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GERSHTEIN-ZELDOVICH BOUND AND COSMOLOGICAL RESTRICTIONS ON NEUTRINO MASS A.D. Dolgov
20-24 December, 2004

COSMOLOGY AND ASTROPHYSICS AT HIGH ENERGIES (dedicated to 90th anniversary of Ya. B. Zeldovich)

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Presently telescopes allow to weight neutrinos more accurately then direct experiments. Tritium decay experiments (Troitsk, Meinz): me < (2 - 3) eV Neutrino oscillation data: m2 ar = (5.4 - 9.5) · 10-5 eV2 sol m2 = (1.2 - 4.8) · 10-3 eV2 atm Hence: m > 5 · 10-2 eV 2 -decay (Heidelberg-Moscow): m > 0.1eV(?) - Ma jorana mass. Astronomy is sensitive now to m (a few)в0.1 eV. based on combined data on LSS and CMBR.
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Derivation of GZ bound. At T > 1 MeV thermal equilibrium between , e± , : n = (3/8)n for each left-handed neutrino flavor, e, µ, . Assumed vanishing charge asymmetry, i.e. n = n . Ї At T me photons are heated by e+e-annihilation, while are decoupled. It leads to the present-day neutrino number density: n + n = (3/11)n = 112/cm3 Ї Hence: ma < 94 eVmh2 a For m < 0.3 and h = 0.7: a ma < 14 eV

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Detailed analysis of LSS observational data together with measured spectrum of density perturbations from the angular fluctuations of CMBR allows to strengthen the bound down to
a ma < (0.4 - 1) eV

The necessary input for this result are GZ calculations of the present day number density of relic neutrinos.

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Historical remarks GZ-bound derived in: REST MASS OF MUONIC NEUTRINO AND COSMOLOGY By S.S. Gershtein, Ya.B. Zeldovich, Pisma Zh. Eksp. Teor. Fiz. v.4, 174, 1966 English translation: JETP Lett. v.4, 120, 1966. Usually quoted: R. Cowsik, J. McClelland AN UPPER LIMIT ON THE NEUTRINO REST MASS Phys. Rev. Lett. v.29, 669, 1972. Effects of photon heating are not accounted for and 2 spin states of neutrinos are assumed to be equally abundant. HEAVY NEUTRINOS: Lee-Weinberg equation, 1971, in Zeldovich, Okun, Pikel'ner, 1964; mass limit LW (1971), VDZ (1971).

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IS IT POSSIBLE TO AVOID GZ BOUND? 1. Thermal equilibrium in the early universe - OK in the standard model. Can be broken if T was never larger than MeV. 2. Nonvanishing lepton asymmetry results in stronger bound. 3. Extra heating of photon plasma below MeV. Care should be taken of BBN and spectrum of CMBR. 4. Neutrino stability at cosmological time scale: tu 1010 years. Difficult to make smaller life-time. New interactions are necessary. 5. New interactions which could lead to strong annihilation of . Very light Ї or massless boson is needed. 6. Right-handed neutrinos - result in stronger bound.

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FORMATION OF LARGE SCALE STRUCTURE Theoretical input: 1. Spectrum of primordial fluctuations. Usually assumed flat, Harrison-Zeldovich type. Predicted by inflation. Confirmed by CMBR at large scales 10M pc. 2. Properties of dark matter. Usually non-interacting CDM+Lambda. 3. Analytical calculations at linear regime, when / 1. Standard physics: GR and hydrodynamics. Is not distorted at large scales accessible to CMBR. 4. Numerical simulations at nonlinear regime, when / 1. Necessary at smaller scales, 10 Mpc. PROBLEMS: M. Kamionkowsky, yesterday talk.

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Neutrino role in LSS formation Very large free-streaming length: lF S 250 Mpc (eV/m ) In neutrino dominated universe the structures below this scale are erased. If neutrinos are sub-dominant they lead to: 1. Suppression of power at small scales. 2. Delayed formation of structures. Ly- clouds at high red-shifts, z 1 are sensitive to light massive neutrinos. An ad tion sp ence of bound hoc modification of perturbaectrum could mimic the presmassive neutrinos and no mass would be obtained

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Growth rate f d log /d log a, in four flat C+HDM models at a = 1 (solid) and 0.1 (dashed) with different neutrino masses: m = 1.2, 2.3, 4.6, 6.9 eV (from top down), corresponding to = 0.05, 0.1, 0.2, 0.3. At small k , the CDM density field grows with the same rate ( a) as in the standard CDM model. At large k , the growth rate is suppressed. (From Ma, 1999.)

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P(k) (arbitrary norm.)

knr
1

high mh2

knr
0.1

m = 0 eV m = 1 eV
0.01 0.01

low mh2
0.1

k (h Mpc ­1)

Effect of 1 eV neutrino on BRG power spectrum compared with expected precision of the SDSS (1 error boxes Upper: m = 1, h = 0.5, bh2 = 0.0125, n = 1 Lower: the same but for an m = 0.2, h = 0.65. (From Hu et al, 1997.) The effect is most pronounced at small scales (and high z , previous figure).

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Massive neutrinos and CMBR Angular spectrum of CMBR does not feel directly presence of light neutrinos - they are practically relativistic at recombination. CMBR allows to measure perturbation spectrum and fit it to spectrum derived from LSS in the common range of wave lengths. Scales available to CMBR: 300 Mpc < d < 10/h Mpc (low accuracy at large d because of cosmic variance).

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LSS: Spectrum at small scales (sensitive to neutrinos) is strongly distorted at non-linear regime. Ly- at z = 2 - 4, scales about 1 Mpc; evolution is non-negligible but manageable. SDSS: 3600 QSO at z > 2.2, high accuracy. BEST LIMIT from combined CMBR, LSS, and Ly-: ma < 0.42 eV

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Various recent limits on the neutrino mass from cosmology and the data sets used in deriving them: 1: WMAP data 2: Other CMB data 3: 2dF data 4: Constraint on 8 5: SDSS data 6: Constraint on H0 7: Constraint from Lyman- forest. WMAP: m <0.69 eV, from 1-4,6, 7 Hannestad: m <1.01 eV, from 1-3,6 Allen et al: m < 0.56+0..3 eV, from 1-4,6 -0 26 SDSS: m <1.8 eV, from 1,5 Barger et al. m < 0.75 eV, from 1-3,5,6 Crotty et al: m < 1.0 eV from 1-3,5 (6) Seljak et al. m <0.42 eV 1,2,4-7 Fogli et al. m <0.5 eV, from 1-7 Hannestad: m <0.65 eV, from 1,5-7.

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CONCLUSION 1. All bounds quoted above are based on the calculations of the present day number density of by GZ. 2. More restrictive results, i.e. m < 1-0.42 eV are based on analysis of LSS at relatively small scales plus CMBR to shift degeneracy of parameters. 3. With almost equal neutrino masses (from oscillation experiments) one can conclude: m < 0.33 - 0.14 eV. 4. Direct experiment pro ject, KATRIN, may approach this accuracy. Do we need it? 5. Possible ways to violate the limit: a) From particle physics: new particles or stronger -interactions; b) From cosmology: unusual spectrum of perturbations at small scales; w = w(t). WHAT ELSE?
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