Achim Kempf, Robert T. W. Martin, Aidan Chatwin-Davies
While a natural ultraviolet cutoff, presumably at the Planck length, is widely assumed to exist in nature, it has proven difficult to implement a minimum length scale covariantly. A key reason is that the presence of a fixed minimum length would seem to contradict the ability of Lorentz transformations to contract lengths. In this paper, we implement a fully covariant Planck scale cutoff by cutting off the spectrum of the d'Alembertian. In this scenario, consistent with Lorentz contractions, wavelengths that are arbitrarily smaller than the Planck length continue to exist. However, the dynamics of modes of wavelengths that are significantly smaller than the Planck length possess a very small bandwidth. This has the effect of freezing the dynamics of such modes. While both, wavelengths and bandwidths, are frame dependent, Lorentz contraction and time dilation conspire to make the freezing of modes of transplanckian wavelengths covariant. In particular, we show that this ultraviolet cutoff can be implemented covariantly also in curved spacetimes. We focus on Friedmann Robertson Walker (FRW) spacetimes and their much-discussed transplanckian question: The physical wavelength of each comoving mode was smaller than the Planck scale at sufficiently early times. What was the mode's dynamics then? Here, we show that in the presence of the covariant UV cutoff, the dynamical bandwidth of a comoving mode is essentially zero up until its physical wavelength starts exceeding the Planck length. In particular, we show that under general assumptions, the number of dynamical degrees of freedom of each comoving mode all the way up to some arbitrary finite time is actually finite. Our results also open the way to calculating the impact of this natural UV cutoff on inflationary predictions for the CMB.
View original:
http://arxiv.org/abs/1210.0750
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