Perturbative approaches to the LSS in CDM and beyond Massimo - - PowerPoint PPT Presentation

Firenze 6/4/2016

Perturbative approaches to the LSS in ΛCDM and beyond

Massimo Pietroni - INFN, Padova

Outline

✤ IR effects on the nonlinear PS ✤ UV effects on the nonlinear PS ✤ Intermediate scales ✤ Putting all together: an

improved TRG

✤ Scalar field (axion-like) DM

Linear and non-linear scales

0.0001 0.001 0.01 0.1 1 10 1x10-7 1x10-6 0.00001 0.0001 0.001 0.01 0.1 1 10 100

∆(k) ≡ k3P(k) 2π2

linear non-linear

k(h/Mpc)

linear Power Spectrum @z=0, ΛCDM

3

increasing z

D(z)2∆(k)

The nonlinear PS

P NL

ab (k, z) = Gac(k, z)Gbd(k, z)P lin cd (k, z) + P MC ab

(k, z)

a, · · · , d = 1

a, · · · , d = 2 density velocity div.

Gab(k; z) = ⌧ δϕa(k, z) δϕb(k, zin) = hϕa(k, z)ϕb(k, zin)i0 P lin(k, zin) + PNG

propagator

intermediate and UV physics IR physics

P P

ab(k; z)

Large scale flows and BAO’s

Padmanabhan et al 1202.0090 reconstruction

110 Mpc/h

O(10 Mpc) displacements

Effect on the Correlation Function

⁄ mn = 0.0 eV; z=0

60 70 80 90 100 110 120 5 10 15

R @MpcêhD R2x

⁄ mn = 0.0 eV; z=1

60 70 80 90 100 110 120 2 4 6

R @MpcêhD R2x

Peloso, MP, Viel, Villaescusa-Navarro, 1505.07477

ξlin ξP

ξMC ξMC

ξP ξlin All the information on the BAO peak is contained in the propagator part The widening of the peak can be reproduced by Zel’dovich approximation (and improvements of it) The widening of the peak contains physical information (not a parameter to marginalize)

(simplified) Zel’dovich approximation

Peloso, MP, Viel, Villaescusa-Navarro, 1505.07477

GZeld(k, z) = e− k2σ2

v(z) 2

P P

11(k, z) = e−

k2σ2 v(z) 2

P lin(k; z)

) is the 1-dimensional velocity di σ2

v(z) = 1

3 Z d3q (2π)3 P lin(q, z) q2 .

linear velocity dispersion: contains information on linear PS, growth factor,…

Z = 1 2π2 Z dq q2 δP lin(q) ✓sin(qR) qR e−q2σ2

v − 1

3 ξ2(R) q2R2 ◆

δξ(R) =

How to include Bulk Motions

hδα(k, τ)δα(k0, τ 0)i = h¯ δα(k, τ)¯ δα(k0, τ 0)iheik·(Dα(τ)Dα(τ 0))i

= h¯ δα(k, τ)¯ δα(k0, τ 0)ie

k2σ2 v(D(τ)D(τ0))2 2

Resummations (~Zel’dovich) take into account the large scale bulk motions

' Dα(τ)

vlong

= 1 3 Z Λ d3q P 0(q) q2

σ2

v =

1 3H2f 2 Z Λ d3qhvi

long(q)vi long(q)i0

Redshift ratios

Peloso, MP, Viel, Villaescusa-Navarro, 1505.07477

⁄ mn = 0.0 eV

z=0.5êz=0 z=1êz=0 z=2êz=0

60 70 80 90 100 110 120 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

R @MpcêhD Ratio xHzLêxHz=0L

Massive neutrinos

Peloso, MP, Viel, Villaescusa-Navarro, 1505.07477

⁄ mn = 0.15 eV; z=0

60 70 80 90 100 110 120 5 10 15

R @MpcêhD R2z

⁄ mn = 0.3 eV; z=0

60 70 80 90 100 110 120 5 10 15

R @MpcêhD R2x

z = 0

60 80 100 120 140 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

R @MpcêhD Ratio x H⁄ mn = 0.15L ê x H⁄ mn = 0L z = 0

60 80 100 120 140 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

R @MpcêhD Ratio x H⁄ mn = 0.3L ê x H⁄ mn = 0L z = 0

60 80 100 120 140 0.5 1.0 1.5

R @MpcêhD Ratio x H⁄ mn = 0.6L ê x H⁄ mn = 0L

P P

11(k, z) = e−

k2σ2 v(z) 2

P lin(k; z)

increasing neutrino masses, Plin decreases, but also damping decreases.

X mν = 0.15 eV

X mν = 0.3 eV

↓ 0.6% ↑ 1.2%

Massive neutrinos

Peloso, MP, Viel, Villaescusa-Navarro, 1505.07477

⁄ mn = 0.0 eV; z=0

60 70 80 90 100 110 120 5 10 15 20 25

R @MpcêhD R2 x s ⁄ mn = 0.15 eV; z=0

60 70 80 90 100 110 120 5 10 15 20 25

R @MpcêhD R2x s ⁄ mn = 0.3 eV; z=0

60 70 80 90 100 110 120 5 10 15 20 25

R @MpcêhD R2x s

⁄ mn = 0.0 eV; z=0 Hreal spaceL

R @MpcêhD R2xhh

⁄ mn = 0.15 eV; z=0 Hreal spaceL

R @MpcêhD R2xhh

⁄ mn = 0.3 eV; z=0 Hreal spaceL

R @MpcêhD R2xhh

Redshift space Halos

Improving over Zel’dovich

PPêPlin, z=1, L=0

PP1loopêPlin PP_CSêPlin PP_TRGêPlin PPêPlin HNbodyL

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.2 0.4 0.6 0.8 1.0

∂ηP P

ab(k; η, η) = ΩacP P cb(k; η, η) ΩbcP P ac(k; η, η)

+ Z

ηin

ds ⇥ Σac(k; η, s)P P

cb(k; s, η) + Σbc(k; η, s)P P ac(k; η, s)

⇤

Exact equation

Peloso, MP, Viel, Villaescusa-Navarro, in preparation Anselmi, Matarrese, MP, 1011.4477

Σab(k; η, s) → Σ1−loop

ab

(k; η, s) for k → 0

Σab(k; η, s) → −k2σ2

v(z)eη+sgab(η; s)

for k → ∞

Mode coupling-Response functions

The nonlinear PS is a functional of the initial one (in a given cosmology and assuming no PNG): SPT is an expansion around P 0(q) = 0

Pab[P 0](k; η) =

∞

X

n=1

1 n! Z d3q1 · · · d3qn δnPab[P 0](k; η) δP 0(q1) · · · δP 0(qn)

• P 0=0

P 0(q1) · · · P 0(qn)

n=1 linear order (= “0-loop”) n=2 “1-loop” …

Mode coupling-Response functions

Let’s instead expand around a reference PS: P 0(q) = ¯

P 0(q)

Pab[P 0](k; η) = Pab[ ¯ P 0](k; η) +

1

X

n=1

1 n! Z d3q1 · · · d3qn δnPab[P 0](k; η) δP 0(q1) · · · δP 0(qn)

• P 0= ¯

P 0

δP 0(q1) · · · δP 0(qn) , = Pab[ ¯ P 0](k; η) + Z dq q Kab(k, q; η) δP 0(q) + · · · , (4)

h δP 0(q) ⌘ P 0(q) ¯ P 0(q) t

Kab(k, q; η) ⌘ q3 Z dΩq δPab[P 0](k; η) δP 0(q)

• P 0= ¯

P 0

Linear response function: Non-perturbative (gets contributions from all SPT orders) Key object for more efficient interpolators ?

UV screening

K(k, q; z) = q δP nl(k; z) δP lin(q; z).

Sensitivity of the nonlinear PS at scale k

• n a change of the initial PS at scale q:

Nishimichi et al 1411.2970

k = 0.161 h Mpc−1

PT overpredicts the effect of UV scales

• n intermediate ones

IR: “Galilean invariance”

K(k, q; z) ∼ q3

Peloso, MP 1302.0223

UV screening

The effect of virialized structures on larger scales is screened (Peebles ’80, Baumann et al 1004.2488, Blas et al 1408.2995). However, the departure from the PT predictions starts at small k’s: is it really a virialization effect?

q, ηin −q, ηin k, η −k, η

e−

q2σ2 v 2

damped propagators! (compare SPT: g=O(1)) memory of initial substructures is largely lost

UV lessons

✤ SPT fails when loop momenta become too high (q ≿ 0.4 h/Mpc) ✤ The real response to modifications in the UV regime is mild ✤ Most of the cosmology dependence is on intermediate scales

Effective approaches to the UV

✤ General idea: take the UV physics from N-body simulations

and use (resummed) PT only for the large and intermediate scales

2π LUV

k

particles

non-linear non-perfect fluid

1 L

“PT” ok coarse-grained sources

Physics at k is independent on L, L_uv (“Wilsonian approach”) Expansion in sources:

hδδiJ = hδδiJ=0 + hδJδiJ=0 + 1 2hδJJδiJ=0 + · · ·

computed in PT with cutoff at 1/L measured from simulations

Vlasov Equation

Liouville theorem+ neglect non-gravitational interactions:

d dτ fmic =  ∂ ∂τ + pi am ∂ ∂xi amri

xφ(x, τ)

• fmic(x, p, τ) = 0

nmic(x, τ) = Z d3pfmic(x, p, τ) vmic(x, τ) = 1 nmic(x, τ) Z d3p p amfmic(x, p, τ)

σij

mic(x, τ) =

1 nmic(x, τ) Z d3p pi am pj amfmic(x, p, τ) − vi

mic(x, τ)vj mic(x, τ)

moments: …

density velocity velocity dispersion

M.P., G. Mangano, N. Saviano, M. Viel, 1108.5203, Carrasco, Hertzberg, Senatore,1206.2976 ....

Buchert, Dominguez, ’05, Pueblas Scoccimarro, ’09, Baumann et al. ’10

fmic(x, p, τ) = X

n

δD(x − xn(τ))δD(p − pn(τ))

Satisfies the “Vlasov eq.”

LUV f(x, p, τ) ≡ 1 V Z d3yW(y/LUV )fmic(x + y, p, τ)

nmic(x, τ) = X

n

δD(x xn(τ)) , vi

n = ˙

xn(τ) , ai

n = ri xφmic(x, τ)

n, vi, φ, σij, . . .

From particles to fluids

Vlasov equation in the L_uv ➞ 0 limit!

φ = hφmiciLUV f = hfmiciLUV

large scales short scales

hgiLUV (x) ⌘ 1 VUV Z d3y W(y/LUV )g(x + y)

 ∂ ∂τ + pi am ∂ ∂xi amri

xφ(x, τ) ∂

∂pi

• f(x, p, τ) =

am  h ∂ ∂pi fmicriφmiciLUV (x, p, τ) ∂ ∂pi f(x, p, τ)ri

xφ(x, τ)

• Coarse-grained Vlasov equation

Taking moments…

∂ ∂τ vi(x) + Hvi(x) + vk(x) ∂ ∂xk vi(x) = ri

xφ(x) Ji σ(x) Ji 1(x)

Ji

σ(x) ≡

1 n(x) ∂ ∂xk (n(x)σki(x))

Ji

1(x) ⌘

1 n(x)

• hnmicriφmici(x) n(x)riφ(x)
• r2φ(x) = 3

2ΩMH2δ(x)

external input

• n UV-physics

needed

{

∂ ∂τ δ(x) + ∂ ∂xi ⇥ (1 + δ(x))vi(x) ⇤ = 0

n(x) = n0(1 + δ(x)) = n0(1 + hδmici(x)) vi(x) = h(1 + δmic)vi

mici(x)

1 + δ(x)

Exact large scale dynamics for density and velocity fields

Lbox = 512 Mpc/h Nparticles = (512)3

W(R/L) = ✓ 2 π ◆3/2 1 L3 e− R2

2L2

LUV = 1, 2, 4 Mpc/h LUV : δ, vi, Ji

1 , Ji σ

L : ¯ δ, ¯ vi, ¯ Ji

1, ¯

Ji

σ

Manzotti, Peloso, MP, Villaescusa-Navarro, Viel, 1407.1342

Measuring the sources in Nbody simulation

COSMOLOGY DEPENDENCE Simulation Suite

Lbox = 512 Mpc/h Nparticles = (512)3

Ratios of UV source correlators

hJδii hJδiREF

From N-body

Scale-independent!!

Rescale using PT information

Amplitude rescaling captured by PT!!

˙ ⇢l + 3H⇢l + 1 a@i(⇢lvi

l) = 0 ,

˙ vi

l + Hvi l + 1

avj

l @jvi l + 1

a@il = − 1 a⇢l @j ⇥ ⌧ ij⇤

Λ .

Relation with EFToLSS

h ⇥ τ ij⇤

Λiδl = pbδij + ρb

 c2

sδlδij c2 bv

Haδij∂kvk

l 3

4 c2

sv

Ha ✓ ∂jvi

l + ∂ivj l 2

3δij∂kvk

l

◆ + ∆τ ij + . . . . (25)

derivative expansion, or expansion in k/k_nl

Ji

1 + Ji σ

coefficients should be scale independent, nice results for simple power law linear PS

Baumann et al 1004.2488 Carrasco et al 1206.2926 …

• P11(k, η) ' P lin

11 (k, η) + P 1−loop ss,11

(k, η) 2(2π) c2

s(1)

k2 k2

NL

P lin(k, η) ,

The PS in 1-loop EFToLSS

0.5 1 1.5 2 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 2 cs

2 (h Mpc-1 / kNL)2

k [h Mpc-1] z=0 z=0.25 z=0.5 z=1 z=1.5

c2

s(1)

higher orders+resummations needed to reduce the scale dependence

(see Senatore Zaldarriaga, 1404.5954)

Putting everything together

∂ηP MC

ab

(k; η, η) = ΩacP MC

cb

(k; η) + Z η ds Σac(k; η, s)P MC

cb

(k; s, η) +eη Z d3qγacd(k, q)BMC

cdb (q, k; η)

hha(k, η)ϕMC

b

(k, η)i +(a $ b)

Improved TRG

Peloso, MP, Viel, Villaescusa-Navarro, in preparation

linear growth IR (propagator) effects Intermediate scales: (resummed) SPT UV sources (from Nbody)

Some results (preliminary)

0.1 0.2 0.3 0.4 0.5 0.6 0.5 1.0 1.5 2.0 2.5 3.0 3.5

PMC/Plin, z=1, L=0 Mpc/h

PMC1loop/Plin PMCSig/Plin (Nbody) PMCSig_eik/Plin PMC/Plin (Nbody)

1-loop +IR

PêPlin, z=1, L=0 Mpcêh

Ptot PtotHNbL Pp PpHNL Pmc PmcHNbL

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.5 1.0 1.5 2.0

PêPlin, z=1, L=4 Mpcêh

Ptot PtotHCL PtotHNbL Pp PpHNL Pmc PmcHCL PmcHNbL

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.5 1.0 1.5 2.0

+UV Intermediate scales at 1-loop SPT no free parameter!

with resummations of the MC part

Anselmi, Lopez-Nacir, Sefusatti, 2014

Anselmi, MP, 1205.2235

Scalar field (axion-like) DM

(2 m2

a) = 0

= (ma p 2)−1( e−imat + ∗eimat)

2 = (1 2V )(@2

t + 3H@t) + a−2(1 + 2V )r2 4 ˙

V @t

ma H

i ˙ 3iH /2 + (2maa2)−1r2 maV = 0.

Shrödinger-Poisson

Perturbations

= ReiS

⇢a = R2 ~ va = (maa)−1rS

Madelung

˙ ¯ ⇢a + 3H ¯ ⇢a = 0 ˙ a + a−1 ~ va · ra + a−1(1 + a)r · ~ va = 0, ˙ ~ va + H ~ va + a−1 (~ va · r) ~ va = a−1r (V + Q)

Q = 1 2m2

aa2

r2p1 + a p1 + a .

“Quantum” term, deviations from CDM

Linear Theory

@a(k, ⌧) @⌧ + ✓(k, ⌧) = 0 @✓(k, ⌧) @⌧ + H(⌧)✓(k, ⌧) + 3 2H2(⌧)a(k, ⌧) k4 4m2

aa2 = 0

kJ =

4

p 6 p maaH ⇡ 1.6a p maH.

Axion Jeans scale

Hlozek, Grin, Marsch, Ferreira 1410.2896

Nonlinear perturbations

∂δa(k, τ) ∂τ + θ(k, τ) + Z d3pd3qδD(k p q)α(q, p)θ(q, τ)δa(p, τ) ∂θ(k, τ) ∂τ + H(τ)θ(k, τ) + 3 2Ωm(τ)H2(τ)δa(k, τ) k4 4m2

aa2δa(k, τ)

+ Z d3pd3qδD(k p q)β(q, p)θ(p, τ)θ(q, τ) + Z d3pd3qδD(k p q)k2(k2 + q2 + p2) 16m2

aa2

δa(q, τ)δa(p, τ)

From expanding Q to 2nd order ~ k^4: UV catastrophe!

| | α(q, p) = (p + q) · q q2 .

β(q, p) = (q + p)2q · p q2p2

SPT fails at all scales

0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 2 3 4 5 6 7

P 1−loop/P lin

k (h/Mpc) ma = 10−26 eV

ma = 10−25 eV

ma = 5 · 10−25 eV

TRG provides the proper UV cutoff (E. Noda, MP,in progress)

Summary

✤ The IR effects are well understood and implemented in most of the

approaches on the market

✤ Widening of the BAO peak well understood, analytically ✤ SPT fails at high loop momenta: UV screening completely missed ✤ Resummations and effective UV approaches must and can be

combined (interpolators from linear response function?)