Cross Terms and Weak Signals
in the CMB Sky
C.Hernández-Monteagudo
MPA
In collaboration with R.A.Sunyaev
Cross Terms and Weak Signals in the CMB Sky – p.1/12
Summary
•
CMB: an Overview & current Status
• Cosmic Variance, (CV)
• Can we detect signals below CV limit?. Our
Approach
• Applications
- tSZ at Low Redshifts
- Resonant Scattering at High Redshifts
- tSZ during Reionization
- kSZ & tSZ at Low Redshifts
Cross Terms and Weak Signals in the CMB Sky – p.2/12
CMB: an Overview (I)
δT (θ, φ) =
X
al,m [Ωm , Ωb , h, nS , ...] Yl,m (θ, φ)
l,m
-CMB field determined
by Power Spectrum:
Cl ≡ hal,m a∗l,m i, l ∼ π/θ
-But, in practice:
l
X
1
al,m a∗l,m
Cl =
2l + 1 m = −l
Cross Terms and Weak Signals in the CMB Sky – p.3/12
CMB: an Overview (and II)
Secondary anisotropies:
• Reionization
• Rees-Sciama effect:
δTRS
2 R
T0
= − c2
dη φ̇(η)
• Gravitational lensing
• Spectrum distortion after
inverse Compton
scattering with hot
electrons, (tSZ effect).
• Doppler kick off electron
plasma in LSS, (kSZ
effect).
• ...
Hu & Dodelson (2002)
Cross Terms and Weak Signals in the CMB Sky – p.4/12
Cosmic Variance, (CV)
1
Cl =
2l + 1
l
X
al,m a∗l,m
m = −l
•
The model predicts average
properties of our universe
•
But we see only one universe,
(one realization).
•
Ergodic theorem: spatial
(angular) averages
⇒ large scales are not well
sampled.
Intrinsic uncertainty in power spectrum =⇒ Cosmic Variance.
2
=
σC
l ,CV
2
Cl2
(2l + 1)fsky
Cross Terms and Weak Signals in the CMB Sky – p.5/12
Cross Terms and Weak Signals in the CMB Sky – p.6/12
Can we detect signal below CV?
- Given two signals in the sky, with different spectral behaviour,
and 1,
Tν (θ, φ) = g1 (ν)t(θ, φ) + g2 (ν)t̃(θ, φ) =
X
[g1 (ν) al,m + g2 (ν) ãl,m ] Yl,m (θ, φ)
l,m
we propose to amplify t̃ (θ, φ) by simply computing:
δCl = Cl (ν1 ) −
g1 (ν1 )
g1 (ν2 )
2
Cl (ν2 ) =
h i
g1 (ν1 ) g2 (ν2 )
∗
∗
g1 (ν1 )g2 (ν1 ) 1 −
hal,m ãl,m +al,m ãl,m i + O 2
g1 (ν2 ) g2 (ν1 )
Cross Terms and Weak Signals in the CMB Sky – p.7/12
δCl = ξ(ν1 , ν2 ) hal,m ã∗l,m i + O
h i
2
This expression:
• ...is not limited by Cosmic Variance
•
...is proportional to , (and not 2 )
•
... contains sign information (not necessarily positive)
The remaining question is...
how large is the cross-correlation hal,m ã∗l,m i?
Cross Terms and Weak Signals in the CMB Sky – p.8/12
h i
δCl = ξ(ν1 , ν2 ) hal,m ã∗l,m i + O 2
Cross Terms and Weak Signals in the CMB Sky – p.9/12
h i
δCl = ξ(ν1 , ν2 ) hal,m ã∗l,m i + O 2
•
If t and t̃ are caused by δ(η1 ) and δ(η2 ) respectively,
Cross Terms and Weak Signals in the CMB Sky – p.9/12
h i
δCl = ξ(ν1 , ν2 ) hal,m ã∗l,m i + O 2
•
•
If t and t̃ are caused by δ(η1 ) and δ(η2 ) respectively,
... and if
R d3 k
δ(η) = D(η) (2π)3 δk eikη ,
∗ i = (2π)3 P (k) δ (k + q),
with hδk δq
D
ψ
Cross Terms and Weak Signals in the CMB Sky – p.9/12
h i
δCl = ξ(ν1 , ν2 ) hal,m ã∗l,m i + O 2
•
•
If t and t̃ are caused by δ(η1 ) and δ(η2 ) respectively,
... and if
R d3 k
δ(η) = D(η) (2π)3 δk eikη ,
∗ i = (2π)3 P (k) δ (k + q),
with hδk δq
D
ψ
htt̃i ∝ hδ(η1 )δ(η2 )i = D(η1 ) D(η2 )
Z
d3 k
ik(η1 −η2 )
P
(k)e
ψ
(2π)3
Cross Terms and Weak Signals in the CMB Sky – p.9/12
h i
δCl = ξ(ν1 , ν2 ) hal,m ã∗l,m i + O 2
•
•
If t and t̃ are caused by δ(η1 ) and δ(η2 ) respectively,
... and if
R d3 k
δ(η) = D(η) (2π)3 δk eikη ,
∗ i = (2π)3 P (k) δ (k + q),
with hδk δq
D
ψ
htt̃i ∝ hδ(η1 )δ(η2 )i = D(η1 ) D(η2 )
=⇒ only modes within k
contribute
<
∼
Z
d3 k
ik(η1 −η2 )
P
(k)e
ψ
(2π)3
kmax =
2π
kη1 −η2 k
will
Cross Terms and Weak Signals in the CMB Sky – p.9/12
h i
δCl = ξ(ν1 , ν2 ) hal,m ã∗l,m i + O 2
•
•
If t and t̃ are caused by δ(η1 ) and δ(η2 ) respectively,
... and if
R d3 k
δ(η) = D(η) (2π)3 δk eikη ,
∗ i = (2π)3 P (k) δ (k + q),
with hδk δq
D
ψ
htt̃i ∝ hδ(η1 )δ(η2 )i = D(η1 ) D(η2 )
Z
d3 k
ik(η1 −η2 )
P
(k)e
ψ
(2π)3
=⇒ only modes within k <
∼ kmax =
contribute
=⇒ δCl will be restricted at
l
< l
∼ max
2π
kη1 −η2 k
will
= kmax · min(η0 − η1 , η0 − η2 )
Cross Terms and Weak Signals in the CMB Sky – p.9/12
An example: tSZ induced by clusters
•
tSZ signal is generated in massive haloes containing hot
electron plasma
⇒ δTtSZ ∝ δm
•
The ISW effect is generated in potential wells, also
described by δm
⇒ δTISW = f (δm )
tSZ
a
=⇒ δCl = [gtSZ (ν1 ) − gtSZ (ν2 )] haISW
l,m
l,m
∗
h i
i + O 2
Cross Terms and Weak Signals in the CMB Sky – p.10/12
δCl ’s at νref = 217 GHz
- Green curve: cross term
(∝ ) in RJ
- Red curve: term in 2 in
RJ
⇒ The final tSZ Power
Spectrum changes at
low l !!
Cross Terms and Weak Signals in the CMB Sky – p.11/12
The Method
- If:
• Given two signals/fields, one is far weaker than the another
• both signals show different dependence on frequency
• both signals are seeded by evolving cosmological perturbations
at different cosmic epochs,
then:
• it is possible to amplify the spectral features of the weakest
signal by the computation of the cross-correlation,
• this amplification takes place on scales larger than the distance
separating the sources of the signals.
Cross Terms and Weak Signals in the CMB Sky – p.12/12