On the absence of gravitational lensing of the cosmic microwave background

The magnification of distant sources by mass clumps at lower ($z \leq 1$) redshifts is calculated analytically. The clumps are initially assumed to be galaxy group isothermal spheres with properties inferred from an extensive survey. The average effect, which includes strong lensing, is exactly counteracted by the beam divergence in between clumps (more precisely, the average reciprocal magnification cancels the inverse Dyer-Roeder demagnification). This conclusion is in fact independent of the matter density function within each clump, and remains valid for arbitrary densities of matter and dark energy. When tested against the CMB, a rather large lensing induced {\it dispersion} in the angular size of the primary acoustic peaks of the TT power spectrum is inconsistent with WMAP observations. The situation is unchanged by the use of NFW profiles for the density distribution of groups. Finally, our formulae are applied to an ensemble of NFW mass clumps or isothermal spheres having the parameters of galaxy {\it clusters}. The acoustic peak size dispersion remains unobservably large, and is also excluded by WMAP. For galaxy groups, two possible ways of reconciling with the data are proposed, both exploiting maximally the uncertainties in our knowledge of group properties. The same escape routes are not available in the case of clusters, however, because their properties are well understood. Here we have a more robust conclusion: neither of the widely accepted models are good description of clusters, or important elements of physics responsible for shaping zero curvature space are missing from the standard cosmological model. When all the effects are accrued, it is difficult to understand how WMAP could reveal no evidence whatsoever of lensing by groups and clusters.Comment: ApJ v628, pp. 583-593 (August 1, 2005

Tav e grandi opere: non è solo un problema di costi

Le grandi opere non solo costano tanto, ma sono spesso inutili. Per giustificarle ci si affida a previsioni di aumento del traffico poco realistiche. Sono necessarie analisi costi-benefici terze, indipendenti e interamente riproducibili

Per un pugno di opere

La svolta decisa dal governo per le infrastrutture medie e piccole dovrebbe essere adottata anche per quelle grandi, abbandonando così l’idea che le opere “strategiche” generino una forte domanda. Altrimenti, dovremo rassegnarci ai cantieri “stop and go”. E al moltiplicarsi dei costi

Regularity bounds by minimal generators and Hilbert function

Let $\rho_C$ be the regularity of the Hilbert function of a projective curve $C$ in \mbox {P}^n_K over an algebraically closed field $K$ and $\beta_1, \ldots, \beta_{n-1}$ be degrees for which there exists a complete intersection of type ($\beta_1, \ldots, \beta_{n-1}$) containing properly $C$. Then the Castelnuovo-Mumford regularity of $C$ is bounded above by max $\{\rho_C +1, \beta 1 + \ldots + \beta_n-1 -(n-1)\}$ .We investigate the sharpness of the above bound, which is achieved by curves algebraically linked to ones having degenerate general hyperplane section