Recombination Thickness as an Uncertainty in Inflationary Observables

Author(s)

Oikonomou, V.K.

Abstract

Standard CMB analysis assumes a direct deterministic mapping between the multipole probed by the CMB $\ell$ and the primordial wavenumber $k$. Since the recombination era has a finite duration, this mapping is probabilistic by construction. We elevate the power spectrum of the primordial perturbations to a probability distribution caused by the finite duration of the recombination era. We show that a finite recombination width introduces a Gaussian smoothing scale in $\ln k$ with $σ_{\ln k} \sim σ_η/ D_*$, leading to a probabilistic mapping from multipoles to inflationary e-folds. This effect is zero in standard power-law inflationary scenarios, but it may become relevant for scenarios with exotic oscillating features of the primordial power spectrum, which will be probed by the future CMB experiments. The observed effective power spectrum is the true primordial spectrum blurred by the uncertainty in scale reconstruction, which is mathematically identical to a Bayesian marginalization over a latent variable, and thus there is a propagation of the measurement error in the independent variable, which is another more formal way to view the smoothing effect. Our results indicate that the smoothing has quantifiable effects on the spectral index and its running, but more importantly the difference between the TT and EE inferred spectral indices, $n_s^{TT}-n_s^{EE}$, is non-trivial, in contrast to standard inflation without smoothing, and might become observable by future cosmic microwave background experiments. Any tension in $n_s^{TT}-n_s^{EE}$ could indicate oscillations in the primordial spectrum and the effects of the power spectrum smoothing. Finally, a minimal Fisher matrix analysis is performed to investigate the observability prospects of the smoothing effect.

Figures

Caption

The fraction $\frac{P'}{P}$ for $P(k)=P_0(k)\left(1+A \cos\left(\omega \ln k+\phi-\omega \ln k_* \right)\right)$, and for $k_*=0.05$Mpc$^{-1}$, $A=0.029$ and $\frac{\phi}{2\pi}=0.698$ in the wavenumber range $k=[0.001,1]$Mpc$^{-1}$, for various frequencies. The range of frequencies which ensures that $\frac{P'}{P}\ll 1$ is $\omega=[0,30]$.

Caption

The fraction $\frac{P'}{P}$ for $P(k)=P_0(k)\left(1+A \cos\left(\omega \ln k+\phi-\omega \ln k_* \right)\right)$, and for $k_*=0.05$Mpc$^{-1}$, $A=0.029$ and $\frac{\phi}{2\pi}=0.698$ in the wavenumber range $k=[0.001,1]$Mpc$^{-1}$, for various frequencies. The range of frequencies which ensures that $\frac{P'}{P}\ll 1$ is $\omega=[0,30]$.

Caption

The fraction $\frac{P'}{P}$ for $P(k)=P_0(k)\left(1+A \cos\left(\omega \ln k+\phi-\omega \ln k_* \right)\right)$, and for $k_*=0.05$Mpc$^{-1}$, $A=0.029$ and $\frac{\phi}{2\pi}=0.698$ in the wavenumber range $k=[0.001,1]$Mpc$^{-1}$, for various frequencies. The range of frequencies which ensures that $\frac{P'}{P}\ll 1$ is $\omega=[0,30]$.

Caption

The signal-to-noise ratio for the smoothing eigenmode, for various CMB experiments.