A Superfluid Universe Kerson Huang Physics Department, MIT, - PowerPoint PPT Presentation

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 1

Dr. Johann Faust (Heidelberg 1509) 2

From Goethe’s Faust , Quoted by Boltzmann on Maxwell’s equations Quoted by Boltzmann on Maxwell s 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? 3

Physics in the 20 th century General relativity • Expanding universe • Dark energy Quantum theory • Superfluidity: Quantum phase coherence • Dynamical vacuum: Quantum field theory 4

Expanding universe • The more distant the galaxy, the faster it moves away from us. • Extrapolated backwards to “big bang” l d b k d “b b ” a(t) Hubble’s law: Velocity proportional to distance 1 1 da da 1 1   Hubble’s parameter: H  9 a dt 15 10 yrs Accelerated expansion: Dark energy Edwin Hubble 5

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 Onnes (1908) 6

Dynamical vacuum ‐‐‐ QFT • Lamb shift in hydrogen: E(2S) – E(2P) = 1060 mhz = 10 ‐ 6 eV E(2S) E(2P) 1060 mhz 10 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 7

Scalar Field Lagrangian density : • The vacuum field fluctuates about 1 1     mean value. We can treat it as a mean value We can treat it as a      2 L V classical field by neglecting 2 flucutuations. • But quantum effect of Potential : renormalization cannot be ignored.   V             2 4 6 2 4 6 • This makes V dependent on the • This makes V dependent on the length scale. Equation of motion : • Especially important for big bang,      when scale changes rapidly. 2 V 0 8

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 9

Renormalization ‐ group (RG) trajectory: V   V     Trajectory of Trajectory of ( , ( ) in function space as scale ) in function space, as scale changes. changes   Fixed point: system scale invariant, = . UV trajectory: Asymptotic freedom IR trajectory: Triviality 10

The Creation • At the big bang       At the big bang . • There was no interaction. V  • Universe was at the Gaussian fixed point ( 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 t l fi ld • 4D generalization of 2D XY model, or sine ‐ Gordon theory. 11

Cosmological equations 1    R g R 8 G T ( E instein's equation)    2      2 V 0 ( S calar field equation) R obertson-W alker m etric (spatial hom ogeneity) ( p g y) G ravity scale = (radius of universe) a  S calar field scale = (cutoff m onentum ) S ince there can be only one scale in the universe,   = /a Dynamical feedback: Dynamical feedback: Gravity provides cutoff to scalar field, which generates gravitational field. 12

Planck units:  3 4  G  5.73  10 − 35 m Planck length  c 3  c 5 4  G  1.91  10 − 43 s Planck time   c 5   8 4  G  3.44  10 18 GeV 5.5 10 Joule Planck energy  We shall put 13

Initial ‐ value problem For illlustration, first use real scalar field. ,   a Ha  k a V       2 H k = curvature parameter = 0, +1, ‐ 1  2 a 3 a  V         Trace anomaly 3 H    k 2 1           2 2 Constraint equation Constraint equation     X X H H V V 0 0   a 3 2  X 0 is a constraint on initial values.  X  Equations guarantee 0. 14

? Time The big bang 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 :   p H t     1 p a exp t Gives dark energy without “fine ‐ tuning” problem 16

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. 17

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 10 18 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    Fe  i    v (Superfluid velocity) Vortex line 2 2 F v = Energy density of superflow           ฀ ฀ ds v ds 2 n C C C C    2 rv 2 n n  v r r Like magnetic field from line current Vortex has cutoff radius of order a(t). Vortex line has energy per unit length. 19

The “worm ‐ hole” cosmos • Replace vortex core by tube. • Scalar field remains uniform outside. The vortex ‐ tube system • Can still use RW metric, represent emergent • but space is multiply ‐ connected. degrees of freedom. 20

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 21

Vortex dynamics Elementary structure is vortex ring Self ‐ induced vortex motion v  1 R 4  R ln R 0 The smaller the radius of curvature R, the faster it moves normal to R. 22

Vortex reconnection Signature: two cusps spring away from each other at very away from each other at very Feynman’s conjecture 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. 23

Magnetic reconnections in sun’s corona Responsible for solar flares. 24

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 , , ( ) 25

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 18 18 844 844 1128 14781 Fractal dimension = 1.6 D. Kivotides, C.F. Barenghi, and D.C. Samuels. Phys. Rev. Lett. 87 , 155301 (2001).