[astro-ph/0512218] Weak Gravitational Lensing of High-Redshi
Posted: December 15 2005
This interesting paper includes a new way for computing the lensing effect on a power spectrum applicable when the power spectrum has lots of small scale power (unlike the CMB, the 21cm spectrum is nearly flat on small scales). They use this and show that the lensing effect on the 21cm power spectrum is very small and hard to detect.
This contrasts with using the shapes of resolved structures observed in 21cm (e.g. astro-ph/0305387), which is claimed to give very powerful constraints. I presume this corresponds to non-Gaussian signals which are not considered in this paper.
Regarding the new method for computing the lensing effect on the power sectrum: they make fully justified criticisms of the series expansion approach, which is indeed very poor in general. However the method of Seljak working with the correlation functions (astro-ph/9505109, reviewed recently in astro-ph/0502425) is in principle non-perturbative and much more accurate. Indeed in Sec 2.5 the authors say "what is physically important is the difference between the deflection angles at the two points", which is exactly what the Seljak method uses: it is based on a calculation of the correlation function of the difference of deflection angles (equivalent to shear and convergence only in the limit of small separation). I would therefore be interested to see whether Seljak's much simpler method would actually work at least as well.
The Seljak result for the lensed power spectrum can be written exactly as
[tex]
\tilde{C}_{l'} = \int r dr \int l dl J_0(l' r) C_l e^{-l^2\sigma^2(r)/2} \sum_{n=-\infty}^\infty I_n(l^2 C_{gl,2}(r)/2) J_{2 n}(l r)
[/tex]
(this form in terms of regular and modified Bessel functions is new - it corresponds to re-summing the series expansion given in the Seljak paper). Here \sigma^2(r) and C_{gl,2}(r) measure the correlation of the deflection angles, and for small [tex]r[/tex] are given by \langle \kappa^2\rangle r^2 and \langle \kappa^2\rangle r^2/2 respectively. This full result is non-perturbative in both terms, though the lowest series expansion given in Seljak is probably accurate at some level up to l ~ 10^4. The number of terms you need to keep in n is I suspect quite small: the Bessel functions are very small at high |n| unless the argument is large, so I'd have thought keeping just a few terms would be sufficient to compute the lensed 21cm spectrum accurately up to l ~ 10^5, though I haven't tried it. In principle the integral from which this result came could also be calculated directly numerically.
Of course the new method is interesting in its own right (though quite complicated!) even if other methods are available. Particularly nice to see the new result for a lensed white spectrum: the ~ 1+ 4\langle \kappa^2 \rangle scaling has been neglected previously.
One further comment: it's not clear to me from Fig. 1 that the convergence power \langle \kappa^2\rangle is actually very small (as stated in 2.2.1), though to a low cut off it would be. Because the spectrum is flattish on small scales in general it depends on what happens on small scales off the right of the figure. I think integrated up values of 0.1 are not unreasonable if very small scale power is included. So it's not neccessarily clear that expansions in \kappa are in general a very good idea unless they are otherwise regularlized (which they probably are).
This contrasts with using the shapes of resolved structures observed in 21cm (e.g. astro-ph/0305387), which is claimed to give very powerful constraints. I presume this corresponds to non-Gaussian signals which are not considered in this paper.
Regarding the new method for computing the lensing effect on the power sectrum: they make fully justified criticisms of the series expansion approach, which is indeed very poor in general. However the method of Seljak working with the correlation functions (astro-ph/9505109, reviewed recently in astro-ph/0502425) is in principle non-perturbative and much more accurate. Indeed in Sec 2.5 the authors say "what is physically important is the difference between the deflection angles at the two points", which is exactly what the Seljak method uses: it is based on a calculation of the correlation function of the difference of deflection angles (equivalent to shear and convergence only in the limit of small separation). I would therefore be interested to see whether Seljak's much simpler method would actually work at least as well.
The Seljak result for the lensed power spectrum can be written exactly as
[tex]
\tilde{C}_{l'} = \int r dr \int l dl J_0(l' r) C_l e^{-l^2\sigma^2(r)/2} \sum_{n=-\infty}^\infty I_n(l^2 C_{gl,2}(r)/2) J_{2 n}(l r)
[/tex]
(this form in terms of regular and modified Bessel functions is new - it corresponds to re-summing the series expansion given in the Seljak paper). Here \sigma^2(r) and C_{gl,2}(r) measure the correlation of the deflection angles, and for small [tex]r[/tex] are given by \langle \kappa^2\rangle r^2 and \langle \kappa^2\rangle r^2/2 respectively. This full result is non-perturbative in both terms, though the lowest series expansion given in Seljak is probably accurate at some level up to l ~ 10^4. The number of terms you need to keep in n is I suspect quite small: the Bessel functions are very small at high |n| unless the argument is large, so I'd have thought keeping just a few terms would be sufficient to compute the lensed 21cm spectrum accurately up to l ~ 10^5, though I haven't tried it. In principle the integral from which this result came could also be calculated directly numerically.
Of course the new method is interesting in its own right (though quite complicated!) even if other methods are available. Particularly nice to see the new result for a lensed white spectrum: the ~ 1+ 4\langle \kappa^2 \rangle scaling has been neglected previously.
One further comment: it's not clear to me from Fig. 1 that the convergence power \langle \kappa^2\rangle is actually very small (as stated in 2.2.1), though to a low cut off it would be. Because the spectrum is flattish on small scales in general it depends on what happens on small scales off the right of the figure. I think integrated up values of 0.1 are not unreasonable if very small scale power is included. So it's not neccessarily clear that expansions in \kappa are in general a very good idea unless they are otherwise regularlized (which they probably are).