| Published Articles | |
| Report number | arXiv:1807.09540 ; TTK-18-28 ; TTK-18-28 |
| Title | Beyond the traditional Line-of-Sight approach of cosmological angular statistics |
| Author(s) | Schöneberg, Nils (RWTH Aachen U.) ; Simonović, Marko (Princeton, Inst. Advanced Study ; CERN) ; Lesgourgues, Julien (RWTH Aachen U.) ; Zaldarriaga, Matias (Princeton, Inst. Advanced Study) |
| Publication | 2018-10-25 |
| Imprint | 2018-07-25 |
| Number of pages | 35 |
| Note | 40 pages, 6 figures; v2: one reference added |
| In: | JCAP 1810 (2018) 047 |
| DOI | 10.1088/1475-7516/2018/10/047 |
| Subject category | Astrophysics and Astronomy ; astro-ph.CO |
| Abstract | We present a new efficient method to compute the angular power spectra of large-scale structure observables that circumvents the numerical integration over Bessel functions, expanding on a recently proposed algorithm based on FFTlog. This new approach has better convergence properties. The method is explicitly implemented in the CLASS code for the case of number count $C_\ell$'s (including redshift-space distortions, weak lensing, and all other relativistic corrections) and cosmic shear $C_\ell$'s. In both cases our approach speeds up the calculation of the exact $C_\ell$'s (without the Limber approximation) by a factor of order 400 at a fixed precision target of 0.1%. |
| Copyright/License | preprint: (License: arXiv nonexclusive-distrib 1.0) |
Corresponding record in: Inspire
Registre creat el 2019-03-16, darrera modificació el 2023-10-04
Enllaços externs:
00006 Auto-correlation spectrum of number count (involving only the density source term) in one redshift bin defined by a Gaussian window function with mean redshift $\bar{z}=1.0$ and width $\Delta z = 0.05$\,.
00002 \textit{(Top)} Auto-correlation spectrum of number count (involving all source contributions) in one redshift bin defined by a Gaussian window function with mean redshift $\bar{z}=1.0$ and width $\Delta z = 0.05$\,. \textit{(Bottom)} Cross-correlation between two redshift bins defined by two Gaussian windows with $(\bar{z}_1, \Delta z_1) = (1.0, 0.05)$ and $(\bar{z}_2, \Delta z_2) = (1.25, 0.05)$.
00004 Auto-correlation spectrum of cosmic shear (or more precisely of the lensing potential $C_\ell^{\phi \phi}$) in one redshift bin defined by a Gaussian window function with mean redshift $\bar{z}=1.0$ and width $\Delta z = 0.05$\,.
00012 \textit{(Top)} Auto-correlation spectrum of number count (involving all source contributions) in one redshift bin defined by a Gaussian window function with mean redshift $\bar{z}=1.0$ and width $\Delta z = 0.05$\,. \textit{(Bottom)} Cross-correlation between two redshift bins defined by two Gaussian windows with $(\bar{z}_1, \Delta z_1) = (1.0, 0.05)$ and $(\bar{z}_2, \Delta z_2) = (1.25, 0.05)$.
00010 Number count spectra involving only density terms for a redshift bin centered at $\bar{z}=1.0$ with width $\Delta z=0.05$\,. \textit{(Top Left)} Total spectra w/o nonlinear corrections from Halofit and massive neutrinos with $M_\nu=1$~eV. \textit{(Top Right)} Impact of these two corrections on the power spectrum, computed as a relative difference (in \%) with respect to the linear spectrum of the massless neutrino model\,. \textit{(Bottom left)} Result of the new method with either the \tquote{full separability} or \tquote{semi-separability} approximations compared to the traditional line-of-sight approach. \textit{(Bottom right)} Relative difference (in \%) between the new and old methods. One can immediately see that the additional effects are well captured and the error remains at the sub-permille level.\\
00014 \textit{(Top)} Auto-correlation spectrum of number count (involving all source contributions) in one redshift bin defined by a Gaussian window function with mean redshift $\bar{z}=1.0$ and width $\Delta z = 0.05$\,. \textit{(Bottom)} Cross-correlation between two redshift bins defined by two Gaussian windows with $(\bar{z}_1, \Delta z_1) = (1.0, 0.05)$ and $(\bar{z}_2, \Delta z_2) = (1.25, 0.05)$.
00005 Number count spectra involving only density terms for a redshift bin centered at $\bar{z}=1.0$ with width $\Delta z=0.05$\,. \textit{(Top Left)} Total spectra w/o nonlinear corrections from Halofit and massive neutrinos with $M_\nu=1$~eV. \textit{(Top Right)} Impact of these two corrections on the power spectrum, computed as a relative difference (in \%) with respect to the linear spectrum of the massless neutrino model\,. \textit{(Bottom left)} Result of the new method with either the \tquote{full separability} or \tquote{semi-separability} approximations compared to the traditional line-of-sight approach. \textit{(Bottom right)} Relative difference (in \%) between the new and old methods. One can immediately see that the additional effects are well captured and the error remains at the sub-permille level.\\
00000 Auto-correlation spectrum of cosmic shear (or more precisely of the lensing potential $C_\ell^{\phi \phi}$) in one redshift bin defined by a Gaussian window function with mean redshift $\bar{z}=1.0$ and width $\Delta z = 0.05$\,.
00007 Auto-correlation spectrum of number count (involving only the density source term) in one redshift bin defined by a Gaussian window function with mean redshift $\bar{z}=1.0$ and width $\Delta z = 0.05$\,.
00008 \textit{(Top)} Auto-correlation spectrum of number count (involving all source contributions) in one redshift bin defined by a Gaussian window function with mean redshift $\bar{z}=1.0$ and width $\Delta z = 0.05$\,. \textit{(Bottom)} Cross-correlation between two redshift bins defined by two Gaussian windows with $(\bar{z}_1, \Delta z_1) = (1.0, 0.05)$ and $(\bar{z}_2, \Delta z_2) = (1.25, 0.05)$.
00001 Number count spectra involving only density terms for a redshift bin centered at $\bar{z}=1.0$ with width $\Delta z=0.05$\,. \textit{(Top Left)} Total spectra w/o nonlinear corrections from Halofit and massive neutrinos with $M_\nu=1$~eV. \textit{(Top Right)} Impact of these two corrections on the power spectrum, computed as a relative difference (in \%) with respect to the linear spectrum of the massless neutrino model\,. \textit{(Bottom left)} Result of the new method with either the \tquote{full separability} or \tquote{semi-separability} approximations compared to the traditional line-of-sight approach. \textit{(Bottom right)} Relative difference (in \%) between the new and old methods. One can immediately see that the additional effects are well captured and the error remains at the sub-permille level.\\
00009 Number count spectra involving only density terms for a redshift bin centered at $\bar{z}=1.0$ with width $\Delta z=0.05$\,. \textit{(Top Left)} Total spectra w/o nonlinear corrections from Halofit and massive neutrinos with $M_\nu=1$~eV. \textit{(Top Right)} Impact of these two corrections on the power spectrum, computed as a relative difference (in \%) with respect to the linear spectrum of the massless neutrino model\,. \textit{(Bottom left)} Result of the new method with either the \tquote{full separability} or \tquote{semi-separability} approximations compared to the traditional line-of-sight approach. \textit{(Bottom right)} Relative difference (in \%) between the new and old methods. One can immediately see that the additional effects are well captured and the error remains at the sub-permille level.\\
00003 Another consequence of the Limber limit: For large $\ell$ the $t_{min}$ parameter behaves as $1/\ell$ (left), and the $|I_\ell(\nu,1)/\ell^{\nu-2}|$ is constant as in equation \ref{eq_Il_limit} (right). Note that the oscillations due to imaginary $\nu$ are correctly captured and the relative size approaches the correct constant. The black lines indicate the behavior for $\nu=-2.1+30i$ and $\epsilon=10^{-4}$, while the grey lines specify asymptotes. On the left, the grey line is $\ell^{-1}$ times an arbitrary constant (here $30/\ell$), while on the right side the constant is fixed by \ref{eq_Il_limit}. The constant for $t_{min}$ is not exactly $\log(1/\epsilon)$ because of the influence of the hypergeometric function.
00011 An illustration of the Limber limit: For large $\ell$ the area under the curve $I_\ell(\nu,t)$ approaches $\pi^2 \ell^{\nu-3}$ when integrated from $0$ to $1$. We see that the factor $\ell^{3-\nu} \, I_\ell(\nu,t)$ approaches the constant $\pi^2$, which is an equivalent statement.
00013 Another consequence of the Limber limit: For large $\ell$ the $t_{min}$ parameter behaves as $1/\ell$ (left), and the $|I_\ell(\nu,1)/\ell^{\nu-2}|$ is constant as in equation \ref{eq_Il_limit} (right). Note that the oscillations due to imaginary $\nu$ are correctly captured and the relative size approaches the correct constant. The black lines indicate the behavior for $\nu=-2.1+30i$ and $\epsilon=10^{-4}$, while the grey lines specify asymptotes. On the left, the grey line is $\ell^{-1}$ times an arbitrary constant (here $30/\ell$), while on the right side the constant is fixed by \ref{eq_Il_limit}. The constant for $t_{min}$ is not exactly $\log(1/\epsilon)$ because of the influence of the hypergeometric function.
00006 Auto-correlation spectrum of number count (involving only the density source term) in one redshift bin defined by a Gaussian window function with mean redshift $\bar{z}=1.0$ and width $\Delta z = 0.05$\,.
00002 \textit{(Top)} Auto-correlation spectrum of number count (involving all source contributions) in one redshift bin defined by a Gaussian window function with mean redshift $\bar{z}=1.0$ and width $\Delta z = 0.05$\,. \textit{(Bottom)} Cross-correlation between two redshift bins defined by two Gaussian windows with $(\bar{z}_1, \Delta z_1) = (1.0, 0.05)$ and $(\bar{z}_2, \Delta z_2) = (1.25, 0.05)$.
00004 Auto-correlation spectrum of cosmic shear (or more precisely of the lensing potential $C_\ell^{\phi \phi}$) in one redshift bin defined by a Gaussian window function with mean redshift $\bar{z}=1.0$ and width $\Delta z = 0.05$\,.
00012 \textit{(Top)} Auto-correlation spectrum of number count (involving all source contributions) in one redshift bin defined by a Gaussian window function with mean redshift $\bar{z}=1.0$ and width $\Delta z = 0.05$\,. \textit{(Bottom)} Cross-correlation between two redshift bins defined by two Gaussian windows with $(\bar{z}_1, \Delta z_1) = (1.0, 0.05)$ and $(\bar{z}_2, \Delta z_2) = (1.25, 0.05)$.
00010 Number count spectra involving only density terms for a redshift bin centered at $\bar{z}=1.0$ with width $\Delta z=0.05$\,. \textit{(Top Left)} Total spectra w/o nonlinear corrections from Halofit and massive neutrinos with $M_\nu=1$~eV. \textit{(Top Right)} Impact of these two corrections on the power spectrum, computed as a relative difference (in \%) with respect to the linear spectrum of the massless neutrino model\,. \textit{(Bottom left)} Result of the new method with either the \tquote{full separability} or \tquote{semi-separability} approximations compared to the traditional line-of-sight approach. \textit{(Bottom right)} Relative difference (in \%) between the new and old methods. One can immediately see that the additional effects are well captured and the error remains at the sub-permille level.\\
00014 \textit{(Top)} Auto-correlation spectrum of number count (involving all source contributions) in one redshift bin defined by a Gaussian window function with mean redshift $\bar{z}=1.0$ and width $\Delta z = 0.05$\,. \textit{(Bottom)} Cross-correlation between two redshift bins defined by two Gaussian windows with $(\bar{z}_1, \Delta z_1) = (1.0, 0.05)$ and $(\bar{z}_2, \Delta z_2) = (1.25, 0.05)$.
00005 Number count spectra involving only density terms for a redshift bin centered at $\bar{z}=1.0$ with width $\Delta z=0.05$\,. \textit{(Top Left)} Total spectra w/o nonlinear corrections from Halofit and massive neutrinos with $M_\nu=1$~eV. \textit{(Top Right)} Impact of these two corrections on the power spectrum, computed as a relative difference (in \%) with respect to the linear spectrum of the massless neutrino model\,. \textit{(Bottom left)} Result of the new method with either the \tquote{full separability} or \tquote{semi-separability} approximations compared to the traditional line-of-sight approach. \textit{(Bottom right)} Relative difference (in \%) between the new and old methods. One can immediately see that the additional effects are well captured and the error remains at the sub-permille level.\\
00000 Auto-correlation spectrum of cosmic shear (or more precisely of the lensing potential $C_\ell^{\phi \phi}$) in one redshift bin defined by a Gaussian window function with mean redshift $\bar{z}=1.0$ and width $\Delta z = 0.05$\,.
00007 Auto-correlation spectrum of number count (involving only the density source term) in one redshift bin defined by a Gaussian window function with mean redshift $\bar{z}=1.0$ and width $\Delta z = 0.05$\,.
00008 \textit{(Top)} Auto-correlation spectrum of number count (involving all source contributions) in one redshift bin defined by a Gaussian window function with mean redshift $\bar{z}=1.0$ and width $\Delta z = 0.05$\,. \textit{(Bottom)} Cross-correlation between two redshift bins defined by two Gaussian windows with $(\bar{z}_1, \Delta z_1) = (1.0, 0.05)$ and $(\bar{z}_2, \Delta z_2) = (1.25, 0.05)$.
00001 Number count spectra involving only density terms for a redshift bin centered at $\bar{z}=1.0$ with width $\Delta z=0.05$\,. \textit{(Top Left)} Total spectra w/o nonlinear corrections from Halofit and massive neutrinos with $M_\nu=1$~eV. \textit{(Top Right)} Impact of these two corrections on the power spectrum, computed as a relative difference (in \%) with respect to the linear spectrum of the massless neutrino model\,. \textit{(Bottom left)} Result of the new method with either the \tquote{full separability} or \tquote{semi-separability} approximations compared to the traditional line-of-sight approach. \textit{(Bottom right)} Relative difference (in \%) between the new and old methods. One can immediately see that the additional effects are well captured and the error remains at the sub-permille level.\\
00009 Number count spectra involving only density terms for a redshift bin centered at $\bar{z}=1.0$ with width $\Delta z=0.05$\,. \textit{(Top Left)} Total spectra w/o nonlinear corrections from Halofit and massive neutrinos with $M_\nu=1$~eV. \textit{(Top Right)} Impact of these two corrections on the power spectrum, computed as a relative difference (in \%) with respect to the linear spectrum of the massless neutrino model\,. \textit{(Bottom left)} Result of the new method with either the \tquote{full separability} or \tquote{semi-separability} approximations compared to the traditional line-of-sight approach. \textit{(Bottom right)} Relative difference (in \%) between the new and old methods. One can immediately see that the additional effects are well captured and the error remains at the sub-permille level.\\
00003 Another consequence of the Limber limit: For large $\ell$ the $t_{min}$ parameter behaves as $1/\ell$ (left), and the $|I_\ell(\nu,1)/\ell^{\nu-2}|$ is constant as in equation \ref{eq_Il_limit} (right). Note that the oscillations due to imaginary $\nu$ are correctly captured and the relative size approaches the correct constant. The black lines indicate the behavior for $\nu=-2.1+30i$ and $\epsilon=10^{-4}$, while the grey lines specify asymptotes. On the left, the grey line is $\ell^{-1}$ times an arbitrary constant (here $30/\ell$), while on the right side the constant is fixed by \ref{eq_Il_limit}. The constant for $t_{min}$ is not exactly $\log(1/\epsilon)$ because of the influence of the hypergeometric function.
00011 An illustration of the Limber limit: For large $\ell$ the area under the curve $I_\ell(\nu,t)$ approaches $\pi^2 \ell^{\nu-3}$ when integrated from $0$ to $1$. We see that the factor $\ell^{3-\nu} \, I_\ell(\nu,t)$ approaches the constant $\pi^2$, which is an equivalent statement.
00013 Another consequence of the Limber limit: For large $\ell$ the $t_{min}$ parameter behaves as $1/\ell$ (left), and the $|I_\ell(\nu,1)/\ell^{\nu-2}|$ is constant as in equation \ref{eq_Il_limit} (right). Note that the oscillations due to imaginary $\nu$ are correctly captured and the relative size approaches the correct constant. The black lines indicate the behavior for $\nu=-2.1+30i$ and $\epsilon=10^{-4}$, while the grey lines specify asymptotes. On the left, the grey line is $\ell^{-1}$ times an arbitrary constant (here $30/\ell$), while on the right side the constant is fixed by \ref{eq_Il_limit}. The constant for $t_{min}$ is not exactly $\log(1/\epsilon)$ because of the influence of the hypergeometric function.