`flatsky` – Flatsky analysis tools

`rec_lens` – quadratic lensing reconstruction

cmblensplus.flatsky.rec_lens.qtt(nx, ny, D, rL, fC, T1, T2, gtype='')

Reconstruct the CMB lensing potential and curl mode from the temperature quadratic estimator.

Parameters

nxint: Number of Fourier grid points in the x direction.
nyint: Number of Fourier grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
fCndarray of float, shape (nx, ny): Temperature power spectrum on the 2D Fourier grid.
T1ndarray of complex, shape (nx, ny): Fourier modes of the first inverse-variance filtered temperature map.
T2ndarray of complex, shape (nx, ny): Fourier modes of the second inverse-variance filtered temperature map.
gtypestr, optional: Type of output. Use 'k' for convergence or '' for lensing potential. Default is ''.

Returns

glmndarray of complex, shape (nx, ny): Fourier modes of the CMB lensing potential.
clmndarray of complex, shape (nx, ny): Fourier modes of the curl mode, or pseudo lensing potential.

cmblensplus.flatsky.rec_lens.qte(nx, ny, D, rL, fC, T, E, gtype='')

Reconstruct the CMB lensing potential and curl mode from the suboptimal TE quadratic estimator.

Parameters

nxint: Number of Fourier grid points in the x direction.
nyint: Number of Fourier grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
fCndarray of float, shape (nx, ny): TE cross-power spectrum on the 2D Fourier grid.
Tndarray of complex, shape (nx, ny): Fourier modes of the inverse-variance filtered temperature map.
Endarray of complex, shape (nx, ny): Fourier modes of the inverse-variance filtered E-mode map.
gtypestr, optional: Type of output. Use 'k' for convergence or '' for lensing potential. Default is ''.

Returns

glmndarray of complex, shape (nx, ny): Fourier modes of the CMB lensing potential.
clmndarray of complex, shape (nx, ny): Fourier modes of the curl mode, or pseudo lensing potential.

cmblensplus.flatsky.rec_lens.qtb(nx, ny, D, rL, fC, T, B, gtype='')

Reconstruct the CMB lensing potential and curl mode from the TB quadratic estimator.

Parameters

nxint: Number of Fourier grid points in the x direction.
nyint: Number of Fourier grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
fCndarray of float, shape (nx, ny): TE cross-power spectrum on the 2D Fourier grid.
Tndarray of complex, shape (nx, ny): Fourier modes of the inverse-variance filtered temperature map.
Bndarray of complex, shape (nx, ny): Fourier modes of the inverse-variance filtered B-mode map.
gtypestr, optional: Type of output. Use 'k' for convergence or '' for lensing potential. Default is ''.

Returns

glmndarray of complex, shape (nx, ny): Fourier modes of the CMB lensing potential.
clmndarray of complex, shape (nx, ny): Fourier modes of the curl mode, or pseudo lensing potential.

cmblensplus.flatsky.rec_lens.qee(nx, ny, D, rL, fC, E1, E2, gtype='')

Reconstruct the CMB lensing potential and curl mode from the EE quadratic estimator.

Parameters

nxint: Number of Fourier grid points in the x direction.
nyint: Number of Fourier grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
fCndarray of float, shape (nx, ny): EE power spectrum on the 2D Fourier grid.
E1ndarray of complex, shape (nx, ny): Fourier modes of the first inverse-variance filtered E-mode map.
E2ndarray of complex, shape (nx, ny): Fourier modes of the second inverse-variance filtered E-mode map.
gtypestr, optional: Type of output. Use 'k' for convergence or '' for lensing potential. Default is ''.

Returns

glmndarray of complex, shape (nx, ny): Fourier modes of the CMB lensing potential.
clmndarray of complex, shape (nx, ny): Fourier modes of the curl mode, or pseudo lensing potential.

cmblensplus.flatsky.rec_lens.qeb(nx, ny, D, rL, fC, E, B, gtype='')

Reconstruct the CMB lensing potential and curl mode from the EB quadratic estimator.

Parameters

nxint: Number of Fourier grid points in the x direction.
nyint: Number of Fourier grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
fCndarray of float, shape (nx, ny): EE power spectrum on the 2D Fourier grid.
Endarray of complex, shape (nx, ny): Fourier modes of the inverse-variance filtered E-mode map.
Bndarray of complex, shape (nx, ny): Fourier modes of the inverse-variance filtered B-mode map.
gtypestr, optional: Type of output. Use 'k' for convergence or '' for lensing potential. Default is ''.

Returns

glmndarray of complex, shape (nx, ny): Fourier modes of the CMB lensing potential.
clmndarray of complex, shape (nx, ny): Fourier modes of the curl mode, or pseudo lensing potential.

cmblensplus.flatsky.rec_lens.qbb(nx, ny, D, rL, fC, B1, B2, gtype='')

Reconstruct the CMB lensing potential and curl mode from the BB quadratic estimator.

Parameters

nxint: Number of Fourier grid points in the x direction.
nyint: Number of Fourier grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
fCndarray of float, shape (nx, ny): BB power spectrum on the 2D Fourier grid.
B1ndarray of complex, shape (nx, ny): Fourier modes of the first inverse-variance filtered B-mode map.
B2ndarray of complex, shape (nx, ny): Fourier modes of the second inverse-variance filtered B-mode map.
gtypestr, optional: Type of output. Use 'k' for convergence or '' for lensing potential. Default is ''.

Returns

glmndarray of complex, shape (nx, ny): Fourier modes of the CMB lensing potential.
clmndarray of complex, shape (nx, ny): Fourier modes of the curl mode, or pseudo lensing potential.

`rec_rot` – quadratic pol. rot. reconstruction

cmblensplus.flatsky.rec_rot.qte(nx, ny, D, rL, fC, T, E)

Reconstruct anisotropic polarization-rotation angles from the TE quadratic estimator.

Parameters

nxint: Number of Fourier grid points in the x direction.
nyint: Number of Fourier grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalently dLx / (2 * pi) and dLy / (2 * pi).
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
fCndarray of float, shape (nx, ny): TE cross-power spectrum on the 2D grid.
Tndarray of complex, shape (nx, ny): 2D Fourier modes of the inverse-variance filtered temperature map.
Endarray of complex, shape (nx, ny): 2D Fourier modes of the inverse-variance filtered E-mode map.

Returns

almndarray of complex, shape (nx, ny): 2D Fourier modes of the anisotropic polarization-rotation angles.

cmblensplus.flatsky.rec_rot.qtb(nx, ny, D, rL, fC, T, B)

Reconstruct anisotropic polarization-rotation angles from the TB quadratic estimator.

Parameters

nxint: Number of Fourier grid points in the x direction.
nyint: Number of Fourier grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalently dLx / (2 * pi) and dLy / (2 * pi).
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
fCndarray of float, shape (nx, ny): TE cross-power spectrum on the 2D grid.
Tndarray of complex, shape (nx, ny): 2D Fourier modes of the inverse-variance filtered temperature map.
Bndarray of complex, shape (nx, ny): 2D Fourier modes of the inverse-variance filtered B-mode map.

Returns

almndarray of complex, shape (nx, ny): 2D Fourier modes of the anisotropic polarization-rotation angles.

cmblensplus.flatsky.rec_rot.qee(nx, ny, D, rL, fC, E1, E2)

Reconstruct anisotropic polarization-rotation angles from the EE quadratic estimator.

Parameters

nxint: Number of Fourier grid points in the x direction.
nyint: Number of Fourier grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalently dLx / (2 * pi) and dLy / (2 * pi).
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
fCndarray of float, shape (nx, ny): EE power spectrum on the 2D grid.
E1ndarray of complex, shape (nx, ny): 2D Fourier modes of the first inverse-variance filtered E-mode map.
E2ndarray of complex, shape (nx, ny): 2D Fourier modes of the second inverse-variance filtered E-mode map.

Returns

almndarray of complex, shape (nx, ny): 2D Fourier modes of the anisotropic polarization-rotation angles.

cmblensplus.flatsky.rec_rot.qeb(nx, ny, D, rL, EE, E, B, BB=0)

Reconstruct anisotropic polarization-rotation angles from the EB quadratic estimator.

Parameters

nxint: Number of Fourier grid points in the x direction.
nyint: Number of Fourier grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalently dLx / (2 * pi) and dLy / (2 * pi).
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
EEndarray of float, shape (nx, ny): EE power spectrum on the 2D grid.
Endarray of complex, shape (nx, ny): 2D Fourier modes of the inverse-variance filtered E-mode map.
Bndarray of complex, shape (nx, ny): 2D Fourier modes of the inverse-variance filtered B-mode map.
BBndarray of float or float, optional: Theory B-mode spectrum on the 2D grid. If a scalar is given, it is applied as a constant. Default is 0.

Returns

almndarray of complex, shape (nx, ny): 2D Fourier modes of the anisotropic polarization-rotation angles.

`rec_tau` – quadratic patchy tau reconstruction

cmblensplus.flatsky.rec_tau.qtt(nx, ny, D, rL, fC, T1, T2)

Reconstruct patchy tau from the temperature quadratic estimator.

Parameters

nxint: Number of Fourier grid points along the x direction.
nyint: Number of Fourier grid points along the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
fCndarray of float, shape (nx, ny): Temperature power spectrum on the 2D Fourier grid.
T1ndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the first inverse-variance filtered temperature field.
T2ndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the second inverse-variance filtered temperature field.

Returns

tlmndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of patchy tau.

cmblensplus.flatsky.rec_tau.qte(nx, ny, D, rL, fC, T, E)

Reconstruct patchy tau from the suboptimal TE quadratic estimator.

Parameters

nxint: Number of Fourier grid points along the x direction.
nyint: Number of Fourier grid points along the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
fCndarray of float, shape (nx, ny): TE cross-power spectrum on the 2D Fourier grid.
Tndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the inverse-variance filtered temperature field.
Endarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the inverse-variance filtered E-mode field.

Returns

tlmndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of patchy tau.

cmblensplus.flatsky.rec_tau.qtb(nx, ny, D, rL, fC, T, B)

Reconstruct patchy tau from the TB quadratic estimator.

Parameters

nxint: Number of Fourier grid points along the x direction.
nyint: Number of Fourier grid points along the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
fCndarray of float, shape (nx, ny): TE cross-power spectrum on the 2D Fourier grid.
Tndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the inverse-variance filtered temperature field.
Bndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the inverse-variance filtered B-mode field.

Returns

tlmndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of patchy tau.

cmblensplus.flatsky.rec_tau.qee(nx, ny, D, rL, fC, E1, E2)

Reconstruct patchy tau from the EE quadratic estimator.

Parameters

nxint: Number of Fourier grid points along the x direction.
nyint: Number of Fourier grid points along the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
fCndarray of float, shape (nx, ny): EE power spectrum on the 2D Fourier grid.
E1ndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the first inverse-variance filtered E-mode field.
E2ndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the second inverse-variance filtered E-mode field.

Returns

tlmndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of patchy tau.

cmblensplus.flatsky.rec_tau.qeb(nx, ny, D, rL, fE, fB, E, B)

Reconstruct patchy tau from the EB quadratic estimator.

Parameters

nxint: Number of Fourier grid points along the x direction.
nyint: Number of Fourier grid points along the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
fEndarray of float, shape (nx, ny): EE power spectrum on the 2D Fourier grid.
fBndarray of float, shape (nx, ny): BB power spectrum on the 2D Fourier grid.
Endarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the inverse-variance filtered E-mode field.
Bndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the inverse-variance filtered B-mode field.

Returns

tlmndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of patchy tau.

`rec_src` – quadratic point-src reconstruction

cmblensplus.flatsky.rec_src.qtt(nx, ny, D, rL, T1, T2)

Reconstruct point-source fields from the temperature quadratic estimator.

Parameters

nxint: Number of grids along the x direction.
nyint: Number of grids along the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi), with bounds 0:1.
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
T1ndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the first inverse-variance filtered temperature map.
T2ndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the second inverse-variance filtered temperature map.

Returns

slmndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the point-source field.

cmblensplus.flatsky.rec_src.qte(nx, ny, D, rL, T, E)

Reconstruct point-source fields from the suboptimal TE quadratic estimator.

Parameters

nxint: Number of grids along the x direction.
nyint: Number of grids along the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi), with bounds 0:1.
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
Tndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the inverse-variance filtered temperature map.
Endarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the inverse-variance filtered E-mode map.

Returns

slmndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the point-source field.

cmblensplus.flatsky.rec_src.qtb(nx, ny, D, rL, T, B)

Reconstruct point-source fields from the TB quadratic estimator.

Parameters

nxint: Number of grids along the x direction.
nyint: Number of grids along the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi), with bounds 0:1.
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
Tndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the inverse-variance filtered temperature map.
Bndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the inverse-variance filtered B-mode map.

Returns

slmndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the point-source field.

cmblensplus.flatsky.rec_src.qee(nx, ny, D, rL, E1, E2)

Reconstruct point-source fields from the EE quadratic estimator.

Parameters

nxint: Number of grids along the x direction.
nyint: Number of grids along the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi), with bounds 0:1.
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
E1ndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the first inverse-variance filtered E-mode map.
E2ndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the second inverse-variance filtered E-mode map.

Returns

slmndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the point-source field.

cmblensplus.flatsky.rec_src.qeb(nx, ny, D, rL, E, B)

Reconstruct point-source fields from the EB quadratic estimator.

Parameters

nxint: Number of grids along the x direction.
nyint: Number of grids along the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi), with bounds 0:1.
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
Endarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the inverse-variance filtered E-mode map.
Bndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the inverse-variance filtered B-mode map.

Returns

slmndarray of complex, shape (nx, ny): Two-dimensional Fourier modes of the point-source field.

`norm_lens` – normalization of quadratic lensing reconstruction

cmblensplus.flatsky.norm_lens.qtt(nx, ny, D, rL, OT, TT, eL)

Return the normalization of the temperature quadratic estimator for the CMB lensing potential and curl mode.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi), with bounds 0:1.
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
OTndarray of float, shape (nx, ny): Inverse observed temperature power spectrum on the 2D grid, with bounds (nx, ny).
TTndarray of float, shape (nx, ny): Theoretical temperature power spectrum on the 2D grid, with bounds (nx, ny).
eLarray_like of int, shape (2,): Minimum and maximum multipoles of the output normalization spectrum.

Returns

Agndarray of complex, shape (nx, ny): Normalization of the CMB lensing potential on the 2D grid.
Acndarray of complex, shape (nx, ny): Normalization of the curl mode, or pseudo lensing potential, on the 2D grid.

cmblensplus.flatsky.norm_lens.n0tt(nx, ny, D, rL, OT0, OT1, TT, eL)

Return the disconnected noise normalization of the temperature quadratic estimator for the CMB lensing potential and curl mode.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi), with bounds 0:1.
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
OT0ndarray of float, shape (nx, ny): First inverse observed temperature power spectrum on the 2D grid.
OT1ndarray of float, shape (nx, ny): Second inverse observed temperature power spectrum on the 2D grid.
TTndarray of float, shape (nx, ny): Theoretical temperature power spectrum on the 2D grid.
eLarray_like of int, shape (2,): Minimum and maximum multipoles of the output normalization spectrum.

Returns

Agndarray of complex, shape (nx, ny): Normalization of the CMB lensing potential on the 2D grid.
Acndarray of complex, shape (nx, ny): Normalization of the curl mode, or pseudo lensing potential, on the 2D grid.

cmblensplus.flatsky.norm_lens.n0ttc(nx, ny, D, rL, OT0, OT1, TT, eL)

Return the disconnected noise normalization of the temperature quadratic estimator for the CMB lensing potential and curl mode, allowing complex inverse observed spectra.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi), with bounds 0:1.
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
OT0ndarray of complex, shape (nx, ny): First inverse observed temperature power spectrum on the 2D grid.
OT1ndarray of complex, shape (nx, ny): Second inverse observed temperature power spectrum on the 2D grid.
TTndarray of float, shape (nx, ny): Theoretical temperature power spectrum on the 2D grid.
eLarray_like of int, shape (2,): Minimum and maximum multipoles of the output normalization spectrum.

Returns

Agndarray of complex, shape (nx, ny): Normalization of the CMB lensing potential on the 2D grid.
Acndarray of complex, shape (nx, ny): Normalization of the curl mode, or pseudo lensing potential, on the 2D grid.

cmblensplus.flatsky.norm_lens.qte(nx, ny, D, rL, OT, OE, TE, eL)

Return the normalization of the TE quadratic estimator for the CMB lensing potential and curl mode.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi), with bounds 0:1.
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
OTndarray of float, shape (nx, ny): Inverse observed temperature power spectrum on the 2D grid.
OEndarray of float, shape (nx, ny): Inverse observed E-mode power spectrum on the 2D grid.
TEndarray of float, shape (nx, ny): Theoretical TE cross spectrum on the 2D grid.
eLarray_like of int, shape (2,): Minimum and maximum multipoles of the output normalization spectrum.

Returns

Agndarray of complex, shape (nx, ny): Normalization of the CMB lensing potential on the 2D grid.
Acndarray of complex, shape (nx, ny): Normalization of the curl mode, or pseudo lensing potential, on the 2D grid.

cmblensplus.flatsky.norm_lens.qtb(nx, ny, D, OT, OB, TE, rL, eL)

Return the normalization of the TB quadratic estimator for the CMB lensing potential and curl mode.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi), with bounds 0:1.
OTndarray of float, shape (nx, ny): Inverse observed temperature power spectrum on the 2D grid.
OBndarray of float, shape (nx, ny): Inverse observed B-mode power spectrum on the 2D grid.
TEndarray of float, shape (nx, ny): Theoretical TE cross spectrum on the 2D grid.
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
eLarray_like of int, shape (2,): Minimum and maximum multipoles of the output normalization spectrum.

Returns

Agndarray of complex, shape (nx, ny): Normalization of the CMB lensing potential on the 2D grid.
Acndarray of complex, shape (nx, ny): Normalization of the curl mode, or pseudo lensing potential, on the 2D grid.

cmblensplus.flatsky.norm_lens.qee(nx, ny, D, OE, EE, rL, eL)

Return the normalization of the EE quadratic estimator for the CMB lensing potential and curl mode.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi), with bounds 0:1.
OEndarray of float, shape (nx, ny): Inverse observed E-mode power spectrum on the 2D grid.
EEndarray of float, shape (nx, ny): Theoretical E-mode power spectrum on the 2D grid.
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
eLarray_like of int, shape (2,): Minimum and maximum multipoles of the output normalization spectrum.

Returns

Agndarray of complex, shape (nx, ny): Normalization of the CMB lensing potential on the 2D grid.
Acndarray of complex, shape (nx, ny): Normalization of the curl mode, or pseudo lensing potential, on the 2D grid.

cmblensplus.flatsky.norm_lens.qeb(nx, ny, D, OE, OB, EE, rL, eL)

Return the normalization of the EB quadratic estimator for the CMB lensing potential and curl mode.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi), with bounds 0:1.
OEndarray of float, shape (nx, ny): Inverse observed E-mode power spectrum on the 2D grid.
OBndarray of float, shape (nx, ny): Inverse observed B-mode power spectrum on the 2D grid.
EEndarray of float, shape (nx, ny): Theoretical E-mode power spectrum on the 2D grid.
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
eLarray_like of int, shape (2,): Minimum and maximum multipoles of the output normalization spectrum.

Returns

Agndarray of complex, shape (nx, ny): Normalization of the CMB lensing potential on the 2D grid.
Acndarray of complex, shape (nx, ny): Normalization of the curl mode, or pseudo lensing potential, on the 2D grid.

`norm_rot` – normalization of quadratic pol. rot. reconstruction

cmblensplus.flatsky.norm_rot.qeb(nx, ny, D, rL, IE, IB, EE, eL, BB=0)

Return the normalization of the EB quadratic estimator for anisotropic polarization-rotation angles.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, or equivalently dLx / (2*pi) and dLy / (2*pi), with bounds 0:1.
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
IEndarray of float, shape (nx, ny): Inverse of the observed E-mode power spectrum on the 2D grid.
IBndarray of float, shape (nx, ny): Inverse of the observed B-mode power spectrum on the 2D grid.
EEndarray of float, shape (nx, ny): Theoretical E-mode power spectrum on the 2D grid.
eLarray_like of int, shape (2,): Minimum and maximum multipoles of the output normalization spectrum.
BBfloat or ndarray of float, optional: Theoretical B-mode power spectrum on the 2D grid. If an array is given, its shape should be (nx, ny). Default is 0.

Returns

Aandarray of complex, shape (nx, ny): Normalization of anisotropic polarization-rotation angles on the 2D grid.

`norm_tau` – normalization of quadratic patchy tay reconstruction

cmblensplus.flatsky.norm_tau.qtt(nx, ny, D, rL, OT, TT, eL)

Return the normalization of the temperature quadratic estimator for patchy optical depth.

Parameters

nxint: Number of grids along the x direction.
nyint: Number of grids along the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi), with bounds 0:1.
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
OTndarray of float, shape (nx, ny): Inverse of the observed temperature power spectrum on the 2D grid, with bounds (0:nx-1, 0:ny-1).
TTndarray of float, shape (nx, ny): Theoretical temperature power spectrum on the 2D grid, with bounds (0:nx-1, 0:ny-1).
eLarray_like of int, shape (2,): Minimum and maximum multipoles of the output normalization spectrum.

Returns

Atndarray of complex, shape (nx, ny): Normalization of patchy optical depth on the 2D grid, with bounds (0:nx-1, 0:ny-1).

cmblensplus.flatsky.norm_tau.qeb(nx, ny, D, rL, IE, IB, EE, eL)

Return the normalization of the EB quadratic estimator for patchy optical depth.

Parameters

nxint: Number of grids along the x direction.
nyint: Number of grids along the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi), with bounds 0:1.
rLarray_like of int, shape (2,): Minimum and maximum CMB multipoles used for reconstruction.
IEndarray of float, shape (nx, ny): Inverse of the observed E-mode power spectrum on the 2D grid, with bounds (0:nx-1, 0:ny-1).
IBndarray of float, shape (nx, ny): Inverse of the observed B-mode power spectrum on the 2D grid, with bounds (0:nx-1, 0:ny-1).
EEndarray of float, shape (nx, ny): Theoretical E-mode power spectrum on the 2D grid, with bounds (0:nx-1, 0:ny-1).
eLarray_like of int, shape (2,): Minimum and maximum multipoles of the output normalization spectrum.

Returns

Atndarray of complex, shape (nx, ny): Normalization of patchy optical depth on the 2D grid, with bounds (0:nx-1, 0:ny-1).

`bispec` – bispectrum tools

cmblensplus.flatsky.bispec.bispec_norm(nx, ny, D, bp, dbin_max=-1, bn=1)

Return the normalization of the binned bispectrum estimator.

Parameters

nxint: Number of Fourier grid points along the x direction.
nyint: Number of Fourier grid points along the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
bparray_like of float, shape (bn + 1,): Multipole bin edges.
dbin_maxint, optional: Maximum bin separation used in the bispectrum calculation. If -1, it is set to bn. Default is -1.
bnint, optional: Number of multipole bins. This value is overwritten by len(bp) - 1. Default is 1.

Returns

normndarray of float: Normalization of the binned bispectrum estimator.

cmblensplus.flatsky.bispec.bispec_bin(kmap, bp, kn=1, bn=1, nx=0, ny=0, dbin_max=-1)

Return the binned bispectrum estimator.

Parameters

kmapndarray of complex, shape (kn, bn, nx, ny): Fourier-space maps used for the bispectrum calculation.
bparray_like of float, shape (bn + 1,): Multipole bin edges.
knint, optional: Number of input map sets. This value is overwritten from kmap. Default is 1.
bnint, optional: Number of multipole bins. This value is overwritten by len(bp) - 1. Default is 1.
nxint, optional: Number of Fourier grid points along the x direction. This value is overwritten from kmap. Default is 0.
nyint, optional: Number of Fourier grid points along the y direction. This value is overwritten from kmap. Default is 0.
dbin_maxint, optional: Maximum bin separation used in the bispectrum calculation. If -1, it is set to bn. Default is -1.

Returns

bispecndarray of float: Binned bispectrum estimator.

cmblensplus.flatsky.bispec.binfilter(nx, ny, D, bp, bn=1)

Return binary masks for multipole bins on two-dimensional Fourier grids.

The mask is 1 inside the corresponding multipole bin and 0 otherwise.

Parameters

nxint: Number of Fourier grid points along the x direction.
nyint: Number of Fourier grid points along the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
bparray_like of float, shape (bn + 1,): Multipole bin edges.
bnint, optional: Number of multipole bins. This value is overwritten by len(bp) - 1. Default is 1.

Returns

bfndarray of float, shape (bn, nx, ny): Binary mask for each multipole bin.

cmblensplus.flatsky.bispec.bispec_norm_1d(nx, ny, D, bfs, bn=1)

Return the normalization of the one-dimensional binned bispectrum estimator.

Parameters

nxint: Number of Fourier grid points along the x direction.
nyint: Number of Fourier grid points along the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
bfsndarray of float, shape (3, bn, nx, ny): Multipole-bin masks on two-dimensional grids, usually obtained from binfilter().
bnint, optional: Number of multipole bins. This value is overwritten from bfs. Default is 1.

Returns

bnormndarray of float, shape (bn,): Normalization of the one-dimensional binned bispectrum estimator at each multipole bin.

cmblensplus.flatsky.bispec.bispec_bin_1d(nx, ny, D, bfs, bnorm, alm, bn=1)

Return the one-dimensional binned bispectrum estimator.

Parameters

nxint: Number of Fourier grid points along the x direction.
nyint: Number of Fourier grid points along the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
bfsndarray of float, shape (3, bn, nx, ny): Multipole-bin masks on two-dimensional grids, usually obtained from binfilter().
bnormarray_like of float, shape (bn,): Normalization of the one-dimensional binned bispectrum estimator at each multipole bin.
almndarray of complex, shape (3, nx, ny): Fourier modes for each leg of the bispectrum.
bnint, optional: Number of multipole bins. This value is overwritten from bfs. Default is 1.

Returns

bispecndarray of float, shape (bn,): One-dimensional binned bispectrum at each multipole bin.

`utils` – other tools

cmblensplus.flatsky.utils.map2alm(nx, ny, D, map)

Perform a discrete Fourier transform of a 2D map.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Side lengths of the map in the x and y directions.
mapndarray of float, shape (nx, ny): Input map on a 2D grid.

Returns

almndarray of complex, shape (nx, ny): Fourier modes on the 2D grid.

cmblensplus.flatsky.utils.alm2map(nx, ny, D, alm)

Perform an inverse discrete Fourier transform of 2D Fourier modes.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Side lengths of the map in the x and y directions.
almndarray of complex, shape (nx, ny): Fourier modes on a 2D grid.

Returns

mapndarray of float, shape (nx, ny): Transformed map on the 2D grid.

cmblensplus.flatsky.utils.el2d(nx, ny, D)

Return the absolute value of the multipole on a 2D Fourier grid.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).

Returns

elsndarray of float, shape (nx, ny): Absolute value of the Fourier mode, sqrt(Lx**2 + Ly**2).

cmblensplus.flatsky.utils.elarrays(nx, ny, D)

Return the Fourier-grid coordinates and their radial quantities.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).

Returns

elxndarray of float, shape (nx, ny): Fourier coordinate Lx.
elyndarray of float, shape (nx, ny): Fourier coordinate Ly.
elsndarray of float, shape (nx, ny): Absolute value of the Fourier mode, sqrt(Lx**2 + Ly**2).
elindarray of float, shape (nx, ny): Inverse of els.

cmblensplus.flatsky.utils.elmask(nx, ny, D, lmin=0, lmax=1000, lxcut=0, lycut=0)

Return a mask in 2D Fourier space.

The mask is unity where lmin <= |L| <= lmax, |Lx| >= lxcut, and |Ly| >= lycut. It is zero elsewhere.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
lminint, optional: Minimum multipole. Default is 0.
lmaxint, optional: Maximum multipole. Default is 1000.
lxcutint, optional: Remove modes with |Lx| < lxcut. Default is 0.
lycutint, optional: Remove modes with |Ly| < lycut. Default is 0.

Returns

lmaskndarray of float, shape (nx, ny): Fourier-space mask.

cmblensplus.flatsky.utils.ulm_flat(nx, ny, D, ulm, lmax=0, alpha=1.0, deapod=True)

Convert spherical harmonic coefficients to a 2D flat-sky grid.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
ulmndarray of complex, shape (lmax + 1, lmax + 1): Harmonic coefficients.
lmaxint, optional: Maximum multipole of ulm. If 0, it is inferred from ulm. Default is 0.
alphafloat, optional: Scaling parameter passed to the backend routine. Default is 1.0.
deapodbool, optional: Whether to apply deapodization. Default is True.

Returns

ul2dndarray of complex, shape (nx, ny): Harmonic coefficients converted onto the 2D grid.

cmblensplus.flatsky.utils.ulm2ulphi(ulm, lmax=0)

Convert spherical harmonic coefficients to u(l, phi_l).

Parameters

ulmndarray of complex, shape (lmax + 1, lmax + 1): Harmonic coefficients.
lmaxint, optional: Maximum multipole of ulm. If 0, it is inferred from ulm. Default is 0.

Returns

ulphindarray of complex, shape (lmax + 1, lmax + 1): Converted harmonic coefficients on the (l, phi_l) grid.

cmblensplus.flatsky.utils.lphi_to_cartesian(nx, ny, D, lmax, ulphi)

Convert coefficients from an (l, phi_l) grid to a Cartesian 2D grid.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths.
lmaxint: Maximum multipole of ulphi.
ulphindarray of complex: Input coefficients on the (l, phi_l) grid.

Returns

array_like: Coefficients converted to a Cartesian 2D grid.

cmblensplus.flatsky.utils.deapodization_bilinear(nx, ny, D, imap, alpha)

Apply bilinear deapodization to a 2D map.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths.
imapndarray of float, shape (nx, ny): Input map.
alphafloat: Deapodization parameter.

Returns

omapndarray of float, shape (nx, ny): Deapodized output map.

cmblensplus.flatsky.utils.alm2bcl(bn, oL, nx, ny, D, alm1, alm2=None, spc='')

Compute a binned angular power spectrum from Fourier modes.

Parameters

bnint: Number of multipole bins.
oLarray_like of int, shape (2,): Minimum and maximum multipoles of the output spectrum.
nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
alm1ndarray of complex, shape (nx, ny): First Fourier-mode array.
alm2ndarray of complex, shape (nx, ny), optional: Second Fourier-mode array. If not given, alm1 is used.
spcstr, optional: Multipole bin spacing. Use '' for linear spacing or 'log' for logarithmic spacing. Default is ''.

Returns

Cbndarray of float, shape (bn,): Binned angular power spectrum.

cmblensplus.flatsky.utils.c2d2bcl(nx, ny, D, c2d, bn, oL, spc='')

Compute a binned 1D angular power spectrum from a 2D power spectrum.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
c2dndarray of float, shape (nx, ny): Input 2D power spectrum.
bnint: Number of multipole bins.
oLarray_like of int, shape (2,): Minimum and maximum multipoles of the output spectrum.
spcstr, optional: Multipole bin spacing. Use '' for linear spacing or 'log' for logarithmic spacing. Default is ''.

Returns

Cbndarray of float, shape (bn,): Binned angular power spectrum.

cmblensplus.flatsky.utils.cl2c2d(nx, ny, D, lmin, lmax, Cl, method='linear')

Interpolate a 1D angular power spectrum onto a 2D Fourier grid.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
lminint: Minimum multipole of Cl to be interpolated.
lmaxint: Maximum multipole of Cl to be interpolated.
Clarray_like of float, shape (lmax + 1,): Input 1D angular power spectrum.
methodstr, optional: Interpolation method. Use 'linear' for linear interpolation or 'step' for step interpolation. Default is 'linear'.

Returns

c2dndarray of float, shape (nx, ny): Interpolated 2D power spectrum.

cmblensplus.flatsky.utils.cb2c2d(bn, bc, nx, ny, D, lmin, lmax, Cb, method='')

Interpolate a binned 1D angular power spectrum onto a 2D Fourier grid.

Parameters

bnint: Number of multipole bins.
bcarray_like of float, shape (bn,): Multipole bin centers.
nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
lminint: Minimum multipole to be interpolated.
lmaxint: Maximum multipole to be interpolated.
Cbarray_like of float, shape (bn,): Binned 1D angular power spectrum.
methodstr, optional: Interpolation method from binned to unbinned spectrum. Use '' for spline interpolation or 'linear' for linear interpolation. Default is ''.

Returns

c2dndarray of float, shape (nx, ny): Interpolated 2D power spectrum.

cmblensplus.flatsky.utils.gauss1alm(nx, ny, D, lmin, lmax, Cl)

Generate a random Gaussian field in 2D Fourier space.

The generated field satisfies the Hermitian condition a_l.conjugate() = a_{-l}.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
lminint: Minimum multipole of Cl to be interpolated.
lmaxint: Maximum multipole of Cl to be interpolated.
Clarray_like of float, shape (lmax + 1,): Input 1D angular power spectrum.

Returns

almndarray of complex, shape (nx, ny): Random Gaussian field on the 2D Fourier plane.

cmblensplus.flatsky.utils.gauss2alm(nx, ny, D, lmin, lmax, TT, TE, EE)

Generate two correlated random Gaussian fields in 2D Fourier space.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
lminint: Minimum multipole of the spectra to be interpolated.
lmaxint: Maximum multipole of the spectra to be interpolated.
TTarray_like of float, shape (lmax + 1,): First 1D auto power spectrum.
TEarray_like of float, shape (lmax + 1,): 1D cross power spectrum.
EEarray_like of float, shape (lmax + 1,): Second 1D auto power spectrum.

Returns

tlmndarray of complex, shape (nx, ny): First random Gaussian field on the 2D Fourier plane.
elmndarray of complex, shape (nx, ny): Second random Gaussian field on the 2D Fourier plane.

cmblensplus.flatsky.utils.window_sin(nx, ny, D, ap=1, cut=1)

Return a sine window function.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths, equivalent to dLx / (2*pi) and dLy / (2*pi).
apfloat, optional: Apodization parameter. The apodized range is (1 - ap) * cut * mapsize. It ranges from 0, full apodization, to 1, no apodization. Default is 1.
cutfloat, optional: Map cut parameter. The cut map size is cut * mapsize. It ranges from 0, full cut, to 1, no cut. Default is 1.

Returns

Wndarray of float, shape (nx, ny): Window function.

cmblensplus.flatsky.utils.window_norm(nx, ny, wind, num)

Return the normalization factor of a window function.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
windndarray of float, shape (nx, ny): Window function.
numint: Power of the window used for the normalization.

Returns

normfloat: Window normalization factor.

cmblensplus.flatsky.utils.window_norm_x(nx, ny, W1, W2, num)

Return the cross-normalization factor of two window functions.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
W1ndarray of float, shape (nx, ny): First window function.
W2ndarray of float, shape (nx, ny): Second window function.
numint: Power used for the normalization.

Returns

normfloat: Cross-window normalization factor.

cmblensplus.flatsky.utils.rotation(nx, ny, rot, QU, rtype='f')

Rotate a two-component field on a 2D grid.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
rotndarray of float, shape (nx, ny): Rotation angle map.
QUndarray of float: Input two-component field.
rtypestr: Rotation type ``l’’ (small angle limit) or ``f’’ (no assumption)

Returns

rotatedndarray: Rotated two-component field.

cmblensplus.flatsky.utils.get_angle(nx, ny, D)

Return the Fourier-space angle on a 2D grid.

Parameters

nxint: Number of grid points in the x direction.
nyint: Number of grid points in the y direction.
Darray_like of float, shape (2,): Map side lengths.

Returns

anglendarray of float, shape (nx, ny): Fourier-space angle on the 2D grid.

cmblensplus.flatsky.utils.cutmap(ox, oy, cx, cy, omap)

Cut out a rectangular region from a 2D map.

Parameters

oxint: Number of grid points in the x direction of the original map.
oyint: Number of grid points in the y direction of the original map.
cxint: Number of grid points in the x direction of the cut map.
cyint: Number of grid points in the y direction of the cut map.
omapndarray: Original input map.

Returns

cmapndarray: Cut-out map.

flatsky – Flatsky analysis tools

rec_lens – quadratic lensing reconstruction

Parameters

Returns

Parameters

Returns

Parameters

Returns

Parameters

Returns

Parameters

Returns

Parameters

Returns

rec_rot – quadratic pol. rot. reconstruction

Parameters

Returns

Parameters

Returns

Parameters

Returns

Parameters

Returns

rec_tau – quadratic patchy tau reconstruction

Parameters

Returns

Parameters

Returns

Parameters

Returns

Parameters

Returns

Parameters

Returns

rec_src – quadratic point-src reconstruction

Parameters

Returns

Parameters

Returns

Parameters

Returns

Parameters

Returns

Parameters

Returns

norm_lens – normalization of quadratic lensing reconstruction

Parameters

Returns

Parameters

Returns

Parameters

Returns

Parameters

Returns

Parameters

Returns

Parameters

Returns

Parameters

Returns

norm_rot – normalization of quadratic pol. rot. reconstruction

Parameters

Returns

norm_tau – normalization of quadratic patchy tay reconstruction

Parameters

Returns

Parameters

Returns

bispec – bispectrum tools

Parameters

Returns

Parameters

Returns

Parameters

Returns

Parameters

Returns

Parameters

Returns

utils – other tools

`flatsky` – Flatsky analysis tools

`rec_lens` – quadratic lensing reconstruction

`rec_rot` – quadratic pol. rot. reconstruction

`rec_tau` – quadratic patchy tau reconstruction

`rec_src` – quadratic point-src reconstruction

`norm_lens` – normalization of quadratic lensing reconstruction

`norm_rot` – normalization of quadratic pol. rot. reconstruction

`norm_tau` – normalization of quadratic patchy tay reconstruction

`bispec` – bispectrum tools

`utils` – other tools