[en] We explore the fundamental limits to which reionization histories can be constrained using only large-scale cosmic microwave background (CMB) anisotropy measurements. The redshift distribution of the fractional ionization x _e(z) affects the angular distribution of CMB polarization. We project constraints on the reionization history of the universe using low-noise full-sky temperature and E-mode measurements of the CMB. We show that the measured TE power spectrum, ${\stackrel{^<!--\; ^\; -->}{C}}_{\mathrm{\ell <!--\; ???\; -->}}^{\mathrm{T}\mathrm{E}}$, has roughly one quarter of the constraining power of ${\stackrel{^<!--\; ^\; -->}{C}}_{\mathrm{\ell <!--\; ???\; -->}}^{\mathrm{E}\mathrm{E}}$ on the reionization optical depth τ, and its addition improves the precision on τ by 20% over using ${\stackrel{^<!--\; ^\; -->}{C}}_{\mathrm{\ell <!--\; ???\; -->}}^{\mathrm{E}\mathrm{E}}$ only. We also use a two-step reionization model with an additional high-redshift step, parameterized by an early ionization fraction $x_e^{min}$, and a late reionization step at z _re. We find that future high signal-to-noise measurements of the multipoles 10 ≤ ℓ < 20 are especially important for breaking the degeneracy between $x_e^{min}$ and z _re. In addition, we show that the uncertainties on these parameters determined from a map with sensitivity 10 μK arcmin are less than 5% larger than the uncertainties in the noiseless case, making this noise level a natural target for future large sky area E-mode measurements.

[en] In this paper we study the small-world network model of Watts and Strogatz, which mimics some aspects of the structure of networks of social interactions. We argue that there is one nontrivial length-scale in the model, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero, giving a normal critical point in this limit. This length-scale governs the crossover from large- to small-world behavior in the model, as well as the number of vertices in a neighborhood of given radius on the network. We derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex-vertex distance on the network, and, using series expansion and Pade approximants, find an approximate analytic form for the scaling function. We calculate the effective dimension of small-world graphs and show that this dimension varies as a function of the length-scale on which it is measured, in a manner reminiscent of multifractals. We also study the problem of site percolation on small-world networks as a simple model of disease propagation, and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms (in epidemiological terms, the point at which an epidemic occurs). The typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph. All our analytic results are confirmed by extensive numerical simulations of the model. (c) 1999 The American Physical Society

[en] We examine the internal consistency of the Planck 2015 cosmic microwave background (CMB) temperature anisotropy power spectrum. We show that tension exists between cosmological constant cold dark matter ($\mathrm{\Lambda <!--\; ??\; -->}\mathrm{C}\mathrm{D}\mathrm{M}$) model parameters inferred from multipoles $\mathrm{\ell <!--\; ???\; -->}<<!--\; <\; -->1000$ (roughly those accessible to Wilkinson Microwave Anisotropy Probe), and from $\mathrm{\ell <!--\; ???\; -->}⩾<!--\; ???\; -->1000$, particularly the CDM density, ${\mathrm{\Omega <!--\; ??\; -->}}_c{h}^2$, which is discrepant at $2.5\mathrm{\sigma <!--\; ??\; -->}$ for a Planck -motivated prior on the optical depth, $\mathrm{\tau <!--\; ??\; -->}=0.07\pm <!--\; ??\; -->0.02$. We find some parameter tensions to be larger than previously reported because of inaccuracy in the code used by the Planck Collaboration to generate model spectra. The Planck $\mathrm{\ell <!--\; ???\; -->}⩾<!--\; ???\; -->1000$ constraints are also in tension with low-redshift data sets, including Planck ’s own measurement of the CMB lensing power spectrum ($2.4\mathrm{\sigma <!--\; ??\; -->}$), and the most precise baryon acoustic oscillation scale determination ($2.5\mathrm{\sigma <!--\; ??\; -->}$). The Hubble constant predicted by Planck from $\mathrm{\ell <!--\; ???\; -->}⩾<!--\; ???\; -->1000$, $H_0=64.1\pm <!--\; ??\; -->1.7$ km s${}^{-<!--\; ???\; -->1}$ Mpc⁻¹, disagrees with the most precise local distance ladder measurement of $73.0\pm <!--\; ??\; -->2.4$ km s${}^{-<!--\; ???\; -->1}$ Mpc⁻¹ at the $3.0\mathrm{\sigma <!--\; ??\; -->}$ level, while the Planck value from $\mathrm{\ell <!--\; ???\; -->}<<!--\; <\; -->1000$, $69.7\pm <!--\; ??\; -->1.7$ km s${}^{-<!--\; ???\; -->1}$ Mpc⁻¹, is consistent within $1\mathrm{\sigma <!--\; ??\; -->}$. A discrepancy between the Planck and South Pole Telescope high-multipole CMB spectra disfavors interpreting these tensions as evidence for new physics. We conclude that the parameters from the Planck high-multipole spectrum probably differ from the underlying values due to either an unlikely statistical fluctuation or unaccounted-for systematics persisting in the Planck data.

[en] Variable-delay Polarization Modulators (VPMs) are currently being implemented in experiments designed to measure the polarization of the cosmic microwave background on large angular scales because of their capability for providing rapid, front-end polarization modulation and control over systematic errors. Despite the advantages provided by the VPM, it is important to identify and mitigate any time-varying effects that leak into the synchronously modulated component of the signal. In this paper, the effect of emission from a 300 K VPM on the system performance is considered and addressed. Though instrument design can greatly reduce the influence of modulated VPM emission, some residual modulated signal is expected. VPM emission is treated in the presence of rotational misalignments and temperature variation. Simulations of time-ordered data are used to evaluate the effect of these residual errors on the power spectrum. The analysis and modeling in this paper guides experimentalists on the critical aspects of observations using VPMs as front-end modulators. By implementing the characterizations and controls as described, front-end VPM modulation can be very powerful for mitigating 1/f noise in large angular scale polarimetric surveys. None of the systematic errors studied fundamentally limit the detection and characterization of B-modes on large scales for a tensor-to-scalar ratio of r = 0.01. Indeed, r < 0.01 is achievable with commensurately improved characterizations and controls