This is a presentation I gave at IIT Indore on my recent work on 21cm Cosmology.

1. Full Sky Bispectrum in Redshift
Space for 21cm Intensity Maps
Rahul Kothari, Post Doctoral Researcher,
University of the Western Cape, South Africa
IIT Indore, 14th September 2020
Collaborators — Ruth Durrer, Mona Jalilvand, Roy Maartens & Francesco Montanari
arXiv:2008.02266 (Accepted in JCAP)

2. Overview
❖ Rudimentary Concepts
❖ HI Intensity Bispectrum
❖ Results’ Summary
❖ Epilogue
❖ Future Plans
❖ Summary
2

3. Image Courtesy: www.pinterest.com
Fundamental
Concepts
❖ Rudimentary Cosmology
concepts
❖ HI Intensity aka 21cm Intensity
❖ Comparing Theory &
Observations
❖ Detectability

4. Crash Course on Cosmology!
❖ Studies universe at its largest length scales of the order of ~ 100 Mpc
❖ Gravity is the predominant force & governed by General Relativity
❖ For there are 10 coupled partial second order differential equations!
❖ Equations are very difficult to solve, Cosmological Principle saves the day!
❖ Universe is statistically isotropic and homogeneous at these length scales
❖ Consequences for the quantities of interest like power spectrum and bispectrum
gμν
Rμν −
1
2
Rgμν = 8πGTμν
4

5. Parallel spins
HI Intensity Mapping
❖ Cosmology as an empirical science uses (a) CMB and (b) LSS surveys
❖ LSS (mostly galaxy number counts) is 3D and hence contains more information
❖ Resort to integrated intensity as resolving galaxies is difficult for high z [MNRAS
464 (2017) 1948]
❖ HI intensity is one such case and can be thought of CMB at different redshifts
❖ Hyperfine transition from parallel to anti-parallel spins of e and p in hydrogen
❖ Very promising as it comes from hydrogen
Image Courtesy: www.cfa.harvard.edu
Courtesy: space.mit.edu
Anti-parallel spins
Transition
5

6. Redshift Space Distortions
❖ In cosmology we measure redshifts, as distances are
very difficult to measure
❖ Galaxies move apart from each other as universe
expands
❖ Redshift measured by an observer depends upon this
velocity which can have peculiar velocity component
❖ In the redshift space, we recover a distorted field
called as Redshift Space Distortions
❖ These won’t be obtained if there we no peculiar
velocities
❖ Peculiar velocity and hence RSD contains a lot of
information about build of structure [Percival & White
arXiv:0808.0003]
Peculiar
Hubble
Actual Distribution
In the Redshift space
to an observer
Ref: Dodelson6

7. Theory & Observations
❖ To relate theory and observations, we almost always seek ensemble averaging of
a given quantity
❖ All statistical information of a gaussian field in 2 point correlation functions (2PCF)
❖ Higher correlators can be calculated using Wick’s theorem
❖ Odd correlators are zero
❖ When non-gaussianities are present due to non-linearities, we need higher points
correlators
❖ Bispectrum (aka 3PCF) is one such correlator and measures non-linearities
⟨X1X2X3X4⟩ = ⟨X1X2⟩⟨X3X4⟩ + ⟨X1X3⟩⟨X2X4⟩ + ⟨X1X4⟩⟨X2X3⟩, Xi ≡ X(zi, ni)
7

8. The Bispectrum
❖ Spherical harmonic decomposition of any given field of interest as
❖ The bispectrum is the three point correlator
❖ Cosmological principle allows a more manageable quantity reduced bispectrum
❖ For the purpose of this talk we also need the angle averaged bispectrum
X(z, n)
B(zi, ni)
8
Bm1m2m3
ℓ1ℓ2ℓ3
(z1, z2, z3) =
(2ℓ1 + 1)(2ℓ2 + 1)(2ℓ2 + 1)
4π (
ℓ1 ℓ2 ℓ3
0 0 0 ) (
ℓ1 ℓ2 ℓ3
m1 m2 m3)
bℓ1ℓ2ℓ3
(z1, z2, z3)
Bm1m2m3
ℓ1ℓ2ℓ3
(z1, z2, z3) =
(
ℓ1 ℓ2 ℓ3
m1 m2 m3)
Bℓ1ℓ2ℓ3
(z1, z2, z3)
X(z, n) =
∞
∑
ℓ=0
m=ℓ
∑
m=−ℓ
Yℓm(n)Xℓm(z)
Bm1m2m3
ℓ1ℓ2ℓ3
(z1, z2, z3) = ⟨Xℓ1m1
(z1) Xℓ2m2
(z2) Xℓ3m3
(z3)⟩
B(zi, ni) =
∑
ℓi,mi
Bm1m2m3
ℓ1ℓ2ℓ3
(z1, z2, z3)Yℓ1m1
(n1)Yℓ2m2
(n2)Yℓ3m3
(n3)

9. Signal to Noise Ratio (SNR)
❖ Signal to Noise Ratio (SNR) can be used to determine
❖ Whether an instrument can detect a given signal or not
❖ What specifications an instrument should be designed with
❖ SNR depends upon the specifications of an experiment
❖ Square Kilometre Array (SKA) — Going to become the world’s largest radio telescope
❖ Hydrogen Intensity and Real time Analysis eXperiment (HIRAX)
❖ Specifications
❖ SKA in single dish mode — Useful to measure large angular scales
❖ HIRAX in interferometric mode — Useful for small angular scales
Survey fsky Nd ttot (Hr) Dd (m) Redshift
SKA 0.48 197 10,000 15 0.3—3
HIRAX 0.36 1024 10,000 6 0.8–2
9

10. Image Courtesy: cfa.harvard.edu
HI Intensity
Bispectrum

11. ObserverSourcePlane
LensPlane
Z
Bispectrum of 21cm Line
❖ Our emphasis is to show the contribution of RSD
❖ Easy to compute at same redshifts where lensing can
be neglected [JCAP 01(2016) 016]
❖ The unlensed HI temperature contrast
❖ First order contribution contains linear RSD term
❖ Second order contribution has second order density
perturbations, non linear RSD etc.
❖ The bispectrum at tree level in the redshift space is
the correlator
❖ Assumes zero primordial non-gaussianities
Δi ≡ Δ(zi, ni)
Actual position
Apparent position
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Δ(z, n) = Δ(1)
(z, n) + Δ(2)
(z, n) − ⟨Δ(2)
⟩(z)
B(zi, ni) = ⟨Δ(1)
1
Δ(1)
2
Δ(2)
3 ⟩ + 2 perms
⟨Δ(1)
1
Δ(1)
2
Δ(1)
3 ⟩ = 0
11

12. ❖ Reduced bispectrum can be calculated now
❖ Contribution from linear RSD
❖ Pure second order RSD
❖ Other from a combination of RSD and density perturbations
The HI Bispectrum
12
bℓ1ℓ2ℓ3 (z1, z2, z3) = bδ(2)
ℓ1ℓ2ℓ3
(z1, z2, z3) + bv(2)′
ℓ1ℓ2ℓ3
(z1, z2, z3) + bδv′
ℓ′2ℓ2ℓ3
(z1, z2, z3)
+bv′2
ℓ1ℓ2ℓ3
(z1, z2, z3) + bδ′v
ℓ1ℓ2ℓ3
(z1, z2, z3) + bv′′v
ℓ1ℓ2ℓ3
(z1, z2, z3)
Bm1m2m3
ℓ1ℓ2ℓ3
(z1, z2, z3) =
(2ℓ1 + 1)(2ℓ2 + 1)(2ℓ2 + 1)
4π (
ℓ1 ℓ2 ℓ3
0 0 0 ) (
ℓ1 ℓ2 ℓ3
m1 m2 m3)
bℓ1ℓ2ℓ3
(z1, z2, z3)

13. Image Courtesy: www.psdgraphics.com
Results’ Summary
❖ Bispectrum Plots
❖ RSD Contribution
❖ Signal to Noise Ratio

14. Bispectrum Contour Plots
14
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❖ The bispectrum would’ve been a 7D plot!
❖ The contour plots choose one specific redshift and
❖ This reduces 7D to mere 3D and colour represents the value
❖ The plots have been normalised appropriately
❖ The triangular region is such that ( and )
❖ Maximises in the squeezed limit
bℓ1ℓ2ℓ3
(z1, z2, z3)
zi = 1 ℓ1 = 200
x = ℓ2/ℓ1 y = ℓ3/ℓ1
{(x, y)|x ≤ y} ∩ {(x, y)|y ≤ 1} ∩ {(x, y)|x + y ≥ 1}
����
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x = y
y = 1
x + y = 1

15. ❖ Panel shows the fractional contribution
❖ “no RSD” has only second order density with no linear RSD term
❖ Estimates all RSD contribution
❖ Thus RSD increases the bispectrum by about a factor of 5
❖ The linear only RSD signal is ~25% of the full
❖ Estimates contribution from non-linear RSD terms
RSD Contribution
15
50 100 150 200
0.72
0.74
0.76
0.78
Equilateral
Squeezed
Folded
50 100 150 200 250 300
0.76
0.78
0.8
0.82
0.84
0.86
Equilateral
Squeezed
Folded
B
B
=
B`1`2`3 [ with RSD] B`1`2`3 [ no RSD]
B`1`2`3
[ with RSD]
B[ all RSD] B[ linear RSD]
B[ all RSD]

16. 0.5 1 1.5 2 2.5 3
100
101
102
103
0 200 400 600
3.5
4
4.5
5
5.5
10-4
HIRAX
Bispectrum Detectability
❖ Expression for SNR for a fixed multipole
configuration
❖ Variance is given by [JCAP 04 (2019) 053]
❖ Theoretical gets modified by adding noise
power spectra
❖ Noise power spectra depends upon
experiment mode
❖ Fixed multipole information can be combined
to get Cumulative SNR
❖ The range of ell for the sum also depends
upon experiment
Cℓ
𝒩ℓ
SNRℓ1ℓ2ℓ3
(z) =
Bℓ1ℓ2ℓ3
(z)
σBℓ1ℓ2ℓ3
(z)
σ2
Bℓ1ℓ2ℓ3
(z) = f−1
sky
˜Cℓ1
(z) ˜Cℓ2
(z) ˜Cℓ3
(z) (1 + 2δℓ1ℓ2
δℓ2ℓ3
+ δℓ1ℓ2
+ δℓ2ℓ3
+ δℓ3ℓ1)
˜Cℓ(z) = Cℓ(z) + 𝒩ℓ(z)
SNR(z)2
=
∑
ℓi
SNRℓ1ℓ2ℓ3
(z)2
0 50 100 150 200
10-4
10-3
10-2
10
-1
SKA
16

17. 50 100 150 200
0
2
4
6
8
10-3
Equilateral
Squeezed
Folded
50 100 150 200
0
2
4
6
8
10
-5
Equilateral
Squeezed
Folded
50 100 150 200
0
0.002
0.004
0.006
0.008
0.01
Equilateral
Squeezed
Folded
❖ SNR becomes almost zero when on account of beam
❖ First four have almost same contributions
❖ Contribution from last two panels can be ignored
ℓ ≳ 150
SNR Plots (Single Dish SKA)
50 100 150 200
0
0.002
0.004
0.006
0.008
0.01
Equilateral
Squeezed
Folded
50 100 150 200
0
0.002
0.004
0.006
0.008
0.01
Equilateral
Squeezed
Folded
50 100 150 200
0
2
4
6
8
10
-6
Equilateral
Squeezed
Folded
17

18. Detectability
*Estimated
z l(min) l(max) CSNR (Lin.) CSNR* (All)
0.5 5 173 5.04 ~20
1.0 5 224 0.69 ~3
1.5 5 179 0.09 ~0.4
CSNR for SKA in Single Dish Mode
z l(min) l(max) CSNR (Lin) CSNR* (All)
1.0 74 366 2.78 ~11
1.5 59 561 1.08 ~4
CSNR for HIRAX in Interferometric Mode
❖ Used Dirac Delta window &
❖ To avoid numerical complexities
❖ SNR depends upon
❖ Study of optimal bin-width and cross
correlations can be pursued in future
❖ For estimating total SNR contribution we
can multiply linear RSD contribution by 4
❖ Cross correlations would increase it
further
❖ Calculation for specific redshifts
❖ Larger signal at smaller z due to larger
non-gaussianities induced by non-linear
gravitational evolution
Δz = 10−4
Δz
18

19. Epilogue ❖ Future Perspectives
❖ Summary
Image Courtesy: www.hwalls.com

20. Future Plans
❖ SNR to be computed using a more realistic window function
❖ SNR also depends upon optimal bin-width which needs to be studied
❖ Cross correlations are to be considered which would increase SNR
❖ Lensing effects are to be studied
❖ Study bispectrum to improve constraints on cosmological parameters
20

21. Summary
Rudimentary
Ideas
21cm
Bispectrum
Results
Future Plans
Cosmological
Principle
21cm
Cosmology
Hyperfine
Transition
Redshift Space
Distortions
Peculiar
Velocity
Observation
& Theory
Power Spectra &
Bispectra
SNR &
Detectability
Lensing
Neglected No Prim. Non
Gauss.
Max in Sq
Limit
RSD
Contribution
Signal
Detectable
Lensing
Effects
Optimal Bin
Width
Realistic
Window Func.

22. Thank You
“Where tireless striving stretches its arms towards perfection.”
— Rabindra Nath Tagore, Gitanjali
Acknowledgements: I’m very grateful to Suman for arranging the talk.

23. Background Material

24. Gaussian Random Fields
❖ Gaussian random variable is defined by the property that its distribution is gaussian
❖ For this distribution, mean is q0
❖ Simple example would be of length measurement
❖ Joint gaussian for N random variables is given by, Mmn is symmetric, non-degenerate and real matrix
❖ Real scalar gaussian random field
❖ Consider a discrete space identified as cube of side ‘L’ and point spacing ‘a’
❖ Points will be identified as (n1,n2,n3), 1≤ni≤L/a, there will be (L/a)3 points
❖ Scalar field is defined at all these points as gaussian random variables
❖ When , becomes gaussian random field
❖ I, Q and U are these kind of fields
Φ(n1, n2, n3)
a → 0 Φ
F (q1,a2,...qN) =
1
q
(2p)N
detM
exp

1
2
qmMmnqn
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F (q) =
1
p
2ps2
exp
"
(q q0)2
2s2
#
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Courtesy— Mathematica
Stack exchange
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