Large-scale structure non-Gaussianities with modal methods (Ascona)

LARGE SCALE STRUCTURE
NON-GAUSSIANITIES
WITH MODAL METHODS
Marcel Schmittfull
DAMTP, University of Cambridge
Collaborators: Paul Shellard, Donough Regan, James Fergusson
Based on 1108.3813, 1207.5678
LSS13, Ascona, 3 July 2013

PRIMORDIAL MOTIVATION
LSS = most promising window for non-Gaussianity after Planck
(3d data; single field inflation can be ruled out with halo bias; BOSS, DES, Euclid, ...)
Local template/linear regime rather well understood
But: Effects beyond local template/linear regime not well studied
Dalal et al. 2008, Jeong/Komatsu 2009, Sefusatti et al.
2009-2012, Baldauf/Seljak/Senatore 2011, ...
Vacuum expectation value of a quantum field perturbation with inflationary Lagrangian
Free theory is Gaussian
Interacting theory is non-Gaussian
possible for all n, k-dependence characterises interactions
➟ Inflationary interactions are mapped to specific types of non-Gaussianity
h⌦| 'k1
· · · 'kn
|⌦i =
(
determined by 2-point function, n even,
0, n odd.
h⌦| 'k1
· · · 'kn
|⌦i 6= 0
h⌦| 'k1 · · · 'kn |⌦i =
R
D[ '] 'k1
· · · 'kn
exp(i
R
C
L( 'k))
R
D[ ']exp(i
R
C
L( 'k))
L ⇠ '2
) ei
R
C
L
L ⇠ '3
, '4
, . . . ) ei
R
C
L
' L

Complementary late-time motivation for LSS bispectrum:
LATE-TIME MOTIVATION
Fry 1994
Verde et al. 1997-2002
Scoccimarro et al. 1998
Sefusatti et al. 2006
But: No LSS bispectrum pipeline
Complicated modelling (non-linear DM, bias, RSD, correlations, systematics, ...)
Ph(k) ⇡ b2
1P (k) Bh(k1, k2, k3) ⇡ b3
1B (k1, k2, k3) + b2
1b2(P (k1)P (k2) + perms)
Halo power spectrum Halo bispectrum
Break b1-!m degeneracy, i.e. can
measure !m from LSS alone
Measure b2
Cannot distinguish
rescaling of P" from b1
➟ b1-!m degeneracy
From (naive) we geth(x) ⇠ b1 (x) +
1
2
b2 (x)2
+ · · ·

BISPECTRUM
Bispectrum
Primary diagnostic for non-Gaussianity because vanishes for Gaussian perturbations
Defined for closed triangles (statistical homogeneity and isotropy)
h (k1) (k2) (k3)i = (2⇡)3
D(k1 + k2 + k3)fNLB (k1, k2, k3)
non-linear
amplitude
bispectrum
k1
k2
k3
sum of the two smaller sides ≥ longest side
(every point corresponds to a triangle config.)
)
k3
kmax
kmax k1
k2
Bispectrum domain = ‘tetrapyd’
(=space of triangle configurations)

BISPECTRUM SHAPES
)
k3
kmax
kmax k1
k2
Squeezed triangles (local shape)
Arises in multifield inflation;
detection would rule out all
single field models!
Different inflation models induce different momentum dependencies (shapes) of B (k1, k2, k3)
k2 ⇡ k3 k1
k1
k2
k3
Equilateral triangles
k1 ⇡ k2 ⇡ k3
k3
kmax
kmax k1
k2
k3
kmax
kmax k1
k2
Typically higher derivative
kinetic terms, e.g. DBI inflation
Folded triangles
k3 = 2k1 = 2k2
k2
k3
k1
E.g. non-Bunch-Davies vacuum

bispectrum drawn on space of
triangle conﬁgurations

INITIAL CONDITIONS
Fast and general non-Gaussian initial conditions for N-body simulations
Arbitrary (including non-separable) bispectra, diagonal-independent trispectra
This is the only method to simulate structure formation for general inflation models to date
Idea:
↵0= +↵1 +↵2 + · · · + ↵nmaxWB
bispectrum drawn on space of
triangle configurations
expansion in separable, uncorrelated basis functions
(around 100 basis functions represent all investigated
bispectra with high accuracy)
Regan, MS, Shellard, Fergusson PRD 86, 123524 (2012), arXiv:1108.3813
• Generate non-Gaussian density
• Convert to initial particle positions and velocities by applying 2LPT to glass
configuration or regular grid (spurious bispectrum at high z decays at low z)
• Feed into Gadget3
Application:

Simplest case: local non-Gaussianity
General case: arbitrary bispectrum
Wagner et al 2010
(=2 in local case)WB(k, k0
, k00
) ⌘
B (k, k0
, k00
)
P (k)P (k0) + P (k)P (k00) + P (k0)P (k00)
INITIAL CONDITIONS: MATHS
Aim: Create non-Gaussian ﬁeld
B
Fergusson, Regan, Shellard PRD 86, 063511 (2012), arXiv: 1008.1730
Regan, MS, Shellard, Fergusson PRD 86, 123524 (2012), arXiv:1108.3813
Requires ~N6 operations in general, but only ~N3 operations if WB was separable:
WB(k, k0
, k00
) = f1(k)f2(k0
)f3(k00
) + perms
full ﬁeld Gaussian
(x) = G(x) + NG(x)
non-Gaussian part
(x) = G(x) + fNL( 2
G(x) h 2
Gi)
Expanding WB in separable basis functions gives N3 scaling for any* bispectrum
*Scoccimarro and Verde groups try to rewrite WB analytically in separable form; this works sometimes, but not in general
NG(k) =
fNL
2
Z
d3
k0
d3
k00
(2⇡)3 D(k k0
k00
)WB(k, k0
, k00
) G(k0
) G(k00
)
N =
#ptcles
dim
⇠ 1000

BISPECTRUM ESTIMATION
Fast and general bispectrum estimator for N-body simulations
MS, Regan, Shellard 1207.5678
Measure ~100 fNL amplitudes of
separable basis shapes, combine them to
reconstruct the full bispectrum
Scales like 100xN3 instead of N6, where
N~1000 (speedup by factor ~107)
Can estimate bispectrum whenever
power spectrum is typically measured
Validated against PT at high z
Useful compression to ~100 numbers
Automatically includes all triangles
Loss of total S/N due to truncation of
basis is only a few percent
(could be improved with larger basis; for ~N3
basis functions the estimator would be exact)
Theory
Expansion
of theory
N-body
Gravity Excess bispectrum
for local NG
z=30

BISPECTRUM ESTIMATION: MATHS
Fergusson, Shellard et al. 2009-2012
Likelihood for fNL given a density perturbation δk
Maximise w.r.t. fNL (given a theoretical bispectrum )
Requires ~N6 operations in general, but only ~N3 operations if was separable
➟ Measure amplitudes of separable basis functions
➟ Combine them to reconstruct full bispectrum from the data:
L /
Z
d3
k1
(2⇡)3
d3
k2
(2⇡)3
d3
k3
(2⇡)3
2
6
41
1
6
h k1 k2 k3
i
| {z }
/fNLB
@
@ k1
@
@ k2
@
@ k3
+ · · ·
3
7
5
1
p
det C
Y
ij
e
1
2
⇤
ki
(C 1
)ij kj
/ P
ˆf
Btheo
NL =
1
NfNL
Z
d3
k1
(2⇡)3
d3
k2
(2⇡)3
d3
k3
(2⇡)3
(2⇡)3
D(k1 + k2 + k3)Btheo
(k1, k2, k3) k1 k2 k3
P (k1)P (k2)P (k3)
Btheo
Btheo
where , ,R
n ⌘
X
m
nm
Q
m Mr(x) ⌘
Z
d3
k
(2⇡)3
eikx qr(k) obs
k
p
kP (k)
Q
m ⌘
Z
d3
xMr(x)Ms(x)Mt(x)
ˆB(k1, k2, k3) =
X
n
R
n
s
P (k1)P (k2)P (k3)
k1k2k3
Rn(k1, k2, k3)
depends on data

3 realisations of 5123 particles in a L = 1600 Mpc/h box with zinit = 49 and k = 0.004 - 0.5 h/Mpc
Plot S/N weighted bispectrum
p
k1k2k3B (k1, k2, k3)/
p
P (k1)P (k2)P (k3)
z=30 Tree level theory N-body
GAUSSIAN SIMULATIONS
COMPARISON WITH TREE LEVEL

p
k1k2k3B (k1, k2, k3)/
p
P (k1)P (k2)P (k3)
Tree level theory N-body
z=2

p
k1k2k3B (k1, k2, k3)/
p
P (k1)P (k2)P (k3)
z=0

ON SMALL SCALES
p
k1k2k3B (k1, k2, k3)/
p
P (k1)P (k2)P (k3)
N-body N-body N-body
z=1 z=0z=2

DM density
COMPARISON WITH DARK MATTER
5123 particles in a L = 400 Mpc/h box with zinit = 49 and k = 0.016 - 2 h/Mpc
Plot DM density in (40 Mpc/h)3 subbox and bispectrum signal
p
k1k2k3B (k1, k2, k3)/
p
P (k1)P (k2)P (k3)
DM bispectrum
(a) Dark matter, z = 4 (b) Bispectrum signal, z = 4
40 Mpc/h
40
M
pc/h
z=4

DM density
p
k1k2k3B (k1, k2, k3)/
p
P (k1)P (k2)P (k3)
DM bispectrum
(c) Dark matter, z = 2 (d) Bispectrum signal, z = 2
40 Mpc/h
40
M
pc/h
z=2

DM density
p
k1k2k3B (k1, k2, k3)/
p
P (k1)P (k2)P (k3)
(e) Dark matter, z = 0 (f) Bispectrum signal, z = 0
. Left: Dark matter distribution in a (40Mpc/h)3 subbox of one of the G512
400 simulations at redshifts z = 4, 2 and
Right: Measured (signal to noise weighted) bispectrum in the range 0.016h/Mpc  k  2h/Mpc, averaged over t
40 Mpc/h
40
M
pc/h
(e) Dark matter, z = 0 (f) Bispectrum signal, z = 0
z=0
DM bispectrum

SUMMARY
Summary Gaussian N-body simulations
z=4
z=0
DM density DM bispectrum
signal
pancakes
enhanced
equilateral
filaments
& clusters
flattened
Measured gravitational DM bispectrum
for all triangles down to k=2hMpc-1
Non-linearities mainly enhance
‘constant’ 1-halo bispectrum
Bispectrum characterises 3d DM
structures like pancakes, filaments,
clusters
Self-similarity
(constant contribution appears towards late
times at fixed length scale, and towards small
scales at fixed time)

3 realisations of 5123 particles in a L = 1600 Mpc/h box with zinit = 49 and k = 0.004-0.5 h/Mpc, fNL=10
Plot S/N weighted excess bispectrum
p
k1k2k3B (k1, k2, k3)/
p
P (k1)P (k2)P (k3)
z=30
LOCAL NON-GAUSSIAN SIMS

p
k1k2k3B (k1, k2, k3)/
p
P (k1)P (k2)P (k3)
z=2

p
k1k2k3B (k1, k2, k3)/
p
P (k1)P (k2)P (k3)
z=0

OTHER SHAPES
Excess DM bispectra for other non-Gaussian initial conditions
multiple fields higher derivative operators non-standard vacuum
z=0, kmax=0.5hMpc-1, 5123 particles
Non-linear regime:
Tree level shape is enhanced by non-linear power spectrum
Additional ~constant contribution to bispectrum signal
Quantitative characterisation with cumulative S/N and 3d shape correlations in 1207.5678
multiple fields
(local)
higher derivatives
(equilateral)
non-standard vacuum
(flattened)

Introduce scalar product , shape correlation and normC
hBi, Bjiest ⌘
V
⇡
Z
VB
dk1dk2dk3
k1k2k3Bi(k1, k2, k3)Bj(k1, k2, k3)
P (k1)P (k2)P (k3)
h·, ·iest
C(Bi, Bj) ⌘
hBi, Bjiest
p
hBi, BiiesthBj, Bjiest
2 [ 1, 1]
if then
estimator for cannot
ﬁnd any and vice versa
Babich et al. 2004, Fergusson, Regan, Shellard 2010
||B|| ⌘
p
hB, Biest
||B||
total integrated S/N
|C(B1, B2)| ⌧ 1
B1
B2
) h ˆfBtheo
NL i = C(Bdata
, Btheo
)
||Bdata
||
||Btheo||
projection of
data on theory
SIMILARITY OF SHAPES

flocal
NL = 10
Btheo
local
ˆBNG
{
NG initial
conditions with
10
20
30
40
floc
NL
k ∈ [0.0039, 0.5]h/Mpc
k ∈ [0.0039, 0.25]h/Mpc
k ∈ [0.0039, 0.13]h/Mpc
0
50
100
¯F
ˆBNG
NL
1
10
||ˆBNG||
0.6
0.8
1
Cα ,th≥ 0.9613
Cβ,α
1 10
0.6
0.8
1
CNG,G
1 + z
ˆflocal
NL
time
z=49today
C( ˆBNG, Blocal)
Btheo
local
ˆBNG
Btheo
local
ˆBNG
}
C( ˆBNG, ˆBGauss)
ˆBNG
ˆBGauss
||B|| / ¯FNL

TIME SHIFT MODEL
Time shift model
Non-Gaussian universe evolves slightly faster
(or slower) than Gaussian universe
Halos form earlier (later) in presence of
primordial non-Gaussianity
Primordial non-Gaussianity gives growth of the
1-halo bispectrum a ‘headstart’ (or delay)
1-halo
bispectrum
at z
1-halo
bispectrum
at z+Δz
Gaussian universeNon-Gaussian universe
=
Motivates simple form of non-Gaussian excess bispectrum:
BNG
=
r
P NL
(k1)P NL
(k2)P NL
(k3)
P (k1)P (k2)P (k3) B (k1, k2, k3)
| {z }
perturbative with non-linear power
+c z@z[Dnh
(z)](k1 + k2 + k3)⌫
| {z }
time shifted 1-halo term
See Valageas for similar philosophy in
Gaussian case (combing PT+halo model)

FITTING FORMULAE
Simple fitting formulae for grav. and primordial DM bispectrum shapes
Valid at 0 ≤ z ≤ 20, k ≤ 2hMpc-1
3d shape correlation with measured shapes is ≥ 94.4% at 0 ≤ z ≤ 20 and ≥ 98 % at z=0
(≥ 99.8% for gravity at 0 ≤ z ≤ 20)
Only ~3 free parameters per inflation model (local, equilateral, flattened, [orthogonal])
Overall amplitude needs
to be rescaled by (poorly
understood) time-
dependent prefactor;
extends Gil-Marin et al
formula to smaller scales
and NG ICs
⌫ ⇡ 1.7
BNG
=
r
P NL
(k1)P NL
(k2)P NL
(k3)
P (k1)P (k2)P (k3) B (k1, k2, k3)
| {z }
perturbative with non-linear power
+c z@z[Dnh
(z)](k1 + k2 + k3)⌫
| {z }
time shifted 1-halo term
10
0
10
1
10
1
10
2
10
3
10
4
1 + z
kBk
(a) Gaussian, kmax = 0.5h/Mpc
10
0
10
1
10
2
10
3
10
4
10
5
1 + z
kBk
(b) Gaussian, kmax = 2h/Mpc
Figure 15. Motivation for using the growth function ¯D in the simple fitting formula (64). The arbitrary weight w(z) in Bo
Bgrav
,NL + w(z)(k1 + k2 + k3)⌫ is determined analytically such that C( ˆB , Bopt
) is maximal (for ⌫ = 1.7). We plot kBgrav
,NLk (black do
kw(z)(k1 + k2 + k3)⌫ k (green) and Bgrav
,const (black dashed) as defined in (14) with fitting parameters given in Table IV, illustrating
w(z) = c1
¯Dnh (z) is a good approximation. The continuous black and red curves show kBfitk from (64) and the estimated bispectrum
k ˆB k, respectively. The overall normalisation can be adjusted with Nfit as explained in the main text.
Simulation L[Mpc
h
] c1,2 n
(prim)
h min
z20
C ,↵ C ,↵(z=0)
G512g 1600 4.1 ⇥ 106
7 99.8% 99.8%
Loc10 1600 2 ⇥ 103
6 99.7% 99.8%
Eq100 1600 8.6 ⇥ 102
6 97.9% 99.4%
Flat10 1600 1.2 ⇥ 104
6 98.8% 98.9%
Orth100 1600 3.1 ⇥ 102
5.5 91.0% 91.0%
G512
400 400 1.0 ⇥ 107
8 99.8% 99.8%
Loc10512
400 400 2 ⇥ 103
dD/da 7 98.2% 99.0%
Eq100512
400 400 8.6⇥102
dD/da 7 94.4% 97.9%
Flat10512
400 400 1.2⇥104
dD/da 7 97.7% 99.1%
Orth100512
400 400 2.6 ⇥ 102
6.5 97.3% 98.9%
Table IV. Fitting parameters c1 and nh for the fit (64) of
the matter bispectrum for Gaussian initial conditions (sim-
which is shown by the dotted line in the lower pane
Fig. 16. While it varies with redshift between 0.7
1.4 for kmax = 2h/Mpc, it deviates by at most 8%
unity for kmax = 0.5h/Mpc. The lower panels also
the measured integrated bispectrum size k ˆBk and
two individual contributions to (64) when the norma
tion factor Nfit is included. These quantities are div
by kBgrav
,NLk for convenience. At high redshifts the
bispectrum size is essentially given by the contribu
from Bgrav
,NL, which equals the tree level prediction fo
gravitational bispectrum in this regime. The cont
tion from Bconst
dominates at z  2 for kmax = 2h/
when filamentary and spherical nonlinear structure
apparent. A similar transition can be seen at later t
on larger scales in Fig. 16a, indicating self-similar
Quality of fit:
all z z=0
10
0
10
1
10
1
10
2
10
3
10
4
1 + z
kBk
(a) Gaussian, kmax = 0.5h/Mpc
10
0
10
1
10
2
10
3
10
4
10
5
1 + z
kBk
(b) Gaussian, kmax = 2h/Mpc
Figure 15. Motivation for using the growth function ¯D in the simple fitting formula (64). The arbitrary weight w(z) in B
Bgrav
,NL + w(z)(k1 + k2 + k3)⌫ is determined analytically such that C( ˆB , Bopt
) is maximal (for ⌫ = 1.7). We plot kBgrav
,NLk (black do
kw(z)(k1 + k2 + k3)⌫ k (green) and Bgrav
,const (black dashed) as defined in (14) with fitting parameters given in Table IV, illustrating
w(z) = c1
¯Dnh (z) is a good approximation. The continuous black and red curves show kBfitk from (64) and the estimated bispectrum
k ˆB k, respectively. The overall normalisation can be adjusted with Nfit as explained in the main text.
Simulation L[Mpc
h
] c1,2 n
(prim)
h min
z20
C ,↵ C ,↵(z=0)
G512g 1600 4.1 ⇥ 106
7 99.8% 99.8%
Loc10 1600 2 ⇥ 103
6 99.7% 99.8%
Eq100 1600 8.6 ⇥ 102
6 97.9% 99.4%
Flat10 1600 1.2 ⇥ 104
6 98.8% 98.9%
Orth100 1600 3.1 ⇥ 102
5.5 91.0% 91.0%
G512
400 400 1.0 ⇥ 107
8 99.8% 99.8%
Loc10512
400 400 2 ⇥ 103
dD/da 7 98.2% 99.0%
Eq100512
400 400 8.6⇥102
dD/da 7 94.4% 97.9%
Flat10512
400 400 1.2⇥104
dD/da 7 97.7% 99.1%
Orth100512
400 400 2.6 ⇥ 102
6.5 97.3% 98.9%
Table IV. Fitting parameters c1 and nh for the fit (64) of
the matter bispectrum for Gaussian initial conditions (sim-
ulations G512g and G512
400) and c2 and nprim
h for the fit (69)
which is shown by the dotted line in the lower pane
Fig. 16. While it varies with redshift between 0.7
1.4 for kmax = 2h/Mpc, it deviates by at most 8%
unity for kmax = 0.5h/Mpc. The lower panels also
the measured integrated bispectrum size k ˆBk and
two individual contributions to (64) when the norma
tion factor Nfit is included. These quantities are div
by kBgrav
,NLk for convenience. At high redshifts the
bispectrum size is essentially given by the contribu
from Bgrav
,NL, which equals the tree level prediction fo
gravitational bispectrum in this regime. The cont
tion from Bconst
dominates at z  2 for kmax = 2h/
when filamentary and spherical nonlinear structure
apparent. A similar transition can be seen at later t
on larger scales in Fig. 16a, indicating self-simila
haviour.

CONCLUSIONS
Conclusions
Efﬁcient and general non-Gaussian N-body initial conditions
Fast estimation of full bispectrum
new standard diagnostic alongside power spectrum
Tracked time evolution of the DM bispectrum in a large suite of non-
Gaussian N-body simulations
Time shift model for effect of primordial non-Gaussianity
New ﬁtting formulae for gravitational and primordial DM bispectra
See 1108.3813 (initial conditions)
1207.5678 (rest)

COMPARISON WITH OTHER ESTIMATORS
Brute force bispectrum estimator
➟ Slow (107 times slower for 10003 simulation)
Taking only subset of triangle conﬁgurations/orientations
➟ Non-optimal estimator: Throw away data
(instead, we compress theory domain, not data domain; and we quantify lost S/N~few %)
Binning
➟ Same as our estimator if basis functions are taken as top-hats
➟ But: Not known how much signal is lost with binning (our monomial basis looses few %)
Correlations between triangles and with power spectrum
➟ Potentially easier to get correlations for 100 than for full 3d bispectrum shape
Comparison with theory
➟ Easier with 100 than for full 3d bispectrum shape
Experimental realism:
➟ Need to try... (Planck started with ~7 bispectrum estimators, 3 survived)
R
n
R
n

Large-scale structure non-Gaussianities with modal methods (Ascona)

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