1

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 .