The bimetric variational principle for General Relativity

The bimetric variational principle is a subtle reinterpretation of general relativity that assumes the spacetime connection to be generated by an independent metric. Unlike the so called Palatini formalism that promotes the connection into a fundamental field, the new variational principle results in a physically distinct theory since the potential for the connection carries new degrees of freedom. The connection-generating metric naturally allows also an antisymmetric component. This sets torsion propagating! It is also shown here that while in the most straightforward generalization of the Einstein-Hilbert action the nonmetric degrees of freedom become ghosts, there exist very simple actions which give rise to viable theories at the linearised level when subjected to the bimetric variational principle. However, the non linear interactions might bring unpleasant features like the Boulware-Deser ghost. This remains to be explored since this new type of bimetric theories does not, in principle, lie in the class of usual bimetric theories where non-linear interactions inevitably come in with new ghost-like degrees of freedom.Comment: 5 pages, no figures. Version 2: corrected sign error, considerably extended discussion

Cosmology with moving dark energy and the CMB quadrupole

We study the consequences of a homogeneous dark energy fluid having a non-vanishing velocity with respect to the matter and radiation large-scale rest frames. We consider homogeneous anisotropic cosmological models with four fluids (baryons, radiation, dark matter and dark energy) whose velocities can differ from each other. Performing a perturbative calculation up to second order in the velocities, we obtain the contribution of the anisotropies generated by the fluids motion to the CMB quadrupole and compare with observations. We also consider the exact problem for arbitrary velocities and solve the corresponding equations numerically for different dark energy models. We find that models whose equation of state is initially stiffer than radiation, as for instance some tracking models, are unstable against velocity perturbations, thus spoiling the late-time predictions for the energy densities. In the case of scaling models, the contributions to the quadrupole can be non-negligible for a wide range of initial conditions. We also consider fluids moving at the speed of light (null fluids) with positive energy and show that, without assuming any particular equation of state, they generically act as a cosmological constant at late times. We find the parameter region for which the models considered could be compatible with the measured (low) quadrupole.Comment: 23 pages, 6 figures. Confidence intervals calculated from WMAP data, new references and comments included. Final version to appear in PR

From coalescing random walks on a torus to Kingman's coalescent

Let $\mathbb{T}^d_N$, $d\ge 2$, be the discrete $d$-dimensional torus with $N^d$ points. Place a particle at each site of $\mathbb{T}^d_N$ and let them evolve as independent, nearest-neighbor, symmetric, continuous-time random walks. Each time two particles meet, they coalesce into one. Denote by $C_N$ the first time the set of particles is reduced to a singleton. Cox [6] proved the existence of a time-scale $\theta_N$ for which $C_N/\theta_N$ converges to the sum of independent exponential random variables. Denote by $Z^N_t$ the total number of particles at time $t$. We prove that the sequence of Markov chains $(Z^N_{t\theta_N})_{t\ge 0}$ converges to the total number of partitions in Kingman's coalescent

Class sizes of prime-power order p'-elements and normal subgroups

We prove an extension of the renowned Itô’s theorem on groups having two class sizes in three different directions at the same time: normal subgroups, p′p′-elements and prime-power order elements. Let NN be a normal subgroup of a finite group GG and let pp be a fixed prime. Suppose that |xG|=1|xG|=1 or mm for every qq-element of NN and for every prime q≠pq≠p. Then, NN has nilpotent pp-complements.We are very grateful to the referee, who provided us a significant simplification of the last step of the proof of the main theorem and for many comments which have contributed to improve the paper. C. G. Shao wants to express his deep gratitude for the warm hospitality he has received in the Departamento de Matematicas of the Universidad Jaume I in Castellon, Spain. This research is supported by the Valencian Government, Proyecto PROMETEO/2011/30, by the Spanish Government, Proyecto MTM2010-19938-C03-02. The third author is supported by the research Project NNSF of China (Grant Nos. 11201401 and 11301218) and University of Jinan Research Funds for Doctors (XBS1335 and XBS1336).Beltrán, A.; Felipe Román, MJ.; Shao, C. (2015). Class sizes of prime-power order p'-elements and normal subgroups. Annali di Matematica Pura ed Applicata. 194(5):1527-1533. https://doi.org/10.1007/s10231-014-0432-4S152715331945Akhlaghi, Z., Beltrán, A., Felipe, M.J.: The influence of $$p$$ p -regular class sizes on normal subgroups. J. Group Theory. 16, 585–593 (2013)Alemany, E., Beltrán, A., Felipe, M.J.: Nilpotency of normal subgroups having two $$G$$ G -class sizes. Proc. Am. Math. Soc. 139, 2663–2669 (2011)Alemany, E., Beltrán, A., Felipe, M.J.: Finite groups with two $$p$$ p -regular conjugacy class lengths II. Bull. Aust. Math. Soc. 797, 419–425 (2009)Beltrán, A., Felipe, M.J.: Normal subgroups and class sizes elements of prime-power order. Proc. Am. Math. Soc. 140, 4105–4109 (2012)Beltrán, A.: Action with nilpotent fixed point subgroup. Arch. Math. (Basel) 69, 177–184 (1997)Camina, A.R.: Finite groups of conjugate rank 2. Nagoya Math. J. 53, 47–57 (1974)Casolo, C., Dolfi, S., Jabara, E.: Finite groups whose noncentral class sizes have the same $$p$$ p -part for some prime $$p$$ p . Isr. J. Math. 192, 197–219 (2012)Huppert, B.: Character Theory of Finite groups, vol. 25. De Gruyter Expositions in Mathemathics, Berlin, New York (1998)Kleidman, P., Liebeck, M.: The Subgroup Structure of The Finite Classical Groups. London Mathematical Society Lecture Note Series, 129. Cambridge University Press, Cambridge (1990)Kurzweil, K., Stellmacher, B.: The Theory of Finite Groups. An Introduction. Springer, New York (2004)The GAP Group, GAP—Groups, Algorithms and Programming, Vers. 4.4.12 (2008). http://www.gap-system.orgVasiliev, A.V., Vdovin, E.P.: An adjacency criterion for the prime graph of a finite simple group. Algebra Logic 44(6), 381–406 (2005

Viability of vector-tensor theories of gravity

We present a detailed study of the viability of general vector-tensor theories of gravity in the presence of an arbitrary temporal background vector field. We find that there are six different classes of theories which are indistinguishable from General Relativity by means of local gravity experiments. We study the propagation speeds of scalar, vector and tensor perturbations and obtain the conditions for classical stability of those models. We compute the energy density of the different modes and find the conditions for the absence of ghosts in the quantum theory. We conclude that the only theories which can pass all the viability conditions for arbitrary values of the background vector field are not only those of the pure Maxwell type, but also Maxwell theories supplemented with a (Lorentz type) gauge fixing term.Comment: 13 pages, 2 figures, 1 table. Final version to appear in JCA