Comments on the Evolution of Strongly Degenerate Neutrinos in the Early Universe

We reconsider the evolution of strongly degenerate neutrinos in the early universe. Our chief concern is the validity of the entropy conservation after the neutrino annihilation process has frozen out (so that the establishment of chemical equilibrium is not trivial). We argue that the entropy indeed conserves because elastic scattering keeps the neutrino and antineutrino distribution functions in the equilibrium form and the sum of their chemical potential keeps zero even after the neutrino annihilation freeze-out. We also simulate the evolution of the degenerate neutrino spectrum to support the argument. We conclude that the change in the neutrino degeneracy parameter when the relativistic degrees of freedom in the universe decreases is calculated using the entropy conservation and the lepton number conservation without worrying about at what temperature the neutrino annihilation process freezes out.Comment: 12 pages, 3 figure

Stress Tensor of the Hydrogen Molecular Ion

The electronic stress tensor of the hydrogen molecule ion H_2^+ is investigated for the ground state (sigma_g 1s) and the first excited state (sigma_u^* 1s) using their exact wave functions. A map of its largest eigenvalue and corresponding eigenvector is shown to be closely related to the nature of chemical bonding. For the ground state, we also show the spatial distribution of interaction energy density to describe in which part of the molecule stabilization and destabilization take place.Comment: 9 pages, 4 figure

Theoretical study of lithium clusters by electronic stress tensor

We study the electronic structure of small lithium clusters Li_n (n=2-8) using the electronic stress tensor. We find that the three eigenvalues of the electronic stress tensor of the Li clusters are negative and degenerate, just like the stress tensor of liquid. This leads us to propose that we may characterize a metallic bond in terms of the electronic stress tensor. Our proposal is that in addition to the negativity of the three eigenvalues of the electronic stress tensor, their degeneracy characterizes some aspects of the metallic nature of chemical bonding. To quantify the degree of degeneracy, we use the differential eigenvalues of the electronic stress tensor. By comparing the Li clusters and hydrocarbon molecules, we show that the sign of the largest eigenvalue and the differential eigenvalues could be useful indices to evaluate the metallicity or covalency of a chemical bond.Comment: 28 pages, 9 figure

Revisiting the Constraint on the Helium Abundance from CMB

We revisit the constraint on the primordial helium mass fraction Yp from observations of cosmic microwave background (CMB) alone. By minimizing chi square of recent CMB experiments over 6 other cosmological parameters, we obtained rather weak constraints as 0.17 < Yp < 0.52 at 1 sigma C.L. for a particular data set. We also study the future constraint on cosmological parameters when we take account of the prediction of the standard big bang nucleosynthesis (BBN) theory as a prior on the helium mass fraction where Yp can be fixed for a given energy density of baryon. We discuss the implications of the prediction of the standard BBN on the analysis of CMB.Comment: 15 pages, 5 figures, published versio