We propose a modification of special relativity in which a physical energy, which may be the Planck energy, joins the speed of light as an invariant, in spite of a complete relativity of inertial frames and agreement with Einstein’s theory at low energies. This is accomplished by a nonlinear modification of the action of the Lorentz group on momentum space, generated by adding a dilatation to each boost in such a way that the Planck energy remains invariant. The associated algebra has unmodified structure constants. We also discuss the resulting modifications of field theory and suggest a modification of the equivalence principle which determines how the new theory is embedded in general relativity.

The hypothesis that the Lorentz transformations may be modified at Planck scale energies is further explored. We present a general formalism for theories which preserve the relativity of inertial frames with a nonlinear action of the Lorentz transformations on momentum space. Several examples are discussed in which the speed of light varies with energy and elementary particles have a maximum momenta and/or energy. Energy and momentum conservation are suitably generalized and a proposal is made for how the new transformation laws apply to composite systems. We then use these results to explain the ultrahigh-energy cosmic ray anomaly and we find a form of the theory that explains the anomaly, and leads also to a maximum momentum and a speed of light that diverges with energy. We finally propose that the spatial coordinates be identified as the generators of translation in Minkowski spacetime. In some examples this leads to a commutative geometry, but with an energy dependent Planck constant.

We consider the cosmological implications of light travelling faster in the early Universe. We propose a prescription for deriving corrections to the cosmological evolution equations while the speed of light $c$ is changing. We then show how the horizon, flatness, and cosmological constant problems may be solved. We also study cosmological perturbations in this scenario and show how one may solve the homogeneity and isotropy problems. As it stands, our scenario appears to most easily produce extreme homogeneity, requiring structure to be produced in the standard big bang epoch. Producing significant perturbations during the earlier epoch would require a rather careful design of the function $c\left(t\right).$ The large entropy inside the horizon nowadays can also be accounted for in this scenario.

We examine previous claims for a preferred axis at $(b,l)\approx (60,-100)$ in the cosmic radiation anisotropy, by generalizing the concept of multipole planarity to any shape preference. Contrary to earlier claims, we find that the amount of power concentrated in planar modes for $l=2,3$ is not inconsistent with isotropy and Gaussianity. The multipoles’ alignment, however, is indeed anomalous, and extends up to $l=5$ rejecting statistical isotropy with a probability in excess of 99.9%. There is also an uncanny correlation of azimuthal phases between $l=3$ and $l=5$. We are unable to blame these effects on foreground contamination or large-scale systematic errors. This reappraisal may be crucial in identifying the theoretical model behind the anomaly.

There is growing observational evidence for a population of supermassive black holes (SMBHs) in galactic bulges. We examine in detail the conditions under which these black holes must have originated from primordial black holes (PBHs). We consider the merging and accretion history experienced by SMBHs to find that, whereas it is possible that they were formed by purely astrophysical processes, this is unlikely and most probably a population of primordial progenitors is necessary. We identify the mass distribution and comoving density of this population and then propose a cosmological scenario producing PBHs with the right properties. Although this is not essential we consider PBHs produced at the end of a period of inflation with a blue spectrum of fluctuations. We constrain the value of the spectral tilt in order to obtain the required PBH comoving density. We then assume that PBHs grow by accreting quintessence, showing that their mass scales like the horizon mass while the quintessence field itself is scaling. We find that if scaling is broken just before nucleosynthesis (as is the case with some attractive nonminimally coupled models) we obtain the appropriate PBH mass distribution. Hawking evaporation is negligible in most cases, but we also discuss situations in which the interplay of accretion and evaporation is relevant.