The growth of gravitational instabilities in an expanding universe - - PowerPoint PPT Presentation
Francis Bernardeau IPhT Saclay, France The growth of gravitational instabilities in an expanding universe IHES november 2012 1 The current model of cosmology A snapshop of the universe 377,000 years after the Big Bang: CMB temperature
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IHES november 2012 Francis Bernardeau IPhT Saclay, France
The current model of cosmology
A snapshop of the universe 377,000 years after the Big Bang: CMB temperature fluctuations
A "concordant" model of cosmology but that contains three puzzling ingredients:
constant responsible for the (recent) acceleration of the universe
low redshift manifestations through the way the large- scale structure of the universe forms and evolves?
What is at stake?
What do we want to learn?
non-Gaussianities
expansion/growth of structure (fifth force)
Nonlinear effects are ubiquitus!
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Francis Bernardeau IPhT Saclay
A self-gravitating expanding dust fluid
The Vlasov equation (collision-less Boltzmann equation) - f(x,p) is the phase space density distribution - are fully nonlinear.
This is what N-body codes aim at simulating...
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df dt = ∂ ∂tf(x, p, t) + p ma2 ∂ ∂xf(x, p, t) − m ∂ ∂xΦ(x) ∂ ∂pf(x, p, t) = 0 ∆Φ(x) = 4πGm a ⇤ f(x, p, t)d3p − ¯ n ⇥
matter dominated universe with a dominant cold dark matter component
Peebles 1980; Fry 1984 FB, Colombi, Gaztañaga, Scoccimarro, Phys. Rep. 2002
The rules of the game: single flow equations
X
Yoo et al., PRD, 2009...
GR correction effects are usually small
∂ ∂tδ(x, t) + 1 a[(1 + δ(x, t))ui(x, t)],i = ∂ ∂tui(x, t) + ˙ a aui(x, t) + 1 auj(x, t)ui,j(x, t) = −1 aΦ,i(x, t) + . . . Φ,ii(x, t) − 4πGρ a2 δ(x, t) =
Francis Bernardeau IPhT Saclay
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The solution (scalar modes)
Connexion with the physics of the early universe (Hu, PhD thesis) δ(x, t) = D+(t)δ+(x) + D−(t)δ−(x)
D+(a) et D-(a)
1 growing and 1 decaying mode
D−(t) ∼ D−3/2
+
(t)
EdS
LCDM
d’expansion k = keq 10
3
10
2
10
1
1 k a = a a = a * a = a
eq
matière mode k /(h/Mpc) horizon
?
facteur radiation
The development of cosmological instabilities across time and scale
Large-scale structure
CMB
Nonlinear growth
Hu, Sugiyama '95, '96
Francis Bernardeau IPhT Saclay
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Eventually objects form and their properties decouple from the global expansion
Davis, Peebles '77
Hierarchical models are based on self- similar growth of correlation functions + stable clustering ansatz. They were popular in the eighties.
Radius time E>0 E=0 E<0 Rmax 1/2 Rmax
The collapse of a spherical object can be computed exactly.
The virialization processes are complex but should lead to the formation of objects roughly half the size of their maximal extension.
Virialization
Hamilton et al. '95 Balian, Schaeffer '89
Francis Bernardeau IPhT Saclay 10
The halo model
The complex matter distribution is replaced by a set of halos characterized by their mass distribution and density profile.
Perturbation theory
Cooray, Sheth '02
Francis Bernardeau IPhT Saclay
gravitational instabilities;
large variety of models, and for a large numbers
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One more rule: it is possible to analytically expand the cosmic fields with respect to initial density fields δ(x, t) = δ(1)(x, t) + δ(2)(x, t) + . . .
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doublet linear propagator
∂ ∂η Φa(k, η) + Ω b
a(η)Φb(k, η) = γ bc a (k1, k2)Φb(k1)Φc(k2)
Φa(k, η) = ✓ δ(k, η) θ(k, η)/f+(η) ◆
cosmological doublet
γ bc
a
(k1, k2) = 8 > > > > > > > > > > < > > > > > > > > > > :
1 2
n 1 + k2·k1
|k2|2
(a, b, c) = (1, 1, 2)
1 2
n 1 + k1·k2
|k1|2
(a, b, c) = (1, 2, 1)
(k1·k2)|k1+k2|2 2|k1|2|k2|2
; (a, b, c) = (2, 2, 2) ;
g b
a (η, η0) = e(η−η0)
5 3 2 3 2
5 −2 2 3 −3
a (η, η0) Φb(k, η0)
linear structure matrix
Ω b
a (η) =
−1 − 3 2f 2 Ωm(η) 3 2f 2 Ωm(η) − 1
Vlasov equation of a single flow pressure-less fluid
Scoccimarro 1997
=
k
g(η) ϕ(k)=
k
=
k
= +
k kNote : detailed effects of baryons versus DM can be taken into account (Somogyi & Smith 2010; FB, Van de Rijt, Vernizzi '12) with a 4-component multiplet, for neutrinos it is more complicated...
Pαβ (k) =
k
Ψα
(1)(k)Ψβ
(1)(k)Pα'β' (k,η0) Ψβ
(1)(k)Ψα
(3)(k,q,-q)q
Ψα
(2)(q,k-q)Ψβ
(2)(q,k-q)+ +
Φa(k, η) = g b
a(η)Φb(k, η = 0) +
Z η dη0g b
a(η − η0)γ cd b (k1, k2)Φc(k1, η0)Φd(k2, η0)
linear evolution mode coupling terms
equilibrium) system, where perturbations grow with time (as ~ power-law). The late time behavior of this system is probably non trivial and there is no known solution to it.
effects, modes being set in the initial conditions.
conservation equation and is independent of the energy content of the universe. Only 2 →1 vertices exist (quadratic couplings). This is not the case generically for modified gravity models (like chameleon, DGP ...)
e.g. ”Renormalizations”, are all finite.
Time-flow (renormalization) equations M. Pietroni ’08
From the field evolution equation to the multi- spectra evolution equation
Anselmi & Pietroni '12
The closure theory
Taruya & Hiramatsu, ApJ 2008, 2009
Valageas P., A&A, 2007
Motion equations for correlators are derived using the Direct-Interaction (DI) approximation in which one separates the field expression in a DI part and a Non-DI part. At leading order in Non- DI >> DI, one gets a closed set of equations, These equations can more rigorously be derived in a large N expansion.
The eikonal approximation
FB, Van de Rijt & Vernizzi 2012
Methods of Field Theory
Effective Theory approaches
Pietroni et al '12, Carrasco et al. '12
Renormalization Perturbation Theory
Crocce & Scoccimarro ’05, 06
Beyond standard PT : "resumming", redefining the series expansions
Inspired by hydro turbulence resummation schemes, see L'vov & Procaccia ’95
The diagram contributing to the power spectrum up to 2-loop order:
linear power spectrum
The key ingredients : the (multipoint) propagators
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Scoccimarro and Crocce PRD, 2005
Gab(k) =
k
FB, Crocce, Scoccimarro, PRD, 2008 Γ(2)
abc(k1, k2, k3) =
Γ(p)
ab1...bp(k1, . . . , kp, η)δD(k − k1...p) = 1
p!
δφb1(k1) . . . δφbp(kp) ⇥
the building blocks
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Sum of positive terms
blocks for higher order moments...
FB, Crocce, Scoccimarro, PRD, 2008
Γ-expansion method
Reconstruction of the power spectrum: from sPT to Multi-point propagator reconstruction
FB, Van de Rijt, Vernizzi 2011 and 2012
photon wavelength is much shorter than any other lengths
p l in
The eikonal approximation :
"Relativistic eikonal expansion", Abarbanel and Itzykson, 1969
dynamics :
∂ ∂η Φa(k, η) + Ω b
a(η)Φb(k, η) = γ bc a (k1, k2)Φb(k1)Φc(k2)
Impact of the long-wave modes into the short wave modes (of interest)
It leads to a "renormalized" theory that takes into account the long wave modes in a nonlinear manner.
assuming linear growing modes and Gaussian initial conditions.
∂ ∂ηΦa(k, η) + Ω b a(η)Φb(k, η) − Ξ b a(k, η)Φb(k, η) = γ bc a (k1, k2)Φb(k1)Φc(k2)|hard domain
Ξ b
a(k, η) =
R d3q
a (q, k) + γ bc a (k, q)
Non trivial k dependence!
The IR modes in the eikonal approximation :
FB, Van de Rijt, Vernizzi 2011
velocity field component only
The "renormalized" theory at linear order
What is in this new term ?
A multi-component fluid analysis with adiabatic modes and iso-curvature/density modes
adiabatic term non-adiabatic term
= Z k.q q2 δd(q)d3q
∂ ∂ηΦa(k, η) + Ω b a(η)Φb(k, η) − Ξ b a(k, η)Φb(k, η) = 0
Ξ b
a(k, η) =
R d3q eik.γ cb
a (q, k) +eik. γ bc a (k, q)
Impact of the adiabatic modes :
(adiabatic) displacement field
Consequences for propagators (building blocks for PT calculations)
Gab(k) =
k
= ⌦ ξ b
a (η)
↵
Ξ = g b a (η) exp
✓ −k2σ2
d(η − η0)2
2 ◆
Ξadiab.
Γ(2)
abc(k1, k2, k3) =
= Γtree
abc (k1, k2, k3) exp
✓ −k2
3σ2 d(η − η0)2
2 ◆
ξ b
a (k, η, η0) = g b a (η, η0) exp
⇥ ik.dadiab.(η0) ⇤
Consequences for equal-time poly-spectra : none
Crocce & Scoccimarro ’05, 06 FB, Crocce & Scoccimarro ’08
c.t. = 1 2 ✓k2σ2
d
2 ◆2
treeΓ b1...bp a
+ k2σ2
d
2
a
A regularization scheme = how to interpolate between n-loop results and the large-k behavior ?
An ad-hoc solution was provided by Crocce and Scoccimarro (RPT) for the
k RegΓ(p) b1...bp a
=
treeΓ b1...bp a
exp ✓ −k2σ2
d
2 ◆ +
a
+ 1 2k2σ2
d treeΓ b1...bp a
✓ −k2σ2
d
2 ◆ + ⇥two−loopΓ b1...bp
a
+ c.t. ⇤ exp ✓ −k2σ2
d
2 ◆
k
FB, Crocce, Scoccimarro '12
The two-point propagator at 1-loop and 2-loop orders
Comparison with numerical simulations at tree and one-loop order for the 3-point propagator
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0.2 0.3 0.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
k @h-1 MpcD GH2LHkê2,kê2,kL
Colinears - z = 0
0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3
k @h-1 MpcD GH2LHk,k,kL
Equilaterals - z = 0
no binning tree
FB, Crocce, Scoccimarro '12
Power spectra up to 1-loop and 2-loop order
computations of power spectra at 2-loop order are now available. http://maia.ice.cat/ crocce/mptbreeze/ http://www- utap.phys.s.u- tokyo.ac.jp/ ~ataruya/ regpt_code.html
within 1% accuracy.
1-loop (std) linear
z=1
0.05 0.10 0.15 0.20 0.25 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 kHh-1MpcL PHkLêPno-wiggleHkL
2-loops (std) 2-loops (RegPT)
0.05 0.10 0.15 0.20 0.25 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 kHh-1MpcL PHkLêPno-wiggleHkL
z=0.35
2-loops (std) 2-loops (RegPT) 1-loop (std)
Taruya , FB, Nishimichi, Codis '12 Crocce, Scocimarro, FB, '12 1st computation of 2-loop order effects in Okamura, Taruya, Matsubara, '11
η2
k
ψa(k, η; Ξadiab.) = ξ b
a (k, η, η0; Ξadiab.)ψb(η0)
Consequence 1: multi-spectra are independent on the large-scale adiabatic modes (in the eikonal limit)
This is a direct consequence of the functional dependance on the large-scale adiabatic displacement field.
Consequence 2: multi-spectra are independent on the large-scale adiabatic modes at any order in standard Perturbation Theory
One-loop correction to power spectrum
+
η2
k
+
η2
k
= 0
k1 k2 k3 q1 k1-q1
FB, Van de Rijt, Vernizzi, '12
adiabatic modes
δ b
a
ξ b
a (k, η, η0; Ξadiab.) = g b a (η, η0) exp
✓ i Z η
η0
dη0 k.vadiab.(η0) ◆
But not necessarily so for all PT schemes...
What is true for adiabatic modes is not true for non-adiabatic modes!
η2
+
η2
k
+
η2
k
≠ 0
Resulting power spectrum in the eikonal limit (beyond one-loop results)
z=40 z=10
non-adiabatic modes
FB, Van de Rijt, Vernizzi, '12 in prep.
modes mainly produced at horizon scale at decoupling
Pδ(k; Ξiso.) = ξ a
1 (k, η, η0; Ξiso.) ξ b 1 (k, η, η0; Ξiso.) P init. ab
(k, η0) D. Tseliakhovich and C. Hirata, PRD, '10
Formation of first structures is modulated and anisotropic
Bad news for biasing: Galaxy formation is potentially modulated by large scale velocity modes (at 100-10 Mpc scales).
Dalal, Pen, Seljak '10 Yoo, Dalal, Seljak '11
In general however non-adiabatic modes have very little (totally negligible ?) impact on modes of interest here.
FB, Van de Rijt, Vernizzi 2011 Somogyi & Smith 2010
Enriching the content of the universe is likely to induce similar effects beyond linear theory results. This is potentially the case for massive neutrinos (whose velocities differ from the velocity
hierarchy of equations in case of massive neutrinos is now
adiabatic modes .
PhD thesis of Nicolas van de Rijt '12
Kernels in Perturbation Theory calculations
k=0.1 h Mpc-1
k=0.5 h Mpc-1
k
k
P ]−loop
NL
(k) = Z dq q K]−loop(k, q) Plin.(q)
FB, Taruya, Nishimichi, '12
0.01 0.05 0.10 0.50 1.00 5.00 10.00 0.001 0.01 0.1 1 10 qêHh Mpc-1L Kernel1-loopHqL PlinHqL
Expression of the density kernel for the propagator at 1-loop order
0.01 0.05 0.10 0.50 1.00 5.00 10.00 0.005 0.010 0.050 0.100 0.500 1.000 qêHh Mpc-1L PHqLK-loopHk,qL
k=0.1 h/Mpc
1-loop 2-loop
Kernels for the 2-point propagators at p-loop order
k
Convergence properties
1-loop
P ]−loop
NL
(k) = Z dq q K]−loop(k, q) Plin.(q)
Should it be regularized or taken into account with Effective Theory approaches?
Pietroni et al. '11, Carrasco et al. '12
ns < −1
k
2-loop
ns < −2
3-loop
q1 k k k+ q1
3-loop
ns < −2.33
comparison with numerical simulations
γabc(k1, k2) =
⇧ 0, (k1+k2).k2
2k2.k2
⌃ ⇧
(k1+k2).k1 2k1.k1
, 0 ⌃ {0, 0} ⇧ 0, k1.k2(k1+k2).(k1+k2)
2k1.k1k2.k2
⌃ ⇥ ⌅
∂ ∂η Φa(k, η) + Ωab(η)Φb(k, η) = γabc(k1, k2)Φb(k1)Φc(k2)
and power counting
γabc(q, k − q) ∼ k2/q2
1 q2 ⇥ q3 Plinear(q) ⇤] loops
for z < 0.5): it is not cleat if it can be obtained from re-summations of contributing diagrams or from extra physical effects (in particular shell crossings, etc...)
change the converging properties of theory. This is suggested by preliminary results obtained in some classes of modified gravity models (with a dynamical dilaton field with Damour-Polyakov mechanism for instance).
problem, that is the impact of the small scale structure on the large
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Large-Scale Structure studies
precision cosmology calculations; An interesting playground for field theory calculations