Includingneutrinosinstandard
perturbationtheory
HélèneDUPUY(GenevaUniversity),
incollabora:onwithFrancisBERNARDEAU
May,26th2016
Nonlinearevolu,onofthelarge-scalestructureofthe
universe:theorymeetsexpecta,ons
1/15
NeffandMνareeverywhere
•  Duringtheradia:on-dominatedera:
"
⇢r = ⇢ + ⇢⌫ = 1 +
7
8
✓
4
11
◆4/3
#
Ne↵ ⇢ .
PhotonsANDneutrinosfixtheexpansionrateatearly:mes.
Neffhasanimpactontheprimordialabundancesoflight
elements.
•  Neutrinoswithamasslyingbetween10-3eVand1eVarerela:vis:c
atmaXer-radia:onequalityandnon-rela:vis:ctoday.
MνhasanimpacteitheronzeqoronΩm,0(dependingonthe
parametersonedecidestokeepfixed).
Inprinciple,theCMBspectrumissensi:vetoMν(background
effects+effectsonsecondaryanisotropies).
2/15
ButtheCMBaloneisnotsufficienttoconstrainsub-eVneutrino
masses.
Massiveneutrinosandthelinear
matterpowerspectrum
Takenfromthe
bookNeutrino
Cosmology
(J.Lesgourgues,
G.Mangano,
G.Mieleand
S.Pastor,2013)
3/15
Massiveneutrinosandthenon-linear
matterpowerspectrum
Linear
predic:on
Numericalsimula:onsshowingtheeffectofmassiveneutrinos
onthenon-linearmaXerpowerspectrum(m=0.15eV)
Authors:S.Birdetal.(arXiv:1109.4416)
4/15
Whyshouldneutrinoperturbationsbe
treatednonlinearly?
Differentapproximateschemesarecomparedtoaoneinwhichnonlineari:esofneutrinosaretakenintoaccount.Thecurvesrepresent
ra:osofnon-linearcontribu:onstothemaXerpowerspectrum.
Authors:D.Blasetal.(arXiv:1408.2995)
5/15
Amulti->lowapproachtostudynoncoldspeciesbeyondthelinearregime
ini:al:me
later:me
ini:almomentum
(labeloftheflow)
tot
X
f~⌧ (⌘, x, p).
•  Totaldistribu:onfunc:on: f (⌘, x, p) =
Z ~⌧
d3 pi f~⌧ (⌘, xi , pi ).
•  Onedensityfieldperflow:nc (⌘, x; ~⌧ ) =
6/15
R 3
d pi f~⌧ (⌘, xi , pi )pi
•  Ineachflow, Pi (⌘, x; ~
⌧) = R 3
.
i
d pi f~⌧ (⌘, x , pi )
arbitraryfunc:on
Z
•  Moregenerally, F [Pi (⌘, x)] nc (⌘, x) = d3 pi f (⌘, xi , pi ) F [pi ] .
•  Thephysicalquan::esofinterestcanbeexpressedintermsofour
fields:
Z
Z
d3 pi f tot (⌘, xi , pi ) F(pi ) = d3 ⌧i nc (⌘, x; ⌧i ) F(Pi (⌘, x; ⌧i )) .
|
{z
} |
{z
}
Boltzmann approach
our approach
7/15
•  Ineachflow,theequa:onofmo:onofthedensityfieldis
@
@
nc +
@⌘
@xi
✓
i
P
nc
0
P
◆
= 0,
i
ij
0
µ
P
=
g
P
and
where
isdefinedsothat
P
Pµ =
P
j
m2 .
T µ⌫ =
•  Ineachflow,
energy-momentumtensor
P µJ ⌫ .
par:clefour-current
Combinedconserva:onlawsimpose
1
P @ ⌫ Pi = P P ⌫ @ i g ⌫ .
2
⌫
8/15
1
0.01
10-4
10-6
0.001
0.01
0.1
1
10
a êaeq.
Rela:vedifferences
(in units of 10-4) in
comparisonwiththe
Boltzmannapproach
Mul:polescomputed
usingourapproach.
Solidline:energy
densitycontrast.
Dashedline:velocity
divergence.
DoXedline:shearstress
100
100
Residuals Hin units of 10-4 L
Multipoles Hin units of yHhin LL
(lmax = 6, Nµ = 12, Nq = N⌧ = 40, k = keq = 0.01h/Mpc, m = 0.3 eV).
1000
5
0
-5
-10
-15
0.1
1
Moredetailscanbefoundin
arXiv:1311.5487(H.DupuyandF.Bernardeau).
10
a êaeq.
9/15
100
1000
USEFULPROPERTIESONSUBHORIZONSCALES
•  InaperturbedFriedmann-Lemaîtremetric,theequa:onsread
✓ i ◆
@
@
P
nc +
nc = 0,
i
0
@⌘
@x P

j
@Pi
P @Pi
1 P jP k
2
0
j
@i hjk .
+ 0 j = a (⌘) P @i A + P @i Bj +
@⌘
P @x
2 P0
•  Inthesubhorizonlimit,theybecome
ini:almomentumoftheflow
D⌘ nc + @i (Vi nc ) = 0,
1 ⌧j ⌧k
@i hjk ,
2 ⌧0
D ⌘ Pi + V j @ j Pi = ⌧ 0 @ i A + ⌧ j @ i B j
q
@
⌧i @
2
2
2
m a + ⌧i , D ⌘ =
with ⌧0 =
@⌘ ⌧0 @xi
Pi ⌧ i
⌧i ⌧j (Pj ⌧j )
peculiarvelocity
and Vi =
+
.
2
⌧0
⌧0
(⌧0 )
10/15
GENERALIZATION1:NOCURLMODESINTHEMOMENTUMFIELD
•  Onsubhorizonscalesthecurlfield,definedas
⌦i = ✏ijk @k Pj ,
obeystheequa:on D⌘ ⌦k + Vi @i ⌦k + @i Vi ⌦k @i Vk ⌦i = 0.
Thecurlfieldisonlysourcedbyitself.
Foradiaba:cini:alcondi:ons,thecomovingmomentum
fieldcanbewriMenasagradient.
11/15
AsthevelocityfieldofcolddarkmaXer,itisenNrely
characterizedbyitsdivergence.
GENERALIZATION2:ALLMODECOUPLINGSAREQUADRATIC
•  ByanalogywithcolddarkmaXer,weintroduce
✓⌧i (⌘, xi ) =
Pi,i (⌘, xi ; ⌧i )
,
maH
⌧i (⌘, x
i
)=
nc (⌘, xi ; ⌧i )
(0)
nc (⌧i )
1.
•  InFourierspace,itgives
✓
✓
a@a
µk⌧
i
H⌧0
◆
~
⌧ (k)
ma
⌧0
@a H
1+a
+ a@a
H
Z
ma
⌧0
✓
2 2
1
µ ⌧
⌧02
◆
✓~⌧ (k) =
d3 k1 d3 k2 ↵R (k1 , k2 ; ~⌧ ) ~⌧ (k1 )✓~⌧ (k2 ),
◆
µk⌧
k2
i
✓~⌧ (k) +
S~⌧ (k) =
H⌧0
maH2
Z
ma
d3 k1 d3 k2 R (k1 , k2 ; ~⌧ )✓~⌧ (k1 )✓~⌧ (k2 ).
⌧0
12/15
•  ConsideringNflows,itisusefultointroducethe2N-uplet
T
a (k) = ( ⌧1 (k), ✓⌧1 (k), . . . , ⌧n (k), ✓⌧n (k)) .
•  Theresul:ngequa:onsis
@ a
a
(k, ⌘) + ⌦ab (k, ⌘) b (k, ⌘) = abc (k1 , k2 , ⌘) b (k1 , ⌘) c (k2 , ⌘).
@a
Therela:vis:cequa:onofmo:onisformallythesameas
theequaNondescribingcolddarkmaMer.
•  ThisstudyispresentedinarXiv:1411.0428(H.Dupuyand
13/15
F.Bernardeau).
5
⌧
Biggestvalueof
qv êqCDM
4
k=0.01h/Mpc
3
2
Smallest
valueof
⌧
1
0.1
1
10
a êaeq.
100
1000
Timeevolu:onofthevelocitydivergencefordifferentvalues
µ = 0 (0.45kB T0 < ⌧ < 9kB T0 ).
of,with
⌧
Solidlines: m = 0.05eV.
m = 0.3eV.
Dashedlines:
14/15
Prospects
•  Solvingthenon-linearsystemofequa:onsintheeikonal
approxima:on(arXiv:1109.3400,F.Bernardeauetal.).
•  Solvingthenon-linearsystemofequa:onsinthegeneralcase.
Whatisthemostsuitabletechnique?Canthe:merenormaliza:ongroupmethod(arXiv:0806.0971,M.Pietroni)
beextendedtothestudyofacollec:onofflows?
•  Usingourresultstovisualizethegravita:onalcollapseof
neutrinosforgivenini:aldistribu:onsofmaXer.Canwelearn
somethingaboutthespacedistribu:onofvoids?
15/15