Analytic Spectra of CMB Anisotropies and Polarization Generated by Scalar Perturbations in Synchronous Gauge

Abstract

We present a detailed analytical calculation of CMB temperature anisotropies α_k and polarization β_k generated by scalar metric perturbations in synchronous gauge, parallel to our previous work with RGW as a generating source. This is realized primarily by an analytic time-integration of Boltzmann's equation, yielding the closed forms of α_k and β_k. Approximations, such as the tight-coupling approximation for photons a prior to the recombination and the long wavelength limit for scalar perturbations are used. The residual gauge modes in scalar perturbations are analyzed and a proper joining condition of scalar perturbations at the radiation-matter equality is chosen, ensuring the continuity of energy perturbation. The resulting analytic expressions of the multipole moments of polarization a^E_l, and of temperature anisotropies a^T_l are explicit functions of the scalar perturbations, recombination time, recombination width, photon free streaming damping factor, baryon fraction, initial amplitude, primordial scalar spectral index, and the running index. These results show that a longer recombination width yields higher amplitudes of polarization on large scales and more damping on small scales, and that a late recombination time shifts the peaks of C^{XX'}_l to larger angular scales. The analytic spectra C^{XX'}_l agree with the numerical ones and with those observed by WMAP on large scales (l \lesssim 500), but deviate considerably from the numerical results on smaller scales, showing the limitations of our approximate analytic calculations. Several possible improvements are pointed out for further studies.

Accepted by Classical and Quantum Gravity. 43 pages, 15 figures

References in corpus (29)

• Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Data Processing, Sky Maps, and Basic Results
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Likelihoods and Parameters from the WMAP data
• Three Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Polarization Analysis
• Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Temperature Analysis
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Angular Power Spectra
• Polarization Observations with the Cosmic Background Imager
• Relic Gravitational Waves in the Accelerating Universe
• Imprints of Relic Gravitational Waves in Cosmic Microwave Background Radiation
• First season QUaD CMB temperature and polarization power spectra
• Analytic spectrum of relic gravitational waves modified by neutrino free streaming and dark energy
• An exact analytic spectrum of relic gravitational waves in an accelerating universe
• Relic Gravitational Waves with A Running Spectral Index and Its Constraints at High Frequencies
• Separating E and B types of polarization on an incomplete sky
• Detecting relic gravitational waves in the CMB: Optimal parameters and their constraints
• The Maximum B-mode Polarization of the Cosmic Microwave Background from Inhomogeneous Reionization
• Modifications by QCD transition and $e^{+}{e}^{-}$ annihilation on analytic spectrum of relic gravitational waves in accelerating universe
• Approximate Analytic Spectra of Reionized CMB Anisotropies and Polarization generated by Relic Gravitational Waves
• The Polarization of the Cosmic Microwave Background Due to Primordial Gravitational Waves
• CMB Temperature Polarization Correlation and Primordial Gravitational Waves
• Constraints upon the spectral indices of relic gravitational waves by LIGO S5
• Cosmological Parameters from the QUaD CMB polarization experiment
• Relic gravitational waves: latest revisions and preparations for new data
• Detecting relic gravitational waves in the CMB: Comparison of different methods
• Stable indications of relic gravitational waves in Wilkinson Microwave Anisotropy Probe data and forecasts for the Planck mission
• Relic Gravitational Waves And CMB Polarization In The Accelerating Universe
• An analytic approach to baryon acoustic oscillations
• Observational Scan Induced Artificial CMB Anisotropy

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