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Dark Energy and Dark matter in
A Superfluid Universe
Kerson Huang
Physics Department, MIT, Cambridge, USA
Institute of Advanced Studies NTU Singapore
Institute of Advanced Studies, NTU, Singapore
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Dr. Johann Faust (Heidelberg 1509)
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From Goethe’s Faust, Quoted by Boltzmann on Maxwell’ss equations
Quoted by Boltzmann on Maxwell
equations
War es ein Gott der diese Zeichen schrieb, Die mit geheimnisvoll verborg’nen Trieb
Die Kräfte der Natur um mich enthüllen
Und mir das Herz mit stiller Freud erfüllen?
Goethe
Was it a god who wrote these signs,
That have calmed yearnings of my soul
That have calmed yearnings of my soul,
And opened to me a secret of Nature?
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Physics in the 20th century
General relativity
• Expanding universe
• Dark energy
Quantum theory
• Superfluidity: Quantum phase coherence
• Dynamical vacuum: Quantum field theory
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Expanding universe
• The more distant the galaxy, the faster it moves away from us.
• Extrapolated backwards to “big bang”
l db k
d
“b b
”
a(t)
Hubble’s law: Velocity proportional to distance Hubble’s parameter:
H 
1 da
1

a dt 15  10 9 yrs
Accelerated expansion:
Dark energy
Edwin Hubble
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Superconductivity superfluidity
Superconductivity, superfluidity
• Quantum phase coherence over macroscopic distances
macroscopic distances
• Phenomenological description:
order parameter = complex scalar field
p
p
x   Fx  e ix
v s  ∇x
∇  
H Kamerlingh Onnes (1908) H. Kamerlingh
(1908)
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Dynamical vacuum ‐‐‐ QFT
• Lamb shift in hydrogen:
E(2S) – E(2P) E(2S) E(2P) = 1060 mhz
1060 mhz = 10
10‐6 eV
• Electron anomalous moment:
(g‐2)/2 = 10‐3
• Vacuum complex scalar field:
Higgs field in standard model
Others in grand unified theories
A vacuum complex scalar field makes the universe a superfluid
A vacuum complex scalar field makes the universe a superfluid. We investigate
• Emergence of vacuum scalar field in big bang
E
f
l fi ld i bi b
• Observable effects
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Scalar Field
Lagrangian density :
1
2
L      V  
2
• The vacuum field fluctuates about mean value We can treat it as a
mean value. We can treat it as a classical field by neglecting flucutuations.
Potential :
• But quantum effect of renormalization cannot be ignored.
V    2 2  4 4  6 6 
Equation of motion :
 2  V   0
• This makes V dependent on the This makes V dependent on the
length scale.
• Especially important for big bang, when scale changes rapidly.
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Renormalization
 In QFT there exist virtual processes.
 Spectrum must be cut off at high momentum .
  is the only scale in the theory.
Ignore
Cutoff  0
Hide
id
Effective cutoff  (Scale)
Renormalization: Renormalization:
Adjusting couplings so as to preserve theory, when scale changes.
Momentum spectrum
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Renormalization‐group (RG) trajectory:
Trajectory of V ( ,  ) in function space,
space as scale  changes.
changes
Fixed point: system scale invariant,  =.
UV trajectory: Asymptotic freedom
IR trajectory: Triviality
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The Creation
•
•
•
•
•
At the big bang .
At
the big bang   
There was no interaction.
Universe was at the Gaussian fixed point
(V  0, massless free field).
It emerges along some direction, on an RG trajectory.
The direction corresponds to a particular form of the potential V.
In the space of all possible theories
p
p
Outgoing trajectory ‐‐‐ Asymptotic freedom Ingoing trajectory ‐‐‐ Triviality (free field)
The only asymptotically free scalar potential is the Halpern‐Huang potential:
• Transcendental function (Kummer function)
• Exponential behavior at large fields
E
ti l b h i
tl
fi ld
• 4D generalization of 2D XY model, or sine‐Gordon theory.
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Cosmological equations
1
R    g   R  8  G T   ( E in s te in 's e q u a tio n )
2
 2  V   0
( S c a la r fie ld e q u a tio n )
R o b e rts o n -W a lk e r m e tric ((s p a tia l h o m o g e n e ity)
y)
G ra v ity s c a le = a (ra d iu s o f u n iv e rs e )
S c a la r fie ld s c a le =  (c u to ff m o n e n tu m )
S in c e th e re c a n b e o n ly o n e s c a le in th e u n iv e rs e ,
 =  /a
Dynamical feedback:
Dynamical
feedback:
Gravity provides cutoff to scalar field,
which generates gravitational field.
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Planck units:
Planck length 
Planck time 
Planck energy 
 4G  5. 73  10 −35 m
c3
 4G  1. 91  10 −43 s
c5
c 5  3. 44  10 18 GeV  5.5  108 Joule
4G
We shall put
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Initial‐value problem
For illlustration, first use real scalar field.
,
a  Ha
k
a V
H  2   2 
a
3 a
V
   3 H  

X  H2
k = curvature parameter = 0, +1,‐1
k 2  1 2

   V   0
a 32

Trace anomaly
Constraint equation
Constraint equation
X  0 is a constraint on initial values.
Equations guarantee X  0.
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?
The big bang
Time
Model starts here O(10‐43 s)
• Initial condition: Vacuum field already present.
• Universe could be created in hot “normal phase”, then make phase transition to “superfluid phase”.
Numerical solutions
Time‐averaged asymptotic behavior :
H  tp

a  exp t 1  p

Gives dark energy without “fine‐tuning” problem
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Comparison of power‐law prediction on galactic redshift with observations
‐‐> earlier epoch
d L = luminosity distance
Different exponents p only affects vertical displacement,
such as A and B.
Horizontal line corresponds to Hubble’s law.
Deviation indicates accelerated expansion (dark energy)
Deviation indicates accelerated expansion (dark energy).
Indication of a crossover transition between two different phase B ‐> A.
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Cosmic inflation
1. Matter creation
How to create enough matter for subsequent nucleogenesis
before universe gets too large
before universe gets too large.
2. Decoupling of matter scale and Planck scale
Matter interactions proceed at nuclear scale of 1 GeV.
p
But equations have built‐in Planck scale of 1018 GeV.
How do these scales decouple in the equations? Model with complete spatial homogeneity fail to answer these questions.
Generalization: Complex scalar field, homogeneous modulus, spatially varying phase.
New physics: Superfluidity in particular quantum vorticity
New physics: Superfluidity, in particular quantum vorticity. 18
Complex scalar field Vortex line
  Fei
v   (Superfluid velocity)
F 2 v 2 = Energy density of superflow

C
ds  v 

C
ds   2 n
2 rv  2 n
v
n
r
Like magnetic field from line current
Vortex has cutoff radius of order a(t).
Vortex line has energy per unit length. 19
• Replace vortex core by tube. • Scalar field remains uniform outside.
• Can still use RW metric,
• but space is multiply‐connected.
The “worm‐hole” cosmos
The vortex‐tube system represent emergent degrees of freedom.
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Nanowires
Vortex tubes in superfluid helium
made visible by adsorption of
metallic powder on surface
metallic powder on surface
(University of Fribourg expt.)
(a) Copper (b) gold Under electron microscope
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Vortex dynamics
Elementary structure is vortex ring
v
1
4R
ln
Self‐induced vortex motion
R
R0
The smaller the radius of curvature R,
the faster it moves normal to R.
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Vortex reconnection
Feynman’s conjecture
Signature: two cusps spring away from each other at very
away from each other at very high speed (due to small radii), creating two jets of energy.
Observed vortex reconnection in liquid helium‐‐ a millisecond event.
D. Lathrop, Physics Today, 3 June, 2010.
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Magnetic reconnections in sun’s corona
Responsible for solar flares.
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Change of topology due to reconnections
Microscopic rings eventually decay.
Quantum turbulence: Steady‐state “vortex tangle” Quantum
turbulence: Steady state “vortex tangle”
when there is steady supply of large vortex rings.
K.W. Schwarz, Phys. Rev. Lett. ,49,283 (1982).
, y
, ,
(
)
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Simulation of quantum turbulence Creation of vortex tangle in presence of “counterflow” (friction).
K W Schwarz Phys Rev B 38, 2398 (1988). K.W. Schwarz, Phys. Rev. B
38 2398 (1988)
Number of reconnections:
Number of reconnections:
0 3
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18 844
844
1128 14781 Fractal dimension = 1.6
D. Kivotides, C.F. Barenghi, and D.C. Samuels. Phys. Rev. Lett. 87, 155301 (2001).
Cosmology with quantum turbulence
a (t )  Radius of universe
F (t )  Modulus of scalar field
((t )  Vortex tube densityy
 (t )  Matter density
• Scalar field has uniform modulus F.
• Phase dynamics manifested via vortex tangle l.
Ph
d
i
if t d i
t t l l
• Matter created in vortex tangle (physically, via reconnections).
Equations for the time derivatives:
Equations for the time derivatives:
a (t ) from Einstein's equation with RW metric.


Source of ggravity:
y Ttot = TF  T

 T
F (t ) from field equation.
 (t ) from Vinen's equations in liquid helium.

 (t ) determined by energy-momentum conservation Ttot;
 = 0.
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Vinen’s equation in liquid helium
  vortex tube density (length per unit spatial volume)
  A 2  B3/2
Growth
Decay
In expanding universe this generalizes to
  3H   A 2  B3/2
• Proposed phenomenologically
Proposed phenomenologically by Vinen
by Vinen (1957).
(1957)
• Derived from vortex dynamics by Schwarz (1988).
• Verified by many experiments.
Put
Ev =a 3 0  (Total vortex energy)
Em =a 3 
((Total matter energy)
gy)
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4G  c    1
Cosmological equations:
Generalized:
Old:
dH  k − 2 dF 2  a ∂V − 1 E  E
m
v
3 ∂a a 3
dt
dt
a2
0 Ev
d 2 F  −3H
dF
3H
−
F − 1 ∂V
2
3
2 ∂F
dt
dt
a
k
a V
H  2  2 
a
3 a
V
  3H  

Constraint:
dE v
k 21

H    2  V   0
a 32

2
d
dE m
d
Essentially constant
3/2
 −E 2v  E v
• Rapid change
• Av. over t
• of order 1018
 0 dF 2
s 1 dt E v

Constraint:
H2 
k
a2
−
2
3
Ḟ 2  V 
1 0
a
3
Ev 
1
a3
Em
0
Decouples into two sets because s1 

t

Planck time scale
Nuclear time scale

Nuclear energy scale
Planck energy scale
18
 10 −18
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Decoupling:
• From the point of view of the cosmic expansion, the From the point of view of the cosmic expansion the
vortex‐matter system is essentially static.
• From the viewpoint vortex‐matter system, cosmic expansion is extremely fast, but its average effect is to give an "abnormally" large rate of matter production.
Inflation scenario:
Inflation scenario:
• Vortex tangle (quantum turbulence) grows and eventually decays.
• All the matter needed for galaxy formation was created in the tangle. • Inflation era = lifetime of quantum turbulence.
After decay of quantum turbulence, the standard hot big After
decay of quantum turbulence the standard hot big
bang theory takes over, but the universe remains a superfluid.
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Era of quantum turbulence
Cosmic inflation:
• Radius increases by factor 10 27
• in 10 ‐30 seconds.
• Matter created = 10 22 sun masses
• Eventually form galaxies outside of vortex cores.
• Large‐scale uniformity
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10‐26 s
Big Bi
bang
105 yrs
Time
Quantum turbulence
CMB
formed
Inflation Validity of this model
Standard hot big bang theory
Plus effects of superfluidity
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Legacies in the post‐inflation universe
Remnant vortex tubes with empty cores grow into cosmic voids in galactic distribution.
The large‐scale structure of the Universe from the CfA2 galaxy survey.
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Reconnection of huge vortex tubes
in the later universe will be rare but spectacular.
Gamma ray burst
• A few events per galaxy per million yrs
• Lasting ms to minutes
• Energy output in 1 s = Sun’s output in entire life
(billions of years)
(billions of years).
Jet of matter 27 light years long
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Dark matter Halo in “bullet cluster”
H
l i “b ll t l t ”
from gravitational lensing (blue)
Galaxyy
Dark matter halo
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“Hair” on black hole Artist’s conception:
g j
p
Rotating object in superfluid
induce vortex filaments.
Observed:
“Non‐thermal filaments" near center of Milky Way.
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Research team at IAS, NTU
Michael Good
H
Hwee‐Boon Low
B
L
Roh‐Suan Tung
Chi Xiong
KH
References:
K. Huang, H.‐B. Low, and R.‐S. Tung,
1. “Scalar field cosmology I: asymptotic freedom and the initial‐value problem”, arXiv:1106.5282 (2011).
arXiv:1106.5282 (2011).
2. “Scalar field cosmology II: superfluidity and quantum turbulence”, arXiv:1106.5283 (2011).
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