NON-LINEAR DESCRIPTION OF MASSIVE NEUTRINOS IN THE...
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NON-LINEAR DESCRIPTION OF MASSIVE NEUTRINOS IN THE FRAMEWORK OF LARGE-SCALE STRUCTURE FORMATION Hélène DUPUY Presentation partly based on JCAP 01 (2014) 030, H.D. & F.Bernardeau [arXiv: 1311.5487] Rencontres de Moriond, Cosmology 2014 March, 27 th 2014 1
Transcript of NON-LINEAR DESCRIPTION OF MASSIVE NEUTRINOS IN THE...
NON-LINEAR DESCRIPTION OF MASSIVE NEUTRINOS
IN THE FRAMEWORK OF LARGE-SCALE STRUCTURE FORMATION
Hélène DUPUY
Presentation partly based on JCAP 01 (2014) 030, H.D. & F.Bernardeau [arXiv: 1311.5487]
Rencontres de Moriond, Cosmology 2014 March, 27th 2014
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Outline
• I-The standard linear description of neutrinos.
• II-The non-linear description of cold dark matter, a model
to follow.
• III-A non-linear alternative to the standard description of
neutrinos.
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I-The standard linear description of neutrinos.
• Neutrinos are considered as a fluid, which can be relativistic. • The fluid is described via its distribution function in phase-
space, whose time evolution is given by the Vlasov equation. • The distribution function is decomposed into
where is an unperturbed quantity and is a first order quantity.
• The expanded form of the Vlasov equation gives for
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f0
@⌘ +
qi
a✏@i +
d log f0(q)
d log q
✓@⌘�� a✏
q2qi@i
◆= 0.
f = f0(1 + ),
I-The standard linear description of neutrinos.
• The quantity is decomposed into Legendre polynomials • It leads to the standard Boltzmann hierarchy
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@⌘ 0(⌘, q) = � qk
3a✏ 1(⌘, q)� @⌘�(⌘)
@⌘ 1(⌘, q) =qk
a✏
✓ 0(⌘, q)�
2
5 2(⌘, q)
◆� a✏k
q (⌘),
@⌘ `(⌘, q) =qk
a✏
`
2`� 1 `�1(⌘, q)�
`+ 1
2`+ 3 `+1(⌘, q)
�(` � 2).
˜
⌘✓d log f0(q)
d log q
◆�1
=X
`
(�i)` ` P`(↵).
I-The standard linear description of neutrinos. • Why not computing a non-linear hierarchy? Introducing the non-linear moments of the Vlasov equation are (see Nicolas Van de Rijt’s PhD thesis “Signatures of the primordial universe in large-scale structure surveys”, École Polytechnique & IPhT, CEA Saclay, 2012)
• Those equations are very difficult to handle. More generally, an exploitable
analytical description of neutrinos in the non-linear regime is still missing.
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Aij...k ⌘Z
d3q
qi
a✏
qj
a✏...qk
a✏
�✏f
a3,
@⌘Ai1...in + (H� @⌘�)
⇥(n+ 3)Ai1...in � (n� 1)Ai1...injj
⇤
+nX
m=1
(@im )Ai1...im�1im+1...in +
nX
m=1
(@im�)Ai1...im�1im+1...injj
+(1 + �+ )@jAi1...inj + [(2� n)@j � (2 + n)@j�]A
i1...inj = 0.
II-The non-linear description of cold dark matter, a model to follow. • Description of cold dark matter: collection of identical point
particles, non-relativistic, sensitive to gravitational interaction only.
• Physics is encoded in the Vlasov-Poisson system (newtonian approximation).
• The Vlasov-Poisson system leads to the continuity and Euler equations:
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@�(x, t)
@t+
1
a[(1 + �(x, t))ui(x, t)],i = 0,
velocity dispersion gravitational potential velocity field density contrast
@ui(x, t)
@t+
a
aui(x, t) +
1
auj(x, t)ui(x, t),j = �1
a�(x, t),i �
(⇢(x, t)�ij(x, t)),ja⇢(x, t)
.
II-The non-linear description of cold dark matter, a model to follow. • Particles are “cold” thus velocity dispersion is negligible:
• This is called the single-flow approximation. • Note: the single-flow approximation breaks down when
shell-crossing occurs, i.e. when gravity makes several flows appear (see the picture below).
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(⇢(x, t)�ij(x, t)),ja⇢(x, t)
.
a one-flow region a multi-flow region
II-The non-linear description of cold dark matter, a model to follow. • In the single-flow approximation, the Euler equation reads
The velocity field is potential so no curl mode can appear. The velocity field can entirely be described by its divergence: . In Fourier space, equations can be written compactly:
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✓(x, t) = 1/(aH)ui(x, t),i
a(k, ⌘) ⌘ (�(k, ⌘),�✓(k, ⌘)),
� 222 (k1,k2) = �D(k� k1 � k2)
|k1 + k2|2 (k1.k2)
2k21k22
,
� bca (ka,kb) = � cb
a (kb,ka),
� 212 (k1,k2) = �D(k� k1 � k2)
(k1 + k2).k1
2k21and otherwise.
@ a(k, ⌘)
@⌘+ ⌦ b
a (⌘) b(k, ⌘) = � bca (k1,k2) b(k1, ⌘) c(k2, ⌘),
� = 0
where
d(aui(x, t))
dt= ��(x, t),i.
II-The non-linear description of cold dark matter, a model to follow. • The compact equation of motion has a formal solution.
• Many developments have been realized starting from this
solution in order to relate it to interesting cosmological observables (see the Les Houches Summer School lecture notes by F.Bernardeau: arXiv 1311.2724).
The simplicity of the motion equation allows to predict relevant results about cold dark matter even in the non-linear regime.
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a(k, ⌘) = g ba (⌘) b(k, ⌘0) +
Z ⌘
⌘0
d⌘0g ba (⌘, ⌘0)� cd
b (k1,k2) c(k1, ⌘0) d(k2, ⌘
0).
initial time Green function
III-A non-linear alternative to the standard description of neutrinos. • Idea: describing neutrinos as a collection of single-flow
fluids in order to take advantage of the single-flow approximation.
• One fluid of the collection = the gathering of all the
neutrinos that have a given velocity at initial time. • Such fluids are actually single flows if we assume that
there is no shell-crossing (which is not a reasonable assumption for the overall neutrino fluid because of velocity dispersion).
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III-A non-linear alternative to the standard description of neutrinos. • Calculations are performed in a perturbed Friedmann-Lemaître metric
• Our variables: - the proper number density measured by an
observer at rest in the metric, denoted - the comoving momentum field with covariant coordinates, denoted
• First motion equation: conservation of the number of particles
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ds2 = a
2 (⌘)⇥� (1 + 2 ) d⌘2 + (1� 2�) dxidxj
�ij
⇤.
@⌘n� (1 + 2�+ 2 )@i(Pi
P0n) = 3n(@⌘��H) + n(2@i � @i�)
Pi
P0.
n,
Pi.
III-A non-linear alternative to the standard description of neutrinos.
• Second motion equation: conservation of the energy-momentum tensor combined with the conservation of the number of particles
• This conservation equation makes only the momentum
variable appear because of the single-flow approximation.
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@⌘Pi � (1 + 2�+ 2 )Pj
P0@jPi = P0@i +
PjPj
P0@i�.
• Zeroth-order of the motion equation of the momentum field: is a good variable to label the different neutrino fluids. We call this variable and we define similarly • With these notations, the linearized equations of motion read
derives from a potential.
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@⌘Pi(0) = 0.
Pi(0)
⌧0 = P (0)0 .
dn(1)
d⌘= 3@⌘�n
(0) � 3Hn(1) + (2@i � @i�)⌧i⌧0
n(0) +
@iPi
(1)
⌧0� ⌧i@iP0
(1)
⌧20
!n(0),
dPi(1)
d⌘= ⌧0@i +
⌧2
⌧0@i�.
III-A non-linear alternative to the standard description of neutrinos.
⌧i
P (1)i
III-A non-linear alternative to the standard description of neutrinos.
• We move to Fourier space. • is potential at linear order so The equations of motion become where
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@⌘�n = iµk⌧
⌧0�n + 3@⌘�+
✓P⌧0
✓1� ⌧2
⌧20µ2
◆� iµk
⌧
⌧0
✓1 +
⌧2
⌧20
◆��
�
@⌘✓P = iµk⌧
⌧0✓P � ⌧0k
2 � ⌧2
⌧0k2�
µ =k.⌧
k⌧.
Pi P (1)i (k) =
�ikik2
✓P (k).
momentum divergence
numerical density contrast
III-A non-linear alternative to the standard description of neutrinos.
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• Hypotheses: - cosmological perturbations are initially adiabatic, - at initial time, neutrinos are already decoupled but still relativistic. • One can show that the condition is in
agreement with the adiabaticity hypothesis.
• The proper number density field is related to the distribution function thanks to
Pi(x, ⌘in) = P (0)i = ⌧i
n(⌘, xi) =
Zd3qi
f(⌘, xi, pi)
a
3=
1 + 3�(⌘, xi)
a
3nc(⌘, x
i).
Comoving number density
III-A non-linear alternative to the standard description of neutrinos. • After decoupling, neutrinos follow a perturbed Fermi-Dirac law
until they become non-relativistic:
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f (⌘in,x, ⌧) /1
1 + exp [⌧(1 + �(x, ⌘in))/akB(T + �T (x, ⌘in))].
• Connecting the number density to the distribution function and using the standard result we deduce that
�T (x, ⌘in)/T (⌘in) = � (x, ⌘in)/2,
�n(⌧,x, ⌘in) = 3�(x, ⌘in) +
✓ (x, ⌘in)
2
+ �(x, ⌘in)
◆d log f0(⌧)
d log ⌧.
temperature fluctuation
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0.1 1 10 100 1000-15
-10
-5
0
5
a êaeq.
ResidualsHinu
nitsof10-4 L
0.001 0.01 0.1 1 10 100 100010-6
10-4
0.01
1
100
a êaeq.
MultipolesHinu
nitsofyHh inLL
Time evolution of the energy multipoles computed using the multi-fluid description. Solid line: energy density contrast. Dashed line: velocity divergence. Dotted line: shear stress. Dot-dashed line: energy density contrast of the CDM component.
Relative differences between energy mulitpoles when they are computed in both approaches, in units of 10-4.
(lmax
= 6, Nµ = 12, Nq
= N⌧ = 40, k = keq
= 0.01h/Mpc,m = 0.3 eV).
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0.001 0.01 0.1 1 10 100 100010-5
0.001
0.1
10
1000
a êaeq.
MultipolesHinu
nitsofyHh inLL
k = 2keq
k = 5keq
(same meaning of the line shapes, same mass).
(same meaning of the line shapes, same mass).
0.001 0.01 0.1 1 10 100 100010-4
0.01
1
100
104
a êaeq.
MultipolesHinu
nitsofyHh inLL
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Time evolution of the velocity divergence for different values of with Solid lines: Dashed lines:
⌧µ = 0
m = 0.05eV. m = 0.3eV.(0.45kBT0 < ⌧ < 9kBT0).
0.1 1 10 100 10001
2
3
4
5
a êaeq.
q vêq CD
M Biggest value of
Smallest value of
⌧
⌧
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0.1 1 10 100 1000-2
0
2
4
6
a êaeq.
d vêd CD
M
Biggest value of
Smallest value of
⌧
⌧
Time evolution of the number density contrast for different values of with Solid lines: Dashed lines:
⌧µ = 0
m = 0.05eV. m = 0.3eV.(0.45kBT0 < ⌧ < 9kBT0).
⌧
Conclusions • We have developed an analytical non-linear description of
massive neutrinos based on perturbation theory.
• This approach is valid up to shell-crossing only (same restriction as in the cold dark matter description).
• Developing - and then implementing - an advanced theory allowing to make analytical predictions about relevant non-linear observables is challenging but a priori doable in this context.
• An intermediate (ongoing) step is the extension of all the equations to a more general perturbed Friedmann-Lemaître metric (with vector and tensor perturbations too).
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