Kination Dominated Reheating and Cold Dark Matter Abundance

We consider the decay of a massive particle under the complete or partial domination of the kinetic energy density generated by a quintessential exponential model and we impose a number of observational constraints originating from nucleosynthesis, the present acceleration of the universe and the dark-energy-density parameter. We show that the presence of kination causes a prolonged period during which the temperature is frozen to a plateau value, much lower than the maximal temperature achieved during the process of reheating in the absence of kination. The decoupling of a cold dark matter particle during this period is analyzed, its relic density is calculated both numerically and semi-analytically and the results are compared with each other. Using plausible values (from the viewpoint of particle models) for the mass and the thermal averaged cross section times the velocity of the cold relic, we investigate scenaria of equilibrium or non-equilibrium production. In both cases, acceptable results for the cold dark matter abundance can be obtained, by constraining the initial energy density of the decaying particle, its decay width, its mass and the averaged number of the produced cold relics. The required plateau value of the temperature is, in most cases, lower than about 40 GeVComment: Final versio

Transcendental equations satisfied by the individual zeros of Riemann ζ\zeta, Dirichlet and modular LL-functions

We consider the non-trivial zeros of the Riemann $\zeta$-function and two classes of $L$-functions; Dirichlet $L$-functions and those based on level one modular forms. We show that there are an infinite number of zeros on the critical line in one-to-one correspondence with the zeros of the cosine function, and thus enumerated by an integer $n$. From this it follows that the ordinate of the $n$-th zero satisfies a transcendental equation that depends only on $n$. Under weak assumptions, we show that the number of solutions of this equation already saturates the counting formula on the entire critical strip. We compute numerical solutions of these transcendental equations and also its asymptotic limit of large ordinate. The starting point is an explicit formula, yielding an approximate solution for the ordinates of the zeros in terms of the Lambert $W$-function. Our approach is a novel and simple method, that takes into account $\arg L$, to numerically compute non-trivial zeros of $L$-functions. The method is surprisingly accurate, fast and easy to implement. Employing these numerical solutions, in particular for the $\zeta$-function, we verify that the leading order asymptotic expansion is accurate enough to numerically support Montgomery's and Odlyzko's pair correlation conjectures, and also to reconstruct the prime number counting function. Furthermore, the numerical solutions of the exact transcendental equation can determine the ordinates of the zeros to any desired accuracy. We also study in detail Dirichlet $L$-functions and the $L$-function for the modular form based on the Ramanujan $\tau$-function, which is closely related to the bosonic string partition function.Comment: Matches the version to appear in Communications in Number Theory and Physics, based on arXiv:1407.4358 [math.NT], arXiv:1309.7019 [math.NT], and arXiv:1307.8395 [math.NT

Observing different quantum trajectories in cavity QED

The experimental observation of quantum jumps is an example of single open quantum systems that, when monitored, evolve in terms of stochastic trajectories conditioned on measurements results. Here we present a proposal that allows the experimental observation of a much larger class of quantum trajectories in cavity QED systems. In particular, our scheme allows for the monitoring of engineered thermal baths that are crucial for recent proposals for probing entanglement decay and also for entanglement protection. The scheme relies on the interaction of a three-level atom and a cavity mode that interchangeably play the roles of system and probe. If the atom is detected the evolution of the cavity fields follows quantum trajectories and vice-versa.Comment: 5 pages, 2 figure