Extended Limber Approximation

By M. Loverde, N. Afshordi. 2008

301 citations recorded by the source corpus.

We develop a systematic derivation for the Limber approximation to the angular cross-power spectrum of two random fields, as a series expansion in $(\ensuremath{\ell}+1/2{)}^{\ensuremath{-}1}$. This extended Limber approximation can be used to test the accuracy of the Limber approximation and to improve the rate of convergence at large $\ensuremath{\ell}$'s. We show that the error in ordinary Limber approximation is $\mathcal{O}({\ensuremath{\ell}}^{\ensuremath{-}2})$. We also provide a simple expression for the 2nd order correction to the Limber formula, which improves the accuracy to $\mathcal{O}({\ensuremath{\ell}}^{\ensuremath{-}4})$. This correction can be especially useful for narrow redshift bins, or samples with small redshift overlap, for which the 0th order Limber formula has a large error. We also point out that using $\ensuremath{\ell}$ instead of $\ensuremath{\ell}+1/2$, as is often done in the literature, spoils the accuracy of the approximation to $\mathcal{O}({\ensuremath{\ell}}^{\ensuremath{-}1})$.

Subfields: Artificial Intelligence, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/630f814168b688b7c26a1800e6b553c39d4bd165

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Citations in the corpus

CFHTLenS revisited: assessing concordance with Planck including astrophysical systematics (2016, methodology): “Using the Limber approximation (Limber 1954, also see Loverde & Afshordi 2008), the convergence power spectrum is then obtained as a weighted integral over the matter power spectrum, C ij GG( ) = ∫ χH 0 dχ qi(χ )qj (χ ) [fK (χ )]2 Pδδ ( + 1/2 fK (χ ) , χ ) , (2) where χ is the comoving distance, χH…”
KiDS-450 + 2dFLenS: Cosmological parameter constraints from weak gravitational lensing tomography and overlapping redshift-space galaxy clustering (2017, methodology): “Employing the extended Limber and flat-sky approximations (Limber 1954; Loverde & Afshordi 2008; Kilbinger et al. 2017), the shear power spectrum is then expressed as C ij GG( ) = ∫ χH 0 dχ qi(χ )qj (χ ) [fK (χ )]2 Pδδ ( + 1/2 fK (χ ) , χ ) , (3) where χ is the comoving distance, to the horizon in…”
Redshift-space distortions in Lagrangian perturbation theory (2020, methodology): “Moreover, for higher n terms the Limber approximation [83, 84] can be used”
KiDS-450: The tomographic weak lensing power spectrum and constraints on cosmological parameters (2017, unknown): “…prediction of the convergence power spectrum per redshift-bin correlation μ, ν in the (extended) Limber approximation (Limber 1953; Kaiser 1992; LoVerde & Afshordi 2008) can be written as CEEμν ( ) = ∫ χH 0 dχ qμ(χ )qν(χ ) f 2K(χ ) Pδ ( k = + 0.5 fK(χ ) ; χ ) , (5) which depends on the…”
The CLASSgal code for relativistic cosmological large scale structure (2013, background): “The transfer module contains a routine allowing to switch to the second-order Limber approximation [34], but we checked that the difference between the two is small when the condition (A.”
KiDS+GAMA : cosmology constraints from a joint analysis of cosmic shear, galaxy-galaxy lensing, and angular clustering (2017, methodology): “For all signals we employ the hybrid approximation proposed by Loverde & Afshordi (2008), which uses + 1/2 in the argument of the matter power spectrum but no additional prefactors.”