![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](https://image.slidesharecdn.com/asconalss13-150124125250-conversion-gate02/85/Large-scale-structure-non-Gaussianities-with-modal-methods-Ascona-2-320.jpg)
![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](https://image.slidesharecdn.com/asconalss13-150124125250-conversion-gate02/85/Large-scale-structure-non-Gaussianities-with-modal-methods-Ascona-25-320.jpg)
![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
MS, Regan, Shellard 1207.5678](https://image.slidesharecdn.com/asconalss13-150124125250-conversion-gate02/85/Large-scale-structure-non-Gaussianities-with-modal-methods-Ascona-26-320.jpg)
⌫
| {z }
time shifted 1-halo term
See Valageas for similar philosophy in
Gaussian case (combing PT+halo model)](https://image.slidesharecdn.com/asconalss13-150124125250-conversion-gate02/85/Large-scale-structure-non-Gaussianities-with-modal-methods-Ascona-27-320.jpg)
![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])
MS, Regan, Shellard 1207.5678
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
z20
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
z20
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.](https://image.slidesharecdn.com/asconalss13-150124125250-conversion-gate02/85/Large-scale-structure-non-Gaussianities-with-modal-methods-Ascona-28-320.jpg)
Large-scale structure non-Gaussianities with modal methods (Ascona)
- 1. 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
- 2. 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
- 3. 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 + · · ·
- 4. 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)
- 5. 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
- 6. bispectrum drawn on space of triangle configurations
- 7. 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:
- 8. 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
- 9. 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
- 10. BISPECTRUM ESTIMATION: MATHS Fergusson, Shellard et al. 2009-2012 MS, Regan, Shellard 1207.5678 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
- 12. MS, Regan, Shellard 1207.5678 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
- 13. MS, Regan, Shellard 1207.5678 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) Tree level theory N-body z=2 GAUSSIAN SIMULATIONS COMPARISON WITH TREE LEVEL
- 14. MS, Regan, Shellard 1207.5678 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) Tree level theory N-body z=0 GAUSSIAN SIMULATIONS COMPARISON WITH TREE LEVEL
- 15. GAUSSIAN SIMULATIONS ON SMALL SCALES MS, Regan, Shellard 1207.5678 3 realisations of 5123 particles in a L = 400 Mpc/h box with zinit = 49 and k = 0.016 - 2.0 h/Mpc Plot S/N weighted bispectrum p k1k2k3B (k1, k2, k3)/ p P (k1)P (k2)P (k3) N-body N-body N-body z=1 z=0z=2
- 16. MS, Regan, Shellard 1207.5678 DM density GAUSSIAN SIMULATIONS 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 (a) Dark matter, z = 4 (b) Bispectrum signal, z = 4 z=4
- 17. MS, Regan, Shellard 1207.5678 DM density GAUSSIAN SIMULATIONS 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 (c) Dark matter, z = 2 (d) Bispectrum signal, z = 2 40 Mpc/h 40 M pc/h (a) Dark matter, z = 4 (b) Bispectrum signal, z = 4 (c) Dark matter, z = 2 (d) Bispectrum signal, z = 2 z=2
- 18. MS, Regan, Shellard 1207.5678 DM density GAUSSIAN SIMULATIONS 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) (c) Dark matter, z = 2 (d) Bispectrum signal, z = 2 (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 (c) Dark matter, z = 2 (d) Bispectrum signal, z = 2 (e) Dark matter, z = 0 (f) Bispectrum signal, z = 0 z=0 DM bispectrum
- 19. GAUSSIAN SIMULATIONS SUMMARY Summary Gaussian N-body simulations z=4 z=0 DM density DM bispectrum signal pancakes enhanced equilateral filaments & clusters flattened MS, Regan, Shellard 1207.5678 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)
- 21. Tree level theory N-body 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 MS, Regan, Shellard 1207.5678
- 22. Tree level theory N-body 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=2 LOCAL NON-GAUSSIAN SIMS MS, Regan, Shellard 1207.5678
- 23. Tree level theory N-body 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=0 LOCAL NON-GAUSSIAN SIMS MS, Regan, Shellard 1207.5678
- 24. OTHER SHAPES Excess DM bispectra for other non-Gaussian initial conditions multiple fields higher derivative operators non-standard vacuum MS, Regan, Shellard 1207.5678 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)
- 25. 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
- 26. 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 MS, Regan, Shellard 1207.5678
- 27. 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 = MS, Regan, Shellard 1207.5678 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)
- 28. 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]) MS, Regan, Shellard 1207.5678 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 z20 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 z20 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.
- 29. 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)
- 30. 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