Download Beyond the limits of cosmological perturbation theory: resummations

Beyond the limits of cosmological
perturbation theory: resummations and
effective approaches
Massimo Pietroni - INFN Padova
Paris, November 25th, 2013
Outline
From particles to fluids
Exact results (consistency relations)
Approximate results (resummations,
effective approaches)
with S. Anselmi, M. Peloso, A. Manzotti, M. Viel, F. Villaescusa-Navarro
Why do we need to study the late
(and non-linear) evolution?
Dark Energy (Baryonic Acoustic
Oscillations)
Neutrino masses
Primordial non-Gaussianity
Cosmic shear, ....
DO IT ACCURATE, FLEXIBLE, FAST!!
(O(%))
(not only LCDM) (<O(mins))
And, if possible, simple to use...
The Eulerian way
+
· [(1 + )v] = 0 ,
@v
+ Hv + (v · r)v =
@⌧
2
3
=
2
r + “sources”
2
H
M
subhorizon scales, newtonian gravity
(for Lagrangian approaches, see Kitaura, Matsubara, Sugiyama,
Tassev- Zaldarriaga, Porto-Senatore-Zaldarriaga, Valageas...)
1) From particles to fluids
Rederiving the fluid equations
Buchert, Dominguez, ’05, Pueblas Scoccimarro, ’09, Baumann et al. ’10
M.P., G. Mangano, N. Saviano, M. Viel, 1108.5203,
Carrasco, Hertzberg, Senatore, 1206.2976
X
nmic (x, ⌧ ) =
xn (⌧ )) ,
D (x
n
vni = ẋn (⌧ ) ,
ain =
fmic (x, p, ⇥ ) =
X
n
D (x
xn (⇥ ))
D (p
pn (⇥ ))
rix
mic (x, ⌧ )
Satisfies the “Vlasov eq.”
Rederiving the fluid equations
Buchert, Dominguez, ’05, Pueblas Scoccimarro, ’09, Baumann et al. ’10
M.P., G. Mangano, N. Saviano, M. Viel, 1108.5203,
Carrasco, Hertzberg, Senatore, 1206.2976
n, v i , ,
ij
,...
LU V
1
f (x, p, ⌧ ) ⌘
V
Z
d3 yW(y/LU V )fmic (x + y, p, ⌧ )
Coarse-Grained Vlasov eq.
large scales

i
@
p @
@
i
+
amrx (x, ⌧ ) i f (x, p, ⌧ ) =
i
am @x
@p
 @⌧
@
@
i
i
am h i fmic r mic iLU V (x, p, ⌧ )
f (x, p, ⌧ )rx (x, ⌧ )
i
@p
@p
hgiLU V (x) ⌘
=h
1
VU V
mic iLU V
Z
short scales
d3 y W(y/LU V )g(x + y)
f = hfmic iLU V
Vlasov in the L_uv ➞ 0 limit!
Short-distance sources
@
@
i
n(x) +
(n(x)v
(x))
=
0
@⌧
@xi
Short-distance
sources
@ i
@ i
i
k
v (x) + Hv (x) + v (x) k v (x)
@⌧
@x
1
@
i
ki
i
= rx (x)
(n(x) (x)) J1 (x)
k
n(x) @x
@
@⌧
ij
+ 2H
ij
@
+v
@xk
k
ij
+
ik
@ j
v +
k
@x
jk
@ i
v
k
@x
1
@
ij
kij
=
n(x)! (x)
J2
k
n(x) @x
ij
and all higher-order moments are
dynamically generated by coarse-graining
Standard Eulerian treatment corresponds to L
(single stream approximation)
0
Short-distance sources
J1i (x)
1
⌘
hnmic ri
n(x)
n(x)ri (x)
mic i(x)
takes into account short-scale density fluctuations
ij
J2 (x)
1
i
⌘
hvmic
nmic rj
n(x)
mic i(x)
v i (x)hnmic rj
mic i(x)
takes into account short-scale velocity fluctuations
...
+ (i $ j)
Exact large scale dynamics for
density and velocity
⇤
@
@ ⇥
i
(x) + i (1 + (x))v (x) = 0
@⌧
@x
@ i
@ i
i
k
v (x) + Hv (x) + v (x) k v (x) =
@⌧
@x
3
2
r (x) = ⌦M H2 (x)
2
external input on
UV-physics needed
{
i
rx
n(x) = n0 (1 + (x))
1
i
J1 (x) ⌘
hnmic ri
n(x)
1
@
J (x) ⌘
(n(x)
k
n(x) @x
i
(x)
mic i(x)
ki
(x))
J (x)
i
i
J1 (x)
n(x)ri (x)
2) Exact results
(extended) galilean invariance and LSS
M. Peloso, M.P.
1302.0223
1310.7915
Kehagias, Riotto, 1302.0130
Kehagias, Norena, Perrier, Riotto 1311.0786
Kehagias, Perrier, Riotto, 1311.5524
Creminelli, Norena, Simonovic, Vernizzi 1309.3557
Creminelli, Gleyzes, Simonovic, Vernizzi, 1311.0290
Valageas, 1311.1236, 1311.4286
(extended) galilean invariance and LSS
M. Peloso, M.P.
1302.0223
1310.7915
Kehagias, Riotto, 1302.0130
Kehagias, Norena, Perrier, Riotto 1311.0786
Kehagias, Perrier, Riotto, 1311.5524
Creminelli, Norena, Simonovic, Vernizzi 1309.3557
Creminelli, Gleyzes, Simonovic, Vernizzi, 1311.0290
Valageas, 1311.1236, 1311.4286
extended galilean transformations
the equations of motions are invariant under
time-dependent, uniform boosts:
0
Rn
physical coordinates
= Rn
Z
t
dt V0 (t )
Vn0 = Vn
0
xn
comoving coordinates
1
d(⌧ ) =
a(⌧ )
Z
⌧
0
vn
d⌧ a(⌧ )V0 (⌧ )
0
0
0
= xn
= vn
0
0
V0 (t)
d(⌧ )
@
d(⌧ )
@⌧
velocity zero-mode: tadpole
all the variables and the sources transform as
scalars (+ zero-modes)
(x ) = (x)
0 0
vi (x ) = vi (x) d˙i (⌧ )
0
0
(x
) = ij (x)
ij
···
0 0 0
ri (x ) = ri (x) + d¨i (⌧ ) + Hd˙i (⌧ )
0 0
Ji (x ) = Ji (x)
0
0
x =x
0
d(⌧ )
the invariance holds at a fully non-perturbative level:
non-linearities+beyond single-stream+beyond CDM
in the equations of motion, the only non-trivial
effect of the zero mode velocity shift comes from
the non-linear terms in continuity and Euler equations
@
@
˙
[(1 + (x))vk (x)] ! dk k (x)
k
@x
@x
@
@
˙
vk (x) k vi (x) ! dk k vi (x)
@x
@x
@
(x)
@⌧
@
vi (x)
@⌧
invariance enforced by cancellation
bet ween linear and non-linear terms
at the same point x: it is a local (UV) property
approximation schemes must respect
the GI constraints at any order
n-th order correlators related to the soft limit of (n+1)th
order ones
Galileo and IR sensitivity
Jain Bertshinger, ’96
Scoccimarro Frieman, ’96
h'a (k, ⌘)'b (k , ⌘ )iV0 = e
0
0
iV0 ·(kT (⌘)+k0 T (⌘ 0 ))
h'a (k, ⌘)'b (k , ⌘ )iV0 =0
0
0
Galileo and IR sensitivity
Jain Bertshinger, ’96
Scoccimarro Frieman, ’96
h'a (k, ⌘)'b (k , ⌘ )iV0 = e
0
0
iV0 ·(kT (⌘)+k0 T (⌘ 0 ))
h'a (k, ⌘)'b (k , ⌘ )iV0 =0
0
GI (+ translation invariance: k’=-k)
(in the single species case)
equal-time PS is insensitive to V0
0
Galileo and IR sensitivity
Jain Bertshinger, ’96
Scoccimarro Frieman, ’96
h'a (k, ⌘)'b (k , ⌘ )iV0 = e
0
0
iV0 ·(kT (⌘)+k0 T (⌘ 0 ))
h'a (k, ⌘)'b (k , ⌘ )iV0 =0
0
0
GI (+ translation invariance: k’=-k)
(in the single species case)
equal-time PS is insensitive to V0
'¯a
O(V0 2 )
'¯a
'¯a
'¯a
'¯a
'¯a
=0
Galileo and IR sensitivity
Jain Bertshinger, ’96
Scoccimarro Frieman, ’96
h'a (k, ⌘)'b (k , ⌘ )iV0 = e
0
0
iV0 ·(kT (⌘)+k0 T (⌘ 0 ))
h'a (k, ⌘)'b (k , ⌘ )iV0 =0
0
0
GI (+ translation invariance: k’=-k)
(in the single species case)
equal-time PS is insensitive to V0
'¯a
O(V0 2 )
'¯a
'¯a
'¯a
'¯a
'¯a
=0
same structure as
1-loop PS with
0
Pab
(q; s, s0 ) ! '¯2 (q, s) '¯2 ( q, s0 )
V0 · q
D (q)
IR divs cancel order by order in standard PT
Galilean Ward identities
at 1-loop...
Valid at all loop: exact relation bet ween the soft limit of a n+1-point
function and a n-point one.
To be satisfied in any resummation (=reorganization of the PT
expansion) scheme.
From Ward to Consistency relations
(k ! 0)
q
vi (k)
k
q
h (q, ⌧ ) (k
0
q
(k ! 0)
vi (k)
k
q
q, ⌧ )ivi (k,⌧ )
00
Take the average over the soft velocity v_i,
constrained to be a solution of the equations of motion
M. Peloso, M.P. 1302.0223
Kehagias,, Riotto 1302.0130
From Ward to Consistency relations
(k ! 0)
q
vi (k)
k
q
h (q, ⌧ ) (k
0
q
(k ! 0)
vi (k)
k
q
q, ⌧ )ivi (k,⌧ )
00
Take the average over the soft velocity v_i,
constrained to be a solution of the equations of motion
(in the single species case)
fully nonlinear bispectrum and power spectrum
M. Peloso, M.P. 1302.0223
Kehagias,, Riotto 1302.0130
all-order result
Spontaneous GI breaking
t wo species with large
scale velocity bias:
✓2 (k)
bv = lim
k!0 ✓1 (k)
✓(k) = ik j vj (k)
Spontaneous GI breaking
t wo species with large
scale velocity bias:
✓2 (k)
bv = lim
k!0 ✓1 (k)
✓(k) = ik j vj (k)
the IR is not GI invariant:
✓2 (k)
✓20 (k)
lim
! lim 0
6= bv
k!0 ✓1 (k)
k!0 ✓1 (k)
Spontaneous GI breaking
t wo species with large
scale velocity bias:
✓2 (k)
bv = lim
k!0 ✓1 (k)
✓(k) = ik j vj (k)
the IR is not GI invariant:
✓2 (k)
✓20 (k)
lim
! lim 0
6= bv
k!0 ✓1 (k)
k!0 ✓1 (k)
the consistency relations get modified by new terms:
exact at all orders, nonvanishing at equal times
order
parameter of
GI breaking
Peloso, MP , 1310.7915
Spontaneous GI breaking: examples
1) new long range force in the DM sector
= 0 for baryons
3
⌦CDM (2
2
2
3
+ 1) + bv ⌦b
2
3 ⌦CDM
2 bv
3
⌦b = 0
2
linear velocity bias
Amendola ’04, Saracco et al ’10,...
Spontaneous GI breaking: examples
1) new long range force in the DM sector
= 0 for baryons
3
⌦CDM (2
2
2
3
+ 1) + bv ⌦b
2
3 ⌦CDM
2 bv
3
⌦b = 0
2
linear velocity bias
Amendola ’04, Saracco et al ’10,...
2) halo/DM velocity bias
Desjacques, ’08,
Elia, Ludlow, Porciani, ’11
3) Approximate results
Exact large scale dynamics for
density and velocity
⇤
@
@ ⇥
i
(x) + i (1 + (x))v (x) = 0
@⌧
@x
@ i
@ i
i
k
v (x) + Hv (x) + v (x) k v (x) =
@⌧
@x
3
2
r (x) = ⌦M H2 (x)
2
external input on
UV-physics needed
{
i
rx
n(x) = n0 (1 + (x))
1
i
J1 (x) ⌘
hnmic ri
n(x)
1
@
J (x) ⌘
(n(x)
k
n(x) @x
i
(x)
mic i(x)
ki
(x))
J (x)
i
i
J1 (x)
n(x)ri (x)
Three strategies
particles
non-linear non-perfect fluid
k
target physics scale
2⇡
LU V
Three strategies
particles
non-linear non-perfect fluid
2⇡
LU V
k
target physics scale
1) Single stream approximation: J =
i
i
J1
= 0,
2⇡/LU V ! 1
Three strategies
particles
non-linear non-perfect fluid
2⇡
LU V
k
target physics scale
1) Single stream approximation: J =
i
i
J1
= 0,
J i (k; LU V ), J1i (k; LU V ), in
2⇡/LU V ! 1
2) EFT of LSS: expand
terms of , v
then measure them at k = k̄ and send 2⇡/LU V ! 1
i
(Baumann et al ’10, Carrasco et al ’12, ’13)
Three strategies
particles
non-linear non-perfect fluid
2⇡
LU V
k
target physics scale
1) Single stream approximation: J =
i
i
J1
= 0,
2⇡/LU V ! 1
J i (k; LU V ), J1i (k; LU V ), in
2) EFT of LSS: expand
terms of , v
then measure them at k = k̄ and send 2⇡/LU V ! 1
i
(Baumann et al ’10, Carrasco et al ’12, ’13)
3) Coarse-grained PT: insert a new scale, L, bet ween the
mildly non-linear and the non linear fluid regime.
MP, Mangano, Saviano, Viel, ’11
particles
non-linear non-perfect fluid
2⇡
LU V
1
L
k
“PT” ok
coarse-grained
sources
Physics at k is independent on L, L_uv (“Wilsonian approach”)
Expansion in J’s:
1
h iJ = h iJ=0 + h J iJ=0 + h JJ iJ=0 + · · ·
2
computed in PT
with cutoff at 1/L
measured from
simulations
Computing the sources
LU V = 1, 2, 4 Mpc/h
LU V : , v i , J1i , J i
Manzotti, Peloso, MP,
Villaescusa-Navarro, Viel, in progress
Lbox = 512 Mpc/h
Nparticles = (512)3
Computing the sources
LU V = 1, 2, 4 Mpc/h
LU V : , v i , J1i , J i
L : ¯, v̄ i , J¯1i , J¯i
✓ ◆3/2
2
1
W(R/L) =
e
3
⇡
L
R2
2L2
Manzotti, Peloso, MP,
Villaescusa-Navarro, Viel, in progress
Lbox = 512 Mpc/h
Nparticles = (512)3
How big are the non-perfect
fluid effects?
1
L=2 Mpc/h
L=4 Mpc/h
0.1
z=0
<delta J>/<delta delta>
D
2
D
0.01
4
<JJ>/<delta delta>
0.001
0.01
genuine nonperturbative
effect
0.1
Manzotti, Peloso, MP, Villaescusa-Navarro, Viel, in progress
How big are the non-perfect
fluid effects?
Pueblas
Scoccimarro,
’09
1
L=2 Mpc/h
L=4 Mpc/h
0.1
z=0
<delta J>/<delta delta>
D
2
D
0.01
4
<JJ>/<delta delta>
0.001
0.01
genuine nonperturbative
effect
0.1
Manzotti, Peloso, MP, Villaescusa-Navarro, Viel, in progress
How big are the non-perfect
fluid effects?
1
L=2 Mpc/h
L=4 Mpc/h
z=1
0.1
<J delta>/<delta delta>
0.01
<J J>/<delta delta>
0.001
0.01
0.1
Compatible with best single-stream resummed PT approaches
Compare with resummations
(Anselmi, MP, ’12)
0.10
0.04
z=0
z = 0.5
z = 1.0
z = 2.0
z = 0.0
z = 0.5
z = 1.0
0.05
HPAP-PEmuLêPEmu
HP-PNBLêPNB
0.02
0.00
0.00
-0.02
Sato & Matsubara N-Body - L=1000 Mpcêh
-0.05
Coyote Emulator - Millennium Cosmology
Wb=0.0448; Wm=0.265; ns=0.936; h=0.71
Wb=0.044; Wm=0.2485; ns=1.0; h=0.7272
w=-1; s8=0.8
w=-1; s8=0.9
-0.04
0.0
0.2
0.4
k HhêMpcL
0.6
0.8
1.0
-0.10
0.0
0.2
0.4
k HhêMpcL
0.6
0.8
1.0
Anselmi,Lopez Nacir, Sefusatti, in progress
Compare with resummations
(Anselmi, MP, ’12)
O(mins) for a full run, only 1-loop type integrals
0.10
0.04
z=0
z = 0.5
z = 1.0
z = 2.0
z = 0.0
z = 0.5
z = 1.0
0.05
HPAP-PEmuLêPEmu
HP-PNBLêPNB
0.02
0.00
0.00
-0.02
Sato & Matsubara N-Body - L=1000 Mpcêh
-0.05
Coyote Emulator - Millennium Cosmology
Wb=0.0448; Wm=0.265; ns=0.936; h=0.71
Wb=0.044; Wm=0.2485; ns=1.0; h=0.7272
w=-1; s8=0.8
w=-1; s8=0.9
-0.04
0.0
0.2
0.4
k HhêMpcL
0.6
0.8
1.0
-0.10
0.0
0.2
0.4
k HhêMpcL
0.6
0.8
1.0
Anselmi,Lopez Nacir, Sefusatti, in progress
Compare with resummations
(Anselmi, MP, ’12)
0.04
0.04
z=0
z = 0.5
z = 1.0
z = 2.0
z=0
z = 0.5
z = 1.0
z = 2.0
0.02
HPAP-PEmuLêPEmu
HPAP-PEmuLêPEmu
0.02
0.00
-0.02
0.00
-0.02
Non-clustering Quintessence
Wbh2 =0.0224; Wmh2 =0.1296; ns=0.97; h=0.648
w=-0.8; s8=0.749
Non-clustering Quintessence
-0.04
-0.04
Wbh2 =0.0224; Wmh2 =0.1296; ns=0.97; h=0.798
w=-1.2; s8=0.847
0.0
0.2
0.4
k HhêMpcL
0.6
0.8
1.0
0.0
0.2
0.4
k HhêMpcL
0.6
0.8
1.0
Anselmi,Lopez Nacir, Sefusatti, in progress
Sources: time and scale dependence
time dep. agrees
with PT at O(1)
scale dep. agrees
with PT for k<0.2
<h phi_in>/<phi_in phi_in>
0.1
z=0
z=1.5
z=3
z=5
PT z=5
<h3 phi>/(D^2 Omega/f^2)
0.01
<h4 phi>/(D^2 Omega/f^2)
L= 8 Mpc/h
Luv = 2 Mpc/h
0.001
0.1
k (h/Mpc)
c2s / < h2,4 '¯1 > order of magnitude of cs fully predictable in plain PT
with no multistream/virialization/etc. !!
Propagator: 1-loop + sources
L=8 Mpc, z=0
1
N-Body
0.8
No sources
G
Sources
0.6
0.4
0.2
0
0
0.1
0.2
k (h/Mpc)
0.3
0.4
0.5
Propagator: RPT+sources
L=8 Mpc, z=0
1
Sources
0.8
No Sources
G
N-Body
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
k (h/Mpc)
PT RESUMMATIONS STILL NEEDED ON THE SMOOTH FIELDS!!
Sources: cosmology dependence
hh3 h3 i
k 4 P (k)
R = 8 Mpc h
1.95 · 10
9
< As < 3.0 · 10
9
1
0.932 < ns < 1
Sources: cosmology dependence
hh3 h3 i
k 4 P (k)I[P (k); R]
R = 8 Mpc h
1.95 · 10
9
< As < 3.0 · 10
9
1
0.932 < ns < 1
PT RECOVERS MOST OF THE COSMOLOGY DEPENDENCE
Summary
Exact results from extended GI: check of
approximation schemes (including N-body), possible
measure of large scale velocity bias;
Non-perfect-fluid effects are O(%) in the BAO region
(z=0), effective approaches needed to systematically
treat them;
Resummation schemes perform (surprisingly?) well,
and are needed in the mildly non-linear regime, even
in effective approaches.