The Microwave Background Bispectrum, Paper I: Basic Formalism

18 Pages·0.17 MB·English

by  David N. Spergel

The Microwave Background Bispectrum
Paper I: Basic Formalism
9
9
9
1 David N. Spergel and David M. Goldberg

n
Princeton University Observatory, Princeton, NJ 08544
a
J

4 February 1, 2008

2
v
2
Abstract
5
2
1 In this paper, we discuss the potential importance of measuring
1 theCMBanisotropy bispectrum. Wedevelop aformalismforcomput-
8
ing the bispectrum and for measuring it from microwave background
9
/ maps. As an example, we compute the bispectrum resulting from
h
the 2nd order Rees-Sciama effect, and find that is undetectable with
p
- current and upcoming missions.
o
r
t
s
a 1 Introduction
:
v
i
X Observations of microwave background fluctuations are a powerful probe of
r the physical conditions in the early universe. Most analyses of the microwave
a
background focus on measuring and interpreting the two-point function. If
the microwave background fluctuations were a purely Gaussian random field,
then the two-point function would completely characterize the microwave
background. The non-linear growth of structure produces non-Gaussian fluc-
tuations. If detectable, these non-Gaussian fluctuations are a new window
into the evolution and growth of structure.
The three-point function and its analog, the bispectrum, are potential
tools for detecting these non-linear effects. At present, there is no clear sig-
nature of such modes. There was no detection of three point correlations in
the COBE data, nor any topological signatures of non-Gaussianity[1]. While
1
there is a reported detection of non-Gaussianity in the COBE bispectrum[2,
3]. Our ability to detect these effects will soon improve by many orders of
magnitudewiththelaunchoftheMAPsatellite[4]andthePLANCK[5]satel-
lite. These upcoming launches motivate our detailed study of the bispectrum
and the three-point function.
In this article, we first discuss why the bispectrum is interesting (§ 2),
develop a general form for calculating the bispectrum (§ 3) and then discuss
the particular case of those perturbations resulting from the Rees-Sciama
effect (§ 4). While the Rees-Sciama bispectrum is not detectable, we show
in a companion paper that the non-Gaussian effects produced at low redshift
througha coupling of gravitationallensing and either the Sunyaev-Zel’dovich
effect or the late time ISW effect produces a detectable signal.
2 Why is the Bispectrum Interesting?
The three-point function and its transform, the bispectrum, are potential
tools for detecting non-Gaussianity in the microwave sky. Just as the mul-
tipole spectrum is a useful statistic for studying the two-point correlation,
the bispectrum is useful in studying the three-point function. To put this in
context, we define the a as the expansion of the temperature anisotropies
lm
into spherical harmonics:
∗
a ≡ dnˆT(nˆ)Y (nˆ) (1)
lm lm
Z
Thus,sincethethreepointfunctionisdefinedasB(ˆl,mˆ,nˆ) ≡ T(ˆl)T(mˆ)T(nˆ),
we define the bispectrum as:
Bm1m2m3 = a a a (2)
l1l2l3 l1m1 l2m2 l3m3
The universe is thought to be rotationally invariant, and hence, a more
useful quantity is the angle-averaged bispectrum:
l l l
B = 1 2 3 Bm1m2m3 (3)
l1l2l3 m m m l1l2l3
m1,m2,m3 1 2 3 !
X
Even if there was no non-Gaussianity in the initial potential fluctuations,
non-linear physics will produce mode-mode couplings. We argue here that
the bispectrum is a particularly powerful tool for detecting these couplings.
2
In inflation, quantum fluctuations produce Gaussian random phase vari-
ations in the gravitational potential:
d3k
Φ(x) = Φ (k)exp(ik·x) (4)
(2π)3 0
Z
with hΦ (k)Φ∗(k′)i = P(k)δ(3)(k−k′), the power spectrum of potential fluc-
0 0
tuations at the current epoch. While processes during inflation may produce
some non-Gaussianity, this is small in most inflationary models [6]. On the
other hand, when the fluctuations reenter the horizon, there are many non-
linear effects that alter these initial conditions. Near decoupling, non-linear
hydrodynamical effects couple modes [7]. After decoupling, non-linear gen-
eral relativistic corrections also alter the microwave background spectrum
[8, 9] as do non-linear coupling between density fluctuations[10, 11]. How-
ever, since potential fluctuations in the early universe are small, hΦ2i ∼ 10−9,
these non-linear effects are usually viewed as undetectable. For example,
Mollerach et al.[10] and Munshi et al.[11] estimated the reduced bispectrum
generated by non-linear evolution of long wavelength modes and showed that
they are undetectable in the COBE maps. In a spatially flat universe, the
microwave background temperature at the surface of last scatter can be lin-
early related to the potential fluctuations at that epoch, giving us what we
shall henceforth call the linear temperature approximation:
d3k
TL(nˆ) = φ(τ ) eik·nˆτrΦ (k)g(k), (5)
r (2π)3 0
Z
where τ is the comoving (conformal) lookback time, φ(τ) is a normaliza-
tion constant for potential perturbations which is equal to unity today, and
g(k) is the linear radiation transfer function and describes the relationship
between potential fluctuations and temperature fluctuations at recombina-
tion. For small k, g(k) = 1/3. For larger k, we need to evolve the coupled
matter-radiation fluid. Codes like CMBFAST compute g(k) for different
cosmological models.
Expanding the linear part of the temperature perturbation into spherical
harmonics:
d3k
aL = φ(τ ) dnˆeik·nˆτΦ (k)g(k)Y∗ (nˆ) . (6)
lm r (2π)3 0 lm
Z
3
Since we will later wish to integrate over the sky, it will be useful to ex-
pand out the exponential into spherical harmonics as well using the Rayleigh
expansion:
eik·r = 4π ilj (kr)Y∗ (ˆr)Y (kˆ) . (7)
l lm lm
lm
X
Thus, the linear approximation of the CMB fluctuations are given by:
d3k
aLlm = 4πφ(τr)(−i)l (2π)3Φ0(~k)Yl,−m(kˆ)jl(kτr)g(k) (8)
Z
InadditiontothetermsproportionaltoΦ (k)(linear),thereareanumber
0
of terms in the temperature fluctuations which are proportional to Φ (k)2
0
(nonlinear). These corrections arise from gravitational couplings, gravita-
tionallensing, and radiationeffects. These terms generically add a correction
to the microwave background:
0
TNL(nˆ) = dτΦ2(nˆ)f(nˆτ,τ) (9)
Zτr
d3k d3k
= dτφ2(τ) 1 2 Φ (k )Φ (k )ei(k1+k2)·nˆτf(k ,k ,τ)
(2π)3(2π)3 0 1 0 2 1 2
Z Z
We may again take the spherical harmonic transform of the non-linear
temperature perturbation to calculate the coefficient, aNL. We will find it
lm
convenient to expand the coupling term, f(k ,k ,τ) into Legendre polyno-
1 2
mials:
4π
f(k ,k ,τ) ≡ f (k ,k ,τ)P (kˆ ·kˆ ) = f (k ,k ,τ)Y (kˆ )Y∗ (kˆ )
1 2 l 1 2 l 1 2 2l+1 l 1 2 lm 1 lm 2
l lm
X X
(10)
Sincethenon-lineartermwillonlycoupletoitselfifk = k ,wecansafely
1 2
ignore this effect, and instead concentrate on the case in which a bispectrum
is produced by:
Bl1l2l3 = aL aL aNL∗ +aL aL aNL∗ +aL aL aNL∗ (11)
m1m2m3 l1m1 l2m2 l3m3 l2m2 l3m3 l1m1 l3m3 l1m1 l2m2
We will only examine the first term in this derivation, and add in the other
two permutation in the end.
4
Looking at aNL∗, we find:
l3m3
(4π)3
aNL∗ = dτφ2(τ) k2dk k2dk Φ∗(k )Φ∗(k ) (12)
l3m3 (2π)6 1 1 2 2 0 1 0 2
Z Z
1
× (−i)l′+l′′ fl(k1,k2,τ)jl′(k1τ)jl′′(k2τ)
2l+1
ll′l′′mm′m′′
X
× dnˆdkˆ1dkˆ2Yl3m3(nˆ)Yl′m′(nˆ)Yl′′m′′(nˆ)Yl∗′m′(kˆ1)Yl∗m(kˆ1)Ylm(kˆ2)Yl∗′′m′′(kˆ2)
Z
Substituting this expression and equation (8) into equation (11), we get:
(4π)5 d3k d3k′
aL aL aNL∗ = φ2(τ ) dτφ2(τ) k2dk k2dk (13)
l1m1 l2m2 l3m3 (2π)6 r (2π)3(2π)3 1 1 2 2
Z Z Z
′ ∗ ∗ ′
× Φ (k)Φ (k)Φ (k )Φ (k )g(k)g(k )
0 0 0 1 0 2
1
′
× 2l+1fl(k1,k2,τ)jl′(k1τ)jl′′(k2τ)jl1(kτr)jl2(k τr)
ll′l′′mm′m′′
X
× dnˆdkˆ1Yl3m3(nˆ)Yl′m′(nˆ)Yl′′m′′(nˆ)Yl∗′m′(kˆ1)Ylm∗ (kˆ1)Yl∗1m1(kˆ)
Z
× dkˆ Y (kˆ )Y∗ (kˆ )Y∗ (kˆ′)
2 lm 2 l′′m′′ 2 l2m2
Z
The integrals over k and k′ give two possible permutations, δ(k−k ) and
1
δ(k − k ). This is patently symmetric, and thus, we will chose the former
2
and multiply the integral by 2. After integrating over k and k′, we find that
we have 3 sets of 3 spherical harmonics. This is precisely the form of the Q
coefficients discussed in the appendix. Thus, integrating over nˆ, kˆ , and kˆ ,
1 2
′ ′′
summing over m,m,m , and convolving with the Wigner 3-j symbol over
m ,m ,m as in equation (3), we find that the first of three terms in the
1 2 3
angle averaged bispectrum is:
1
B(1) = φ2(τ ) dτφ2(τ) dk dk k2k2P(k )P(k )g(k )g(k ) (14)
l1l2l3 2π4 r 1 2 1 2 1 2 1 2
Z Z
× (2l′ +1)(2l′′ +1)Qll1l′l′2l′l3fl(k1,k2,τ)jl′(k1τ)jl′′(k2τ)jl1(k1τr)jl2(k2τr)
ll′l′′
X
3 Measuring the Bispectrum
Measuring this effect from observations is quite straightforward. If there is
complete sky coverage and no sky cuts, the the angle averaged bispectrum
5
can be computed by rewriting the Wigner 3-j symbols as integrals over the
sky,
l l l l l l l l l
1 2 3 B = 1 2 3 1 2 3 a a a
0 0 0 l1l2l3 0 0 0 m m m l1m1 l2m2 l3m3
! m1,m2,m3 ! 1 2 3 !
X
(4π)3
=
v(2l +1)(2l +1)(2l +1)
u 1 2 3
u
t
× dqˆY (qˆ)Y (qˆ)Y (qˆ)
l1m1 l2m2 l3m3
m1,m2,m3Z
X
× dˆldmˆdnˆY∗ (ˆl)Y∗ (mˆ)Y∗ (nˆ)T(ˆl)T(mˆ)T(nˆ)
l1m1 l2m2 l3m3
Z
= dqˆe (qˆ)e (qˆ)e (qˆ) (15)
l1 l2 l3
Z
where the sky map is averaged over rings centered around point qˆ:
2l+1
e (qˆ) = dˆlT(ˆl)P qˆ·ˆl (16)
l l
s 4π
Z (cid:16) (cid:17)
Sincethebispectrum signalisrather weak, thedominant sourceofnoiseis
the cosmic variance ofthe dominant Gaussiansignal. Fora Gaussian random
field, ha a a i = 0. However, its variance, h(a a a )2i =
l1m1 l2m2 l3m3 l1m1 l2m2 l3m3
c c c for l 6= l , l 6= l , l 6= l [12]. When we include detector noise, then
l1 l2 l3 1 2 2 3 1 3
h(a a a )2i = c +σ2w−2 c +σ2w−2 c +σ2w−2 (17)
l1m1 l2m2 l3m3 l1 0 l1 l2 0 l2 l3 0 l3
(cid:16) (cid:17)(cid:16) (cid:17)(cid:16) (cid:17)
where σ is the detector noise and w is the experimental window function.
0 l
Similarly, for a Gaussian field,
hB i = 0
l1l2l3
Bl1l2l3Bl1′l2′l3′ = cl1 +σ02wl−12 cl2 +σ02wl−22 cl3 +σ02wl−32 (18)
D E (cid:16) × δl1′l2′l3′(cid:17)+(cid:16)δl1′l2′l3′ +δl1′l2′(cid:17)l3′(cid:16)+δl1′l2′l3′ +δl(cid:17)1′l2′l3′ +δl1′l2′l3′
l1l2l3 l1l3l2 l2l1l3 l2l3l1 l3l1l2 l3l2l1
(cid:16) (cid:17)
for l 6= l , l 6= l , l 6= l . Because of this symmetry, we restrict ourselves to
1 2 2 3 1 3
bispectrum terms where l < l < l .
1 2 3
6
4 The Bispectrum from the Rees-Sciama Ef-
fect
Several groups[10, 11] have discussed the possibility of detecting the non-
linearsignatureoftheRees-Sciama effectintheCOBEdata. Theyconcluded
that the signal was undetectable on COBE scales. In their analysis, they
focus only on the diagonal terms in the bispectrum Here, we compute the
predicted signal on smaller angular scales and compute all of the bispectrum
terms.
The second order Rees-Sciama effect arises from the non-linear growth of
density fluctuations[10]:
∂
TNL(nˆ) = 2 dτ ΦNL(nˆτ,τ) (19)
∂τ
Z
∂ d3k
= 2 dτ ΦNL(kˆ,τ)eik·nˆτ
∂τ (2π)3
Z Z
and give the second-order potential perturbation as:
(τ −τ)2 (d3k′)
ΦNL(k,τ) = − 0 φ2(τ) Φ (k−k′)Φ (k′) 3k2k′2 +7k2k·k′ −10(k·k′)2
84k2 (2π)3 0 0
Z h i
(20)
This second -order approximation is valid for k < 1h Mpc−1. [13].
Since we expect much of the signal to come from high redshift contribu-
˙
tions, and since in a flat, Ω = 1 universe, φ(τ) = 0, we will take φ(τ) is
m
constant for the remainder of this derivation. Additionally, we will redefine
our variables of integration: k ≡ k−k′ and k = k′. This yields:
1 2
d3k d3k
TNL(nˆ) = dτφ2(τ) 1 2 Φ (k )Φ (k )ei(k1+k2)nˆτ (21)
(2π)3(2π)3 0 1 0 2
Z Z
(τ −τ)
× 0 10k2k2 +7(k2 +k2)k k µ+4k2k2µ2
21(k +k )2 1 2 1 2 1 2 1 2
1 2
h i
where µ ≡ kˆ ·kˆ .
1 2
Thesecondpartoftheequationisthecouplingfunction, f(kˆ ,kˆ ,τ). The
1 2
vectors in the denominator make it difficult to decompose it into Legendre
polynomials, however, this may be simplified by noting that:
aNL ∝ dnˆei(k1+k2)nˆτY∗ (nˆ) (22)
lm lm
Z
7
Thus, taking the Laplacian of aNL, we find:
lm
d3k d3k
l(l+1)aNL∗ = dnˆ dτφ2(τ) 1 2 Φ (k )Φ (k )ei(k1+k2)·nˆτYNL(nˆ)
lm (2π)3(2π)3 0 1 0 2 lm
Z Z Z
× f(k ,k ,τ)(k +k )2τ2 (23)
1 2 1 2
We now define a new coupling function:
f˜(k ,k ,τ) ≡ f(k ,k ,τ)(k +k )2τ2 , (24)
1 2 1 2 1 2
which can be easily substituted into the definition of the bispectrum, with
˜
B ⇒ l (l +1)B and f ⇒ f .
l1l2l3 3 3 l1l2l3 l l
For the Rees-Sciama effect, this is decomposed into only the l = 0,l = 1,
and l = 2 Legendre coefficients:
2(τ −τ)τ2 17
˜ 0
f (k ,k ,τ) = k k k k (25)
0 1 2 1 2 1 2
3 21
(cid:18) (cid:19)
2(τ −τ)τ2 1
f˜(k ,k ,τ) = 0 k k (k2 +k2) (26)
0 1 2 3 1 2 2 1 2
(cid:18) (cid:19)
2(τ −τ)τ2 4
˜ 0
f (k ,k ,τ) = k k k k (27)
2 1 2 1 2 1 2
3 21
(cid:18) (cid:19)
(28)
so,
1
l (l +1)B = φ2(τ ) dτφ2(τ)(τ −τ)τ2 (29)
3 3 l1l2l3 3π4 r 0
Z
× dk dk k3k3P(k )P(k )g(k )g(k )j (k τ )j (k τ )
1 2 1 2 1 2 1 2 l1 1 r l2 2 r
Z
′ ′′
× jl′(k1τ)jl′′(k2τ)(2l +1)(2l +1)
l′l′′
X
17 1 4
× Ql1l2l3k k + Ql1l2l3(k2 +k2)+ Ql1l2l3k k
21 0l′′l′ 1 2 2 1l′′l′ 1 2 21 2l′′l′ 1 2
(cid:20) (cid:21)
As discussed in the appendix, (2l′+1)(2l′′+1)Ql1l2l3 = (2l +1)(2l +
l′l′′ ll′′l′ 1 2
1)Ql1l2l3, and thus, if l is small (in this case, it only takes on the values 0,1,2),
0l1l2 P
′ ′′
and hl′l′′ is a function which varies slowly with l and l , we may make the
approximation:
(2l′ +1)(2l′′ +1)Qll1l′l′2l′l3hl′l′′ ≃ (2l1 +1)(2l2 +1)Ql01ll12ll23hl2l1 (30)
l′l′′
X
8
Under this assumption, the bispectrum reduces to:
4φ2(τ )I
l (l +1)B = r l1l2l3 dτφ2(τ)(τ −τ)τ2 (31)
3 3 l1l2l3 3π5/2 0
Z
× dk dk k3k3P(k )P(k )g(k )g(k )
1 2 1 2 1 2 1 2
Z
× j (k τ )j (k τ )j (k τ)j (k τ)(k +k )2
l1 1 r l2 2 r l2 1 l1 2 1 2
where
l l l
I = (2l +1)(2l +1)(2l +1) 1 2 3 (32)
l1l2l3 1 2 3 0 0 0
!
q
In order to deal with the spherical Bessel functions, we will first note that
in the asymptotic limit, if x > l, j (x) ≃ sin(x−πl/2)/x. The lower limit of
l
integration will be taken to be:
l l
1 2
τ = max , (33)
min
k2 k1!
Likewise, the lower limit of integration on k is l /τ and k is l /τ . In this
1 1 r 2 2 r
limit, we can expand out the four spherical Bessel functions as:
1 cos[(k −k )τ )]cos[(k −k )τ)]
j (k τ )j (k τ )j (k τ)j (k τ) ≃ (−1)l1+l2 1 2 r 1 2
l1 1 r l2 2 r l2 1 l1 2 4 k2k2τ2τ2
1 2 r
(34)
plus additional terms which oscillate much more quickly.
We can now approximate the the time integral:
τr
dττ2(τ −τ)φ2(τ)j (k τ )j (k τ )j (k τ)j (k τ) (35)
0 l1 1 r l2 2 r l2 1 l1 2
Zτmin
1 τr cos[(k −k )τ]
≃ (−1)l1+l2 cos[(k −k )τ ] dτ(τ −τ)φ2(τ) 1 2
4 1 2 r 0 k2k2τ2
Zτmin 1 2 r
πδ(k −k )
≃ 1 2 (−1)l1+l2φ2(τ )(τ −τ )
4 k2k2τ2 min 0 min
1 2 r
where we have approximated φ(τ)(τ −τ) as a slowly varying function which
0
is maximally weighted near τ = τ , and sin(x)/x is approximated as a
min
delta function.
9
Thus, the bispectrum can be approximated as:
φ2(τ ) 4 I ∞
B(1) ≃ r l1l2l3 dkk4P2(k)g2(k)φ2(τ )(τ −τ )
l1l2l3 τ2 3π3/2l (l +1) min 0 min
r 3 3 Zlm/τr
(36)
where we have defined l ≡ max(l ,l ).
m 1 2
In equation (8), we defined g(k) in terms of aL . Defining c ≡ haL aL∗i,
lm l lm lm
we find:
2
c = k2dkP(k)j2(kτ )g2(k) (37)
l π l r
Z
For large values the spherical Bessel functions are very close to zero for
kτ < l and since k2P(k) is a monotonically decreasing function for all values
r
of k, we may approximate j2(x) = πδ(l−x)/2. With this approximation:
l
φ2(τ )
c ≃ r P(k)g2(k)k2 (38)
kτr τ (2k/τ +1)
r r
Substituting this into the previous expression yields:
4 I ∞
B(1) ≃ l1l2l3 dkk2P(k)c (2kτ +1)φ2(τ )(τ −τ )
l1l2l3 3π3/2τ l (l +1) kτr r min 0 min
r 3 3 Zkmin
4 I l
≃ l1l2l3 l2(2l+1)P c φ2(τ l /l)(τ −τ l /l)
3π3/2τr4l3(l3 +1) lX=lm τr! l r m 0 r m
4 I
≡ l1l2l3 b (39)
3π3/2τ4l (l +1) lm
r 3 3
In Figure 1 we have plotted the coefficient b for flat cosmologies where
lm
Ω = 0.1, 0.3, and 1.0.
m
Will we beableto detect thiseffect? We canestimate thisby determining
whether ina Ω = 0.3,Ω = 0.7 universe, we canuse the MAP data to reject
m Λ
the hypothesis that there is no non-Gaussianity in the CMB maps:
hB i2
χ2 = l1,l2,l3 (40)
hB2 i
l1X,l2,l3 l1,l2,l3
InFigure2weshow theamount of“information”gainedbyincreasing the
maximum index (l in our notation). Note that since all of the signal occurs
3
around l < 600, and the MAP and PLANCK satellites are cosmic variance
10