lensing

[Barun Pal]

Dear all,
I am recently studying correlation function method to calculate
lensed C_l following the calculation of arXiv: astro-ph/0502425 . In that paper the flat sky correlation function for temperature anisotropy is given by equation(17) which is an integral over "l". The integrand consists of [tex][ l/(2\pi) C_l \exp(-l^2 \sigma^2(r)/2)(.......)][/tex]. My question is how to perform the numerical integration for a given value of "r" as C_ls are known at integer points (l=2,3,4...) only.
Also what should be the range of [l] in that integration and when we calculate the lensed C_l's we need to perform an integration over "r"
what should be range of r.

I have looked into CAMB: lensing.f90 CorrFuncFlatSky subroutine
but did not understand the program specifically how the evaluation of
"sigmasq", "cgl2" are done

Code: Select all

do i=1,npoints-1

      theta = i * dtheta 
      sigmasq =0
      Cgl2=0
      fac = theta /dbessel
     
      do l=lmin,CP%Max_l

!Interpolate the Bessel functions, and compute sigma^2 and C_{gl,2} 
        b0 = l*fac
        b_lo = int(b0) +1 
        a0=  b_lo - b0                
        b0=  1._dl - a0 
        fac1 = a0*b0*dbessfac
        fac2 = fac1*(a0-2)
        fac1 = fac1*(b0-2)

        Bessel0(l) = a0*Bess0(b_lo)+ b0*Bess0(b_lo+1) +fac1*ddBess0(b_lo) &
                       +fac2*ddBess0(b_lo+1)
        sigmasq = sigmasq + (1-Bessel0(l))*Cphil3(l) 


        Bessel2(l) = a0*Bess2(b_lo)+ b0*Bess2(b_lo+1) +fac1*ddBess2(b_lo) &
                      +fac2*ddBess2(b_lo+1)
        Cgl2 =  Cgl2 + Bessel2(l)*Cphil3(l)

        Bessel4(l) = a0*Bess4(b_lo)+ b0*Bess4(b_lo+1) +fac1*ddBess4(b_lo) &
                      +fac2*ddBess4(b_lo+1)
        Bessel6(l) = a0*Bess6(b_lo)+ b0*Bess6(b_lo+1) +fac1*ddBess6(b_lo) &
                      +fac2*ddBess6(b_lo+1)

      end do

Thank you in advance for your reply.

[Antony Lewis]

I think the [tex]l[/tex] integration is done using integer sampling. The r integral in principle is over \pi, though the large angles don't actually contribute significantly.

[Barun Pal]

Dear Prof Lewis thank you very much for your reply.

So it is more like a summation over multipoles "l" rather than integration over "l".