1. BBN bounds on active-sterile neutrino oscillations
F.L. Villante – Università di Ferrara and INFN
NOW 2006 – 10 September 2006
Based on work done in collaboration with A.D. Dolgov

2. The Physics of BBN GW / H = 1 Baryon density Bh2 ≈ 3.7 10 7 η
The abundances of 4He, D, 3He, 7Li produced by BBN depends on the following quantities:
Deuterium bottleneck
Baryon density
Bh2 ≈ η
Hubble expansion rate
GW = Weak rate (ne +n  p+e)
Weak interaction freeze-out
GW / H = 1
Non standard neutrinos may affect BBN via modifications of:
- Expansion rate of the universe (extra neutrinos, nm or nt degeneracy …)
Weak interaction rates (ne degeneracy, distorsion of ne spectrum …) ……

3. Active-sterile n-oscillations
Kinetic + chemical equilibrium
Ann./creat. neutrino reactions
nn  l l
Kinetic equilibrium
Elastic scattering
n l  n l
nmt ne
act - s oscillations before chemical decoupling  ns brought into equilibrium
 boost the Universe exp. rate
Kinetic + Chemical eq.
Kinetic eq.
e - s oscillations after chemical decoupling  e number density reduction
 affect n/p interconversion rate.
e - s oscillations after kinetic decoupling  e spectral distorsion
Energy density increase
ne depletion

4. n-oscillations in the early universe: the basic equations
Neutrino oscillations in the early universe are described by kinetic equations:
Coherence breaking terms due to:
- Annihilation nn  l l ni ni  nh nh
- Elastic scattering n l  n l
Generally dominant term in potential
U = neutrino mixing matrix
Among all the references:
Dolgov,Barbieri 1990

5. Neutrino oscillations: 1 active + 1 sterilex = m T y E
Analytic estimates are possible: H = n e u t r i o h a m l H x @ a = i s ( ) I c o l [ ] + = n e u t r i o a c : s
For dm2 > 10-6 eV2, sterile neutrino production occurs at “high” temperatures: T p r o d ' 1 M e V ( m 2 = ) 6
Neutrino interaction rates (gas) are large compared to hubble expansion rate (H)
 Quasi stationary approximation for off-diagonal components: a s = H ( ) i
In presence of “late” resonance (Haa – Hss =0) the behaviour of ras through resonace can be obtained by saddle-point integration.
Dolgov and Villante 2004

6. Non-resonance case Gs = ( Gact / 4 ) sin2(2 matter)
Energy density +
e depletion
ms > ma in the small mixing angle limit
The rate of ns production is:
Gs = ( Gact / 4 ) sin2(2 matter)
Log(dm2/eV2)
nmt- ns
This gives:
 - s mixing
(m2 / eV2) sin4(2) < 1.7 * 10-5 (N)2
e - s mixing
Log(dm2/eV2)
ne-ns
(m2 / eV2) sin4(2) < 3.2 * 10-5 (N)2
Dolgov 2001
e number density reduction.
Relevant effect for small mass differences.
Log(sin2(2Q))

7. Resonance Case (m2 / eV2) sin4(2) < 1.9 * 10-9 (N)2
Energy density +
e depletion
ms < ma in the small mixing angle limit
Our anlytic results coincide (in the proper limit) with the Landau-Zener description of resonance crossing:
nmt- ns
Log(dm2/eV2)
 - s mixing
(m2 / eV2) sin4(2) < 1.9 * 10-9 (N)2
e - s mixing
(m2 / eV2) sin4(2) < 5.9 * (N)2
Dolgov and FLV 2004
ne-ns
Our calculations show larger effects respect to previous estimates
(e.g. Enqvist et a. 1992, Shi et al. 1993)
Log(dm2/eV2)
ne depl. effect
included in numerical
and analyt. est.
Log(sin2(2Q))

8. Neutrino oscillations: 3 active + 1 sterile
Active neutrinos are now known to be mixed.
Their mixing should be taken into account together with nact - ns mixing: 4
(dm2)14 = variable 3
(dm2)23 = (dm2)atmo 2
(dm2)12 = (dm2)solar 1
ns mixing with one flavour eigenstate (e.g. h1 ≠ 0, h2,h3 = 0)
 three different dm2 (e1,e2,e3 ≠ 0), new resonances ….
ns mixing with one mass eigenstate (e.g. e1 ≠ 0, e2,e3 = 0)
 one dm2, oscillation into mixed flavours (h1,h2,h3 ≠ 0)
N.B. Mixing among active neutrinos cannot be rotated away, because BBN is flavour sensitive.

9. Neutrino oscillations: 3 active + 1 sterile
Problem partially simplificated considering that early universe does not distinguish nm and nt
Complete numerical calculation
“ne-ns mixing”
Analytic estimates: h3’
Log(dm2/eV2)
Analytic estimates: h1’ and h2’
Log(dm2/eV2)
All results in Dolgov and Villante 2004
See also:
Cirelli, Marandella, Strumia, Vissani 2004
Dolgov and Villante 2003
Log((2h1) 2)

10. Comparing with observations …
Main messagge of the previous slides:
BBN can rule-out large regions of nact-ns mass-mixing parameter space ... if
a robust bound DNn << 1 is obtained from observations
However:
Systematic errors and evolutionary effects makes difficult the interpretation of the observational results.
Focusing on 4He  Note that ΔYp  ΔNn
To obtain ΔNn < 1.0 at 2, we need ΔYp < or better
At present (ΔYp)syst > (or larger: PDG2006  (ΔYp)syst = 0.009)

11. In conclusion, …
We performed a complete analysis of active-sterile neutrino oscillations in early universe, providing:
Analytical understanding (resonant and non resonant oscillations);
Accurate and complete numerical description;
3 active + 1 sterile scenario.
At this point … we need better observational data.
N.B. An accurate verification of BBN is important not only for n-sterile
(e.g. CMB/LSS - BBN consistency).