A general, gauge-invariant analysis of the cosmic microwave anisotropy

Abbott, L. F.; Schaefer, R. K.

doi:10.1086/164525

Cited by 137 publications

(186 citation statements)

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“…The final expression agrees with the expression given in [21], which was obtained using the radial decomposition of the tensor eigenfunctions [25]. Although the final result is not new, the simplicity of the derivation presented here demonstrates the utility of this approach and will in fact be used to derive tensor polarization power spectra.…”

Section: (T )supporting

confidence: 77%

All-sky analysis of polarization in the microwave background

1997

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Using the formalism of spin-weighted functions we present an all-sky analysis of polarization in the Cosmic Microwave Background (CMB). Linear polarization is a second-rank symmetric and traceless tensor, which can be decomposed on a sphere into spin ±2 spherical harmonics. These are the analog of the spherical harmonics used in the temperature maps and obey the same completeness and orthogonality relations. We show that there exist two linear combinations of spin ±2 multipole moments which have opposite parities and can be used to fully characterize the statistical properties of polarization in the CMB. Magnetic-type parity combination does not receieve contributions from scalar modes and does not cross-correlate with either temperature or electric-type parity combination, so there are four different power spectra that fully characterize statistical properties of CMB. We present their explicit expressions for scalar and tensor modes in the form of line of sight integral solution and numerically evaluate them for a representative set of models. These general solutions differ from the expressions obtained previously in the small scale limit both for scalar and tensor modes. A method to generate and analyze all sky maps of temperature and polarization is given and the optimal estimators for various power spectra and their corresponding variances are discussed.98.70.V, 98.80.C

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Section: (T )supporting

confidence: 77%

All-sky analysis of polarization in the microwave background

1997

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“…We can provide a formal solution to equation (3.32) for the intensity multipoles by noting that the homogeneous equations (obtained by setting n e = 0 and σ k = 0) are solved by the functions ℓ(ℓ − 1)Φ ν ℓ (x)/ sinh 2 (x) in an open universe, where x ≡ √ |K|(η R − η) with η R the conformal time at our current position R. Here, Φ ν ℓ (x) are the ultra-spherical Bessel functions [6]; see Appendix A also. Defining the optical depth back to cosmic time t, with t = t R at R:…”

Section: Analytic Solutions For E Ab σ Ab and I A ℓmentioning

confidence: 99%

“…Early ‡ A.D.Challinor@mrao.cam.ac.uk calculational schemes were based on linear perturbation theory in the synchronous gauge [20], with Legendre expansions of the relativistic distribution functions [3,5]. Some technical improvements in these schemes resulted from the use of gauge-invariant variables [6,21], and, more recently, from improved normal mode expansions of the distribution functions [22,23]. Adopting gauge-invariant variables frees one from the need to keep track of the implications of any residual gauge freedom in the calculation (the so called gauge problem; see e.g.…”

Section: Introductionmentioning

confidence: 99%

Microwave background anisotropies from gravitational waves: the 1 + 3 covariant approach

Challinor

2000

Class. Quantum Grav.

149

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Abstract. We present a 1+3 covariant discussion of the contribution of gravitational waves to the anisotropy of the cosmic microwave background radiation (CMB) in an almost-Friedmann-Robertson-Walker (FRW) universe. Our discussion is based in the covariant approach to perturbations in cosmology, which provides a physically transparent and gauge-invariant methodology for CMB physics. Applying this approach to linearised gravitational waves, we derive a closed set of covariant equations describing the evolution of the shear and the Weyl tensor, and the angular multipoles of the CMB intensity, valid for an arbitrary matter description and background spatial curvature. A significant feature of the present approach is that the normal mode expansion of the radiation distribution function, which arises naturally here, preserves the simple quadrupolar nature of the anisotropic part of the Thomson scattering source terms, and provides a direct characterisation of the power in the CMB at a given angular multipole, as in the recently introduced total angular momentum method. We provide the integral solution of the multipole equations, and analytic solutions for the shear and the Weyl tensor, for models with arbitrary spatial curvature. Numerical results for the CMB power spectrum in open models are also presented, which provide an independent verification of the calculations of other groups.

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“…ϕ χ becomes ϕ in the zero-shear gauge (often called the Newtonian gauge) which sets χ ≡ 0, etc. Using Bardeen's notation in [13] we have α χ ≡ Φ A and ϕ χ ≡ Φ H , [14]. * For the gauge transformation properties, see eq.…”

Section: Temperature Anisotropy In An Arbitrary Gaugementioning

confidence: 99%

1/3 factor in the CMB Sachs-Wolfe effect

Hwang

Padmanabhan

Lahav³

et al. 2002

Phys. Rev. D

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We point out that a pseudo-Newtonian interpretation of the (1/3) factor in the Sachs-Wolfe effect, which relates the fluctuations in temperature and potential, (δT /T ) = (1/3)δΦ, is not supported by the General Relativistic analysis. Dividing the full gravitational effect into separate parts depends on the choice of time slicing (gauge) and there exist infinitely many different choices. More importantly, interpreting the parts as being due to the gravitational redshift and the time dilation is not justified in the rigorous relativistic perturbation theory. One should regard the (1/3) factor as the General Relativistic result which applies in a restricted situation of adiabatic perturbation in the K = 0 = Λ model with the last scattering occuring in the matter dominated era. For an isocurvature initial condition the corresponding result, (δT /T ) = 2δΦ, has a different numerical coefficient. PACS number(s): 98.70.Vc, 98.80.Hw

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A general, gauge-invariant analysis of the cosmic microwave anisotropy

Cited by 137 publications

References 0 publications

All-sky analysis of polarization in the microwave background

All-sky analysis of polarization in the microwave background

Microwave background anisotropies from gravitational waves: the 1 + 3 covariant approach

1/3 factor in the CMB Sachs-Wolfe effect

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