Decoupling in an expanding universe: backreaction barely constrains short distance effects in the CMB

TL;DR

The paper investigates transplanckian effects on the CMB by employing a boundary effective field theory to encode the cosmological vacuum ambiguity via a boundary action. The main approach is to compute one-loop backreaction arising from irrelevant boundary operators, establishing that backreaction constraints on the leading operator are mild and that the inflationary power spectrum is the most sensitive observable to initial-state corrections, potentially allowing percent-level effects for $H/M$ not far below unity. A key result is the identification of an earliest time after which the EFT is valid and the realization that the leading corrections induce oscillatory features in the power spectrum, with the amplitude scaling as $H/M$ and the maximal effect occurring near the highest observable momenta. The significance lies in providing a self-consistent, quantitative framework for transplanckian physics in cosmology, showing that current and near-future CMB data could reveal or tightly constrain high-energy initial-state effects without destabilizing the inflationary background.

Abstract

We clarify the status of transplanckian effects on the cosmic microwave background (CMB) anisotropy. We do so using the boundary effective action formalism of hep-th/0401164 which accounts quantitatively for the cosmological vacuum ambiguity. In this formalism we can clearly 1) delineate the validity of cosmological effective actions in an expanding universe. The corollary of the initial state ambiguity is the existence of an earliest time. The inability of an effective action to describe physics before this time demands that one sets initial conditions on the earliest time hypersurface. A calculation then shows that CMB anisotropy measurements are generically sensitive to high energy corrections to the initial conditions. 2) We compute the one-loop contribution to the stress-tensor due to high-energy physics corrections to an arbitrary cosmological initial state. We find that phenomenological bounds on the backreaction do not lead to strong constraints on the coefficient of the leading boundary irrelevant operator. Rather, we find that the power spectrum itself is the quantity most sensitive to initial state corrections. 3) The computation of the one-loop backreaction confirms arguments that irrelevant corrections to the Bunch-Davies initial state yield non-adiabatic vacua characterized by an energy excess at some earlier time. However, this excess only dominates over the classical background at times before the `earliest time' at which the effective action is valid. We conclude that the cosmological effective action with boundaries is a fully self-consistent and quantitative approach to transplanckian corrections to the CMB.

Figures (2)

• Figure 1: A refined estimate of the sensitivity of the CMB to new physics.$\,$ The left panel shows the change in the (amplitude of the) power spectrum due to the presence of the leading order irrelevant operator $\frac{\beta}{M} (\partial_i\phi)^2$ as a function of the physical momentum in units of the size of the horizon at the earliest time. (Only for one specific choice is the full oscillatory Bessel function behavior plotted.) This graph should be read as follows. Given the scale of new physics $M$ and the Hubble constant $H$ during inflation (or more precisely at the time when the highest mode $k_{max}$ of interest exits the horizon) the earliest time up to which we can trust the effective action is $y_{0,max}\equiv k_{max}/a_{0,min}H =M/H$ (see section \ref{['sec:an-earliest-time']}). Anything to the right of $y_{0,max}$ should be discarded as untrustworthy. The observed CMB stretches to four orders of magnitude smaller momenta from $10^{-4}y_{0,max}$ to $y_{0,max}$. Precisely at $y_{0,max}$ the change in the power spectrum is linearly dependent on the value of $\beta$. The values of $M/H$ and $\beta$ corresponding to the various curves can thus be read off from the intersection of the plumblines to the upper and right axis. The right panel shows an exclusion plot for $\beta$ as a function of $H/M$. The $45^o$ lines correspond to the backreaction bounds derived in section \ref{['sec:backreacion']} (continuous for zeroth order in slow roll, dashed for first order in slow roll, dotted for second order in slow roll). The $60^o$ lines correspond to the order of magnitude estimate for the backreaction Porrati:2004gz; they are equivalent to an estimate based on dimensional analysis. The upper horizontal line is an order of magnitude estimation of the current error to which we have a nearly scale invariant spectrum WMAP. The lower horizontal line is an order of magnitude estimate of the cosmic variance limitations of resolution. Finally the vertical line denotes a maximal value of $H/M$ consistent with observation using a value for $H$ extracted from the allowed scalar/tensor ratio and $M \equiv 10^{16}$ GeV. This leaves the shaded region as the window of opportunity to observe transplanckian physics in the CMB.
• Figure 2: Exact first order correction to stress-tensor due to the irrelevant operator $\beta$ (absolute value of $\bar{K}$ in units of the classical density $\rho=3H^2M_{pl}^2$; solid) compared to the exponential scaling $e^{-2M^2(t-t_0)^2}$ (dashed) derived in the high $k$ approximation.