Compensated isocurvature perturbations in the curvaton model

Primordial fluctuations in the relative number densities of particles, or isocurvature perturbations, are generally well constrained by cosmic microwave background (CMB) data. A less probed mode is the compensated isocurvature perturbation (CIP), a fluctuation in the relative number densities of cold dark matter and baryons. In the curvaton model, a subdominant field during inflation later sets the primordial curvature fluctuation $\zeta$. In some curvaton-decay scenarios, the baryon and cold dark matter isocurvature fluctuations nearly cancel, leaving a large CIP correlated with $\zeta$. This correlation can be used to probe these CIPs more sensitively than the uncorrelated CIPs considered in past work, essentially by measuring the squeezed bispectrum of the CMB for triangles whose shortest side is limited by the sound horizon. Here, the sensitivity of existing and future CMB experiments to correlated CIPs is assessed, with an eye towards testing specific curvaton-decay scenarios. The planned CMB Stage 4 experiment could detect the largest CIPs attainable in curvaton scenarios with more than 3$\sigma$ significance. The significance could improve if small-scale CMB polarization foregrounds can be effectively subtracted. As a result, future CMB observations could discriminate between some curvaton-decay scenarios in which baryon number and dark matter are produced during different epochs relative to curvaton decay. Independent of the specific motivation for the origin of a correlated CIP perturbation, cross-correlation of CIP reconstructions with the primary CMB can improve the signal-to-noise ratio of a CIP detection. For fully correlated CIPs the improvement is a factor of $\sim$2$-$3.Comment: 20 pages, 8 figures, minor changes matching publicatio

Sample Complexity of Sample Average Approximation for Conditional Stochastic Optimization

In this paper, we study a class of stochastic optimization problems, referred to as the \emph{Conditional Stochastic Optimization} (CSO), in the form of \min_{x \in \mathcal{X}} \EE_{\xi}f_\xi\Big({\EE_{\eta|\xi}[g_\eta(x,\xi)]}\Big), which finds a wide spectrum of applications including portfolio selection, reinforcement learning, robust learning, causal inference and so on. Assuming availability of samples from the distribution \PP(\xi) and samples from the conditional distribution \PP(\eta|\xi), we establish the sample complexity of the sample average approximation (SAA) for CSO, under a variety of structural assumptions, such as Lipschitz continuity, smoothness, and error bound conditions. We show that the total sample complexity improves from \cO(d/\eps^4) to \cO(d/\eps^3) when assuming smoothness of the outer function, and further to \cO(1/\eps^2) when the empirical function satisfies the quadratic growth condition. We also establish the sample complexity of a modified SAA, when $\xi$ and $\eta$ are independent. Several numerical experiments further support our theoretical findings. Keywords: stochastic optimization, sample average approximation, large deviations theoryComment: Typo corrected. Reference added. Revision comments handle