Higher-order supersymmetric quantum mechanics

We review the higher-order supersymmetric quantum mechanics (H-SUSY QM), which involves differential intertwining operators of order greater than one. The iterations of first-order SUSY transformations are used to derive in a simple way the higher-order case. The second order technique is addressed directly, and through this approach unexpected possibilities for designing spectra are uncovered. The formalism is applied to the harmonic oscillator: the corresponding H-SUSY partner Hamiltonians are ruled by polynomial Heisenberg algebras which allow a straight construction of the coherent states.Comment: 42 pages, 12 eps figure

Unraveling the origin of black holes from effective spin measurements with LIGO-Virgo

We investigate how to use information on the effective spin parameter of binary black hole mergers from the LIGO-Virgo gravitational wave detections to discriminate the origin of the merging black holes. We calculate the expected probability distribution function for the effective spin parameter for primordial black holes. Using LIGO-Virgo observations, we then calculate odds ratios for different models for the distribution of black holes' spin magnitude and alignment. We evaluate the posterior probability density for a possible mixture of astrophysical and primordial black holes as emerging from current data, and calculate the number of future merger events needed to discriminate different spin and alignment models at a given level of statistical significance.Comment: 22 pages, 10 figure

Comment on "keV Neutrino Dark Matter in a Fast Expanding Universe" by Biswas et al

Biswas et al. found that the thermal relic density of a dark matter particle freezing out while the universe's energy density is dominated by a non-standard extra component $\phi$, whose energy density redshifts faster than radiation, can be greatly suppressed. Here we show that this result, which contradicts extensive previous literature, is incorrect: the mistake lies with the assumption that the (decoupled) extra component $\phi$ contributes to the entropic degrees of freedom relevant for dark matter freeze out. If this were the case, a completely different approach would be needed to calculate the dark matter relic abundance, with dramatically different results

Discrete second order constrained Lagrangian systems: first results

We briefly review the notion of second order constrained (continuous) system (SOCS) and then propose a discrete time counterpart of it, which we naturally call discrete second order constrained system (DSOCS). To illustrate and test numerically our model, we construct certain integrators that simulate the evolution of two mechanical systems: a particle moving in the plane with prescribed signed curvature, and the inertia wheel pendulum with a Lyapunov constraint. In addition, we prove a local existence and uniqueness result for trajectories of DSOCSs. As a first comparison of the underlying geometric structures, we study the symplectic behavior of both SOCSs and DSOCSs.Comment: 17 pages, 6 figure

When the Universe Expands Too Fast: Relentless Dark Matter

We consider a modification to the standard cosmological history consisting of introducing a new species $\phi$ whose energy density red-shifts with the scale factor $a$ like $\rho_\phi \propto a^{-(4+n)}$. For $n>0$, such a red-shift is faster than radiation, hence the new species dominates the energy budget of the universe at early times while it is completely negligible at late times. If equality with the radiation energy density is achieved at low enough temperatures, dark matter can be produced as a thermal relic during the new cosmological phase. Dark matter freeze-out then occurs at higher temperatures compared to the standard case, implying that reproducing the observed abundance requires significantly larger annihilation rates. Here, we point out a completely new phenomenon, which we refer to as $\textit{relentless}$ dark matter: for large enough $n$, unlike the standard case where annihilation ends shortly after the departure from thermal equilibrium, dark matter particles keep annihilating long after leaving chemical equilibrium, with a significant depletion of the final relic abundance. Relentless annihilation occurs for $n \geq 2$ and $n \geq 4$ for s-wave and p-wave annihilation, respectively, and it thus occurs in well motivated scenarios such as a quintessence with a kination phase. We discuss a few microscopic realizations for the new cosmological component and highlight the phenomenological consequences of our calculations for dark matter searches.Comment: 14 pages, 9 figure

On the Sobolev trace Theorem for variable exponent spaces in the critical range

In this paper we study the Sobolev Trace Theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. Then we give local conditions on the exponents and on the domain (in the spirit of Adimurthy and Yadava) in order to satisfy such conditions, and therefore to ensure the existence of extremals.Comment: 21 pages, submitte