2019/11/20 Theoretical aspects of non-Gaussianity from modern perspectives@YITP

Constraining
Primordial non-Gaussianitywith future galaxy surveys
Daisuke Yamauchi
Kanagawa University

Current CMB constraint on PNG◆Planck 2018

fNLlocal = -0.9 +- 5.1
fNLequil = -26 +- 47
fNLortho =-38 +- 24
gNLlocal = (-5.8 +- 6.5)×104

CMB experiments are already close to CV-limited ones.

[Komatsu+Spergel(2000)]

Current LSS constraint on PNG◆Galaxy power spectrum

fNLlocal = -113 +-154 (1σ) [Ho+(2013)]

fNLlocal = 5 +- 21 (1σ) [Giannaotonio+(2013)]
fNLlocal = -15+-36 (2σ) [Castorina+(2019)][Giannaotonio+(2013)]

How do we constrain local
PNG with galaxy surveys?

Galaxy bias

Luminous sources should betreated as biased tracer of LSS.

matter overdensity δm[Kazuhiro’s talk]δcr
Short-wavelength
perturbation
Long-wavelength
perturbation

position

Galaxy bias
Galaxy number density contrast δgal does not exactly
coincide with matter overdensity field δm, but …

δgal = F[δm,…]= b1 δm + b2 δm2/2+…
Linear bias Nonlinear bias
(which is naturally induced bynonlinear gravitational evolution)✓ In the Gaussian case, the bias on large scales is scale-invariant.

fNLlocal induces scale-dependence(1)[Dalal+Dore+Huterer+Shirokov(2008) ]◆“Rough” estimation[Kazuhiro’s talk]ΦNG = φG + fNL(φG)2Acting the Laplacian on ΦNG…

2
∇Φ

NG2
=∇ φ2
G + 2fNLφG∇ φG +・・・

∝ δNG

∝δ

∝ δ/k2∝δδNG ≃ (1+ 2fNLφG)δ

fNLlocal induces scale-dependence(2)[Smith+LoVerde(2011), Smith+Ferraro+LoVerde(2011) ]◆To be precise, let us decompose the potential into

Φ = φshort + φlong[Yuichiro’s talk]ΔΦ = (1 + 2fNLφlong) ΔΦshort +…
◆The halo number density on large scales is given bynhalo = nbg + (dn/dδlong) δlong+ 2fNL(dn/dΔΦ) φlong +…

fNLlocal induces scale-dependence(3)➢ Local-type PNG induces Δb∝1/k2 dependence
such that the effect dominates at very large scales:[Dalal+(2008), Desjacques+(2009)]

fNL=+100fNL=+500

fNL=-100fNL=-500[Dalal+(2008)]

fNLlocal induces scale-dependence(3)Pgg = (b1 + bNG(f) fNLlocal/k2)2 Pmm
Heavier galaxysample

Lighter galaxysample[Ferremacho+(2014)]

fNL , gNL and τNL
fNLlocal → Δb = bNG(f) fNLlocal /k2
gNLlocal → Δb = bNG(g) gNLlocal /k2➢ fNL + τNL case

Pgg = [(b1 + bNG(f) fNL/k2)2+{(5/6)2 τNL ｰ fNL2}(bNG(f))2/k4 ] Pmm≧ 0 (Suyama-Yamaguchi inequality)

Constraining local PNG with galaxy survey[Ferraro+Smith(2014)]

V=25Gpc3/hz=0.7
bg=2.5

PlanckPlanckEuclid SKA2Planck

High noise ←→ Low noise[ Sekiguchi+Takahashi+Tashiro+Yokoyama(2018)]

Constraining local PNG with minihalosσ(gNL)/103=
18 [COrE(CMB)]
2.3 [SKA]
0.79 [FFTT]

σ(τNL)=
160 [COrE(CMB)]
28 [SKA]
0.59 [FFTT]

fNLequil , fNLfold and fNLortho[e.g., Matsubara(2012)]

fNLlocal → Δb ∝ fNLlocal /k2
Relatively weak k-dependence

fNLequil → Δb ∝ fNLequil /k0
fNLfold → Δb ∝ fNLfold /k1

fNLortho → Δb ∝ fNLortho /k1
It is difficult to constrain non-local PNG with scale-dep bias…

[Yoo(2010), Challinor+Lewis(2011), Bruni+(2011),…]

Note: Relativistic corrections◆ The observed galaxy overdensity on superhorizon
scales depends on the gravitational potentials:

standardlensing

GR effect∝δ/k2 (Poisson eq.)
In order not to bias the estimation of PNG, it will be
crucial to correctly model the relativistic corrections !

Note: Relativistic corrections[Jeong+Schmidt+Hirata(2011)][Fonseca+(2015)]

There is a degeneracy
between fNLlocal and GR effects.

Multi-tracer cosmology

Multi-tracer cosmology
Cosmic variance: fundamental limit to large-scale obs.
• Only a finite number of Fourier modes in our Hubble volume• Biased populations probe the same DM field → deterministic• Tracer-dependent quantities are no CV-limited [Seljak(2008)]

Tracer1

Tracer2

δ1 = (b1 + f μ2) δDM
Tracers are stochastic2δ2 = (b2 + f μ ) δDM

δ1/δ2 = (b1 + f μ2)/(b2 + f μ2)
Ratio is deterministic
Noise on MT quantities scales like shot noise→ Need high source density & large bias ratio

Multi-tracer cosmology➢Error on cosmological parameter θσ(θ) = (Fθθ)-1/2 = ( dP/dθ・Cov[P,P]-1・dP/dθ)-1/2➢Single tracer caseσ( lnP ) = ( P・(P+N-1)-2・P )-1/2 →1+O(1/PN)
Even when N→∞,
finite error remains.
:Cosmic Variance[Seljak (2009), Hamaus+Seljak+Desjacques(2011)]

Multi-tracer cosmology
All galaxy samples

Ntot,
δgal = btot δLight galaxies
N1δgal,1 = b(1) δHeavy galaxies
N2δgal,2= b(2) δσ(θ) = (Fθθ)-1/2 = ( dP/dθ・Cov[P,P]-1・dP/dθ)-1/2

[Seljak (2009), Hamaus+Seljak+Desjacques(2011)]

Multi-tracer cosmology
All galaxy samples

Ntot,δgal = btot δLight galaxies
N1δgal,1 = b(1) δHeavy galaxies
N2
δgal,2= b(2) δ

P = { P(11), P(12), P(22) }= { α2P2, rαP2, P2 }σ(θ) = (Fθθ)-1/2 = ( dP/dθ・Cov[P,P]-1・dP/dθ)-1/2
Cov =(α2P2+N1-1)2(α2P2+N1-1)rαP2

r2α2P22*((α2P2+N1-1)(P2+N2-1)+r2α2P22)/2(P2+N2-1)rαP2**(P2+N2-1)2

[Seljak (2009), Hamaus+Seljak+Desjacques(2011)]

Multi-tracer cosmology
All galaxy samples

Ntot,
δgal = btot δLight galaxies
N1δgal,1 = b(1) δHeavy galaxies
N2
δgal,2= b(2) δP = { P(11), P(12), P(22) }= { α2P2, rαP2, P2 }σ(α) = (Fαα)-1/2 = ( dP/dα・Cov[P,P]-1・dP/dα)-1/2σ(α =b1(1)/b1(2)) → ((P2N1)-1 + α2(P2N2)-1)1/2
Error on ratio is determined only by shot noise!

amplitude

bhbh/blbl
Shape of CV noise is same…

Pgal,h= (bh)2PDM
Pgal,l= (bl)2PDM

PDMwave number

Constraining power of MT[Ferraro+Smith(2014)]

MT is effective for low shot-noise region.

Low noise ←→ High noiseConstraining fNLlocal with SKA+Euclid[DY+Takahashi+Oguri(2014)]

Constraining local PNG with SKA+Euclid[DY+Takahashi(2015)]

σ(gNL)/103=
21 [SKA1]
7.4 [SKA2]
18 [Euclid](65 [Planck])σ(τNL)=
43 [SKA1]
23 [SKA2]
62 [Euclid](2800 [Planck])fiducial τNLCan we confirm SY-inequality?

prohibitedprohibitedτNL<((6/5)fNL)2
fiducial fNL

Can we confirm SY-inequality?

fiducial τNLFor large τNL, the constraining power on fNL
decreases, because the correction from τNL
to the bias dominates.

It is generally
difficult to detect
small fNL and/or τNL.

prohibitedprohibitedτNL<((6/5)fNL)2

fiducial fNL

Confirming Suyama-Yamaguchi ineq.
[DY+Takahashi(2015)]

The region where both fNL and τNL are detected at 1σ

Can we constrain non-local
PNG with galaxy surveys?

Local-type/non-local type PNG➢ Local type → Can be determined by scale-dependent bias

BΦlocal(k1,k2,k3) = 2 fNL (PΦ(k1)PΦ(k2)+cyc)➢ Non-local type
• Equilateral• OrthogonalBΦorth(k1,k2,k3) = 6 fNL [-3(PΦ(k1)PΦ(k2)+cyc)- 8 (PΦ(k1)PΦ(k2) PΦ(k3))2/3+ 3 (PΦ1/3(k1)PΦ2/3(k2) PΦ(k3)+5perm) ]Can galaxy survey
constrain these?

BΦeq(k1,k2,k3) = 6 fNL [- (PΦ(k1)PΦ(k2)+cyc)- 2 (PΦ(k1)PΦ(k2) PΦ(k3))2/3+ (PΦ1/3(k1)PΦ2/3(k2) PΦ(k3)+5perm) ]non-local type PNG is NOT sensitive
to scale-dependent bias
fNLloc → Δb[fNL loc] ∝ 1/k2
gNLloc → Δb[gNLloc] ∝ 1/k2

τNLloc → Δb[τNLloc] ∝ 1/k4

Strong low-k dependence→ Scale-dependent bias
in galaxy power spectrum

fNLeq → Δb[fNLeq] ∝ 1/k0
fNLfol → Δb[fNLfol] ∝ 1/k1

Relatively weak

fNLorth → Δb[fNLorth] ∝ 1/k1

→ Galaxy bispectrum[e.g., Matsubara(2012)]

Galaxy bispectrum• Galaxy power spectrum for tracer (a) and (b)Ps(a)(b) = (b1(a) +fμ12) (b1(b) +fμ22) PL(k)• Galaxy bispectrum for tracer (a), (b), and (c)Bs,grav(a)(b)(c) = (1/6)[(b1(a) +fμ12)(b1(b) +fμ22)
×(b2(c) +2b1(c) F2(k1,k2))+(perm) ]PL(k1)P(k2) +cyc.
Nonlinearbias

2nd order
kernelincludesPNG contribution

[DY+Yokoyama+Takahashi(2017)]

MT for galaxy bispectrum➢ For simplicity, let us consider equilateral bispectrum

B(abc) (k,k,k) = ( b1(a)b1(b)b2(c) + (perm) )PL2 (k)σ(θ) = (Fθθ)-1/2 = ( dB/dθ・Cov[B,B]-1・dB/dθ)-1/2
B = { B(111), B(112), B(122), B(222) }= { α2γB2, (α2+2αγ)B2/3, (2α+γ)B2/3, B2 }

[DY+Yokoyama+Takahashi(2017)]

MT for galaxy bispectrum➢ For simplicity, let us consider equilateral bispectrum

B(abc) (k,k,k) = ( b1(a)b1(b)b2(c) + (perm) )PL2 (k)σ(γ) = (Fγγ)-1/2 = ( dB/dγ・Cov[B,B]-1・dB/dγ)-1/2

σ(γ = b2(1)/b2(2))→ (3P23/B22)1/2 ((P2N1)-1 + α2(P2N2)-1)1/2
Ratio of nonlinear bias factor is also determined onlyby shot noise!

[DY+Yokoyama+Takahashi(2017)]

Error on fNL from galaxy bispectrum
MT is effective for low shot-noise region.

Euclid SKA2High noise ←→ Low noise[DY+Yokoyama+Takahashi(2017)]

non-local fNL with galaxy bispectrum

1-tracer2-tracer

1-tracer2-tracer

Planck(43)(43)

Planck(24)(24)

Euclid30.4

30.0

Euclid13.6

13.1

SKA2

25.1

23.0

SKA2

12.4

10.2

[DY+S.Yokoyama+T.Takahashi, work in progress]

Scale-dependent PNG➢ Genelarized local-type bispectrum[Shandere+Dalal+Huterer (2011)]

BΦ(k1,k2,k3) = 2 fNLeff [ξs(k3)ξm(k1)ξm(k2)PΦ(k1)PΦ(k2)+cyc. ]with ξs,m(k)=(k/kpiv)

nfNL(s,m)✓ Galaxy bispectrum is useful to break the degeneracy!
✓ Scale-dependent linear bias is also induced.
[Matsubara(2012), Desjacques+ (2011),Shandere+ (2011)]→ Combined analysis of galaxy power- and bi-spectrum

[DY+S.Yokoyama+T.Takahashi, work in progress]

Constraining scale-dependent PNG
SKA2-like survey

Can we reach fNL~0.01?

Far future 21cm line survey[Cooray(2006)]

If we use multi-freq
information
between z=100 and
z=30, we can probe(S/N) = 100 fNL→ fNL = 0.01!

<alm2al’m’*> = Xl δll’δmm’

Cumulative S/NShort summary◆ Primoridial non-Gaussianity can beconstrained by using future LSS surveys!
✓ Local-type PNG → galaxy power spectrumwith scale-dependent bias
◆ We may confirm (or reject) SY-inequality in near future(?).

✓ Non-local PNG → galaxy bispectrum

What’s next?