Keeping an eye on DBI: power-counting for small-c_s cosmology

Abstract

Inflationary mechanisms for generating primordial fluctuations ultimately compute them as the leading contributions in a derivative expansion, with corrections controlled by powers of derivatives like the Hubble scale over Planck mass: H/Mp. At face value this derivative expansion breaks down for models with a small sound speed, cs, to the extent that cs ≪ 1 is obtained by having higher-derivative interactions like Leff ∼ (∂ Φ)4 compete with lower-derivative propagation. This concern arises more generally for models whose lagrangian is given as a function P(X) for X = −∂μ Φ ∂μ Φ—including in particular DBI models for which P(X) ∝ √1−kX—since these keep all orders in ∂ Φ while dropping ∂n Φ for n > 1. We here find a sensible power-counting scheme for DBI models that gives a controlled expansion in powers of three types of small parameters: H/Mp, slow-roll parameters (possibly) and cs ≪ 1. We do not find a similar expansion framework for generic small-cs or P(X) models. Our power-counting result quantifies the theoretical error for any prediction (such as for inflationary correlation functions) by fixing the leading power of these small parameters that is dropped when not computing all graphs (such as by restricting to the classical approximation); a prerequisite for meaningful comparisons with observations. The new power-counting regime arises because small cs alters the kinematics of free fluctuations in a way that changes how interactions scale at low energies, in particular allowing 1−cs to be larger than derivative-measuring quantities like (H/Mp)2.

Cosmological Cutting Rules

“…Power counting schemes analogous to NDA were developed for inflation in[50,51] (see also[52,53] for theories with small cs in particular).14 Note that while NDA was generalised to d dimensions in[55], this assumed d-dimensional Lorentzinvariant kinetic terms for the fields -in our case, although the loop integrals are done over only spatial momenta in d = 3, the underlying field theory is four dimensional, and so we retain the 4π counting of (3 + 1) dimensions.…”

Non-Gaussianities in the Extended EFT of Inflation

“…in the Lagrangian for the Goldstone boson. As we will see, such ghosts are generated in the Extended EFT of inflation too which limits the size and value of their corresponding coupling constants 4 . The Lagrangian of the EFToI (2.3) in the decoupling limit, where Ḣ → 0 and M 2 Pl Ḣ is kept fixed, results into the following quadratic and cubic expressions for π [11,17]:…”

“…In fact, as we do not yet have a well understood theory of quantum gravity, every inflationary model should be viewed as an Effective Field Theory in the semi-classical gravity regime. Therefore, it is important to make sure the effective field theory description is self consistent in its regime of its validity [4,5]. Meanwhile, as a healthy scientific exercise we should also explore if there are other sensible alternative scenarios [6,7] to inflationary paradigm that can describe the initial conditions of the universe.…”

“…The scaling power of M 0,2,2 ∝ (∂ i π) 2 and M 0,4,2 ∝ (∂ 2 i π) 2 which modify the dispersion relation, as we had pointed out earlier are respectively −4/3 and −2/3. The scaling power of 4 scale as −2/3 as well. The term M 0,4,4 ∝ (∂ i π) 4 contributes to trispectrum which we will not discuss in this paper 10 .…”

“…The scaling power of 4 scale as −2/3 as well. The term M 0,4,4 ∝ (∂ i π) 4 contributes to trispectrum which we will not discuss in this paper 10 . Therefore, to calculate the strongest contributions to bispectrum we only need to consider operators (0, 4, 3) and (1, 2, 3), which are…”