The covariant perturbative approach to cosmic microwave background anisotropies

The Ehlers-Ellis 1+3 formulation of covariant hydrodynamics, when supplemented with covariant radiative transport theory, gives an exact, physically transparent description of the physics of the cosmic microwave background radiation (CMB). Linearisation around a Friedmann-Robertson-Walker (FRW) universe provides a very direct and seamless route through to the linear, gauge-invariant perturbation equations for scalar, vector and tensor modes in an almost-FRW model. In this contribution we review covariant radiative transport theory and its application to the perturbative method for calculating and understanding the anisotropy of the CMB. Particular emphasis is placed on the inclusion of polarization in a fully covariant manner. With this inclusion, the covariant perturbative approach offers a complete description of linearised CMB physics in an almost-FRW universe.Comment: To appear in proceedings of SARS99 meeting in honour of G.F.R.Elli

The radiative transfer for polarized radiation at second order in cosmological perturbations

This article investigates the full Boltzmann equation up to second order in the cosmological perturbations. Describing the distribution of polarized radiation by using a tensor valued distribution function, the second order Boltzmann equation, including polarization, is derived without relying on the Stokes parameters.Comment: 4 pages, no figure; replaced to match published versio

Nonlinear Effects in the Cosmic Microwave Background

Major advances in the observation and theory of cosmic microwave background anisotropies have opened up a new era in cosmology. This has encouraged the hope that the fundamental parameters of cosmology will be determined to high accuracy in the near future. However, this optimism should not obscure the ongoing need for theoretical developments that go beyond the highly successful but simplified standard model. Such developments include improvements in observational modelling (e.g. foregrounds, non-Gaussian features), extensions and alternatives to the simplest inflationary paradigm (e.g. non-adiabatic effects, defects), and investigation of nonlinear effects. In addition to well known nonlinear effects such as the Rees-Sciama and Ostriker-Vishniac effects, further nonlinear effects have recently been identified. These include a Rees-Sciama-type tensor effect, time-delay effects of scalar and tensor lensing, nonlinear Thomson scattering effects and a nonlinear shear effect. Some of the nonlinear effects and their potential implications are discussed.Comment: Invited contribution to Relativistic Cosmology Symposium (celebrating the 60th year of GFR Ellis); to appear Gen. Rel. Gra

Lensed CMB power spectra from all-sky correlation functions

Weak lensing of the CMB changes the unlensed temperature anisotropy and polarization power spectra. Accounting for the lensing effect will be crucial to obtain accurate parameter constraints from sensitive CMB observations. Methods for computing the lensed power spectra using a low-order perturbative expansion are not good enough for percent-level accuracy. Non-perturbative flat-sky methods are more accurate, but curvature effects change the spectra at the 0.3-1% level. We describe a new, accurate and fast, full-sky correlation-function method for computing the lensing effect on CMB power spectra to better than 0.1% at l<2500 (within the approximation that the lensing potential is linear and Gaussian). We also discuss the effect of non-linear evolution of the gravitational potential on the lensed power spectra. Our fast numerical code is publicly available.Comment: 16 pages, 4 figures. Changes to match PRD version including new section on non-linear corrections. CAMB code available at http://camb.info

Cosmic microwave background anisotropies in the CDM model: a covariant and gauge-invariant approach

We present a fully covariant and gauge-invariant calculation of the evolution of anisotropies in the cosmic microwave background (CMB) radiation. We use the physically appealing covariant approach to cosmological perturbations, which ensures that all variables are gauge-invariant and have a clear physical interpretation. We derive the complete set of frame-independent, linearised equations describing the (Boltzmann) evolution of anisotropy and inhomogeneity in an almost Friedmann-Robertson-Walker (FRW) cold dark matter (CDM) universe. These equations include the contributions of scalar, vector and tensor modes in a unified manner. Frame-independent equations for scalar and tensor perturbations, which are valid for any value of the background curvature, are obtained straightforwardly from the complete set of equations. We discuss the scalar equations in detail, including the integral solution and relation with the line of sight approach, analytic solutions in the early radiation dominated era, and the numerical solution in the standard CDM model. Our results confirm those obtained by other groups, who have worked carefully with non-covariant methods in specific gauges, but are derived here in a completely transparent fashion.Comment: Two new sections added to earlier version. To appear in Astrophys.

A covariant and gauge-invariant analysis of cosmic microwave background anisotropies from scalar perturbations

We present a new, fully covariant and manifestly gauge-invariant expression for the temperature anisotropy in the cosmic microwave background radiation resulting from scalar perturbations. We pay particular attention to gauge issues such as the definition of the temperature perturbation and the placing of the last scattering surface. In the instantaneous recombination approximation, the expression may be integrated up to a Rees-Sciama term for arbitrary matter descriptions in flat, open and closed universes. We discuss the interpretation of our result in the baryon-dominated limit using numerical solutions for conditions on the last scattering surface, and confirm that for adiabatic perturbations the dominant contribution to the anisotropy on intermediate scales (the location of the Doppler peaks) may be understood in terms of the spatial inhomogeneity of the radiation temperature in the baryon rest frame. Finally, we show how this term enters the usual Sachs-Wolfe type calculations (it is rarely seen in such analyses) when subtle gauge effects at the last scattering surface are treated correctly.Comment: 20 pages, 2 Postscript figure

CLOVER - A new instrument for measuring the B-mode polarization of the CMB

We describe the design and expected performance of Clover, a new instrument designed to measure the B-mode polarization of the cosmic microwave background. The proposed instrument will comprise three independent telescopes operating at 90, 150 and 220 GHz and is planned to be sited at Dome C, Antarctica. Each telescope will feed a focal plane array of 128 background-limited detectors and will measure polarized signals over angular multipoles 20 < l < 1000. The unique design of the telescope and careful control of systematics should enable the B-mode signature of gravitational waves to be measured to a lensing-confusion-limited tensor-to-scalar ratio r~0.005.Comment: 4 pages, 5 figures. To appear in the proceedings of the XXXVIXth Rencontres de Moriond "Exploring the Universe

Influences of increasing temperature on Indian wheat: quantifying limits to predictability

As climate changes, temperatures will play an increasing role in determining crop yield. Both climate model error and lack of constrained physiological thresholds limit the predictability of yield. We used a perturbed-parameter climate model ensemble with two methods of bias-correction as input to a regional-scale wheat simulation model over India to examine future yields. This model configuration accounted for uncertainty in climate, planting date, optimization, temperature-induced changes in development rate and reproduction. It also accounts for lethal temperatures, which have been somewhat neglected to date. Using uncertainty decomposition, we found that fractional uncertainty due to temperature-driven processes in the crop model was on average larger than climate model uncertainty (0.56 versus 0.44), and that the crop model uncertainty is dominated by crop development. Simulations with the raw compared to the bias-corrected climate data did not agree on the impact on future wheat yield, nor its geographical distribution. However the method of bias-correction was not an important source of uncertainty. We conclude that bias-correction of climate model data and improved constraints on especially crop development are critical for robust impact predictions

Comparing the effects of calibration and climate errors on a statistical crop model and a process-based crop model

Understanding the relationship between climate and crop productivity is a key component of projections of future food production, and hence assessments of food security. Climate models and crop yield datasets have errors, but the effects of these errors on regional scale crop models is not well categorized and understood. In this study we compare the effect of synthetic errors in temperature and precipitation observations on the hindcast skill of a process-based crop model and a statistical crop model. We find that errors in temperature data have a significantly stronger influence on both models than errors in precipitation. We also identify key differences in the responses of these models to different types of input data error. Statistical and process-based model responses differ depending on whether synthetic errors are overestimates or underestimates. We also investigate the impact of crop yield calibration data on model skill for both models, using datasets of yield at three different spatial scales. Whilst important for both models, the statistical model is more strongly influenced by crop yield scale than the process-based crop model. However, our results question the value of high resolution yield data for improving the skill of crop models; we find a focus on accuracy to be more likely to be valuable. For both crop models, and for all three spatial scales of yield calibration data, we found that model skill is greatest where growing area is above 10-15 %. Thus information on area harvested would appear to be a priority for data collection efforts. These results are important for three reasons. First, understanding how different crop models rely on different characteristics of temperature, precipitation and crop yield data allows us to match the model type to the available data. Second, we can prioritize where improvements in climate and crop yield data should be directed. Third, as better climate and crop yield data becomes available, we can predict how crop model skill should improve

An improved approach to measuring H_0 using X-ray and SZ observations of galaxy clusters

We present an improved method for predicting the Sunyaev-Zeldovich (SZ) effect in galaxy clusters from spatially-resolved, spectroscopic X-ray data. Using the deprojected electron density and temperature profiles measured within a fraction of the virial radius, and assuming a Navarro, Frenk & White (1995) mass model, we show how the pressure profile of the X-ray gas can be extrapolated to large radii, allowing the Comptonization parameter profile for the cluster to be predicted precisely. We apply our method to Chandra observations of three X-ray luminous, dynamically relaxed clusters with published SZ data: RX J1347.5-1145, Abell 1835 and Abell 478. Combining the predicted and observed SZ signals, we determine improved estimates for the Hubble constant from each cluster and obtain a weighted mean of H_0=69\pm8km/s/Mpc for a cosmology with Omega_m=0.3 and Omega_Lambda=0.7. This result is in good agreement with independent findings from the Hubble Key Project and the combination of cosmic microwave background and galaxy cluster data.Comment: 9 pages, 2 figures, 4 tables. Accepted for publication in MNRA