NOW 2014
Neutrino oscillation workshop
Conca Specchiulla, 07-14 September 2014
DAMPING THE FLAVOR PENDULUM
BY BREAKING HOMOGENEITY
(Alessandro MIRIZZI, Hamburg U.)
(Based on work in collaboration with G. Mangano & N. Saviano, 1403.1892)
DENSITY MATRIX FOR THE NEUTRINO ENSEMBLE
Diagonal elements related
to flavor content
 
  ee
 *
    e
 
 e
 e
 
 * 
 e
 






Off-diagonal elements
responsible for flavor
conversions
F  ( E , r )
F ( E, r )
In 2 scenario. Decompose density matrix over Pauli matrices to get the “polarization” (Bloch) vector P.
Survival probability Pee =1/2(1+Pz) . Pz = -1 -> Pee =0 ; Pz = 0 -> Pee =1/2 (flavor decoherence)
Alessandro Mirizzi
NOW 2014
Conca Specchiulla, 8 September 2014
EQUATIONS OF MOTION FOR A DENSE NEUTRINO GAS
(Sigl & Raffelt, 1992)
t  p, x  v p  x  p, x  p   p  p, x  i[ p, x ,  p, x ]
Liouville operator
Hamiltonian
t  p, x
Explicit time evolution
v p   x  p, x
Drift term due to space inhomogeneities
p   p  p, x
 p, x  vac  matt  
Force term acting on neutrinos (negligible)
7-dimensional problem. Never solved in its complete form. Symmetries have been used to reduce the
complexity of the problem.
Alessandro Mirizzi
NOW 2014
Conca Specchiulla, 8 September 2014
SPACE/TIME HOMOGENEITIY
Space Homogeneity:
t  p, x  v p  x  p, x  i[ p, x ,  p, x ]
Pure temporal evolution (Neutrinos in Early Universe)
Time Homogeneity:
t  p, x  v p  x  p, x  i[ p, x ,  p, x ]
Stationary space evolution (SN neutrinos)
However, small deviations from these symmetries have to be expected. Can these act as seed for new
instabilities?
Alessandro Mirizzi
NOW 2014
Conca Specchiulla, 8 September 2014
TOY MODEL: PENDULUM IN FLAVOR SPACE
 t P  [B  L  D]  P
 t P  [B  L  D]  P
1
2
 p  (1  P   ) Two-flavor polarization vectors
m 2

2E
Vacuum oscillation frequency
  2GF ne
Matter potential. Large HOMOGENEOUS  can be rotated away !
  2GF n
 potential
D  PP
Alessandro Mirizzi
NOW 2014
Conca Specchiulla, 8 September 2014
FLAVOR OSCILLATIONS AS SPIN PRECESSION
Slide from G. Raffelt
Alessandro Mirizzi
NOW 2014
Conca Specchiulla, 8 September 2014
HOMOGENEOUS PENDULUM
[Hannestad et al, astro-ph/0608695]
 e , e
((
q
  , 
For homogeneous , >>
Periodic  e e   x x pair conversions in IH
Alessandro Mirizzi
NOW 2014
Conca Specchiulla, 8 September 2014
NON-HOMOGENEOUS BACKGROUNDS
( x   t ) P( x, t )  [B   ( x, t ) L   ( x, t ) D( x, t )] P( x, t )
(1D spatial motion)
The partial differential equation can be transformed into a tower of ordinary differential equations
for the Fourier modes

Pk (t )   dxP( x, t )eikx

k (t )  FT[ ( x, t )]
k (t )  FT[ ( x, t )]
MONOCHROMATIC MATTER INHOMOGENEITY
   cos(k0 x)
  
 k   [ (k  k0 )   (k  k0 )]
FT
k  2 (k )
  const
kn  nk0
Pn  Pkn
Alessandro Mirizzi
NOW 2014
Conca Specchiulla, 8 September 2014
EOMs for the n=0,1 modes
Starting from homogeneous initial condition: only P0  0 , n ≥ 1 modes are excited in sequence
Alessandro Mirizzi
NOW 2014
Conca Specchiulla, 8 September 2014
DAMPING THE FLAVOR PENDULUM
1403.1892
 0
  107
  103
k0    2
pendulum oscillation frequency
A small seed of inhomogeneity is enough to produce a run-away from the stable pendulum behavior. The
average P0 tends towards the flavor equilibrium.
Alessandro Mirizzi
NOW 2014
Conca Specchiulla, 8 September 2014
TRAJECTORIES OF THE FLAVOR PENDULUM
 0
  103
((
Stable pendulum
Alessandro Mirizzi
NOW 2014
Unstable pendulum
Conca Specchiulla, 8 September 2014
EVOLUTION OF DIFFERENT FOURIER MODES
  103
n=1
n=2
n=3 n=4
After P1 starts rising, the higher Fourier modes are also rapidly excited in sequence reaching | Pn|~0.1
Alessandro Mirizzi
NOW 2014
Conca Specchiulla, 8 September 2014
DEPENDENCE ON PERTURBATION WAVE-NUMBER
k0  c
c  102
c  102
c 1
c 1
k0  k The flavor decoherence is approached earlier
c  102
Longer perturbation wave-length. The system needs more cycles to feel inhomogeneities
c  102
Perturbations are averaged during an oscillation cycle. The effect is shifted al later times
Alessandro Mirizzi
NOW 2014
Conca Specchiulla, 8 September 2014
NON-TRIVIAL SPACE BEHAVIOR

P( x, t )   dkPk (t )eikx

homogeneous solution
inhomogeneous
solutions
Alessandro Mirizzi
NOW 2014
Conca Specchiulla, 8 September 2014
DECLINING NEUTRINO DENSITY
  0 exp(t /  )
Quick decoherence. Similar to the case of
constant 
Lowering  the system requires more time to
decohere. Decoherece is not complete.
For a too fast  decline, the system has not
enough time to decohere
Alessandro Mirizzi
NOW 2014
Conca Specchiulla, 8 September 2014
CONCLUSIONS
We studied the effects of small inhomogeneities on the self-induced evolution of a dense neutrino
gas, by Fourier transforming the EOMs
We found that the neutrino flavor pendulum is not stable under the effects of small
inhomogeneities
However, a declining neutrino potential can suppress the effect of the inhomogeneities
The effect on the flavor evolutions of neutrinos in SN or in the Early Universe needs further
investigations with more realistic models.
Alessandro Mirizzi
NOW 2014
Conca Specchiulla, 8 September 2014