cmb_modules.py

import numpy as np
import matplotlib
import sys
import matplotlib.cm as cm
import matplotlib.mlab as mlab
import matplotlib.pyplot as plt
import astropy.io.fits as fits



def make_CMB_T_map(N,pix_size,ell,DlTT):
    "makes a realization of a simulated CMB sky map given an input DlTT as a function of ell,"
    "the pixel size (pix_size) required and the number N of pixels in the linear dimension."
    #np.random.seed(100)
    # convert Dl to Cl
    ClTT = DlTT * 2 * np.pi / (ell*(ell+1.))
    ClTT[0] = 0. # set the monopole and the dipole of the Cl spectrum to zero
    ClTT[1] = 0.
    
    # make a 2D real space coordinate system
    onesvec = np.ones(N)
    inds  = (np.arange(N)+.5 - N/2.) /(N-1.) # create an array of size N between -0.5 and +0.5
    # compute the outer product matrix: X[i, j] = onesvec[i] * inds[j] for i,j
    # in range(N), which is just N rows copies of inds - for the x dimension
    X = np.outer(onesvec,inds)
    # compute the transpose for the y dimension
    Y = np.transpose(X)
    # radial component R
    R = np.sqrt(X**2. + Y**2.)
    
    # now make a 2D CMB power spectrum
    pix_to_rad = (pix_size/60. * np.pi/180.) # going from pix_size in arcmins to degrees and then degrees to radians
    ell_scale_factor = 2. * np.pi /pix_to_rad  # now relating the angular size in radians to multipoles
    ell2d = R * ell_scale_factor # making a fourier space analogue to the real space R vector
    ClTT_expanded = np.zeros(int(ell2d.max())+1)
    # making an expanded Cl spectrum (of zeros) that goes all the way to the size of the 2D ell vector
    ClTT_expanded[0:(ClTT.size)] = ClTT # fill in the Cls until the max of the ClTT vector
    
    # the 2D Cl spectrum is defined on the multiple vector set by the pixel scale
    CLTT2d = ClTT_expanded[ell2d.astype(int)]
    #plt.imshow(np.log(CLTT2d))
    
    
    # now make a realization of the CMB with the given power spectrum in real space
    random_array_for_T = np.random.normal(0,1,(N,N))
    FT_random_array_for_T = np.fft.fft2(random_array_for_T)   # take FFT since we are in Fourier space
    
    FT_2d = np.sqrt(CLTT2d) * FT_random_array_for_T # we take the sqrt since the power spectrum is T^2
    #plt.imshow(np.real(FT_2d))
    
    
    ## make a plot of the 2D cmb simulated map in Fourier space, note the x and y axis labels need to be fixed
    #Plot_CMB_Map(np.real(np.conj(FT_2d)*FT_2d*ell2d * (ell2d+1)/2/np.pi),0,np.max(np.conj(FT_2d)*FT_2d*ell2d * (ell2d+1)/2/np.pi),ell2d.max(),ell2d.max())  ###
    
    # move back from ell space to real space
    CMB_T = np.fft.ifft2(np.fft.fftshift(FT_2d))
    # move back to pixel space for the map
    CMB_T = CMB_T/(pix_size /60.* np.pi/180.)
    # we only want to plot the real component
    CMB_T = np.real(CMB_T)
    
    ## return the map
    return(CMB_T)
###############################

def Plot_CMB_Map(Map_to_Plot,c_min,c_max,X_width,Y_width):
    from mpl_toolkits.axes_grid1 import make_axes_locatable
    print("map mean:",np.mean(Map_to_Plot),"map rms:",np.std(Map_to_Plot))
    plt.gcf().set_size_inches(10, 10)
    im = plt.imshow(Map_to_Plot, interpolation='bilinear', origin='lower',cmap=cm.RdBu_r)
    im.set_clim(c_min,c_max)
    ax=plt.gca()
    divider = make_axes_locatable(ax)
    cax = divider.append_axes("right", size="5%", pad=0.05)
    
    cbar = plt.colorbar(im, cax=cax)
    #cbar = plt.colorbar()
    im.set_extent([0,X_width,0,Y_width])
    plt.ylabel('angle $[^\circ]$')
    plt.xlabel('angle $[^\circ]$')
    cbar.set_label('tempearture [uK]', rotation=270)
    
    plt.show()
    return(0)
###############################


def Poisson_source_component(N,pix_size,Number_of_Sources,Amplitude_of_Sources):
    "makes a realization of a naive Poisson-distributed point source map"
    PSMap = np.zeros([np.int(N),np.int(N)])
    i = 0
    print('Number of sources required: ', Number_of_Sources)
    
    while (i < int(Number_of_Sources)):
        pix_x = np.int(N*np.random.rand())
        pix_y = np.int(N*np.random.rand())
        PSMap[pix_x,pix_y] += np.random.poisson(Amplitude_of_Sources)
        i = i + 1

    return(PSMap)    
  ############################### 

def Exponential_source_component(N,pix_size,Number_of_Sources_EX,Amplitude_of_Sources_EX):
    N=int(N)
    "makes a realization of a naive exponentially-distributed point source map"
    PSMap = np.zeros([N,N])
    i = 0
    while (i < Number_of_Sources_EX):
        pix_x = int(N*np.random.rand() )
        pix_y = int(N*np.random.rand()) 
        PSMap[pix_x,pix_y] += np.random.exponential(Amplitude_of_Sources_EX)
        i = i + 1

    return(PSMap)    
  ############################### 

def SZ_source_component(N,pix_size,Number_of_SZ_Clusters,Mean_Amplitude_of_SZ_Clusters,SZ_beta,SZ_Theta_core,do_plots):
    "makes a realization of a nieve SZ map"
    N=int(N)
    SZMap = np.zeros([N,N])
    SZcat = np.zeros([3,Number_of_SZ_Clusters]) ## catalogue of SZ sources, X, Y, amplitude
    # make a distribution of point sources with varying amplitude
    i = 0
    while (i < Number_of_SZ_Clusters):
        pix_x = np.int(N*np.random.rand())
        pix_y = np.int(N*np.random.rand() )
        pix_amplitude = np.random.exponential(Mean_Amplitude_of_SZ_Clusters)*(-1.)
        SZcat[0,i] = pix_x
        SZcat[1,i] = pix_y
        SZcat[2,i] = pix_amplitude
        SZMap[pix_x,pix_y] += pix_amplitude
        i = i + 1
    if (do_plots):
        hist,bin_edges = np.histogram(SZMap,bins = 50,range=[SZMap.min(),-10])
        plt.semilogy(bin_edges[0:-1],hist)
        plt.xlabel('source amplitude [$\mu$K]')
        plt.ylabel('number or pixels')
        plt.show()
    
    # make a beta function
    beta = beta_function(N,pix_size,SZ_beta,SZ_Theta_core)
    
    # convovle the beta funciton with the point source amplitude to get the SZ map
    FT_beta = np.fft.fft2(np.fft.fftshift(beta))
    FT_SZMap = np.fft.fft2(np.fft.fftshift(SZMap))
    SZMap = np.fft.fftshift(np.real(np.fft.ifft2(FT_beta*FT_SZMap)))
    
    # return the SZ map
    return(SZMap,SZcat)    
  ############################### 

def beta_function(N,pix_size,SZ_beta,SZ_Theta_core):
  # make a beta function
    N=int(N)
    ones = np.ones(N)
    inds  = (np.arange(N)+.5 - N/2.) * pix_size
    X = np.outer(ones,inds)
    Y = np.transpose(X)
    R = np.sqrt(X**2. + Y**2.)
    
    beta = (1 + (R/SZ_Theta_core)**2.)**((1-3.*SZ_beta)/2.)

    # return the beta function map
    return(beta)
  ############################### 

def convolve_map_with_gaussian_beam(N,pix_size,beam_size_fwhp,Map):
    "convolves a map with a gaussian beam pattern.  NOTE: pix_size and beam_size_fwhp need to be in the same units" 
    # make a 2d gaussian 
    gaussian = make_2d_gaussian_beam(N,pix_size,beam_size_fwhp)
  
    # do the convolution
    FT_gaussian = np.fft.fft2(np.fft.fftshift(gaussian))
    FT_Map = np.fft.fft2(np.fft.fftshift(Map))
    convolved_map = np.fft.fftshift(np.real(np.fft.ifft2(FT_gaussian*FT_Map)))
    
    # return the convolved map
    return(convolved_map)
  ###############################   

def make_2d_gaussian_beam(N,pix_size,beam_size_fwhp):
     # make a 2d coordinate system
    N=int(N)
    ones = np.ones(N)
    inds  = (np.arange(N)+.5 - N/2.) * pix_size
    X = np.outer(ones,inds)
    Y = np.transpose(X)
    R = np.sqrt(X**2. + Y**2.)
  
    # make a 2d gaussian 
    beam_sigma = beam_size_fwhp / np.sqrt(8.*np.log(2))
    gaussian = np.exp(-.5 *(R/beam_sigma)**2.)
    gaussian = gaussian / np.sum(gaussian)
 
    # return the gaussian
    return(gaussian)
  ###############################  

def make_noise_map(N,pix_size,white_noise_level,atmospheric_noise_level,one_over_f_noise_level):
    "makes a realization of instrument noise, atmosphere and 1/f noise level set at 1 degrees"
    ## make a white noise map
    N=int(N)
    white_noise = np.random.normal(0,1,(N,N)) * white_noise_level/pix_size
 
    ## make an atmosperhic noise map
    atmospheric_noise = 0.
    if (atmospheric_noise_level != 0):
        ones = np.ones(N)
        inds  = (np.arange(N)+.5 - N/2.) 
        X = np.outer(ones,inds)
        Y = np.transpose(X)
        R = np.sqrt(X**2. + Y**2.) * pix_size /60. ## angles relative to 1 degrees  
        mag_k = 2 * np.pi/(R+.01)  ## 0.01 is a regularizaiton factor
        atmospheric_noise = np.fft.fft2(np.random.normal(0,1,(N,N)))
        atmospheric_noise  = np.fft.ifft2(atmospheric_noise * np.fft.fftshift(mag_k**(5/3.)))* atmospheric_noise_level/pix_size

    ## make a 1/f map, along a single direction to illustrate striping 
    oneoverf_noise = 0.
    if (one_over_f_noise_level != 0): 
        ones = np.ones(N)
        inds  = (np.arange(N)+.5 - N/2.) 
        X = np.outer(ones,inds) * pix_size /60. ## angles relative to 1 degrees 
        kx = 2 * np.pi/(X+.01) ## 0.01 is a regularizaiton factor
        oneoverf_noise = np.fft.fft2(np.random.normal(0,1,(N,N)))
        oneoverf_noise = np.fft.ifft2(oneoverf_noise * np.fft.fftshift(kx))* one_over_f_noise_level/pix_size

    ## return the noise map
    noise_map = np.real(white_noise + atmospheric_noise + oneoverf_noise)
    return(noise_map)
  ###############################
def Filter_Map(Map,N,N_mask):
    N=int(N)
    ## set up a x, y, and r coordinates for mask generation
    ones = np.ones(N)
    inds  = (np.arange(N)+.5 - N/2.) 
    X = np.outer(ones,inds)
    Y = np.transpose(X)
    R = np.sqrt(X**2. + Y**2.)  ## angles relative to 1 degrees  
    
    ## make a mask
    mask  = np.ones([N,N])
    mask[np.where(np.abs(X) < N_mask)]  = 0

    return apply_filter(Map,mask)


def apply_filter(Map,filter2d):
    ## apply the filter in fourier space
    FMap = np.fft.fftshift(np.fft.ifft2(np.fft.fftshift(Map)))
    FMap_filtered = FMap * filter2d
    Map_filtered = np.real(np.fft.fftshift(np.fft.fft2(FMap_filtered)))
    
    ## return the output
    return(Map_filtered)



def cosine_window(N):
    "makes a cosine window for apodizing to avoid edges effects in the 2d FFT" 
    # make a 2d coordinate system
    ones = np.ones(N)
    inds  = (np.arange(N)+.5 - N/2.)/N *np.pi ## eg runs from -pi/2 to pi/2
    X = np.outer(ones,inds)
    Y = np.transpose(X)
  
    # make a window map
    window_map = np.cos(X) * np.cos(Y)
   
    # return the window map
    return(window_map)
  ###############################


def average_N_spectra(spectra,N_spectra,N_ells):
    avgSpectra = np.zeros(N_ells)
    rmsSpectra = np.zeros(N_ells)
    
    # calcuate the average spectrum
    i = 0
    while (i < N_spectra):
        avgSpectra = avgSpectra + spectra[i,:]
        i = i + 1
    avgSpectra = avgSpectra/(1. * N_spectra)
    
    #calculate the rms of the spectrum
    i =0
    while (i < N_spectra):
        rmsSpectra = rmsSpectra +  (spectra[i,:] - avgSpectra)**2
        i = i + 1
    rmsSpectra = np.sqrt(rmsSpectra/(1. * N_spectra))
    
    return(avgSpectra,rmsSpectra)

def calculate_2d_spectrum(Map,delta_ell,ell_max,pix_size,N,Map2=None):
    "calculates the power spectrum of a 2d map by FFTing, squaring, and azimuthally averaging"
    import matplotlib.pyplot as plt
    # make a 2d ell coordinate system
    N=int(N)
    ones = np.ones(N)
    inds  = (np.arange(N)+.5 - N/2.) /(N-1.)
    kX = np.outer(ones,inds) / (pix_size/60. * np.pi/180.)
    kY = np.transpose(kX)
    K = np.sqrt(kX**2. + kY**2.)
    ell_scale_factor = 2. * np.pi 
    ell2d = K * ell_scale_factor
    
    # make an array to hold the power spectrum results
    N_bins = int(ell_max/delta_ell)
    ell_array = np.arange(N_bins)
    CL_array = np.zeros(N_bins)
    
    # get the 2d fourier transform of the map
    FMap = np.fft.ifft2(np.fft.fftshift(Map))
    if Map2 is None: FMap2 = FMap.copy()
    else: FMap2 = np.fft.ifft2(np.fft.fftshift(Map2))
    
#    print FMap
    PSMap = np.fft.fftshift(np.real(np.conj(FMap) * FMap2))
 #   print PSMap
    # fill out the spectra
    i = 0
    while (i < N_bins):
        ell_array[i] = (i + 0.5) * delta_ell
        inds_in_bin = ((ell2d >= (i* delta_ell)) * (ell2d < ((i+1)* delta_ell))).nonzero()
        CL_array[i] = np.mean(PSMap[inds_in_bin])
        i = i + 1


    CL_array_new = CL_array[~np.isnan(CL_array)]
    ell_array_new = ell_array[~np.isnan(CL_array)]
    # return the power spectrum and ell bins
    return(ell_array_new,CL_array_new*np.sqrt(pix_size /60.* np.pi/180.)*2.)