Dynamics of Fuzzy Dark Matter

Dongsu Bak

May 2020

Dongsu Bak FDM Dynamics

Schrodinger-Newton system

I Galaxy clusters are roughly consisting of dark matter halos (∼90%), intra-cluster gases (∼ 9%), and the remaining fraction ofcompact objects (mainly stars) made out of ordinary matters. In ourmodel, the dark matter halos mainly consists of solitonic coreswhere each solitonic core may be considered as a ground state ofSchrodinger-Newton system.

I These solitonic cores have typically masses of about 108M� with asize of one kpc scale. Indeed in the simulation of cosmologicalstructure formation with our ultra-light DM, granular structureshave been numerically found where each granule may be consideredas a solitonic core of the Schrodinger-Newton system.

Dongsu Bak FDM Dynamics

Schrodinger-Newton system

I The Schrodinger-Newton system is described by

i~∂tψ(x, t) = − ~2

2m∇2ψ(x, t) + mV (x, t)ψ(x, t)

∇2V (x, t) = 4πGMtot |ψ|2(x, t)

where Mtot is the total mass of the system and the wave function isnormalized by

∫d3x |ψ|2 = 1.

Dongsu Bak FDM Dynamics

Schrodinger-Newton system

I Redefine√Mtotψ → ψ.

I The Schrodinger-Newton system then becomes

i~∂tψ(x, t) = − ~2

2m∇2ψ(x, t) + mV (x, t)ψ(x, t)

∇2V (x, t) = 4πG |ψ|2(x, t)

where Mtot =∫d3x |ψ|2.

I Hence, this new |ψ|2 can be interpreted as the FDM mass density.

I ψ is describing “condensate” of FDM particles moving coherently.

Dongsu Bak FDM Dynamics

Solitonic ground state and dSphs

I Dwarf spheroidal galaxies (dSph) are relatively small galaxies withspherical shape.

I They typically have mass order of 108 M� and size of one kpc scale.

Dongsu Bak FDM Dynamics

More dSphs

I Note almost spherical shape of dwarf spheroidal galaxies.

I They are known to be dark-matter dominated. RDM > 99%!

Dongsu Bak FDM Dynamics

Ghost galaxy

Dongsu Bak FDM Dynamics

DM galaxies

Although at fainter luminosities of dwarf spheroidal galaxies, it is notuniversally agreed upon how to differentiate between a dwarf spheroidalgalaxy and a star cluster; however, many astronomers decide thisdepending on the object’s dynamics: if it seems to have more darkmatter, then it is likely that it is a dwarf spheroidal galaxy rather than afaint star cluster. In the current predominantly accepted Lambda colddark matter cosmological model, the presence of dark matter is oftencited as a reason to classify dwarf spheroidal galaxies as a different classof object from globular clusters, which show little to no signs of darkmatter. Because of the extremely large amounts of dark matter in dwarfspheroidal galaxies, they may deserve the title “most darkmatter-dominated galaxies.”

Dongsu Bak FDM Dynamics

Soliton cores in the Schrodinger-Newton system

I The time independent Schrodinger-Newton system then becomes

Enψ(x) = − ~2

2m∇2ψ(x) + mV (x)ψ(x)

∇2V (x) = 4πG |ψ|2(x)

with Mtot =∫d3x |ψ|2.

I Assuming the spherical symmetry (ψ(r)), the eigenvalue equationcan be solved numerically. The ground state corresponds to thesoliton core.

I All the excite states turn out to be unstable.

Dongsu Bak FDM Dynamics

Shape of the ground state

Dongsu Bak FDM Dynamics

Code space units

I In order to map into the code space, we introduce dimensionlessvariables by the rescaling

t → τc t =~3

m3

1

(GM)2t

x→ `c x =~2

m2

1

GMx

ψ → αψ ψ =m3

~3(GM)

32

(Mtot

M

) 12

ψ

V → αV V =m2

~2(4πGM)2 V

I This leads to a form suited for the numerical analysis

Dongsu Bak FDM Dynamics

Code space units

I This leads to a form suited for the numerical analysis

i ∂tψ(x, t) = −1

2∇2ψ(x, t) + V (x, t)ψ(x, t)

∇2V (x, t) = 4π |ψ|2(x, t)

I Note that the normalization of the wave function in the code spacebecomes Mtot =M

∫d3x |ψ|2.

I FDM particle mass: m = 10−22eV/c2.

I We shall take our reference mass scale as M = 4π × 107M�.

I This then fixes the time and length scales as

τc ∼ 2.3584× 107 yr`c ∼ 0.68000 kpc

I Hence the unit velocity in the code space corresponds to

vc = `c/τc ∼ 28.194 km/s

in the real space.

Dongsu Bak FDM Dynamics

Soliton ground state-numerics

I We take the soliton mass as Ms and M = 4π × 107M�.

I Then the half-mass radius r1/2 = MMs`c fn and the energy eigenvalue

E = −εn~/τc(Ms

M)2

with

τc ∼ 2.3584× 107 yr`c ∼ 0.68000 kpc

Dongsu Bak FDM Dynamics

Shape of the ground state

Dongsu Bak FDM Dynamics

Madelung - reformulation SN equation in terms of fluid

I Introduce new variables ρ and S such that

ψ(x, t) = ρ12 (x, t)e

i~S(x,t)

I Then the SN equation can be rewritten as

ρ+∇ · v = 0v + vi∇vi = −∇(V + Q)

where the velocity is defined by v = 1m∇S and the so-called

quantum pressure is given by

Q = − ~2

2m2ρ−

12∇2ρ

12

I The first is the continuity equation representing conservation ofFDM particle number. V is the Newtonian potential and Q is theextra pressure from quantum effect.

Dongsu Bak FDM Dynamics