THE ANALYTICAL STUDY OF COSMOLOGICAL STRUCTURE
FORMATION
Alvaro Domı́nguez, Universidad de Sevilla
Abstract:
The theoretical study of cosmological structure formation by gravitational instability has relied basically on the so-called “dust model” (pressureless self–gravitating
fluid). This model, however, is ill posed and the solutions become singular in a
finite time. In this talk I present the idea of the small–size expansion (SSE): it is
the mathematical implementation of the notion that the coupling between scales
is relatively subdominant during cosmological structure formation. The SSE allows
to derive an improved model without resort to phenomenological approximations.
The SSE model contains the dust model as the early–time limit. I show how the
correction to the dust model leads to the “adhesion model”, which is a well–known
phenomenological proposal to prevent the formation of singularities. More generally,
the SSE model offers an explanation of the physics behind the adhesion model and
an extension thereof. The SSE correction is also a source of vorticity, whose effect
is computed perturbatively. Finally, I discuss briefly directions of promising future
work.
Bibliography:
A.
A.
T.
A.
Domı́nguez, Phys. Rev. D 62 (2000) 103501
Domı́nguez, Mon. Not. RAS 334 (2002) 435
Buchert, A. Domı́nguez, preprint (2004)
Knebe, A. Domı́nguez, R. Domı́nguez–Tenreiro, work in progress
THE ANALYTICAL STUDY OF COSMOLOGICAL
STRUCTURE FORMATION
Alvaro Domı́nguez, Universidad de Sevilla
Outline
• Motivation
• Closure ansatz: Small-Size Expansion
• Corrections to the dust model: – Adhesion model
– Generation of vorticity
• Conclusions and outlook
MOTIVATION
• Structure formation in Cold Dark Matter by gravitational instability

Particle–level description, {xα (t), uα(t)}α=1,...N 

Kinetic description, f (x, u, t)
⇒ Too detailed because


– No direct observational access to DM spatial and kinematical
distribution with sufficient resolution
– Limited resolution of numerical simulations
• Use instead a hydrodynamic–like description, {%(x, t), u(x, t)}
density and velocity fields smoothed over a length scale L
– Change L in order to probe the evolution at different scales
MICROSCOPIC MODEL
• Homogeneous Universe: Friedmann–Lemaı̂tre model ⇒ H, Ωm , Λ
• Initial conditions: tiny fluctuations about homogeneous expansion
• Many-body self-gravitating system: Vlasov–Poisson equations
f (x, u, t) = one–particle phase–space density
∂f
1
∂f
∂f
+ u·
+ (w − H u) ·
= 0,
∂t
a
∂x
∂u
∇ · w(x, t) = −4πGa m
Z
du f (x, u, t) − %b ,
∇ × w(x, t) = 0.
• Comoving coordinates
x :=
1
r,
a
u := v − H r,
w := g −
Λ − 4πG%b
r,
3
HYDRODYNAMIC DESCRIPTION
• Average over velocities
Z
%mic(x, t) := m du0 f (x, u0, t)
Z
%mic umic(x, t) := m du0 u0 f (x, u0, t)
• Spatial coarsening over a comoving scale L
dy
x−y
mass density: %(x, t) :=
W
%mic (y, t)
L3
L
Z
dy
x−y
center-of-mass velocity: % u(x, t) :=
W
%mic umic(y, t)
3
L
L
Z
R
dx W (x) = 1
W (x)
replacements
∼1
x
normalized window
EVOLUTION EQUATIONS
• Mass and momentum conservation
∂%
1
+ ∇ · (% u) + 3H% = 0
∂t
a
∂(% u)
1
1
+ ∇ · (% u u) + 4H% u = % w + F − ∇ · Π
∂t
a
a
∇ · w = −4πGa(% − %b),
∇×w =0
• Correction to mean–field gravity (higher–than–monopole terms)
F(x) =
Z
dy
W
L3
x−y
L
%mic (y)[wmic (y) − w(x)]
• Velocity dispersion
Π(x) =
Z
dy
W
3
L
x−y
L
%mic (y)[umic (y) − u(x)][umic (y) − u(x)]
• The fields F and Π couple large scales (> L) to small scales (< L)
Goal: to express F and Π in terms of % and u ⇒ Closure ansatz
The fields F and Π
• The microscopic gravitational field wmic is defined by the equations
∇ · wmic (x, t) = −4πGa [%mic (x, t) − %b] ,
∇ × wmic (x, t) = 0.
The field F is due to the deviation of the actual gravitational field wmic on scales
< L from the mean–field w:
Z
x−y
dy
W
%mic(y)[wmic(y) − w(x)].
F(x) =
L3
L
The source of the field w is % = mass monopole of the coarsening cell defined
by W (·). The source of F are the higher multipole mass moments of the cell.
• The second–rank tensor field Π measures the deviation of the actual velocity on
scales < L from the coarse–grained velocity u:
Z
Z
dy
x−y
m du0 f (x, u0, t)[u0 − u(x)][u0 − u(x)]
Π(x) =
W
3
L
L
Z
Z
dy
x−y
0
0
0
0
=
m
d
u
f
(
x
,
u
,
t)[
u
−
u
(
y
)][
u
− umic (y)] +
W
mic
L3
L
Z
x−y
dy
W
%mic(y)[umic (y) − u(x)][umic(y) − u(x)]
L3
L
– The first term is the coarse–grained velocity dispersion at each point, as
described by the kinetic distribution f (x, u, t).
– The second term is the velocity dispersion inside the cell due to coarse–
graining.
In a hierarchical scenario of structure formation, the first term is negligible
compared to the second one when the scale length L is of cosmological relevance.
EXAMPLE: EULER & NAVIER–STOKES EQUATIONS
• Fluid with dominant short–ranged forces: w → 0
• Local thermal equilibrium
Πij →
pideal δij
→ −∇ pnon−ideal
peq = equilibrium pressure
F



⇒ F − ∇ · Π → −∇ peq


Euler equation
• “Small” departure from equilibrium
F − ∇ · Π → −∇ peq + η∇2u + ζ∇(∇ · u)
η, ζ = viscosity coefficients
Navier–Stokes equation
SMALL-SIZE EXPANSION
• Bottom-up structure formation ⇒ nested matter distribution
Large scales ( L)
weakly coupled
to small scales ( L)
Mode–mode coupling estimated
by an expansion in (L ∇)
⇓
• Small-Size Expansion
Πij → B L2 % (∂k ui)(∂k uj )
Fi
→ B L2 (∂k %)(∂k wi)
CLOSURE
ANSATZ
+ o(L4 )
(local strains)
+ o(L4 )
(local tidal forces)
high–orders do not add into a relevant contribution
so that the expansion can be truncated
SMALL-SIZE EXPANSION in detail
• The density in Fourier space:
Z
dy
x−y
%(x) =
%mic (y)
W
L3
L
⇒
%(k) = W (Lk)%mic (k).
And the correction to mean field (using Poisson’s equation for w and wmic):
Z
dq iq
F(k) = −4πGa
%mic(q)%mic (k − q)[W (Lk) − W (Lq)W (L(k − q))].
(2π)3 q 2
• Taylor expansion of the window function:
1
W (Lk) = 1 − BL2k2 + o(Lk)4,
2
1
B :=
3
Z
dy y 2 W (y).
• Formal expansion in powers of L of the definition of F:
Z
dq iq
2
4
%
(
q
)%
(
k
−
q
)
BL
q
·
(
k
−
q
)
+
o(L
).
F(k) = −4πGa
mic
mic
(2π)3 q 2
Back in physical space:
F = BL2 (∇% · ∇)w + o(L4),
in terms of the local tidal forces given by ∇w.
• Similarly for the velocity dispersion tensor:
Π = BL2 % (∂k u)(∂k u) + o(L4),
in terms of the local strains given by ∇u. This expression for Π is known as
gradient model in the context of simulations of turbulent flow.
• Closure ansatz: the contribution of the modes with k, q L−1 in the definitions
of F and Π is subdominant
o(L0 ): DUST MODEL
∂%
1
+ ∇ · (% u) + 3H% = 0
∂t
a
∂(% u)
1
+ ∇ · (% u u) + 4H% u = % w
∂t
a
∇ · w = −4πGa (% − %b) ,
∇×w =0
• Dynamics on scales < L neglected
– No substructure ⇒ mean-field gravity
– No velocity dispersion ⇒ pressureless fluid
• “Truncated” dust model: coarsening length is a defining ingredient
• Density singularities (shell crossing)
• Kelvin theorem ⇒ no source of vorticity
o(L2 ): GENERAL CONSIDERATIONS
∂%
1
+ ∇ · (% u) + 3H% = 0
∂t
a
1
∂(% u) 1
2
+ ∇·(% u u)+4H% u = % w+B L (∇ % · ∇)w − ∇ · [% (∂iu)(∂iu)]
∂t
a
a
∇ · w = −4πGa (% − %b) ,
∇×w =0
• The corrections are nonlinear in the fields %, u, w
• The equations are reversible in time
• The amplitude of the corrections depends explicitly on
– the coarsening length L
– the window shape (through the constant B)
o(L2 ): ADHESION MODEL
• Assumption:
dust evolution correct almost everywhere
L2–correction relevant near “dust singularities”
Zel’dovich approximation:
w∝u
Rescaled time b and velocity v





⇒




∂v
+ (v · ∇)v = µ |∇ · v| ∇2 v
∂b
∇×v = 0
µ = 3 B L2
density
dust
PSfrag replacements
• Boundary layer theory ⇒
singularities are regularized
(> 0)
adhesion
∼L
spatial coord.
• In collapsing regions, L2–corrections behave like a sink of energy
The correction due to velocity dispersion Π is dominant
o(L2): ADHESION MODEL in detail
• “Dust singularities” are locally plane–parallel: near a singularity on the plane
perpendicular to the unit vector n we have approximately (∇ × u = 0 for dust)
u(x, t) ≈ u(n, t)n,
∂iuj ≈ (∇ · u) ninj ,
∂iwj ≈ (∇ · w) ninj .
• Zel’dovich’s approximation to the dust evolution: parallelity of u and w
b
b
⇒
% ≈ %b 1 − (∇ · u) .
w ≈ 4πG%b u
ḃ
ḃa
Here, b(t) := asymptotically growing solution of b̈ + 2H ḃ − 4πG%bb = 0.
• The L2–correction term is evaluated with these two approximations:
1
∂%
1 ∂
(∇ % · ∇)w − ∇ · [% (∂i u)(∂iu)] ≈ n
(∇ · w) −
[% (∇ · u)2]
a
∂n
a ∂n
b %b
2a
ḃ
b
2
2
3(∇
·
u
)
−
+
4πGa%
(∇
·
u
)
∇
u.
≈
b
b
a2 ḃ
ḃ
Near a singularity, ∇ · u → −∞ and the first term dominates. The resulting
equation for u is simplified by using b as temporal variable and introducing the
scaled velocity v = u/aḃ:
∂u 1
b
3
+ (u·∇)u+H u ≈ 4πG%b u+ B L2|∇·u|∇2u
∂t a
a
ḃ
⇒
∂v
+(v·∇)v ≈ 3B L2 |∇·v| ∇2v.
∂b
• Boundary layer theory: near the singularity, vn := v · n is given by (∂vn /∂n ≤ 0)
2
∂vn
∂v
∂ vn
∂vn
n
vn
≈ 3B L2
,
lim
= 0.
n→±∞ ∂n
∂n
∂n ∂n2
o(L2 ): VORTICITY IN PERTURBATION THEORY
• Expansion in the small amplitude of the initial fluctuations
%
− 1 = δ(1) + δ(2) + · · ·
u = u(1) + u(2) + · · ·
δ :=
%b
• First order: δ(1) and u(1) follow the linear regime of dust
(because the L2 –correction is nonlinear in the fields)
In particular, Ω(1) = 1a ∇ × u(1) → 0 asymptotically in time
• Second order: equation for Ω(2) = 1a ∇ × u(2) (taking Ω(1) = 0):
∂Ω(2)
1
1
+2H Ω(2) = B L2 ∇× (∇ δ(1) · ∇)w(1) − ∇ · [(∂iu(1))(∂i u(1))]
∂t
a
a
• Vorticity generation by tidal torques and shear stretching
q
• σΩ (R, t) := hΩ2iR = variance of Ω in balls of radius R
Initial power spectrum of density fluctuations: P (k) ∝ k n , −3 < n
σΩ (R L) ∼
(
R−n−5,
R−7/2,
−3 < n < −3/2
,
−3/2 < n
σΩ (R L) ∼
r
2
hδ(1)
iL
CONCLUSIONS
• SSE yields a set of evolution equations for % and u
without phenomenological assumptions
• Better understanding of the dust model and the adhesion model
– Derivation of apparent dissipation from a reversible model
• SSE predicts vorticity generated by tidal torques and shear
OUTLOOK
• Systematic study of the “weak mode–mode coupling” hypothesis
• Improvement of the
adhesion model:
– beyond shell crossing and inside pancakes
– role of vorticity in nonlinear collapse
• Improvement of simulations: influence of unresolved scales
– SSE can be generalized to adaptive smoothing L(x, t)