In [1]:
# Notebook demonstrating the use of quantities defined incamb.symbolic, and examples of usage
# for defining custom sources and plotting different quantities.

# Set of scalar equations implemented in CAMB, and calculation of the line-of-sight sources
# indices are: 
# g - photons, r- massless neutrinos, c - CDM, b - baryons, de - dark energy, nu - massive neutrinos

# kappa = 8 pi G, Pi is anisotropic stress, q =(rho+p)v the heat flux 
# (and for components rho_i q_i = (rho_i+p_i)v_i)
# z is the perturbation to the expansion, h perturbation to the scale factor, sigma the shear
# phi the Weyl potential, and eta the 3-curvature. Equations are in a general gauge,
# and are implemented in CAMB in the CDM frame (synchronous gauge, but using variables above). 
# There are functions to convert into Newtonian and synchronous gauge metric variables

%matplotlib inline
%config InlineBackend.figure_format = 'retina'
import sys, platform, os
from matplotlib import pyplot as plt
import numpy as np
from IPython.display import display
import warnings
warnings.filterwarnings('ignore', category=DeprecationWarning, module='.*/IPython/.*')
print('Using CAMB installed at %s'%(os.path.realpath(os.path.join(os.getcwd(),'..'))))
import camb
from camb.symbolic import *
print('CAMB: %s, Sympy: %s'%(camb.__version__,sympy.__version__))
Using CAMB installed at c:\work\dist\git\camb
CAMB: 1.1.1, Sympy: 1.4
In [2]:
#can use tot_eqs as combination of total_eqs + pert_eqs + background_eqs
$\displaystyle \left[ \frac{d}{d t} a{\left(t \right)} = H{\left(t \right)} a{\left(t \right)}, \ \frac{d}{d t} H{\left(t \right)} = - \frac{\kappa \left(3 P{\left(t \right)} + \rho{\left(t \right)}\right) a^{2}{\left(t \right)}}{6}, \ \frac{d}{d t} \operatorname{exptau}{\left(t \right)} = \operatorname{visibility}{\left(t \right)}\right]$
$\displaystyle \left[ Kf_{1} k^{3} \phi{\left(t \right)} + \frac{k \kappa \left(Kf_{1} \Pi{\left(t \right)} + \delta{\left(t \right)}\right) a^{2}{\left(t \right)}}{2} + \frac{3 \kappa H{\left(t \right)} a^{2}{\left(t \right)} q{\left(t \right)}}{2}, \ k^{2} \eta{\left(t \right)} + 2 k H{\left(t \right)} z{\left(t \right)} - \kappa a^{2}{\left(t \right)} \delta{\left(t \right)}, \ \frac{2 k^{2} \left(- Kf_{1} \sigma{\left(t \right)} + z{\left(t \right)}\right)}{3 a^{2}{\left(t \right)}} + \kappa q{\left(t \right)}, \ - \frac{k z{\left(t \right)}}{3} + A{\left(t \right)} H{\left(t \right)} + \dot{h}{\left(t \right)}\right]$
$\displaystyle \left\{ \dot{h}{\left(t \right)} : \frac{- k^{2} \eta{\left(t \right)} + \kappa a^{2}{\left(t \right)} \delta{\left(t \right)} - 6 A{\left(t \right)} H^{2}{\left(t \right)}}{6 H{\left(t \right)}}, \ \phi{\left(t \right)} : - \frac{\kappa \left(Kf_{1} k \Pi{\left(t \right)} + k \delta{\left(t \right)} + 3 H{\left(t \right)} q{\left(t \right)}\right) a^{2}{\left(t \right)}}{2 Kf_{1} k^{3}}, \ \sigma{\left(t \right)} : \frac{k \left(- k^{2} \eta{\left(t \right)} + \kappa a^{2}{\left(t \right)} \delta{\left(t \right)}\right) + 3 \kappa H{\left(t \right)} a^{2}{\left(t \right)} q{\left(t \right)}}{2 Kf_{1} k^{2} H{\left(t \right)}}, \ z{\left(t \right)} : \frac{- k^{2} \eta{\left(t \right)} + \kappa a^{2}{\left(t \right)} \delta{\left(t \right)}}{2 k H{\left(t \right)}}\right\}$
$\displaystyle q{\left(t \right)} = \frac{2 k^{2} \left(Kf_{1} \sigma{\left(t \right)} - \frac{- k^{2} \eta{\left(t \right)} + \kappa a^{2}{\left(t \right)} \delta{\left(t \right)}}{2 k H{\left(t \right)}}\right)}{3 \kappa a^{2}{\left(t \right)}}$
$\displaystyle \left[ \frac{d}{d t} z{\left(t \right)} = Kf_{1} k A{\left(t \right)} - H{\left(t \right)} z{\left(t \right)} + \frac{3 \kappa \left(P{\left(t \right)} + \rho{\left(t \right)}\right) A{\left(t \right)} a^{2}{\left(t \right)}}{2 k} - \frac{\kappa \left(\delta{\left(t \right)} + 3 \delta_{P}{\left(t \right)}\right) a^{2}{\left(t \right)}}{2 k}, \ \frac{d}{d t} \sigma{\left(t \right)} = k \left(A{\left(t \right)} + \phi{\left(t \right)}\right) - H{\left(t \right)} \sigma{\left(t \right)} - \frac{\kappa \Pi{\left(t \right)} a^{2}{\left(t \right)}}{2 k}, \ \frac{d}{d t} \eta{\left(t \right)} = - \frac{2 K z{\left(t \right)} + 2 Kf_{1} k A{\left(t \right)} H{\left(t \right)} + \kappa a^{2}{\left(t \right)} q{\left(t \right)}}{k}, \ \frac{d}{d t} \phi{\left(t \right)} = - H{\left(t \right)} \phi{\left(t \right)} + \frac{\kappa \left(k \left(P{\left(t \right)} + \rho{\left(t \right)}\right) \sigma{\left(t \right)} + k q{\left(t \right)} - H{\left(t \right)} \Pi{\left(t \right)} - \frac{d}{d t} \Pi{\left(t \right)}\right) a^{2}{\left(t \right)}}{2 k^{2}}\right]$
$\displaystyle \left[ \frac{d}{d t} q{\left(t \right)} = - \frac{2 Kf_{1} k \Pi{\left(t \right)}}{3} - k \left(P{\left(t \right)} + \rho{\left(t \right)}\right) A{\left(t \right)} + k \delta_{P}{\left(t \right)} - 4 H{\left(t \right)} q{\left(t \right)}, \ \frac{d}{d t} \delta{\left(t \right)} = - k q{\left(t \right)} - 3 \left(P{\left(t \right)} + \rho{\left(t \right)}\right) \dot{h}{\left(t \right)} - 3 \left(\delta{\left(t \right)} + \delta_{P}{\left(t \right)}\right) H{\left(t \right)}\right]$

Newtonian gauge variables $\Psi_N$ and $\Phi_N$ (not used in CAMB but may be useful) defined for metrix sign choices so flat metric is $$ds^2 = a(\eta)^2\left( (1+2\Psi_N)d\eta^2 - (1-2\Phi_N)\delta_{ij}dx^idx^j\right)$$ (default, as defined by Ma and Bertschinger, number count and 21cm papers, etc.) Definitions are also valid for non-flat.

The alternative definition $$ds^2 = a(\eta)^2\left( (1+2\Psi_N)d\eta^2 - (1+2\Phi_N)\delta_{ij}dx^idx^j\right)$$ is used by Hu, lensing review, etc, corresponding to a sign change in $\Phi_N$.

In [3]:
#Relatations for going to/from the Newtonian gauge
$\displaystyle \left[ \Psi_{N}{\left(t \right)} = - A{\left(t \right)} + \frac{H{\left(t \right)} \sigma{\left(t \right)} + \frac{d}{d t} \sigma{\left(t \right)}}{k}, \ \Phi_{N}{\left(t \right)} = - \frac{H{\left(t \right)} \sigma{\left(t \right)}}{k} - \frac{\eta{\left(t \right)}}{2 Kf_{1}}\right]$
$\displaystyle \left[ A{\left(t \right)} = - \Psi_{N}{\left(t \right)}, \ \frac{d^{2}}{d t^{2}} \sigma{\left(t \right)} = 0, \ \frac{d}{d t} \sigma{\left(t \right)} = 0, \ \sigma{\left(t \right)} = 0, \ \phi{\left(t \right)} = \frac{\Phi_{N}{\left(t \right)}}{2} + \frac{\Psi_{N}{\left(t \right)}}{2}, \ \eta{\left(t \right)} = - 2 Kf_{1} \Phi_{N}{\left(t \right)}, \ \dot{h}{\left(t \right)} = - \frac{d}{d t} \Phi_{N}{\left(t \right)}, \ z{\left(t \right)} = \frac{3 \left(- H{\left(t \right)} \Psi_{N}{\left(t \right)} - \frac{d}{d t} \Phi_{N}{\left(t \right)}\right)}{k}\right]$
In [4]:
#e.g. get the Newtonian gauge equation for diff(Phi,t) + H*Psi 
#[eq 23b of Ma and Bertschinger, 
# noting that their (rho+P)theta = k q since q = (rho+P)v]
                    (diff(Phi_N,t) + H*Psi_N)).simplify().doit()))
$\displaystyle \frac{\kappa a^{2}{\left(t \right)} q{\left(t \right)}}{2 k}$
In [5]:
#CDM frame is used by CAMB, corresponding to zero acceleration or CDM velocity
display('cdm_subs', cdm_subs)
$\displaystyle \left[ \frac{d}{d t} A{\left(t \right)} = 0, \ A{\left(t \right)} = 0, \ \frac{d}{d t} \operatorname{v_{c}}{\left(t \right)} = 0, \ \operatorname{v_{c}}{\left(t \right)} = 0\right]$

Define synchonous gauge variables in Ma and Bertschinger notation (generalized to non-flat)

In terms of Hu et al variables $h_L+ h_T/3 = \eta_s$ and $h_L = -h_s/6$

In [6]:
#General gauge-invariant form of synchronous gauge metric variables
$\displaystyle \left[ \eta_{s}{\left(t \right)} = \frac{\frac{Kf_{1} H{\left(t \right)} \operatorname{v_{c}}{\left(t \right)}}{k} - \frac{\eta{\left(t \right)}}{2}}{Kf_{1}}, \ \dot{h}_{s}{\left(t \right)} = 6 A{\left(t \right)} H{\left(t \right)} + 6 \dot{h}{\left(t \right)} + \frac{6 \left(\frac{Kf_{1} k^{2}}{3} + \frac{\kappa \left(P{\left(t \right)} + \rho{\left(t \right)}\right) a^{2}{\left(t \right)}}{2}\right) \operatorname{v_{c}}{\left(t \right)}}{k}\right]$
In [7]:
#To convert from general to synchronous variables can use these
$\displaystyle \left[ \eta{\left(t \right)} = - 2 Kf_{1} \eta_{s}{\left(t \right)}, \ \dot{h}{\left(t \right)} = \frac{\dot{h}_{s}{\left(t \right)}}{6}, \ \phi{\left(t \right)} = - \frac{\kappa \left(Kf_{1} k \Pi{\left(t \right)} + k \delta{\left(t \right)} + 3 H{\left(t \right)} q{\left(t \right)}\right) a^{2}{\left(t \right)}}{2 Kf_{1} k^{3}}, \ z{\left(t \right)} = \frac{\dot{h}_{s}{\left(t \right)}}{2 k}, \ \sigma{\left(t \right)} = \frac{\dot{h}_{s}{\left(t \right)} + 6 \frac{d}{d t} \eta_{s}{\left(t \right)}}{2 k}, \ \frac{d}{d t} A{\left(t \right)} = 0, \ A{\left(t \right)} = 0, \ \frac{d}{d t} \operatorname{v_{c}}{\left(t \right)} = 0, \ \operatorname{v_{c}}{\left(t \right)} = 0\right]$
In [8]:
#Alternative pure-metric expression for phi
Eq(phi,subs(Eq(Pi,-(3*diff(eta_s,t,t)+diff(hdot_s,t)/2 + 2*(3*H*diff(eta_s,t) + H*hdot_s/2) - k**2*eta_s)/kappa/a**2),
 synchronous_gauge(subs(solve(constraints[1:],[delta,q,A]), phi_sub).rhs)).simplify())
$\displaystyle \phi{\left(t \right)} = \frac{2 k^{2} \eta_{s}{\left(t \right)} + 6 \frac{d^{2}}{d t^{2}} \eta_{s}{\left(t \right)} + \frac{d}{d t} \dot{h}_{s}{\left(t \right)}}{4 k^{2}}$
In [9]:
#Check the four synchronous gauge equations
synchronous_gauge(subs(var_subs,subs(synchronous_vars, K_fac*k**2*eta_s-H*hdot_s/2)).simplify())
$\displaystyle - \frac{\kappa a^{2}{\left(t \right)} \delta{\left(t \right)}}{2}$
In [10]:
synchronous_gauge(subs(pert_eqs,subs(synchronous_vars, -K/2*hdot_s/k**2 + K_fac*diff(eta_s,t)).doit()))
$\displaystyle \frac{\kappa a^{2}{\left(t \right)} q{\left(t \right)}}{2 k}$
In [11]:
        subs(synchronous_vars, -(diff(hdot_s,t) + H*hdot_s)/6).doit()).doit())).simplify().expand())
$\displaystyle \frac{\kappa \left(\delta{\left(t \right)} + 3 \delta_{P}{\left(t \right)}\right) a^{2}{\left(t \right)}}{6}$
In [12]:
#Seems to be a factor of 2 missing in last line of eq A8 of Hu et al.
    subs(tot_eqs+background_eqs, subs(var_subs,subs(pert_eqs,subs(synchronous_vars,
    3*diff(eta_s,t,t)+diff(hdot_s,t)/2 + 2*(3*H*diff(eta_s,t) + H*hdot_s/2) - k**2*eta_s).doit()).doit()).doit()))
$\displaystyle - \kappa \Pi{\left(t \right)} a^{2}{\left(t \right)}$
In [13]:
#Fluid components
display('delta_eqs', delta_eqs)
#can use component_eqs as combination of density_eqs + delta_eqs + vel_eqs
$\displaystyle \left[ \frac{d}{d t} \rho_{b}{\left(t \right)} = - 3 \left(\operatorname{p_{b}}{\left(t \right)} + \rho_{b}{\left(t \right)}\right) H{\left(t \right)}, \ \frac{d}{d t} \rho_{c}{\left(t \right)} = - 3 H{\left(t \right)} \rho_{c}{\left(t \right)}, \ \frac{d}{d t} \rho_{g}{\left(t \right)} = - 4 H{\left(t \right)} \rho_{g}{\left(t \right)}, \ \frac{d}{d t} \rho_{r}{\left(t \right)} = - 4 H{\left(t \right)} \rho_{r}{\left(t \right)}, \ \frac{d}{d t} \rho_{\nu}{\left(t \right)} = - 3 \left(\operatorname{p_{\nu}}{\left(t \right)} + \rho_{\nu}{\left(t \right)}\right) H{\left(t \right)}, \ \frac{d}{d t} \rho_{de}{\left(t \right)} = - 3 \left(\operatorname{w_{de}}{\left(t \right)} + 1\right) H{\left(t \right)} \rho_{de}{\left(t \right)}\right]$
$\displaystyle \left[ \frac{d}{d t} \Delta_{r}{\left(t \right)} = - k \operatorname{q_{r}}{\left(t \right)} - 4 \dot{h}{\left(t \right)}, \ \frac{d}{d t} \Delta_{g}{\left(t \right)} = - k \operatorname{q_{g}}{\left(t \right)} - 4 \dot{h}{\left(t \right)}, \ \frac{d}{d t} \Delta_{b}{\left(t \right)} = \left(k \operatorname{v_{b}}{\left(t \right)} + 3 \dot{h}{\left(t \right)}\right) \left(- \frac{\operatorname{p_{b}}{\left(t \right)}}{\rho_{b}{\left(t \right)}} - 1\right) + \left(- 3 \operatorname{c^{2}_{sb}}{\left(t \right)} + \frac{3 \operatorname{p_{b}}{\left(t \right)}}{\rho_{b}{\left(t \right)}}\right) \Delta_{b}{\left(t \right)} H{\left(t \right)}, \ \frac{d}{d t} \Delta_{c}{\left(t \right)} = - k \operatorname{v_{c}}{\left(t \right)} - 3 \dot{h}{\left(t \right)}, \ \frac{d}{d t} \Delta_{\nu}{\left(t \right)} = - k \operatorname{q_{\nu}}{\left(t \right)} + \left(- \frac{3 \operatorname{p_{\nu}}{\left(t \right)}}{\rho_{\nu}{\left(t \right)}} - 3\right) \dot{h}{\left(t \right)} + 3 \left(- \Delta_{P \nu}{\left(t \right)} + \frac{\Delta_{\nu}{\left(t \right)} \operatorname{p_{\nu}}{\left(t \right)}}{\rho_{\nu}{\left(t \right)}}\right) H{\left(t \right)}, \ \frac{d}{d t} \Delta_{de}{\left(t \right)} = - k \left(\operatorname{w_{de}}{\left(t \right)} + 1\right) \operatorname{v_{de}}{\left(t \right)} - 3 \left(\Delta_{de}{\left(t \right)} + \frac{3 \left(\operatorname{w_{de}}{\left(t \right)} + 1\right) H{\left(t \right)} \operatorname{v_{de}}{\left(t \right)}}{k}\right) \left(\hat{c}^{2}_{sde}{\left(t \right)} - \operatorname{w_{de}}{\left(t \right)}\right) H{\left(t \right)} + \left(- 3 \operatorname{w_{de}}{\left(t \right)} - 3\right) \dot{h}{\left(t \right)} - \frac{3 H{\left(t \right)} \operatorname{v_{de}}{\left(t \right)} \frac{d}{d t} \operatorname{w_{de}}{\left(t \right)}}{k}\right]$
$\displaystyle \left[ \frac{d}{d t} \operatorname{q_{r}}{\left(t \right)} = - \frac{2 Kf_{1} k \pi_{r}{\left(t \right)}}{3} - \frac{4 k A{\left(t \right)}}{3} + \frac{k \Delta_{r}{\left(t \right)}}{3}, \ \frac{d}{d t} \operatorname{q_{g}}{\left(t \right)} = - \frac{2 Kf_{1} k \pi_{g}{\left(t \right)}}{3} - \frac{4 k A{\left(t \right)}}{3} + \frac{k \Delta_{g}{\left(t \right)}}{3} + \left(- \operatorname{q_{g}}{\left(t \right)} + \frac{4 \operatorname{v_{b}}{\left(t \right)}}{3}\right) \operatorname{opacity}{\left(t \right)}, \ \frac{d}{d t} \operatorname{v_{b}}{\left(t \right)} = - k A{\left(t \right)} + \frac{k \Delta_{b}{\left(t \right)} \operatorname{c^{2}_{sb}}{\left(t \right)} \rho_{b}{\left(t \right)} - \left(- \operatorname{q_{g}}{\left(t \right)} + \frac{4 \operatorname{v_{b}}{\left(t \right)}}{3}\right) \operatorname{opacity}{\left(t \right)} \rho_{g}{\left(t \right)}}{\operatorname{p_{b}}{\left(t \right)} + \rho_{b}{\left(t \right)}} - H{\left(t \right)} \operatorname{v_{b}}{\left(t \right)} - \frac{\operatorname{v_{b}}{\left(t \right)} \frac{d}{d t} \operatorname{p_{b}}{\left(t \right)}}{\operatorname{p_{b}}{\left(t \right)} + \rho_{b}{\left(t \right)}}, \ \frac{d}{d t} \operatorname{v_{c}}{\left(t \right)} = - k A{\left(t \right)} - H{\left(t \right)} \operatorname{v_{c}}{\left(t \right)}, \ \frac{d}{d t} \operatorname{q_{\nu}}{\left(t \right)} = - \frac{k \left(2 Kf_{1} \pi_{\nu}{\left(t \right)} - 3 \Delta_{P \nu}{\left(t \right)}\right)}{3} - k \left(\frac{\operatorname{p_{\nu}}{\left(t \right)}}{\rho_{\nu}{\left(t \right)}} + 1\right) A{\left(t \right)} - \left(- \frac{3 \operatorname{p_{\nu}}{\left(t \right)}}{\rho_{\nu}{\left(t \right)}} + 1\right) H{\left(t \right)} \operatorname{q_{\nu}}{\left(t \right)}, \ \frac{d}{d t} \operatorname{v_{de}}{\left(t \right)} = - k A{\left(t \right)} + \frac{k \Delta_{de}{\left(t \right)} \hat{c}^{2}_{sde}{\left(t \right)}}{\operatorname{w_{de}}{\left(t \right)} + 1} - \left(1 - 3 \hat{c}^{2}_{sde}{\left(t \right)}\right) H{\left(t \right)} \operatorname{v_{de}}{\left(t \right)}\right]$
In [14]:
#e.g. can check we recover standard Newtonian gauge equations 
#(note all equations above are valid in any frame)
$\displaystyle \left[ \frac{d}{d t} \Delta_{r}{\left(t \right)} = - k \operatorname{q_{r}}{\left(t \right)} + 4 \frac{d}{d t} \Phi_{N}{\left(t \right)}, \ \frac{d}{d t} \Delta_{g}{\left(t \right)} = - k \operatorname{q_{g}}{\left(t \right)} + 4 \frac{d}{d t} \Phi_{N}{\left(t \right)}, \ \frac{d}{d t} \Delta_{b}{\left(t \right)} = - \frac{\left(k \operatorname{v_{b}}{\left(t \right)} - 3 \frac{d}{d t} \Phi_{N}{\left(t \right)}\right) \left(\operatorname{p_{b}}{\left(t \right)} + \rho_{b}{\left(t \right)}\right) + 3 \left(\operatorname{c^{2}_{sb}}{\left(t \right)} \rho_{b}{\left(t \right)} - \operatorname{p_{b}}{\left(t \right)}\right) \Delta_{b}{\left(t \right)} H{\left(t \right)}}{\rho_{b}{\left(t \right)}}, \ \frac{d}{d t} \Delta_{c}{\left(t \right)} = - k \operatorname{v_{c}}{\left(t \right)} + 3 \frac{d}{d t} \Phi_{N}{\left(t \right)}, \ \frac{d}{d t} \Delta_{\nu}{\left(t \right)} = \frac{- k \operatorname{q_{\nu}}{\left(t \right)} \rho_{\nu}{\left(t \right)} - 3 \left(\Delta_{P \nu}{\left(t \right)} \rho_{\nu}{\left(t \right)} - \Delta_{\nu}{\left(t \right)} \operatorname{p_{\nu}}{\left(t \right)}\right) H{\left(t \right)} + 3 \left(\operatorname{p_{\nu}}{\left(t \right)} + \rho_{\nu}{\left(t \right)}\right) \frac{d}{d t} \Phi_{N}{\left(t \right)}}{\rho_{\nu}{\left(t \right)}}, \ \frac{d}{d t} \Delta_{de}{\left(t \right)} = \frac{k \left(- k \operatorname{v_{de}}{\left(t \right)} + 3 \frac{d}{d t} \Phi_{N}{\left(t \right)}\right) \left(\operatorname{w_{de}}{\left(t \right)} + 1\right) - 3 \left(k \Delta_{de}{\left(t \right)} + 3 \left(\operatorname{w_{de}}{\left(t \right)} + 1\right) H{\left(t \right)} \operatorname{v_{de}}{\left(t \right)}\right) \left(\hat{c}^{2}_{sde}{\left(t \right)} - \operatorname{w_{de}}{\left(t \right)}\right) H{\left(t \right)} - 3 H{\left(t \right)} \operatorname{v_{de}}{\left(t \right)} \frac{d}{d t} \operatorname{w_{de}}{\left(t \right)}}{k}\right]$
In [15]:
#Can use make_frame_invariant function to get explicit gauge-invariant equations for Newtonian (or other gauge) quantities
delta_N, v_b_N, sigma_N, eta_N = make_frame_invariant([delta,v_b, sigma, eta], 'Newtonian')
delta_sync, v_b_sync, sigma_sync, eta_sync = make_frame_invariant([delta,v_b, sigma, eta], 'CDM')

display('Gaude-dependent:', [delta, v_b, sigma, eta])
display('Newtonian:', [delta_N, v_b_N, sigma_N, eta_N]) 
display('Synchronous (CDM frame):', [delta_sync, v_b_sync, sigma_sync, eta_sync]) 
$\displaystyle \left[ \delta{\left(t \right)}, \ \operatorname{v_{b}}{\left(t \right)}, \ \sigma{\left(t \right)}, \ \eta{\left(t \right)}\right]$
$\displaystyle \left[ \delta{\left(t \right)} - \frac{3 \left(P{\left(t \right)} + \rho{\left(t \right)}\right) H{\left(t \right)} \sigma{\left(t \right)}}{k}, \ \sigma{\left(t \right)} + \operatorname{v_{b}}{\left(t \right)}, \ 0, \ \frac{2 Kf_{1} H{\left(t \right)} \sigma{\left(t \right)}}{k} + \eta{\left(t \right)}\right]$
'Synchronous (CDM frame):'
$\displaystyle \left[ \delta{\left(t \right)} + \frac{3 \left(P{\left(t \right)} + \rho{\left(t \right)}\right) H{\left(t \right)} \operatorname{v_{c}}{\left(t \right)}}{k}, \ \operatorname{v_{b}}{\left(t \right)} - \operatorname{v_{c}}{\left(t \right)}, \ \sigma{\left(t \right)} + \operatorname{v_{c}}{\left(t \right)}, \ - \frac{2 Kf_{1} H{\left(t \right)} \operatorname{v_{c}}{\left(t \right)}}{k} + \eta{\left(t \right)}\right]$
In [16]:
def show_gauges(x):
    print('Newtonian gauge version:')
    print('CDM frame version:')
    print('Synchronous gauge variable version:')

def check_equation(camb_eq, view = True, p_b_zero=False):
    if view:
    res = simplify(subs(background_eqs +component_eqs,camb_eq.simplify()))
    res = subs(var_subs,res).simplify()
    res = subs(Friedmann_Kfac_subs, res).simplify()
    if p_b_zero: res = subs(Eq(p_b,0),res).doit().simplify()
    if res==0:
        print('Non-zero, equal to:')

#For example CAMB implements the equation d Delta_g/dt= -k*(4/3*z+qg) in synchronous gauge. Check this works.
Delta_g_sync, z_sync, q_g_sync = make_frame_invariant([Delta_g, z, q_g], 'CDM')
check_equation(diff(Delta_g_sync,t) + k*(4*z_sync/3 + q_g_sync)) 
Newtonian gauge version:
$\displaystyle \frac{\frac{k^{2} \left(4 Kf_{1} \operatorname{v_{c}}{\left(t \right)} + 3 \operatorname{q_{g}}{\left(t \right)} - 4 \operatorname{v_{c}}{\left(t \right)}\right)}{3} - 4 k \left(H{\left(t \right)} \Psi_{N}{\left(t \right)} + \frac{d}{d t} \Phi_{N}{\left(t \right)}\right) + k \frac{d}{d t} \Delta_{g}{\left(t \right)} + 2 \kappa \left(P{\left(t \right)} + \rho{\left(t \right)}\right) a^{2}{\left(t \right)} \operatorname{v_{c}}{\left(t \right)} + 4 H{\left(t \right)} \frac{d}{d t} \operatorname{v_{c}}{\left(t \right)} + 4 \operatorname{v_{c}}{\left(t \right)} \frac{d}{d t} H{\left(t \right)}}{k}$
CDM frame version:
$\displaystyle k \operatorname{q_{g}}{\left(t \right)} + \frac{4 k z{\left(t \right)}}{3} + \frac{d}{d t} \Delta_{g}{\left(t \right)}$
Synchronous gauge variable version:
$\displaystyle k \operatorname{q_{g}}{\left(t \right)} + \frac{2 \dot{h}_{s}{\left(t \right)}}{3} + \frac{d}{d t} \Delta_{g}{\left(t \right)}$
In [17]:
csq_b_sync, Delta_b_sync = make_frame_invariant([csq_b, Delta_b], 'CDM')
Newtonian gauge version:
$\displaystyle \frac{- k \Delta_{b}{\left(t \right)} \operatorname{c^{2}_{sb}}{\left(t \right)} \rho_{b}{\left(t \right)} - \frac{\left(3 \operatorname{q_{g}}{\left(t \right)} - 4 \operatorname{v_{b}}{\left(t \right)}\right) \operatorname{opacity}{\left(t \right)} \rho_{g}{\left(t \right)}}{3} + \left(\left(\operatorname{v_{b}}{\left(t \right)} - \operatorname{v_{c}}{\left(t \right)}\right) H{\left(t \right)} + \frac{d}{d t} \operatorname{v_{b}}{\left(t \right)} - \frac{d}{d t} \operatorname{v_{c}}{\left(t \right)}\right) \rho_{b}{\left(t \right)} + \operatorname{v_{c}}{\left(t \right)} \frac{d}{d t} \operatorname{p_{b}}{\left(t \right)}}{\rho_{b}{\left(t \right)}}$
CDM frame version:
$\displaystyle \frac{- \frac{\left(3 \operatorname{q_{g}}{\left(t \right)} - 4 \operatorname{v_{b}}{\left(t \right)}\right) \operatorname{opacity}{\left(t \right)} \rho_{g}{\left(t \right)}}{3} + \left(- k \Delta_{b}{\left(t \right)} \operatorname{c^{2}_{sb}}{\left(t \right)} + H{\left(t \right)} \operatorname{v_{b}}{\left(t \right)} + \frac{d}{d t} \operatorname{v_{b}}{\left(t \right)}\right) \rho_{b}{\left(t \right)}}{\rho_{b}{\left(t \right)}}$
Synchronous gauge variable version:
$\displaystyle \frac{- \frac{\left(3 \operatorname{q_{g}}{\left(t \right)} - 4 \operatorname{v_{b}}{\left(t \right)}\right) \operatorname{opacity}{\left(t \right)} \rho_{g}{\left(t \right)}}{3} + \left(- k \Delta_{b}{\left(t \right)} \operatorname{c^{2}_{sb}}{\left(t \right)} + H{\left(t \right)} \operatorname{v_{b}}{\left(t \right)} + \frac{d}{d t} \operatorname{v_{b}}{\left(t \right)}\right) \rho_{b}{\left(t \right)}}{\rho_{b}{\left(t \right)}}$
In [18]:
#can see how a synchronous gauge variable used in CAMB can be obtained in Newtonian gauge, e.g.
Newtonian gauge version:
$\displaystyle - \frac{2 Kf_{1} \left(k \Phi_{N}{\left(t \right)} + H{\left(t \right)} \operatorname{v_{c}}{\left(t \right)}\right)}{k}$
CDM frame version:
$\displaystyle \eta{\left(t \right)}$
Synchronous gauge variable version:
$\displaystyle - 2 Kf_{1} \eta_{s}{\left(t \right)}$
In [19]:
Delta_g_N = make_frame_invariant(Delta_g, 'Newtonian')
Newtonian gauge version:
$\displaystyle \Delta_{g}{\left(t \right)}$
CDM frame version:
$\displaystyle \Delta_{g}{\left(t \right)} - \frac{4 H{\left(t \right)} \sigma{\left(t \right)}}{k}$
Synchronous gauge variable version:
$\displaystyle \frac{k^{2} \Delta_{g}{\left(t \right)} - 2 \left(\dot{h}_{s}{\left(t \right)} + 6 \frac{d}{d t} \eta_{s}{\left(t \right)}\right) H{\left(t \right)}}{k^{2}}$
In [20]:
#Relations between components and totals
$\displaystyle \left[ \rho{\left(t \right)} = \rho_{b}{\left(t \right)} + \rho_{c}{\left(t \right)} + \rho_{de}{\left(t \right)} + \rho_{g}{\left(t \right)} + \rho_{\nu}{\left(t \right)} + \rho_{r}{\left(t \right)}, \ P{\left(t \right)} = \operatorname{p_{b}}{\left(t \right)} + \operatorname{p_{\nu}}{\left(t \right)} + \rho_{de}{\left(t \right)} \operatorname{w_{de}}{\left(t \right)} + \frac{\rho_{g}{\left(t \right)}}{3} + \frac{\rho_{r}{\left(t \right)}}{3}\right]$
$\displaystyle \left[ \Pi{\left(t \right)} = \pi_{g}{\left(t \right)} \rho_{g}{\left(t \right)} + \pi_{\nu}{\left(t \right)} \rho_{\nu}{\left(t \right)} + \pi_{r}{\left(t \right)} \rho_{r}{\left(t \right)}, \ \delta{\left(t \right)} = \Delta_{b}{\left(t \right)} \rho_{b}{\left(t \right)} + \Delta_{c}{\left(t \right)} \rho_{c}{\left(t \right)} + \Delta_{de}{\left(t \right)} \rho_{de}{\left(t \right)} + \Delta_{g}{\left(t \right)} \rho_{g}{\left(t \right)} + \Delta_{\nu}{\left(t \right)} \rho_{\nu}{\left(t \right)} + \Delta_{r}{\left(t \right)} \rho_{r}{\left(t \right)}, \ q{\left(t \right)} = \left(\operatorname{p_{b}}{\left(t \right)} + \rho_{b}{\left(t \right)}\right) \operatorname{v_{b}}{\left(t \right)} + \left(\operatorname{w_{de}}{\left(t \right)} + 1\right) \rho_{de}{\left(t \right)} \operatorname{v_{de}}{\left(t \right)} + \operatorname{q_{g}}{\left(t \right)} \rho_{g}{\left(t \right)} + \operatorname{q_{\nu}}{\left(t \right)} \rho_{\nu}{\left(t \right)} + \operatorname{q_{r}}{\left(t \right)} \rho_{r}{\left(t \right)} + \rho_{c}{\left(t \right)} \operatorname{v_{c}}{\left(t \right)}, \ \delta_{P}{\left(t \right)} = \left(\Delta_{de}{\left(t \right)} \hat{c}^{2}_{sde}{\left(t \right)} + \frac{3 \left(\operatorname{w_{de}}{\left(t \right)} + 1\right) \left(\hat{c}^{2}_{sde}{\left(t \right)} - \operatorname{w_{de}}{\left(t \right)} + \frac{\frac{d}{d t} \operatorname{w_{de}}{\left(t \right)}}{3 \left(\operatorname{w_{de}}{\left(t \right)} + 1\right) H{\left(t \right)}}\right) H{\left(t \right)} \operatorname{v_{de}}{\left(t \right)}}{k}\right) \rho_{de}{\left(t \right)} + \Delta_{P \nu}{\left(t \right)} \rho_{\nu}{\left(t \right)} + \Delta_{b}{\left(t \right)} \operatorname{c^{2}_{sb}}{\left(t \right)} \rho_{b}{\left(t \right)} + \frac{\Delta_{g}{\left(t \right)} \rho_{g}{\left(t \right)}}{3} + \frac{\Delta_{r}{\left(t \right)} \rho_{r}{\left(t \right)}}{3}\right]$
In [21]:
#First few equations in Boltzmann hierarchies for L>=2 (J photon, G neutrino)
$\displaystyle \left[ \frac{d}{d t} \pi_{g}{\left(t \right)} = - \frac{k \left(3 \operatorname{J_{3}}{\left(t \right)} {Kf}_{2} - 2 \operatorname{q_{g}}{\left(t \right)}\right)}{5} + \frac{8 k \sigma{\left(t \right)}}{15} - \operatorname{opacity}{\left(t \right)} \pi_{g}{\left(t \right)} + \operatorname{opacity}{\left(t \right)} \operatorname{polter}{\left(t \right)}, \ \frac{d}{d t} \pi_{r}{\left(t \right)} = - \frac{k \left(3 \operatorname{G_{3}}{\left(t \right)} {Kf}_{2} - 2 \operatorname{q_{r}}{\left(t \right)}\right)}{5} + \frac{8 k \sigma{\left(t \right)}}{15}, \ \frac{d}{d t} \operatorname{E_{2}}{\left(t \right)} = - \frac{k \operatorname{E_{3}}{\left(t \right)} {Kf}_{2}}{3} - \operatorname{E_{2}}{\left(t \right)} \operatorname{opacity}{\left(t \right)} + \operatorname{opacity}{\left(t \right)} \operatorname{polter}{\left(t \right)}, \ \frac{d}{d t} \operatorname{J_{3}}{\left(t \right)} = - \frac{k \left(4 \operatorname{J_{4}}{\left(t \right)} {Kf}_{3} - 3 \pi_{g}{\left(t \right)}\right)}{7} - \operatorname{J_{3}}{\left(t \right)} \operatorname{opacity}{\left(t \right)}, \ \frac{d}{d t} \operatorname{G_{3}}{\left(t \right)} = - \frac{k \left(4 \operatorname{G_{4}}{\left(t \right)} {Kf}_{3} - 3 \pi_{r}{\left(t \right)}\right)}{7}, \ \frac{d}{d t} \operatorname{E_{3}}{\left(t \right)} = - \frac{k \left(- 3 \operatorname{E_{2}}{\left(t \right)} + 3 \operatorname{E_{4}}{\left(t \right)} {Kf}_{3}\right)}{7} - \operatorname{E_{3}}{\left(t \right)} \operatorname{opacity}{\left(t \right)}, \ \frac{d}{d t} \operatorname{J_{4}}{\left(t \right)} = - \frac{k \left(- 4 \operatorname{J_{3}}{\left(t \right)} + 5 \operatorname{J_{5}}{\left(t \right)} {Kf}_{4}\right)}{9} - \operatorname{J_{4}}{\left(t \right)} \operatorname{opacity}{\left(t \right)}, \ \frac{d}{d t} \operatorname{G_{4}}{\left(t \right)} = - \frac{k \left(- 4 \operatorname{G_{3}}{\left(t \right)} + 5 \operatorname{G_{5}}{\left(t \right)} {Kf}_{4}\right)}{9}, \ \frac{d}{d t} \operatorname{E_{4}}{\left(t \right)} = - \frac{k \left(- 4 \operatorname{E_{3}}{\left(t \right)} + \frac{21 \operatorname{E_{5}}{\left(t \right)} {Kf}_{4}}{5}\right)}{9} - \operatorname{E_{4}}{\left(t \right)} \operatorname{opacity}{\left(t \right)}\right]$
In [22]:
# polter is the quadrupole source. Need its derivatives for line-of-sight solution.
polterdot = subs(hierarchies,diff(polter_t,t))
$\displaystyle - \frac{k \left(3 \operatorname{J_{3}}{\left(t \right)} {Kf}_{2} - 2 \operatorname{q_{g}}{\left(t \right)}\right)}{50} - \frac{k \operatorname{E_{3}}{\left(t \right)} {Kf}_{2}}{5} + \frac{4 k \sigma{\left(t \right)}}{75} - \frac{3 \operatorname{E_{2}}{\left(t \right)} \operatorname{opacity}{\left(t \right)}}{5} - \frac{\operatorname{opacity}{\left(t \right)} \pi_{g}{\left(t \right)}}{10} + \frac{7 \operatorname{opacity}{\left(t \right)} \operatorname{polter}{\left(t \right)}}{10}$
In [23]:
polterddot = subs(phi_sub,diff(polterdot,t).subs(diff(sigma,t),dsigma).simplify()).simplify()
$\displaystyle \frac{8 Kf_{1} k^{3} A{\left(t \right)} - 8 Kf_{1} k^{2} H{\left(t \right)} \sigma{\left(t \right)} + Kf_{1} k \left(- 3 k \left(3 \frac{d}{d t} \operatorname{J_{3}}{\left(t \right)} {Kf}_{2} - 2 \frac{d}{d t} \operatorname{q_{g}}{\left(t \right)}\right) - 30 k \frac{d}{d t} \operatorname{E_{3}}{\left(t \right)} {Kf}_{2} - 4 \kappa \Pi{\left(t \right)} a^{2}{\left(t \right)} - 90 \operatorname{E_{2}}{\left(t \right)} \frac{d}{d t} \operatorname{opacity}{\left(t \right)} - 90 \operatorname{opacity}{\left(t \right)} \frac{d}{d t} \operatorname{E_{2}}{\left(t \right)} - 15 \operatorname{opacity}{\left(t \right)} \frac{d}{d t} \pi_{g}{\left(t \right)} + 105 \operatorname{opacity}{\left(t \right)} \frac{d}{d t} \operatorname{polter}{\left(t \right)} - 15 \pi_{g}{\left(t \right)} \frac{d}{d t} \operatorname{opacity}{\left(t \right)} + 105 \operatorname{polter}{\left(t \right)} \frac{d}{d t} \operatorname{opacity}{\left(t \right)}\right) - 4 \kappa \left(Kf_{1} k \Pi{\left(t \right)} + k \delta{\left(t \right)} + 3 H{\left(t \right)} q{\left(t \right)}\right) a^{2}{\left(t \right)}}{150 Kf_{1} k}$
In [24]:
monopole_source, ISW, doppler, quadrupole_source = get_scalar_temperature_sources()
In [25]:
#These are definitions used in CAMB to get the various sources for the temperature
print(camb_fortran(dphi, 'phidot'))
print(camb_fortran(dsigma, 'sigmadot'))
print(camb_fortran(diff(polter_t,t), 'polterdot'))
print(camb_fortran(polterddot, 'polterddot'))
print(camb_fortran(monopole_source, 'monopole_source'))
print(camb_fortran(doppler, 'doppler'))
print(camb_fortran(quadrupole_source, 'quadrupole_source'))
phidot = (1.0d0/2.0d0)*(-adotoa*dgpi - 2*adotoa*k**2*phi + dgq*k &
    -diff_rhopi + k*sigma*(gpres + grho))/k**2
sigmadot = -adotoa*sigma - 1.0d0/2.0d0*dgpi/k + k*phi
polterdot = (1.0d0/10.0d0)*pigdot + (3.0d0/5.0d0)*Edot(2)
polterddot = -2.0d0/25.0d0*adotoa*dgq/(k*Kf(1)) - &
    4.0d0/75.0d0*adotoa*k*sigma - 4.0d0/75.0d0*dgpi - &
    2.0d0/75.0d0*dgrho/Kf(1) + dopacity*(-1.0d0/10.0d0*pig + &
    (7.0d0/10.0d0)*polter - 3.0d0/5.0d0*E(2)) &
    -3.0d0/50.0d0*k*octgdot*Kf(2) + (1.0d0/25.0d0)*k*qgdot - &
    1.0d0/5.0d0*k*Edot(3)*Kf(2) + opacity*(-1.0d0/10.0d0*pigdot + &
    (7.0d0/10.0d0)*polterdot - 3.0d0/5.0d0*Edot(2))
ISW = 2*exptau*phidot
monopole_source = (1.0d0/4.0d0)*visibility*(-4*etak + k*(clxg + &
doppler = (dvisibility*(sigma + vb) + visibility*(sigmadot + vbdot))/k
quadrupole_source = (5.0d0/8.0d0)*(3*ddvisibility*polter + &
    6*dvisibility*polterdot + visibility*(k**2*polter + &
In [26]:
In [27]:
pars = camb.set_params(H0=67.5, ombh2=0.022, omch2=0.122, As=2e-9, ns=0.95, tau=0.055)
from matplotlib import rcParams
rcParams.update( {'axes.labelsize': 14,
              'font.size': 14,
              'legend.fontsize': 14,
              'xtick.labelsize': 13,
              'ytick.labelsize': 13})
cl_label= r'$\ell(\ell+1)C_\ell/2\pi\quad [\mu {\rm K}^2]$'
In [28]:
#Example of plotting the time evolution of Newtonian gauge variables and the monopole sources 

data= camb.get_background(pars)
conformal_times = np.linspace(1, 800, 300)
ks = [0.01,0.05]
Delta_g_N = make_frame_invariant(Delta_g, 'Newtonian')
display('Temperature monopole source in general', monopole_source)
display('Temperature monopole source in Newtonian gauge', newtonian_gauge(monopole_source))

ev = data.get_time_evolution(ks, conformal_times, ['delta_photon', Delta_g_N, Psi_N, monopole_source, ISW])
_, axs= plt.subplots(1,2, figsize=(14,6))
for i, ax in enumerate(axs):
    ax.plot(conformal_times,ev[i,:, 0])
    ax.plot(conformal_times,ev[i,:, 1])
    ax.plot(conformal_times,ev[i,:, 2])

    ax.plot(conformal_times,ev[i,:, 3]*1000)
    ax.plot(conformal_times,ev[i,:, 4]*1000)

    ax.set_title('$k= %s$'%ks[i])
    ax.set_xlim(conformal_times[0], conformal_times[-1]);
plt.legend([r'$\Delta_\gamma (CDM)$', r'$\Delta_\gamma (Newtonian)$',r'$\Psi_N$',
            r'Monopole (x1000)', r'ISW (x1000)']);
'Temperature monopole source in general'
$\displaystyle \left(\frac{\Delta_{g}{\left(t \right)}}{4} + 2 \phi{\left(t \right)} + \frac{\eta{\left(t \right)}}{2 Kf_{1}}\right) \operatorname{visibility}{\left(t \right)}$
'Temperature monopole source in Newtonian gauge'
$\displaystyle \frac{\left(\Delta_{g}{\left(t \right)} + 4 \Psi_{N}{\left(t \right)}\right) \operatorname{visibility}{\left(t \right)}}{4}$
In [29]:
# You can also calculate power spectra for custom source functions.
# For example, let's split up the standard temperature result into the various sub-terms,
# and see how they contribute to the total 

early_ISW = sympy.Piecewise( (ISW, 1/a-1> 30),(0, True))  #redshift > 30
late_ISW = ISW - early_ISW

names = ['mon','ISW','eISW','LISW','dop', 'Q']
pars.set_custom_scalar_sources([monopole_source, ISW,early_ISW, late_ISW,doppler,quadrupole_source], 
        source_names =names)

data= camb.get_results(pars)
dic = data.get_cmb_unlensed_scalar_array_dict(CMB_unit='muK')
In [30]:
ls =np.arange(dic['TxT'].shape[0])
plt.semilogx(ls,dic['monxmon'], color='C0')
plt.semilogx(ls,dic['LISWxLISW'], color='C1')
plt.semilogx(ls,dic['eISWxeISW'], ls='--', color='C1')
plt.semilogx(ls,dic['QxQ'], color='C3')
plt.semilogx(ls,dic['TxT'], lw=2, color='k')
plt.xlim(2, ls[-1])
plt.legend(['Monopole','Late ISW','Early ISW','Doppler','Quadrupole', 'Total'], loc = 'upper left');
In [31]:
# The monopole sources can also be decomposed in various different ways, 
# e.g. in terms of comoving-frame quantities: 

# a comoving photon density term (important on sub-horizon scales),
dg = make_frame_invariant(Delta_g/4*visibility, frame='comoving') 

# plus a sum of potential and comoving curvature terms, and ISW
rest=make_frame_invariant(monopole_source - dg, frame=q) +ISW

names = ['dg','rest','mon']
pars.set_custom_scalar_sources([dg, rest, monopole_source+ISW], source_names =names)
data= camb.get_results(pars)
mondic = data.get_cmb_unlensed_scalar_array_dict(CMB_unit='muK')

ls =np.arange(dic['TxT'].shape[0])

plt.semilogx(ls,mondic['restxrest'], color='r')
plt.semilogx(ls,mondic['restxdg'], color='m')
plt.semilogx(ls,mondic['monxmon'], color='k', lw=2)

plt.xlim(2, ls[-1])
plt.legend([r'$\bar{\Delta}_\gamma$',r'$2\phi+\bar{\eta}/2 + {\rm ISW}$',
             'Cross term', 'Monopole total']);

We now demonstrate how to calculate non-standard sources, e.g. as needed to calculate corrections to the standard lensing result from emission angle and time delay effects (see arXiv:1706.02673, as now implemeted in camb in the camb.emission_angle module).

In [32]:
#Various sources for emission angle and time delay, from recombination and reionization

chi = tau0-t #assume flat

emission_sources = {
  'vperp' :  -(sigma+v_b)*visibility/k/chi,
  'emit'  : 15*diff(polter*visibility,t)/8/k**2/chi,
  'delay' : 15*diff(polter*visibility,t)/8/k**2/chi**2 *(tau0 - tau_maxvis),
  'E'     : scalar_E_source}

sources ={}
for key, source in list(emission_sources.items()):
    sources[key+'1'] = sympy.Piecewise((source,1/a-1>30),(0, True))  #recombination
    sources[key+'2'] =  sympy.Piecewise((source,1/a-1<=30),(0, True))  #reionization  
pars.set_custom_scalar_sources(sources, source_ell_scales={'E1':2,'E2':2})
data= camb.get_results(pars)
dic = data.get_cmb_unlensed_scalar_array_dict(CMB_unit='muK')
In [33]:
#Plot polariazation potentials

ls = np.arange(dic['ExE'].shape[0])
lfac = np.sqrt(ls*(ls+1))
plt.loglog(dic['emit1xE1']*ls, color='C2')
plt.loglog(-dic['emit1xE1']*ls,ls ='--',color='C2')

plt.legend(['$C_\ell^E$',r'$\ell^2 C_\ell^{\psi_\zeta}$',r'$\ell C_\ell^{E\psi_\zeta}$'],
           loc='upper left', frameon=False)
In [34]:
#temperature velocity potentials

plt.legend([r'$\ell^2 C^{\psi_v}_\ell$',r'$\ell C^{T\psi_v}_\ell$',r'$C^{T}_\ell$'],
In [35]:
#The camb.emission_angle module uses the above to calculate BB from emission angle and time delay
#e.g. compare BB from lensing, field rotation and emission angle/time delay
from camb import emission_angle
from camb import postborn
%time BB = emission_angle.get_emission_delay_BB(pars, lmax=3500)
%time Bom=postborn.get_field_rotation_BB(pars, lmax=3500)
Wall time: 33.2 s
Wall time: 55.5 s
In [36]:
ls = np.arange(2001)
plt.loglog(ls, data.get_lensed_scalar_cls(2000, CMB_unit = 'muK')[:,2])
plt.xlim([10, 2000])
plt.legend([r'$\kappa\kappa$', r'$\omega\omega$','emission + delay']);
plt.title('BB power spectra');