=18.60

200Mpc

0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20

Tb[mK]

z=20.22 Peaks grow BUT at the cost of valleys

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Epoch of Reionization

Morphology of δnl

20 40 ncon, hole

con hole z = 10.26 z = 13.28 z = 16.41

100 200 rcon, hole(Mpc)

• 2

2

ν 0.4 0.6 0.8 βcon, hole 20 40 ntot

z = 10.26 z = 13.28 z = 16.41 10 20 30 40

rtot(Mpc)

• 2

2

ν 0.55 0.60 0.65 βtot 60 80 100 120 140 Ncon, hole, tot 16 18 20 rch

con, hole, tot

6 8 10 12 14 16 18 z

0.55 0.60 0.65 βch

con, hole, tot

tot hole con

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Epoch of Reionization

Morphology of xHI field

← − − − − − − − − − − − − − − − 200Mpc − − − − − − − − − − − − − − − → z = 16 ← − − − − − − − − − − − − − − − 200Mpc − − − − − − − − − − − − − − − → z = 12 ← − − − − − − − − − − − − − − − 200Mpc − − − − − − − − − − − − − − − → z = 9 ← − − − − − − − − − − − − − − − 200Mpc − − − − − − − − − − − − − − − → z = 7 0.0 0.2 0.4 0.6 0.8 1.0 xHI 0.0 0.2 0.4 0.6 0.8 1.0 xHI 0.0 0.2 0.4 0.6 0.8 1.0 xHI 0.0 0.2 0.4 0.6 0.8 1.0 xHI

To define a connected region or hole as neutral or ionized: Holes: If νmax > 0 then νhigh = 0 Connected Regions: If νmin < 0 then νlow = 0

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Epoch of Reionization

Morphology of xHI field : Progress of ionization

The rate of Formation of sources The rate of growth of Bubbles The rate of mergers of Bubbles zfrag− → Rate of source formation=Merger rate of Bubbles z0.5− → Nc = Nh ze − → Nc starts decreasing

20 40 60 80 100 Ncon, hole

z0.5 ze zfrag

40 80 120 rch

con, hole(Mpc)

6 8 10 12 14 16 18

z 0.5 0.6 0.7 βcon, hole

Con Hole

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Epoch of Reionization

Morphology of xHI field: Model Comparison

ζfcoll(x, z, R) ≥ 1 Number of ionizing photons > number of neutral hydrogen atoms Fiducial model: ζ = 17.5 , ζX = 2 × 1056, Tvir = 3 × 104 K Fiducial model with ζ = 10.9, Tvir = 1 × 104 K Fiducial model with ζ = 23.3, Tvir = 5 × 103 K ζfcoll(x, z, R) ≥ 1 + nrec(x, z, R)

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Epoch of Reionization

Morphology of xHI field: Model Comparison

Low Tvir → Less efficient but more numerous sources –Frequent mergers

10 20 30 Ncon 40 80 120 rch

con(Mpc)

6 8 10 12 14 16 18

z 0.2 0.4 0.6 0.8 βch

con

15 20 25 30 15 25 35

7.0 7.5

z 0.58 0.59

Fiducial ζ = 10.9, Tvir = 1 × 104K ζ = 23.3, Tvir = 5 × 104K 20 40 60 80 100 Nhole 20 40 60 80 rch

hole(Mpc)

6 8 10 12 14 16 18

z 0.55 0.65 βch

hole

15 20 25 30 6 8 10

7.0 7.5

z 0.58 0.59

Fiducial ζ = 10.9, Tvir = 1 × 104K ζ = 23.3, Tvir = 5 × 104K 27 / 36

Epoch of Reionization

Morphology of Ts field

T−1

s

= T−1

γ

+ (xc + xα)T−1

k

1 + xc + xα

Fluctuations in xc at these redshifts can be ignored. Therefore only fluctuations in xα and Tk determine Ts fluctuations. Before X-ray heating, Tk ∝ 1/(1 + z)−2 → no fluctuations in Tk,

• nly fluctuations in xα will determine the fluctuations in Ts.

If xα is high, TS is closer to TK than it is to Tγ, hence Ts will be closer to TK evolution in such regions. xα ∝ (1 + z)−1 Jα which is the Ly − α background flux which depends directly upon the rate of appearance of Ly − α sources. Soon xα will saturate and highest density regions will now host X-ray efficient sources → Tk is now fluctuating component and determines the fluctuations in Ts.

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Epoch of Reionization

Highest Density regions have higher xα and hence lower Ts The same regions will be the first places where X-ray sources will appear at later times Flipping between a valley and a peak Density Ts

200Mpc

0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20

Tb[mK]

200Mpc

16.50 17.32 18.14 18.96 19.79 20.61 21.43 22.25 23.07 23.89 24.71 25.54 26.36 27.18 28.00

Ts[K]

z=20.22

200Mpc

0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20

Tb[mK]

200Mpc

12.00 12.86 13.71 14.57 15.43 16.29 17.14 18.00 18.86 19.71 20.57 21.43 22.29 23.14 24.00

Ts[K]

z=18.60

200Mpc

0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20

Tb[mK]

200Mpc

7.00 7.93 8.86 9.79 10.71 11.64 12.57 13.50 14.43 15.36 16.29 17.21 18.14 19.07 20.00

Ts[K]

z=16.41

200Mpc

0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20

Tb[mK]

200Mpc

6.00 7.71 9.43 11.14 12.86 14.57 16.29 18.00 19.71 21.43 23.14 24.86 26.57 28.29 30.00

Ts[K]

z=15.73

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Epoch of Reionization

Morphology Ts Field : Model Comparison

ζX → Number of X-ray photons produced per solar mass Lower Tvir → Less efficient sources

20 40 60 Ncon 20 40 rch

con(Mpc)

6 8 10 12 14 16 18 20

z 0.55 0.65 βch

con

Fiducial ζX = 2 × 1056, Tvir = 1 × 104K ζX = 2 × 1056, Tvir = 5 × 104K 20 40 60 Nhole 20 40 60 rch

hole(Mpc)

6 8 10 12 14 16 18 20

z 0.55 0.65 βch

hole

Fiducial ζX = 2 × 1056, Tvir = 1 × 104K ζX = 2 × 1056, Tvir = 5 × 104K 30 / 36

Epoch of Reionization

Morphology Ts Field : Model Comparison

ζX → Number of X-ray photons produced per solar mass Lower ζX correspond to less efficient sources

20 40 60 Ncon 10 20 rch

con(Mpc)

8 10 12 14 16 18 20

z 0.7 0.8 βch

con

Fiducial ζX = 1 × 1057

20 40 Nhole 20 40 rch

hole(Mpc)

6 8 10 12 14 16 18 20

z 0.6 0.7 βch

hole

Fiducial ζX = 1 × 1057 31 / 36

Epoch of Reionization

Morphology of δTb Field

200Mpc

210 180 150 120 90 60 30 30

δTb [mK]

z=19.40

200Mpc

210 180 150 120 90 60 30 30

δTb [mK]

z=11.68

200Mpc

210 180 150 120 90 60 30 30

δTb [mK]

z=17.11

200Mpc

210 180 150 120 90 60 30 30

δTb [mK]

z=9.82

200Mpc

210 180 150 120 90 60 30 30

δTb [mK]

z=16.41

200Mpc

210 180 150 120 90 60 30 30

δTb [mK]

z=7.88

200Mpc

210 180 150 120 90 60 30 30

δTb [mK]

z=15.08

200Mpc

210 180 150 120 90 60 30 30

δTb [mK]

z=6.88

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Epoch of Reionization

Morphology of δTb Field

Regime 1: z > ztr Ts & δnl Regime 2: zfrag < z < ztr Ts & xHI with Ts dominating Regime 3: zEoR < z < zfrag Ts & xHI with xHI dominating Regime 4:z < zEoR Tb follows xHI

101 102 Ncon

z0.5 ze zEoR zfrag ztr

20 40 rch

con(Mpc)

6 8 10 12 14 16 18 20

z 0.55 0.60 0.65 βch

con

101 102 Nhole

z0.5 ze zEoR zfrag ztr

20 40 60 rch

hole(Mpc)

6 8 10 12 14 16 18 20

z 0.60 0.65 βch

hole

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Epoch of Reionization

50 100 Ncon 10 20 30 rch

con(Mpc)

6 8 10 12 14 16 18 20 z 0.55 0.60 0.65 βch

con

Fiducial ζ = 10.9, Tvir = 1 × 104K ζ = 23.3, Tvir = 5 × 104K 50 100 Ncon 15 30 45 rch

con(Mpc)

6 8 10 12 14 16 18 20 z 0.55 0.60 0.65 βch

con

Fiducial ζX = 1 × 1057 50 100 Nhole 20 40 60 80 rch

hole(Mpc)

6 8 10 12 14 16 18 20 z 0.55 0.60 0.65 βch

hole

Fiducial ζ = 10.9, Tvir = 1 × 104K ζ = 23.3, Tvir = 5 × 104K 50 100 Nhole 30 60 rch

hole(Mpc)

8 10 12 14 16 18 20 z 0.60 0.65 0.70 βch

hole

Fiducial ζX = 1 × 1057

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Epoch of Reionization

Summary

Kapahtia, Chingangbam et al (arXiv:1904.06840)-Under review

Model zfrag z0.5 ze zre τre Fiducial ∼ 11.69 ∼ 7.407 ∼ 6.58 ∼ 6.28 ∼ 0.054 Tvir = 1 × 104K ∼ 13.857 ∼ 7.698 ∼ 6.58 ∼ 6.00 ∼ 0.058 Tvir = 5 × 104K ∼ 11.194 ∼ 7.32 ∼ 6.58 ∼ 6.00 ∼ 0.052 ζX = 1 × 1057 ∼ 12.73 ∼ 7.5 ∼ 6.58 ∼ 6.28 ∼ 0.034 Recombination ∼ 12.2 ∼ 6.8 – < 6.00 Model r ch

z0.5 (Mpc)

zEoR ¯ xEoR

HI

ztr Fiducial ∼ 20.5 ± 0.78 ∼ 8.7 ∼ 0.73 ∼ 17.11 Tvir = 1 × 104K ∼ 15 ± 0.424 ∼ 9.1 ∼ 0.71 ∼ 19.4 Tvir = 5 × 104K ∼ 22.5 ± 0.96 ∼ 8.6 ∼0.77 ∼ 15.7 ζX = 1 × 1057 ∼ 20 ± 0.689 ∼ 9.12 ∼0.77 ∼ 18.6

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Epoch of Reionization

Conclusion

The number, size and shape of structures of excursion set of the fields exhibit clear evolution as a function of redshift. This evolution gives the important time and length scales of EoR This allows us to discriminate different EoR models Ongoing and future Work Sensitivity and signal to noise measures of Minkowski functionals for SKA like interferometers. Performing Bayesian analysis to obtain constraints on different models

• f reionization

Extension to Minkowski tensors in 3-D.

36 / 36