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A solution of A solution of the cusp problem the cusp problem in relaxed halos in relaxed halos E.V. Mikheeva in collaboration with A.G.Doroshkevich and V.N.Lukash Astro Space Centre of P.N.Lebedev Physics Institute Cusp problem: density

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A solution of A solution of the cusp problem the cusp problem in relaxed halos in relaxed halos

E.V. Mikheeva

in collaboration with

A.G.Doroshkevich and V.N.Lukash Astro Space Centre of P.N.Lebedev Physics Institute

SLIDE 2

Diemand et al. 2004

Cusp problem: density profiles

(r) r- at r0 < 1 - core 1 - cusp

bserv

servatio tions N-b

body sim

simula latio tions

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The problem of cusp can be solved in the framework of standard CDM

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Idea: to take into account the small scale part

f initial background perturbations that

transforms into random velocities of DM particles in the process of relaxation

Method: total entropy = initial (given) +

generated (gained during relaxation)

Idea and method

SLIDE 5

for DM particles with isotropic velocity distribution in relaxed halos: p = <v2> = nT = Fn5/3

ne-dimensional peculiar velocity

Entropy function

entropy function Entropy profiles related to density profiles in DM halos

SLIDE 6

p= C1 + C2 r2(1-)

+ Hydrostatic equilibrium:

r ) r ( GM dr dp 1

=
Power-law density profiles: (0, 2)

(r) r-,

=

r dr r ) r ( ) r ( M M

=1 is a critical value

0< < 1 - core (finite pressure in the centre) 1 < 2 - cusp (infinite pressure in the centre)

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cr cr

6 5 1

2 1

= = =

=
)

, ( , /

, 2 1 2 1

3 1 3 3 2 1

=

M C M C ) M ( F

2 1

re :

cusp:

5 3 2 6 5

2 1

, , ,

10 6 5 3 4 6 5

2 1

, , ,

=

Entropy mass function

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Linear field of density perturbations

Displacement Velocity Density perturbation Coordinates:

Euler Lagrange x r S r r r

r V & r r =

S div r =

)

r , t ( v

) x , ( v

(

)

dx dx , B q a d r d a dt

2 1 2 1 2 1 (

2 2 2 2 2

=

) ( ) ( ) r

1 = a

Initial entropy

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DM halo formation

(Zel’dovich approximation)

nditional perturbations

[transform (adiabatically at least) into microscopic particles’ motion in halo] local background – protohalo with linear scale R [collapsing into virialized halo by z0 with compression factor ~ 5(1+z0)]

, )] x ( S ) z ( g x [ ) z ( ) x , z ( r r r r r r

=

1 ) x ( S ) x ( S , ) z ( . : R | x x |

R R R

1 69 1 r r r r r r

+
<
)

x ( ) x ( ) x ( , ) x ( S ) x ( S ) x ( S

* R * R

r r r r r r r r r

= + = )] x ( S ) z ( g / x [ ) z ( H ) x , z ( V

r r r r r

1

2 1

= dk )] kR ( W )[ k ( P ) R ( S , S

* * * * 2 2 2

1 r

Mpc h S

1 2

11 = r

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) / M ( log n M ln M ln d ) M ( F ln d

=

1 3 1 2

0.67

0.57 0.5 0.3

7 10 n M

83 6 5 .

Critical value for

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Violent relaxation entropy (special analytical models)

Isotermal shere

(Fillmore & Goldreich 1984)

3 4 2 / g

M ~ F , r ~ M , r ~

1 7 1 . . ~

Collapse of ellipsoide

(Gurevich, Zybin 1988, 1995)

Generated entropy

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6 5 / â , M ~ ) M ( F

b b

<

M Ñ M Ñ ) M ( F

g b

+ =

6 5 / â , M ~ ) M ( F

g g

)

, ( F Fb 1

=
Analytically modeled halos

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g =4/3 (isothermal sphere) g =5/6 (NFW) r/rmax Burkert

Generated rotation curves

Fb~F Fb<<F

NFW

b=0.333 b=0.567 b=0.667 b=0.333 b=0.567 b=0.667

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