Cosmological evolution of warm dark matter fluctuations II: Solution from small to large scales and keV sterile neutrinos

We solve the cosmological evolution of warm dark matter (WDM) density fluctuations with the Volterra integral equations of paper I. In the absence of neutrinos, the anisotropic stress vanishes and the Volterra equations reduce to a single integral equation. We solve numerically this equation both for DM fermions decoupling at equilibrium and DM sterile neutrinos decoupling out of equilibrium. We give the exact analytic solution for the density fluctuations and gravitational potential at zero wavenumber. We compute the density contrast as a function of the scale factor a for a wide range of wavenumbers k. At fixed a, the density contrast grows with k for k k_c, (k_c ~ 1.6/Mpc). The density contrast depends on k and a mainly through the product k a exhibiting a self-similar behavior. Our numerical density contrast for small k gently approaches our analytic solution for k = 0. For fixed k < 1/(60 kpc), the density contrast generically grows with a while for k > 1/(60 kpc) it exhibits oscillations since the RD era which become stronger as k grows. We compute the transfer function of the density contrast for thermal fermions and for sterile neutrinos in: a) the Dodelson-Widrow (DW) model and b) in a model with sterile neutrinos produced by a scalar particle decay. The transfer function grows with k for small k and then decreases after reaching a maximum at k = k_c reflecting the time evolution of the density contrast. The integral kernels in the Volterra equations are nonlocal in time and their falloff determine the memory of the past evolution since decoupling. This falloff is faster when DM decouples at equilibrium than when it decouples out of equilibrium. Although neutrinos and photons can be neglected in the MD era, they contribute in the MD era through their memory from the RD era.Comment: 27 pages, 6 figures. To appear in Phys Rev

A method for solve integrable A2A_2 spin chains combining different representations

A non homogeneous spin chain in the representations $\{3 \}$ and $\{3^*\}$ of $A_2$ is analyzed. We find that the naive nested Bethe ansatz is not applicable to this case. A method inspired in the nested Bethe ansatz, that can be applied to more general cases, is developed for that chain. The solution for the eigenvalues of the trace of the monodromy matrix is given as two coupled Bethe equations different from that for a homogeneous chain. A conjecture about the form of the solutions for more general chains is presented. PACS: 75.10.Jm, 05.50+q 02.20 SvComment: PlainTeX, harvmac, 13 pages, 3 figures, to appear in Phys. Rev.

Hardness of Graph Pricing through Generalized Max-Dicut

The Graph Pricing problem is among the fundamental problems whose approximability is not well-understood. While there is a simple combinatorial 1/4-approximation algorithm, the best hardness result remains at 1/2 assuming the Unique Games Conjecture (UGC). We show that it is NP-hard to approximate within a factor better than 1/4 under the UGC, so that the simple combinatorial algorithm might be the best possible. We also prove that for any $\epsilon > 0$, there exists $\delta > 0$ such that the integrality gap of $n^{\delta}$-rounds of the Sherali-Adams hierarchy of linear programming for Graph Pricing is at most 1/2 + $\epsilon$. This work is based on the effort to view the Graph Pricing problem as a Constraint Satisfaction Problem (CSP) simpler than the standard and complicated formulation. We propose the problem called Generalized Max-Dicut($T$), which has a domain size $T + 1$ for every $T \geq 1$. Generalized Max-Dicut(1) is well-known Max-Dicut. There is an approximation-preserving reduction from Generalized Max-Dicut on directed acyclic graphs (DAGs) to Graph Pricing, and both our results are achieved through this reduction. Besides its connection to Graph Pricing, the hardness of Generalized Max-Dicut is interesting in its own right since in most arity two CSPs studied in the literature, SDP-based algorithms perform better than LP-based or combinatorial algorithms --- for this arity two CSP, a simple combinatorial algorithm does the best.Comment: 28 page

Multi-String Solutions by Soliton Methods in De Sitter Spacetime

{\bf Exact} solutions of the string equations of motion and constraints are {\bf systematically} constructed in de Sitter spacetime using the dressing method of soliton theory. The string dynamics in de Sitter spacetime is integrable due to the associated linear system. We start from an exact string solution $q_{(0)}$ and the associated solution of the linear system $\Psi^{(0)} (\lambda)$, and we construct a new solution $\Psi(\lambda)$ differing from $\Psi^{(0)}(\lambda)$ by a rational matrix in $\lambda$ with at least four poles $\lambda_{0},1/\lambda_{0},\lambda_{0}^*,1/\lambda_{0}^*$. The periodi- city condition for closed strings restrict $\lambda _{0}$ to discrete values expressed in terms of Pythagorean numbers. Here we explicitly construct solu- tions depending on $(2+1)$-spacetime coordinates, two arbitrary complex numbers (the 'polarization vector') and two integers $(n,m)$ which determine the string windings in the space. The solutions are depicted in the hyperboloid coor- dinates $q$ and in comoving coordinates with the cosmic time $T$. Despite of the fact that we have a single world sheet, our solutions describe {\bf multi- ple}(here five) different and independent strings; the world sheet time $\tau$ turns to be a multivalued function of $T$.(This has no analogue in flat space- time).One string is stable (its proper size tends to a constant for $T\to\infty$, and its comoving size contracts); the other strings are unstable (their proper sizes blow up for $T\to\infty$, while their comoving sizes tend to cons- tants). These solutions (even the stable strings) do not oscillate in time. The interpretation of these solutions and their dynamics in terms of the sinh- Gordon model is particularly enlighting.Comment: 25 pages, latex. LPTHE 93-44. Figures available from the auhors under reques

An exact solution of the inelastic Boltzmann equation for the Couette flow with uniform heat flux

In the steady Couette flow of a granular gas the sign of the heat flux gradient is governed by the competition between viscous heating and inelastic cooling. We show from the Boltzmann equation for inelastic Maxwell particles that a special class of states exists where the viscous heating and the inelastic cooling exactly compensate each other at every point, resulting in a uniform heat flux. In this state the (reduced) shear rate is enslaved to the coefficient of restitution $\alpha$, so that the only free parameter is the (reduced) thermal gradient $\epsilon$. It turns out that the reduced moments of order $k$ are polynomials of degree $k-2$ in $\epsilon$, with coefficients that are nonlinear functions of $\alpha$. In particular, the rheological properties ($k=2$) are independent of $\epsilon$ and coincide exactly with those of the simple shear flow. The heat flux ($k=3$) is linear in the thermal gradient (generalized Fourier's law), but with an effective thermal conductivity differing from the Navier--Stokes one. In addition, a heat flux component parallel to the flow velocity and normal to the thermal gradient exists. The theoretical predictions are validated by comparison with direct Monte Carlo simulations for the same model.Comment: 16 pages, 4 figures,1 table; v2: minor change

Scattering of Plane Waves in Self-Dual Yang-Mills Theory

We consider the classical self-dual Yang-Mills equation in 3+1-dimensional Minkowski space. We have found an exact solution, which describes scattering of $n$ plane waves. In order to write the solution in a compact form, it is convenient to introduce a scattering operator $\hat{T}$. It acts in the direct product of three linear spaces: 1) universal enveloping of $su(N)$ Lie algebra, 2) $n$-dimensional vector space and 3) space of functions defined on the unit interval.Comment: 16 pages, LaTeX fil

Spinor techniques for massive fermions with arbitrary polarization

We present a new variant of the spinor techniques for calculating the amplitudes of processes involving massive fermions with arbitrary polarization. It is relatively simple and leads to basic spinor products. Our procedure is not more complex than CALCUL spinor techniques for massless fermions. We obtain spinor Chisholm identities for massive fermions. As an illustration, expressions are given for the amplitudes of electron-positron annihilation into fermions-pairs for several polarizations.Comment: 14 pages, 2 figure

Quantum spin chain with "soliton non-preserving" boundary conditions

We consider the case of an integrable quantum spin chain with "soliton non-peserving" boundary conditions. This is the first time that such boundary conditions have been considered in the spin chain framework. We construct the transfer matrix of the model, we study its symmetry and we find explicit expressions for its eigenvalues. Moreover, we derive a new set of Bethe ansatz equations by means of the analytical Bethe ansatz method.Comment: 12 pages, LaTeX, two appendices added, minor correction

Excitations and S-matrix for su(3) spin chain combining 3{3} and ${3^{*}}

The associated Hamiltonian for a su(3) spin chain combining ${3}$ and ${3^{*}}$ representations is calculated. The ansatz equations for this chain are obtained and solved in the thermodynamic limit, and the ground state and excitations are described. Thus, relations between the number of roots and the number of holes in each level have been found . The excited states are characterized by means of these quantum numbers. Finally, the exact S matrix for a state with two holes is found.Comment: 17 pages, plaintex, harvmac (to be published in J. of Phys. A