Tzihong Chiueh
National Taiwan University
in collaboration with
H.Y. Schive, T. Broadhurst, U.H, Zhang, H.H. Chang, T.P. Woo , S.R. Chen , K.H. Leong
Woo & Chiueh, ApJ, (2009), Schive et al., Nature Phys.
(2014), Schive, et al, PRL (2014), Schive et al. ApJ (2016),
Chen et al. MNRAS (2017), De Martino, et al, PRL (2017),
Zhang & Chiueh, PRD (2017 a, b), Schive & Chiueh
MNRAS (2017), Leong et at., MNRAS (2019)⚫ Axion DM as Bose-Einstein Condensate

𝑚𝑎 ~ 10−22 eV
𝑛0 ~1025 /cc, 𝑇𝑐𝑟𝑖𝑡 /(1+z)~10 keVAxion DM (a<𝑎𝑒𝑞 )Free-particle wave DM (a>𝑎𝑒𝑞 )Schroedinger Eq. for DM dynamics (ψDM)⚫ Small-scale astronomical problems⚫ ψDM halo: dynamical simulation⚫ Lyα forest Problem (z~2.5-4.5)---Extreme Axion⚫ Additional observation tests, e.g., pulsar timing,
21cm tomography Small-scale (~ kpc) problems:(a) Missing satellite dwarf galaxies around Milky Way> 100 satelites for CDM(b) Flat-cores in dwarf spheroidal galaxies as opposed to
CDM bound halo: ρ (r) = 𝜌0 / r(1+ c 𝑟 2 )Using Jeans Eq. : d(ρ𝜎 2 )/d r = - ρGM(r)/𝑟 2

⚫ Dark matter described by a single wave function ψ

ΔxΔv = ħ/ma ~(10km/s)kpc : astronomical scale⚫ Expect that the uncertainty principle respected in
Milky Way: Δv ~200km/s
Δx ~0.1kpc, small scale!
⚫ However, G & Δv demands large scale ~100kpc!
Solution has coexisting small & large scales!
⚫

 What if G and ħ/ma balance and share the same scale?
Δx = (ħ/ma)2/G
for a particular massBlack hole analogy, G balances c:=G/c2

Dynamical simulations:Reproduce CDM galaxy distribution > 30kpc scale
But small scales ….

1 Mpc

Small-scale linear power suppressed at
Compton (Jeans) length
𝑘𝑐𝑜𝑚𝑝 𝑘_𝑐
Linear matter
spectrum at z=0 r s 𝑜𝑚𝑝⚫ Wavelength enters horizon kc/a ~ H
when particles becomes realkc/a ~ 𝑚𝑎 𝑐 2⚫ 𝑘𝑐𝑜𝑚𝑝 = 𝑘𝐽𝑒𝑎𝑛𝑠 at 𝑎𝑒𝑞

Hu et al., PRL, (2000)
Woo & Chiueh, ApJ (2009)
Zhang & Chiueh, PRD (2017a)

CDM simulation of a major halo
several 100s of dwarf galaxies,
but only several 10s are detected

Jeans wavenumber kJ2 ~ aG/cs2 ~ a1/2
with cs2 replaced by v2 ~ (ħ𝑘𝐽 /ma)2

A soliton in a galaxy haloz = 0 dwarf galaxies

CDM

Schive at al.
2014, PRLRedshift evolution

χ2 ~ 1
(CDM)

Fornax dwarf

Schive et al. Nature Phys (2014)• Given observed stars densityand velocity profiles, determine
𝑚𝑎 ~10-22 eV using Ms rs~ ħ/ma
• Confirm 𝑚𝑎 with 4 more dSphs,
Chen et al. (2016)

can be explained byequilibration ofsoliton ”temperature”& halo “temperature”

1/3

𝑀𝑠 ~ 𝑀ℎΔvsol = (ħ/ma)/rsol ~ (Eh/Mh)1/2 = Δvh(1+z)1/2

Compare ψDM and CDM halos
ψDM
CDM
ψDM
CDM

R/RHL =

1

• a stronger soliton with the addition of stars• velocity dispersion increases by 2 inward from R=RHL

2

3

Vanlenti et al. A&A (2018), Portail, et al. MNRAS (2017)

1kpc
Gas measurements by Sofue et al.
2009

2Rsol=250pc

H absorption (Lyα forest) power spectra
quasarH-poor

BOSS data andsimulation curves

quasarH-richt• Free ψDM of 𝑚𝑎 = 10−22 eV has too little small-scale power!
(Armengaud et al. 2017; Irsic et al. 2017)• Extreme axion DM gives better agreement with the data than CDM𝛿𝜃0 = 90𝑜 − θ0

few 1010 𝑀𝑠𝑢𝑛

k2

CDMδθ0 ~ 90𝑜θZhang & Chiueh(b), PRD (2017); Schive &Chiueh, MNRAS (2017)

•
•
•
•
•

Parametric instability, δx with time
Delayed oscillation avoids Hubble friction
Growth factor ~ k2 since long waves enter horizon lateFirst galaxies ~ 1010 𝑀𝑠𝑢𝑛
Best-fit δθ0 ~2.5𝑜 from Lyman-α forest data (Leong et at. MNRAS 2019) P(0)(t)= ρ(0) cos (2m t+2S)
 Poisson gauge, ds2 = - (1+2Φ) dt2 + (1-2Ψ) dx2

Khmelnitsky & Rubako
JCAP (2014)~ O( v4/c4)(1) mass oscillation, (2) proportional to local mass density

O(v/c) smaller=δν/ν~ ρ(0) cos (2mt+2S) , ν~ 1 yr -1 for m ~ 10-22 eVΨ(1) ~ 10 -13 at the soliton (ρ ~ 1-2 TeV/cc)~ 10 -16 at 1 kpc inner halo (ρ ~ 1GeV/cc)Ψ

De Martino, etal, PRL, 2017

SKA
limit

 Review on basic wave dark matter (ψDM) which solves

small-scale problems with matter spectrum and solitonCentral soliton + granules in any halo:
soliton mass & halo mass relation

Adding stars enhances soliton; cusp in star velocity
dispersion due to the central mass lump

Extreme axion ψDM has enhanced small scale power about
1010 𝑀𝑠𝑢𝑛 , modifying the universe landscape in z > 4
Pulsar timing is within reach to test ψDM 21cm tomography probed by SKA