Superfluid Dark Matter

Lasha Berezhiani

Department of Physics, Princeton University

May 16, 2017 It faces some challenges at galactic scales.

Introduction

ΛCDM works remarkably well on large scales. Introduction

ΛCDM works remarkably well on large scales.

It faces some challenges at galactic scales. 4 vc Mb = GNa0

MOND fails on cosmological scales.

MOND: Milgrom ’83

( aN aN  a0 a = √ aNa0 aN  a0

a0 ∼ H0 MOND fails on cosmological scales.

MOND: Milgrom ’83

( aN aN  a0 a = √ aNa0 aN  a0

a0 ∼ H0

4 vc Mb = GNa0 MOND: Milgrom ’83

( aN aN  a0 a = √ aNa0 aN  a0

a0 ∼ H0

4 vc Mb = GNa0

MOND fails on cosmological scales. LB, Justin Khoury ’15

Unification can be achieved through superfluidity.

2M2 3/2 pl h 2i LMOND = − (∇~ ϕ) − ϕρb 3a0

vs

(∇~ ϕ)2 L = P(X ) with X ≡ µ +ϕ ˙ − superfluid 2m

P depends on the properties of the superfluid.

Unified Framework 2M2 3/2 pl h 2i LMOND = − (∇~ ϕ) − ϕρb 3a0

vs

(∇~ ϕ)2 L = P(X ) with X ≡ µ +ϕ ˙ − superfluid 2m

P depends on the properties of the superfluid.

Unified Framework LB, Justin Khoury ’15

Unification can be achieved through superfluidity. vs

(∇~ ϕ)2 L = P(X ) with X ≡ µ +ϕ ˙ − superfluid 2m

P depends on the properties of the superfluid.

Unified Framework LB, Justin Khoury ’15

Unification can be achieved through superfluidity.

2M2 3/2 pl h 2i LMOND = − (∇~ ϕ) − ϕρb 3a0 Unified Framework LB, Justin Khoury ’15

Unification can be achieved through superfluidity.

2M2 3/2 pl h 2i LMOND = − (∇~ ϕ) − ϕρb 3a0

vs

(∇~ ϕ)2 L = P(X ) with X ≡ µ +ϕ ˙ − superfluid 2m

P depends on the properties of the superfluid. Condensation takes place when 1 λdB = √ > ` mT At this point the particles become indistinguishable and the classical statistics breaks down.

The critical temperature for this phase transition is estimated to be

n2/3 T = c m

Bose-Einstein Condensate Bose-Einstein Condensate

Condensation takes place when 1 λdB = √ > ` mT At this point the particles become indistinguishable and the classical statistics breaks down.

The critical temperature for this phase transition is estimated to be

n2/3 T = c m At finite temperature we have a mixture of 2 fluids:

BEC + Normal fluid

BEC exhibits superfluidity if it is a condensate of interacting (repulsive) particles.

Bose-Einstein Condensate

At T < Tc

N  T 3/2 k=0 = 1 − Ntot Tc BEC exhibits superfluidity if it is a condensate of interacting (repulsive) particles.

Bose-Einstein Condensate

At T < Tc

N  T 3/2 k=0 = 1 − Ntot Tc

At finite temperature we have a mixture of 2 fluids:

BEC + Normal fluid Bose-Einstein Condensate

At T < Tc

N  T 3/2 k=0 = 1 − Ntot Tc

At finite temperature we have a mixture of 2 fluids:

BEC + Normal fluid

BEC exhibits superfluidity if it is a condensate of interacting (repulsive) particles. For the contact 2-point interactions with coupling constant λ λn c2 = s m3

Superfluid

The sound waves of the superfluid (so called ’phonons’) propagate with the dispersion relation

k4 ω2 ' c2k2 + k s 4m2 Superfluid

The sound waves of the superfluid (so called ’phonons’) propagate with the dispersion relation

k4 ω2 ' c2k2 + k s 4m2

For the contact 2-point interactions with coupling constant λ λn c2 = s m3 Consider an impurity moving through the fluid with velocity v. The dissipation happens through the process

Impurity → Impurity + phonon

Landau’s criterion for superfluidity

v < cs

Superfluidity

For small momenta

ωk = cs k The dissipation happens through the process

Impurity → Impurity + phonon

Landau’s criterion for superfluidity

v < cs

Superfluidity

For small momenta

ωk = cs k

Consider an impurity moving through the fluid with velocity v. Landau’s criterion for superfluidity

v < cs

Superfluidity

For small momenta

ωk = cs k

Consider an impurity moving through the fluid with velocity v. The dissipation happens through the process

Impurity → Impurity + phonon Superfluidity

For small momenta

ωk = cs k

Consider an impurity moving through the fluid with velocity v. The dissipation happens through the process

Impurity → Impurity + phonon

Landau’s criterion for superfluidity

v < cs e.g.

λ L = −|∂Ψ|2 − m2|Ψ|2 − |Ψ|4 2

We are looking for a state with large number of particles in k = 0 state.

This corresponds to the homogeneous classical solution.

Superfluid

It is a degenerate state of interacting bosons. We are looking for a state with large number of particles in k = 0 state.

This corresponds to the homogeneous classical solution.

Superfluid

It is a degenerate state of interacting bosons.

e.g.

λ L = −|∂Ψ|2 − m2|Ψ|2 − |Ψ|4 2 This corresponds to the homogeneous classical solution.

Superfluid

It is a degenerate state of interacting bosons.

e.g.

λ L = −|∂Ψ|2 − m2|Ψ|2 − |Ψ|4 2

We are looking for a state with large number of particles in k = 0 state. Superfluid

It is a degenerate state of interacting bosons.

e.g.

λ L = −|∂Ψ|2 − m2|Ψ|2 − |Ψ|4 2

We are looking for a state with large number of particles in k = 0 state.

This corresponds to the homogeneous classical solution. This is the solution as long as

µ˜2 − m2 − λv 2 = 0

v is fixed by the concentration

n = 2µ˜v 2

Superfluid

Let’s try

Ψ = veiµ˜t v is fixed by the concentration

n = 2µ˜v 2

Superfluid

Let’s try

Ψ = veiµ˜t

This is the solution as long as

µ˜2 − m2 − λv 2 = 0 Superfluid

Let’s try

Ψ = veiµ˜t

This is the solution as long as

µ˜2 − m2 − λv 2 = 0

v is fixed by the concentration

n = 2µ˜v 2 Easy to show that

mρ = 2m and mϕ = 0

We can integrate out ρ to get

(∇~ ϕ)2 L ∝ X 2 with X ≡ µ +ϕ ˙ − 2m For generic potential we get a generic function of X .

Fluctuations

Ψ = (v + ρ)ei(˜µt+ϕ) We can integrate out ρ to get

(∇~ ϕ)2 L ∝ X 2 with X ≡ µ +ϕ ˙ − 2m For generic potential we get a generic function of X .

Fluctuations

Ψ = (v + ρ)ei(˜µt+ϕ)

Easy to show that

mρ = 2m and mϕ = 0 For generic potential we get a generic function of X .

Fluctuations

Ψ = (v + ρ)ei(˜µt+ϕ)

Easy to show that

mρ = 2m and mϕ = 0

We can integrate out ρ to get

(∇~ ϕ)2 L ∝ X 2 with X ≡ µ +ϕ ˙ − 2m Fluctuations

Ψ = (v + ρ)ei(˜µt+ϕ)

Easy to show that

mρ = 2m and mϕ = 0

We can integrate out ρ to get

(∇~ ϕ)2 L ∝ X 2 with X ≡ µ +ϕ ˙ − 2m For generic potential we get a generic function of X . Unified Framework

3/2 h 2i LMOND ∼ (∇~ ϕ)

vs

(∇~ ϕ)2 L = P(X ) with X ≡ µ +ϕ ˙ − superfluid 2m

P depends on the properties of the superfluid. Unitary Fermi Gas is described by

5/2 LUFG ∼ X

Fractional Powers?

3/2 Lsuperfluid ∼ X Fractional Powers?

3/2 Lsuperfluid ∼ X

Unitary Fermi Gas is described by

5/2 LUFG ∼ X 2 2 2 6 L = −|∂µΦ| − m |Φ| − λ|Φ|

Upon BEC, it exhibits superfluidity.

EFT for phonons is

L ∼ X 3/2

X 3/2? Upon BEC, it exhibits superfluidity.

EFT for phonons is

L ∼ X 3/2

X 3/2?

2 2 2 6 L = −|∂µΦ| − m |Φ| − λ|Φ| X 3/2?

2 2 2 6 L = −|∂µΦ| − m |Φ| − λ|Φ|

Upon BEC, it exhibits superfluidity.

EFT for phonons is

L ∼ X 3/2 Thermalization requirement

−1 Γ >∼ tdyn

BEC of Dark Matter

For degeneracy

1 m1/3 λdB ∼ > ` ≡ mv ρ BEC of Dark Matter

For degeneracy

1 m1/3 λdB ∼ > ` ≡ mv ρ

Thermalization requirement

−1 Γ >∼ tdyn Therefore, the region above some critical density will transform into the superfluid.

Density Profile

At virialization we expect NFW profile

4

3

2

1

0.5 1.0 1.5 2.0

As density increases, it becomes easier to satisfy the condensation criteria. Density Profile

At virialization we expect NFW profile

4

3

2

1

0.5 1.0 1.5 2.0

As density increases, it becomes easier to satisfy the condensation criteria.

Therefore, the region above some critical density will transform into the superfluid.  σ/m 1/4 Galaxies: Rcore > R/3 ⇒ m < 1.5 eV 0.5 cm2/g  σ/m 1/4 Clusters: Rcore < R/10 ⇒ m > 0.75 eV 0.5 cm2/g

Density Profile

NFW

Superfluid  σ/m 1/4 Clusters: Rcore < R/10 ⇒ m > 0.75 eV 0.5 cm2/g

Density Profile

NFW

Superfluid

 σ/m 1/4 Galaxies: Rcore > R/3 ⇒ m < 1.5 eV 0.5 cm2/g Density Profile

NFW

Superfluid

 σ/m 1/4 Galaxies: Rcore > R/3 ⇒ m < 1.5 eV 0.5 cm2/g  σ/m 1/4 Clusters: Rcore < R/10 ⇒ m > 0.75 eV 0.5 cm2/g DM particles are generated when H ∼ m, corresponding to p Tbaryons ∼ mMpl ∼ 50 TeV

As soon as it is generated, DM becomes superfluid with

 T   T  ' 10−28 vs ' 10−2 Tc cosmo Tc MW

Cosmology:

DM with eV mass must be produced out of equilibrium. For instance, through axion-like vacuum displacement mechanism. As soon as it is generated, DM becomes superfluid with

 T   T  ' 10−28 vs ' 10−2 Tc cosmo Tc MW

Cosmology:

DM with eV mass must be produced out of equilibrium. For instance, through axion-like vacuum displacement mechanism.

DM particles are generated when H ∼ m, corresponding to p Tbaryons ∼ mMpl ∼ 50 TeV  T  vs ' 10−2 Tc MW

Cosmology:

DM with eV mass must be produced out of equilibrium. For instance, through axion-like vacuum displacement mechanism.

DM particles are generated when H ∼ m, corresponding to p Tbaryons ∼ mMpl ∼ 50 TeV

As soon as it is generated, DM becomes superfluid with

 T  ' 10−28 Tc cosmo Cosmology:

DM with eV mass must be produced out of equilibrium. For instance, through axion-like vacuum displacement mechanism.

DM particles are generated when H ∼ m, corresponding to p Tbaryons ∼ mMpl ∼ 50 TeV

As soon as it is generated, DM becomes superfluid with

 T   T  ' 10−28 vs ' 10−2 Tc cosmo Tc MW We conjecture that the superfluid has MOND-like action

P(X ) = Λm3/2X p|X |

To mediate MOND force, phonons must couple to baryons Λ Lint = −α ϕρb Mpl

Superfluid Properties:

Low energy degrees of freedom are phonons. In general

(∇~ ϕ)2 L = P(X ), with X ≡ µ +ϕ ˙ − 2m To mediate MOND force, phonons must couple to baryons Λ Lint = −α ϕρb Mpl

Superfluid Properties:

Low energy degrees of freedom are phonons. In general

(∇~ ϕ)2 L = P(X ), with X ≡ µ +ϕ ˙ − 2m We conjecture that the superfluid has MOND-like action

P(X ) = Λm3/2X p|X | Superfluid Properties:

Low energy degrees of freedom are phonons. In general

(∇~ ϕ)2 L = P(X ), with X ≡ µ +ϕ ˙ − 2m We conjecture that the superfluid has MOND-like action

P(X ) = Λm3/2X p|X |

To mediate MOND force, phonons must couple to baryons Λ Lint = −α ϕρb Mpl The equation of state is

ρ3 P = 12Λ2m6

q 2µ The sound speed is given by cs = m

Superfluid Properties:

The pressure of the condensate is

P(µ) = Λ(mµ)3/2 Superfluid Properties:

The pressure of the condensate is

P(µ) = Λ(mµ)3/2

The equation of state is

ρ3 P = 12Λ2m6

q 2µ The sound speed is given by cs = m For fiducial values m = eV and Λ = 0.04 meV, for 12 MDM = 10 M , we have

R ' 120 kpc

The superfluid scenario provides the simple resolution to the cusp-core and "too-big-to-fail" problems.

Core Profile:

Assuming hydrostatic equilibrium π r ρ(r) ' ρ cos1/2 ( ); r ≤ R 0 2 R Core Profile:

Assuming hydrostatic equilibrium π r ρ(r) ' ρ cos1/2 ( ); r ≤ R 0 2 R For fiducial values m = eV and Λ = 0.04 meV, for 12 MDM = 10 M , we have

R ' 120 kpc

The superfluid scenario provides the simple resolution to the cusp-core and "too-big-to-fail" problems. In order to have w  1 at matter radiation equality, we need

Λ  0.1 eV

This is five orders of magnitude larger than the fiducial value Λ = 0.04 meV assumed in galaxies.

Recall that  T   T  ' 10−28, ' 10−2 Tc cosmo Tc MW

Cosmology

P ρ2 w = = ρ 12Λ2m6 Recall that  T   T  ' 10−28, ' 10−2 Tc cosmo Tc MW

Cosmology

P ρ2 w = = ρ 12Λ2m6

In order to have w  1 at matter radiation equality, we need

Λ  0.1 eV

This is five orders of magnitude larger than the fiducial value Λ = 0.04 meV assumed in galaxies. Cosmology

P ρ2 w = = ρ 12Λ2m6

In order to have w  1 at matter radiation equality, we need

Λ  0.1 eV

This is five orders of magnitude larger than the fiducial value Λ = 0.04 meV assumed in galaxies.

Recall that  T   T  ' 10−28, ' 10−2 Tc cosmo Tc MW a -8 a 2. × 10 MOND1.02

] -8 1.00

2 1.5 × 10

s

/ 0.98 -8 [cm 1. × 10 0.96

a 5. × 10-9 0.94

0.2 0.4 r 0.6 0.8 1.0 0.2 0.4 r 0.6 0.8 1.0 R R

Phonon-Mediated Force Between Baryons

In the limit of large phonon gradients s 3 2 α Λ GNMb(r) aφ(r) ' 2 MPl r Phonon-Mediated Force Between Baryons

In the limit of large phonon gradients s 3 2 α Λ GNMb(r) aφ(r) ' 2 MPl r

a -8 a 2. × 10 MOND1.02

] -8 1.00

2 1.5 × 10

s

/ 0.98 -8 [cm 1. × 10 0.96

a 5. × 10-9 0.94

0.2 0.4 r 0.6 0.8 1.0 0.2 0.4 r 0.6 0.8 1.0 R R superfluid MOND a0 < a0

Closer Look at Rotation Curves LB, Benoit Famaey, Justin Khoury, to appear

9 IC 2574: Mb = 3.4 × 10 M .

a[cm/s2]

2.× 10 -9

1.5× 10 -9 aphonons a 1.× 10 -9 N-total aMOND

5.× 10 -10

r[kpc] 2 4 6 8 10 12 Closer Look at Rotation Curves LB, Benoit Famaey, Justin Khoury, to appear

9 IC 2574: Mb = 3.4 × 10 M .

a[cm/s2]

2.× 10 -9

1.5× 10 -9 aphonons a 1.× 10 -9 N-total aMOND

5.× 10 -10

r[kpc] 2 4 6 8 10 12

superfluid MOND a0 < a0 Closer Look at Rotation Curves

LB, Benoit Famaey, Justin Khoury, to appear

superfluid MOND a0 = 0.73a0

a[cm/s2]

2.× 10 -9

1.5× 10 -9 aphonons+aN-total -9 1.× 10 aMOND

5.× 10 -10

r[kpc] 2 4 6 8 10 12 Rotation Curves

LB, Benoit Famaey, Justin Khoury, to appear

v[km/s] v[km/s] 80 350

60 300

40 250

20 200

r[kpc] r[kpc] 0 2 4 6 8 10 20 30 40 50 60

9 11 Mb = 3.4 × 10 M Mb = 1.6 × 10 M

Mtotal = 10Mb Mtotal = 10Mb

Rb = 12kpc Rb = 60kpc

Rcore = 49kpc Rcore = 65kpc

Rvir = 49kpc Rvir = 126kpc Validity of EFT and the Solar System

MOND regime corresponds to large phonon gradients

ϕ02  µ 2m

In terms of superfluid velocity vs = |∇~ ϕ|/m, this becomes

vs  cs

Thus, violating Landau’s criterion.

This is not surprising, since we are interested in generating large coherent phonon background.

High derivative terms need to be small. This is so as long as −1 r  Λs . Validity of EFT and the Solar System

We should also verify that

 ρ 1/3 v  v ≡ s c m4 Leading to

 1/2  −5/6 Mb  m −1/2 Λ r  0.2 11 kpc 10 M eV meV Applied to the Sun

 m −1/3  Λ −5/9 r  250 AU , eV meV In our case we have DM, unlike TeVeS, and hence we may even get away with γ = −1.

Gravitational Lensing

We might need to couple baryons to

2α   g˜µν ' gµν − ϕ γgµν + (1 + γ)uµuν Mpl

In TeVeS γ = 1. Gravitational Lensing

We might need to couple baryons to

2α   g˜µν ' gµν − ϕ γgµν + (1 + γ)uµuν Mpl

In TeVeS γ = 1.

In our case we have DM, unlike TeVeS, and hence we may even get away with γ = −1. If the infall velocity is sub-sonic, the superfluid components should pass through each other with negligible friction, however the normal components should be slowed down.

Merging Clusters:

In general, the outcome depends on relative fraction of superfluid vs normal components in the clusters. Merging Clusters:

In general, the outcome depends on relative fraction of superfluid vs normal components in the clusters.

If the infall velocity is sub-sonic, the superfluid components should pass through each other with negligible friction, however the normal components should be slowed down. Results: DM halos are superfluids at finite temperature, with interesting phenomenology.

Next Step: Finding a viable microscopic theory.

Summary

Assumptions:

I Bose-Einstein condensation

3 I Superfluid with equation of state p ∼ ρ

I Phonon-mediated force

I Temperature dependance of parameters Next Step: Finding a viable microscopic theory.

Summary

Assumptions:

I Bose-Einstein condensation

3 I Superfluid with equation of state p ∼ ρ

I Phonon-mediated force

I Temperature dependance of parameters

Results: DM halos are superfluids at finite temperature, with interesting phenomenology. Summary

Assumptions:

I Bose-Einstein condensation

3 I Superfluid with equation of state p ∼ ρ

I Phonon-mediated force

I Temperature dependance of parameters

Results: DM halos are superfluids at finite temperature, with interesting phenomenology.

Next Step: Finding a viable microscopic theory.