Boundary Effective Field Theory and Trans-Planckian Perturbations: Astrophysical Implications

TL;DR

This work compares two trans-Planckian frameworks for inflationary perturbations: Boundary Effective Field Theory (BEFT) and New Physics Hypersurface (NPH). BEFT imposes initial conditions on a single spacelike boundary and encodes corrections via higher-dimension operators, yielding $P_{BEFT}(k) = P_{BD}(k)[1 + \beta (H_i/M) y_i \sin(2 y_i)]$ with $y_i = k/(a_i H_i)$, while NPH normalizes modes when their physical wavelength first exceeds the cutoff, giving $P_{NPH}(k) = P_{BD}(k)[1 + (1/2 y_c) \sin(2 y_c/(1-\epsilon))]$ with $y_c \sim k^{\epsilon}$. The authors show these approaches produce qualitatively different observational predictions for the CMB and large-scale structure and discuss how to restore time invariance in BEFT or reconcile the two. They also map the implications to astrophysical probes and outline necessary theoretical improvements for a self-consistent EFT treatment in an expanding universe.

Abstract

We contrast two approaches to calculating trans-Planckian corrections to the inflationary perturbation spectrum: the New Physics Hypersurface [NPH] model, in which modes are normalized when their physical wavelength first exceeds a critical value, and the Boundary Effective Field Theory [BEFT] approach, where the initial conditions for all modes are set at the same time, and modified by higher dimensional operators enumerated via an effective field theory calculation. We show that these two approaches -- as currently implemented -- lead to radically different expectations for the trans-Planckian corrections to the CMB and emphasize that in the BEFT formalism we expect the perturbation spectrum to be dominated by quantum gravity corrections for all scales shorter than some critical value. Conversely, in the NPH case the quantum effects only dominate the longest modes that are typically much larger than the present horizon size. Furthermore, the onset of the breakdown in the standard inflationary perturbation calculation predicted by the BEFT formalism is likely to be associated with a feature in the perturbation spectrum, and we discuss the observational signatures of this feature in both CMB and large scale structure observations. Finally, we discuss possible modifications to both calculational frameworks that would resolve the contradictions identified here.

Figures (4)

• Figure 1: Schematic diagram of the initial conditions hypersurfaces for the BEFT and NPH models of trans-Planckian physics. For the NPH case, the initial conditions are set when the wavelength $a/k$ is equal to the minimum length $1/M$, and all modes are normalized on a vertical timelike hypersurface. Conversely, in the BEFT case the initial conditions for all the modes are set on the same spacelike hypersurface, where $a=a_i$. The physical wavelength of a given mode is measured by $\lambda$.
• Figure 2: Modification to the primordial power spectrum for four choices of parameters for the Boundary Effective Field Theory, chosen so that the modulation has an impact on the CMB. The plots show $\beta=1$, $a_i M = 1.4\ \hbox{Mpc}^{-1}$ and $M/H = 1$ (black), 10 (blue), 100 (purple), 1000 (orange).
• Figure 4: Modification to the linear theory matter power spectrum for three choices of parameters for the Boundary Effective Field Theory. The plots show $\beta=1$, $a_i M = 1.4\ \hbox{Mpc}^{-1}$ and $M/H = 1$ (black), 10 (blue), and 100 (purple). The single decade of $k$ values for which the modulation is measurable and where the BEFT spectrum is reliable is $0.01 < k < 0.1$.
• Figure 5: Modification to the linear theory matter power spectrum for four choices of parameters for the Boundary Effective Field Theory, plotted along with error bars from SDSS. Parameter choices match those of Figure \ref{['fig:PSmod']}. As a rough guide, WMAP Bennett:2003bz probes $k < 0.06$ Mpc$^{-1}$, SDSS probes $0.015$ Mpc$^{-1} < k < 0.3$ Mpc$^{-1}$, and the Lyman-$\alpha$ power spectrum probes $0.2$ Mpc$^{-1} < k < 4$ Mpc$^{-1}$. Note the observations of the Lyman-$\alpha$ forest for close quasar pairs may probe scales as short as 25 Mpc$^{-1}$Lidz:2003fv.