The case for a sterile neutrino as warm dark matter

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The case for a sterile neutrino as warm dark matterHey! CDM do you have a small scale problem? WDM: Sterile neutrinos come to the rescue!! Any sightings of (El) s lately? abundance phase space density small scales: cutoff in P(k)+features

Properties:

Constraints and bounds Production and decoupling: models Small scale transfer function A cosmic roadmap

1

Cold (CDM): small velocity dispersion: small structure forms first,

bottom-up hierarchical merger WIMPsHot (HDM) : large velocity dispersion: big structure forms first,

top-down, fragmentationlight neutrinos m eVWarm (WDM): ``in between sterile neutrinos ms ~ few keV

CDM

Concordance Model:

CMB+ LSS + N-body: DM is COLD and COLLISIONLESS``clumpy halo, large number of satellite galaxies

N-body:

(r) ~1/ r , 1 1.5 (NFW)

2

Observational Evidence:

Too few ``satellite galaxies dSphS: cores instead of cusps 34 ~ 1 kpc1/r (NFW)

100

Cusps vs Cores ??101

(M pc3 )

Ursa Minor Draco 102 LeoII LeoI Carina Sextans 1/r 103 101 r (kpc) 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 proles is striking, as is their shallow prole, and their similar central mass densities. Also shown is an r 1 density prole, predicted by many CDM numerical simulations (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)).

3

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.)

4

And the plot thickens..

Another over abundance problem--``mini voids: too many small haloes in CDM voids are too small compared to observations: Tikhonov, Klypin, Tikhonov, et. al.

WDM with m ~ keVCDM

may be a possible solution (Tikhonov et.al)

overpredicts the number of DM haloes with masses >

1010 M Sawala et.al.

WDM ``may offer a viable possibility [of resolution..] (Sawala et. al.) (Millenium II)

A keV Sterile Neutrino fits 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? V 2 1 ]2 A: Larger velocity dispersion, larger free-streaming length fs [ H om cuts-off power spectrum at small scales < f s5

Is there any evidence? May be..

Sterile neutrinos and the X-ray background: the evidence

2 1

W W 2 l W 1 2 l l 1

From Kusenko and Lowenstein

6

Constraints from sterile neutrino decays (see Kusenko)

10 12

20.010 10 10 8

15.0

pulsar kicks(allowed)

10.010 6

excluded

ms (keV)

ms(keV)

7.0 5.0

excluded region if 100% of dark matter

10 4 10 2

3.012

10

10

4

LSS Xray BBN CMB Lyman15

SN1987A Accelerator Decays Atmospheric Beam 1011

2.0 1.5

pulsar kicks (allowed)

10

6

1.01014

10

10

13

10

12

10

10

10

9

10

8

10

7

10

6

10

5

10

4

10

3

10

2

10

1

1011

1010

sin21

sin2

109

108

7

Microphysics:Collisionless DM= decoupled particles

>H T(t) Td3 (t ) = mn(t ) = mg f ( Pf , T ) = mg 2 3 y 2 f ( y )dy 3 (2 ) 2 a (t ) 0

d 3 Pf

V = 2

P f2 m2

=

(2 )3

d 3 P f P f23

m3

2

f [ a ( t ) Pf ]

( 2 )

d Pf

f [ a ( t ) Pf ]

Td = m a (t )

2

0 0

y 4 f ( y ) dy y 2 f ( y )dy

Constraints:1) ABUNDANCE: Upper bound

DM h2 = 0.105# of Relativistic d.o.f at decoupling

ma 2.695 g

2 gd (3)

(eV)9

0

y 2 f ( y)dy

2) Phase space density: LOWER BOUNDFor decoupled non-relativistic particles the phase space density is CONSERVED

D =

n (t ) p 2 f 3 2

=

g 2 2

[ [

0 0

y 2 f ( y ) dy ] y f ( y ) dy ]4

5 2 3 2

M DM m 4 = 6.611 10 8 [ D ][ ] 3 DM keV (kpc km / s) 3one dimensional velocity dispersion

NR-DM phase space density

OBSERVATIONS: dSphs: OBSERVATIONS

(km/ s)3 0.9 [104 3 ] 20 3 M / (kpc)Thm: phase space density diminishes in ``violent relaxation (mergers)10

[62.36 eV ][10D1 4

4

2 ( km / s ) 3 1 ]4 < m 2.695 g d2 (3) (eV) 3 M / ( kp c) 3 g y f ( y ) dy0

Lower bound from phase space density of dSphs

Upper bound from abundance

BOUNDS FOR THERMAL RELICS 1) Relativistic at decoupling: Fermions or Bosons m ~ 1 keV consistent with CORED profiles for Td for cusps gd 2000 (beyond SM!!) 2) Non-relativistic at decoupling : Wimps

100-300 GeV

M ~ 100 GeV, Td ~ 10 MeV

) = 3 Wimp

) 1015 3 cusp ) 1018 3 cored

CAN THIS RELAXATION BE POSSIBLE??11

CONSTRAINTS: SUMMARY ARBITRARY DECOUPLED DISTRIBUTION FUNCTION ABUNDANCE UPPER BOUND LOWER BOUND 100-300

dSphs (DM dominated) PHASE SPACE

m ~ keV THERMAL RELICS decoupled when relativistic 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 2 k fs (teq ) = a cutoff determined by the free streaming wave vector: fs

fs (teq ) = 616 (

2 ) gd

1 3

( keV ) m

0

y 4 f ( y ) dy y 2 f ( y ) dy (kpc) smaller for

larger gd (colder species) species f(y) larger for small y

0

12

Transfer function and power spectrum:

Boltzmann-Einstein eqn for (DM+rad.) density + gravitational perturbation Radiation-matter dominated cosmology (z > 2) All scales relevant for structure formation k keq ~ 0.01(Mpc)1

Whats out? Baryons: modify T(k) ~ few %. BAO on scales ~ 150 Mpc (acoustic horizon) (interested in MUCH smaller scales!!) Anisotropic stresses Why?

Study arbitrary distribution functions, couplings, masses Semi-analytical understanding of small scale properties No tinkering with codes13

f ( p, x, ) = f0 ( p) + F1 ( p, x, )Unperturbed decoupled distribution (DM) perturbation

No anisotropic stresses

( k , ) = ( k , )Newtonian potential Spatial curvature

Solution of Linearized Boltzmann eqn.

F1 (k , p ;) = F1 (k , p ;i )e

ik l ( p, ,i )

d f0 ( p) p( )i d eik l ( p,, )[d(k , ) i k (k , )] dp d v( p, )

l ( p,,) = d v( p, ) ; v( p, ) =

p p + m a ()2 2 2

Free streaming distance between

,

= k p

from 00-Einstein equation

14

keV DM: Three stages: i) R.D;relativistic: relativistic free streaming l ii) R.D; no-relativistic: N.R. free streaming l ln( ) iii) M.D: no-relativistic.In i) + ii) gravitational perturbations determined by radiation fluid

( z ) = 3 i ( k )[

(

z ) co s( z ) sin ( z ) 3 3 3 ] ; z = k z 3 ( ) 3

acoustic oscillations of radiation fluid

In iii) 00-Einstein at small scalesgravitational potential

Poissons eqn.

2 3 keq aeq m (k , ) = 2 (k , ) 4 k a( )

DM density perturbation15

Strategydf0 ( p) 1 ) ; ki 1 a) Initial conditions: adiabatic: F (k , p;i ) = i (k ) p ( 1 dp 2b) Integrate BE during stage i) R.D.rel. use the result as initial condition for stage ii) R.D.non. rel. repeat for stage iii).Stage i): R.D. relativistic: time dependence of from acoustic oscillations of radiation fluid ISW effect: enhancement of perturbation

02

Z

51

01

5

0

5.0-

0.2 5.1 0.1 5.0 0.0

)0(o )z(o

monopole of DM density perts.

5.2

sound horizon

Wavelengths larger than sound horizon amplified by ISW.

= k

16

Stage iii): Boltzmann-Poisson inhomogeneous differential integral equation for density perturbations,initial conditions + inhomogeneity determined by history during stages i) + ii). Fredholm solution, leading order (in free streaming): Born approximation: WDM fluid description.

d 2 (2 + 3a ) d 3 2 + + 2 = I [ k , ] 2 da 2a (1 + a ) da 2a (1 + a ) 4a (1 + a ) 6 k k fs

I + initial conditions determined by past history during stages i + ii) : R.D.

k fs =

11.17

gd 1 m ( )( )3 (Mpc)1 y 2 keV 2

a=

a 1 aeq sinh[u]

= 0 = CDMFor I [k , ] = 0 Meszaros equation for WDM: can be solved EXACTLY:

WDM acoustic oscillations17

``growing modes

Transfer function in Born approximation

T (k ; ) =

2 30 keq

k (1 + )(4 + 2 2

2

) (k ) i

0

ueq

Q ( , u )a (u ) I [k ; ; u ]du E 7 2

45 k 4 2ke TCDM ( k ) = ln [ 4 k 3 k eq

2 eq 2

]18

u

``decaying modes

1.0-

6.31 = 2.03.0-

3.6 =

4.0-

SEDOM-P5.06.0-

01 = 7.08.0-

9.05-

2 1 0