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Hey! do you have a small scalesmall scale problem?
WDM: Sterile neutrinos come to the rescue!!
Any sightings of (El) lately?
abundanceabundance Properties: phase space densityphase space density
small scales: cutoff in small scales: cutoff in P(k)+featuresP(k)+features
Constraints and bounds Production and decoupling: models
Small scale transfer function
A cosmic roadmap
The case for a sterile neutrino as warm dark matter
CDMΛ
sν
2
Cold (CDM): small velocity dispersion: small structure forms first,
bottom-up hierarchical merger –WIMPsWIMPsHot (HDM) : large velocity dispersion: big structure forms first,
top-down , fragmentation—light neutrinos Warm (WDM): ``in between’’ –sterile neutrinos
CDMΛ Concordance Model:
CMB+ LSS + N-body :
DM is COLD and COLLISIONLESS
N-body :``clumpy halo’’, large number of satellite galaxies
(NFW)
eVmν ≪
~ few keVsm
( ) ~1/ , 1 1.5r rβρ β≲ ≲
3
– 34 –
r (kpc)
ρ(M
⊙pc
−3
)
Ursa Minor
Draco
LeoII
LeoI
Carina
Sextans
1/r
10−1 100
10−3
10−2
10−1
100
Fig. 4.— Derived inner mass distributions from isotropic Jeans’ equation analyses for six
dSph galaxies. The modelling is reliable in each case out to radii of log (r)kpc∼ 0.5. The
unphysical behaviour at larger radii is explained in the text. The general similarity of the
inner mass profiles is striking, as is their shallow profile, and their similar central mass
densities. Also shown is an r−1 density profile, predicted by many CDM numerical simula-
tions (eg Navarro, Frenk & White 1997). The individual dynamical analyses are described
in full as follows: Ursa Minor (Wilkinson et al. (2004)); Draco (Wilkinson et al. (2004));
LeoII (Koch et al. (2007)); LeoI (Koch et al. (2006)); Carina (Wilkinson et al. (2006),
and Wilkinson et al in preparation); Sextans (Kleyna et al. (2004)).
Observational Evidence: Too few ``satellite’’ galaxiesdSphS: cores instead of cusps
~ 1 kpc
1/r (NFW)Cusps vsCores ??
4
A common mass for Dwarf Spheroidals (dSphs)??
From Strigari et.al.
``A warm DM candidate with m > 1 keV would yield a DM halo consistent with observations..’’ (Strigari. Et.al.)
5
And the plot thickens…..
Another over abundance problemAnother over abundance problem ----``mini voids``mini voids ’’’’ : too many small haloes in : too many small haloes in voids are too small compared to observations: voids are too small compared to observations: TikhonovTikhonov , , KlypinKlypin , , TikhonovTikhonov , et. al., et. al.
may be a possiblmay be a possibl e solution (e solution ( TikhonovTikhonov et.alet.al ))
overpredictsoverpredicts the number of DM haloes with masses > the number of DM haloes with masses > SawalaSawala et.alet.al ..
WDMWDM ``may offer a viable possibility [of resolution..]``may offer a viable possibility [of resolution..] ’’’’ ((SawalaSawala et. al.) (et. al.) ( MilleniumMillenium II)II)
1010 M⊙
CDMΛ
CDMΛ
with ~ keVWDM mΛ
A A keVkeV Sterile NeutrinoSterile Neutrino fits the billfits the bill ……
Q: What IS a Sterile Neutrino?A: A massive neutrino without electroweak interactions, couples to active neutrinos via a (seesaw) mass matrix. Q: Why/how does it help?A: Larger velocity dispersion, larger free-streaming lengthcuts-off power spectrum at small scales <
122[ ]fs
o m
V
Hλ ⟨ ⟩∝
Ωf sλ
6
From Kusenko and Lowenstein
ν2 ν2l l lν1 ν1
W W
W
γ
γ
Sterile neutrinos and the X-ray background: the evidence 2 1ν ν γ→
Is there anyany evidence? May be..
7
sin2 θ
msif 100% of dark matterexcluded region
(keV)
10−11 10−10 10−9 10−8
1.0
10.0
5.0
2.0
20.0
3.0
1.5
15.0
7.0
(allowed)
excluded
pulsar kicks
pulsar
(allowed)
kicks
10−6
10−4
10−2
1
10 2
10 4
10 6
10 8
10 10
10 12
10−15
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
sin2θ1µ
ms (
keV
)
LSS
α
X−rayBBNCMBLyman−
SN1987AAcceleratorDecaysAtmosphericBeam
Constraints from sterile neutrino decays (see Kusenko)
8
Collisionless DM= decoupled particles
LTE: Eq. distribution, FD, BE,MB
Out of LTE: frozen distribution
Frozen distribution obeys collisionless BE:
pcTd Dimensionless const.(microphysics)
1 2,, , )( xf y x ⋯Distribution function Of decoupled particle
HΓ >
HΓ <
( ; ) 0 f ff
Hp f p tt p
∂ ∂− =∂ ∂
( ; ) ( ( ) ) ( )f f cf p t f a t p f p= =
Microphysics:Microphysics:
9
Constraints:
Non-relativistic particles: m > T(t)
3 3
3 22
3 0( ) ( ) ( , )
(2 2( )
) ( )f d
f
d P Tt mn t mg f P T mg
ay y
tf dyρ
π π∞
= = = ∫∫
3 2
22 3 22
32
3
4
0
2
0
[ ( ) ](2 )
( )[ ( ) ]
(
( )
( )2 )
f ff
f d
ff
d P Pf a t P
P TmV
d Pm m
y f y dy
y f y da tP yf a t
π
π
∞
∞
= = =
∫∫
∫⟨ ⟩ ⟨ ⟩
∫
1) ABUNDANCE: Upper boundUpper bound
2
0
2 (3)2.695 (eV
()
)
da
y f y dy
gm
g
ξ∞≤∫
# of Relativisticd.o.f at decoupling
2 0.105DMhΩ =
10
2) Phase space density: LOWER BOUND2) Phase space density: LOWER BOUND
For decoupled non-relativistic particles the phase space density is CONSERVEDCONSERVED
one dimensional velocity dispersion
ThmThm : phase space density : phase space density diminishesdiminishes in ``violentin ``violent ’’’’ relaxation (mergers)relaxation (mergers)
NRNR--DM phase spaceDM phase spacedensity density
OBSERVATIONSOBSERVATIONS: dSphs:
34
3 3
km/s0.9 10 20
/ kpc
( )[ ]( )M
ρσ
−≤ ≤⊙
8 43 3
6.611 10keV (kpc km / s)
[ ] [ ]DM
DM
Mmρσ
= × ⊙D
52 2
03 32
2 42 2
0
( ) (
2
)
( )
[ ]
[ ]f
fy dyn t g
p y y dy
y
fπ
∞
∞= =⟨ ⟩
∫
∫D
11
3 14 4
1 3 34
2
0
km / s62.36 eV 2 (3)10 2.695 (eV)
/ kp ( )c
( )[ ] [ ]( )
d
y f
g
M gm
y dy
ξρσ ∞
− < ≤∫⊙D
Upper bound from abundanceLower bound from phase space density of dSphs
BOUNDS FOR THERMALTHERMAL RELICS
1) Relativistic at decoupling: Fermions or Bosons
m ~ 1 keV consistent with COREDCORED profiles for Td ≤ 100-300 GeV
for cusps gd ≥ 20002000 (beyond SM!!)
2) Non-relativistic at decoupling : Wimps M ~ 100 GeV, Td ~ 10 MeV
3)Wimp
ρσ =
153
10)cusp
ρσ
×
183
10)cored
ρσ
×CAN THIS RELAXATION BE POSSIBLE??CAN THIS RELAXATION BE POSSIBLE??
12
CONSTRAINTS: SUMMARY
ARBITRARY DECOUPLED DISTRIBUTION FUNCTION
ABUNDANCE UPPER BOUND
dSphs (DM dominated) PHASE SPACE LOWER BOUND
m ~ keV THERMAL RELICS decoupled when relativistic 100-300 GeV consistent with CORES
Wimps with m ~ 100 GeV, Td ~ 10 MeV PSD ~ 1018-1015 x (dSphs)!!
≲
Power spectrum of density perturbations during matter era: P(k) features
a cutoff determined by the free streaming wave vector: 2
( )fs eqfs
k tπ
λ=
4103
2
0
( )2 keV( ) 616 (kpc)
( )( ) ( )fs eq
d
y f y dyt
g m y f y dyλ
∞
∞= ∫∫
smaller forsmaller forlarger gd (colder speciescolder species )
f(y) larger for small y
13
Transfer function and power spectrum:
Boltzmann-Einstein eqn for ((DM+radDM+rad.).) density + gravitational perturbationdensity + gravitational perturbation Radiation-matter dominated cosmology (z > 2) All scales relevant for structure formation
What’s out?
Baryons: modify T(k) ~ few %. BAO on scales ~ 150 Mpc(acoustic horizon) (interested in MUCH smaller scales!!)
Anisotropic stresses
Why?
Study arbitraryarbitrary distribution functions, couplings, masses
Semi-analytical understanding of small scale properties
No tinkering with codes
1~ 0.01( )eqk Mpck −≫
14
Unperturbed decoupleddistribution
(DM) perturbation
Solution of Linearized Boltzmann eqn.
0 1( , , ) ( ) ( , , )f p x f p F p xη η= +
( , ) ( , )k kφ η ψ η=
No anisotropic stresses
Newtonian potential
Spatial curvature
( , , ) ( , , )01 1
( ) ( , )( , ; ) ( , ; ) ( , )
( , )( ) [ ]i
i
ik l p ik l pi
d f p d k kF k p F k p e p d e i k
dp d v p
ηµ η η µ η τη
φ τ µη η τ φ ττ τ
− −= − −∫
2 2 2( , , ) ( , ) ; ( , )
( )
pl p d v p v p
p m a
η
ηη η τ τ τ
η′′ = =
+∫ Free streaming distance
between ,η η ′
ɵ ɵ·µ = k pfrom 00-Einstein equation φ
15
keV DM: Three stages:• i) R.D;relativistic: relativistic free streaming• ii) R.D; no-relativistic: N.R. free streaming• iii) M.D: no-relativistic.
l η∝ln( )l η∝
In i) + ii) gravitational perturbations determined by radiation fluid
3
cos( ) sin( )3 3 3( ) 3 ( ) ;
( )3
( )[ ]i
z z z
z k z kz
φ φ η−
= − =
In iii) 00-Einstein at small scales Poisson’s eqn .
acoustic oscillations of radiation fluid
2
2
3( , ) ( , )
4 ( )eq eq
m
k ak k
k aφ η δ η
η= −gravitational potential DM density perturbation
16
Strategy
a) Initial conditions: adiabatic : 01
( )1( , ; ) ( ) ; 1
2( )i i i
df pF k p k p k
dpη φ η=
≪
b) Integrate BE during stage i) R.D.—rel. use the res ult as initial condition for stage ii) R.D.—non. rel. repeat for sta ge iii).
Stage i): R.D. relativistic: time dependence of from acoustic oscillations of radiation fluid ISW effect: enhancement of perturbation
φ
Z0 5 10 15 20
θo(z)
-0.50.00.51.01.52.02.5θo(0)
kη=
monopole of DM density perts. sound horizon
Wavelengths larger Wavelengths larger than sound horizon than sound horizon
amplified by ISWamplified by ISW .
17
Stage iii): Boltzmann-Poisson inhomogeneous differential integral equation for density perturbations,initial conditions + inhomogeneity determined by history during stages i) + ii). Fredholmsolution, leading order (in free streaming): Born approximation : WDM fluid description.
2 2
2 2
(2 3 ) 3[ , ]
2 (1 ) 2 (1 ) 4 (1 )
d a dI k
da a a da a a a a
δ δ δ κ δ η++ − + =+ + +ɶ
ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ
I + initial conditions determined by past history during stages i + ii) : R.D .
6
f s
k
kκ ≡
113
2
11.17(Mpc)
keV 2( )( )d
fs
gmk
y
−=
For Meszaro’s equation for WDM: can be solved EXACTLY :
WDM acoustic oscillations
0[ ],I k η =
1
sinh[ ]eq
aa
a u= ≡ −ɶ
0 CDMκ = =
18
u-0.8 -0.6 -0.4 -0.2 0.0
Q(κ,u)
-10-505
10 Q-MODESκ = 6.3 (fundamental)
κ = 10 κ = 13.6u-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1
P(κ,u)
-5-4-3-2-1012 P-MODES
κ = 13.6κ = 10κ = 6.3
``growing modes’’``decaying modes’’
Transfer function in Born approximation2
0
2 2 2
30( ; ) ( , ) ( ) [ ; ; ]
(1 )(4 ) ( ) eq
eq
ui
kT k Q u a u I k u du
k kκ κ κ
κ κ φ′ ′ ′ ′=
+ + ∫ ɶ
72 2
2
45 4 2 ( ) ln
4 3[ ]
E
eqCDM
eq
k k eT k
k k
γ −
=
19
Sterile neutrinos: a Sterile neutrinos: a NONNON--ThermalThermal DM candidateDM candidate
Non resonant Dodelson Widrow: production via active-sterile mixing
aν sνf ( )
1yy
e
β=+
150MeVdT ≃
Constrained from abundance for
1M p cfsλ ≃
independent of β
2~10β −
~keVsm
Main ingredient: distribution function at decouplin g of DM partiMain ingredient: distribution function at decouplin g of DM parti cle:cle:
20
Sterile neutrinos II: a different nonSterile neutrinos II: a different non --resonant resonant production mechanism : Boson decayproduction mechanism : Boson decay
= Scalar, gauge singlet,could be Higgs
= ``sterile neutrino’’ SU(2) singlet
Production: Scalar + Vector boson decay, all with t hermal abundaProduction: Scalar + Vector boson decay, all with t hermal abunda nce at 100 nce at 100 GeVGeV
τ0 2 4 6 8 10
n(y;τ)/Λ
012345678 y=0.2
y=0.5y=1
2~10−Λ
( )
M
T tτ =
Freeze-out
†1 2 [ ] h.cSM s s s s si Y H l Yν ν ν ν ν= + ∂ − − Φ + Φ +ɶL L L
sν 0
( )HH
H −=ɶ ( )al
f
ν= Φ
0~ ~100GeVH⟨Φ⟩ ⟨ ⟩ 8 512
2
~10 ~ keV ; sin ~10s
YY m
Yθ− −→ =
Φ
ν2 ν2
ν2ν2
ν1
f
Z0
W−
21
Frozen distributionFrozen distribution
5
20 1
2
( )
( ) 2
g y
f y
y
π= Λ y = p/Td
Strong enhancement of small momentumStrong enhancement of small momentum
Decoupling at ~ 100 GeV
Abundance + phase space density constraints:Abundance + phase space density constraints:
560 eV 1330 eVm≲ ≲ Consistent with model ``beyond SM’’
k (Mpc)-10 2 4 6 8 10
TB(k)
0.00.20.40.60.81.0 DW; gd = 30 m = 1,2 keVm= 2 keV
m=1 keVk (Mpc)-15 10 15 20 25 30
T(k)
0.00.20.40.60.81.0 BD; gd=100 ; m = 1,2 keV
m=1keVm=2keV
colder species
( )( )
( )cdm
T kT k
T k=
22
k (Mpc)-15 10 15 20
TISW(k)
-0.0050.0000.0050.0100.0150.020 BD; gd=100 ; m =1 keV
DW; gd=30 ; m =1 keV
Enhancement at small k by ISW: remnant from early RD-rel. stage i)
WDM acoustic oscillations at small scales:WDM acoustic oscillations at small scales:scale determined by ``fundamentalscale determined by ``fundamental ’’’’ QQ--mode.mode.
k (Mpc)-110 11 12 13 14 15
T(k)
0.00000.00050.00100.00150.00200.00250.0030 DW; gd = 30 ; m = 1 keV
k (Mpc)-130 31 32 33 34 35
T(k)
0.00060.00080.00100.00120.00140.00160.0018 BD; gd = 100 ; m = 1 keVcolder species
23
Roadmap: from micro to macroRoadmap: from micro to macro
1) Microphysics:Microphysics: Particle physics model, kinetics of production, decoupling
= decoupled distribution function, y=p/T0,d
2) ConstrainConstrain mass, couplings, T0,d from abundance + phase space density
Upper bound from abundance
( )f y
Lower bound from phase Space density of dSphs
1 24
0
100 eV6.5 eV
( )
dgm
g y f y dy∞≤ ≤∫D
52 2
032
4 2
0
( )
2( )
[ ]
[ ]
y f y dyg
y f y dyπ
∞
∞= ∫
∫D;
Thermal relics or non-thermal sterile neutrinos that decouple relativistically 3~ 2 10−×D m ~ keV
3) T(kT(k):): Solve BE-equation—cutoff on scale fsλ
Born approximation: easily implementable for generi c Born approximation: easily implementable for generi c distribution function: scales, distribution function: scales, WDMAOWDMAO’’ss
24
Executive SummaryExecutive Summary LCDMLCDM May May have problems at small scales: cores vs cusps, over
abundance of satellites, minivoids and massive halo es.
A keV particle May May provide a solution.
A few keV sterile neutrino is a suitable candidate: d irect detection unlikely but indirect: X-ray background may alreadyhint.
Microscopic physics (beyond standard model): product ion and freeze out distribution function constraints from abundances, phase space densities and power spectrum (free streaming scale). Several possible mechanisms: DW, bo son decay at EW scales…
B-E: obtain T(k) for generic distribution. Assess smal l scale properties, cutoff and WDMAO’s scale.
Systematic program: from micro to macro .
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