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pktools.py
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pktools.py
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import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import quad
import cosmotools
########################################################################
# Get dimensions of close-fitting cuboid for given range of (R.A., #
# Dec., redshift). #
########################################################################
def boxsize(zmin,zmax,rmin,rmax,dmin,dmax):
rmin,rmax,dmin,dmax = np.radians([rmin,rmax,dmin,dmax])
rcen = rmin + (rmax - rmin)/2; dcen = dmin + (dmax - dmin)/2
zcen = zmin + (zmax - zmin)/2
lx = (rmax - rmin) * np.cos(dcen) * cosmotools.D_com(zcen)
ly = (dmax - dmin) * cosmotools.D_com(zcen)
lz = cosmotools.D_com(zmax) - cosmotools.D_com(zmin)
return lx,ly,lz
def AngularDistance(r1,r2,d1,d2):
return np.arccos( np.sin(d1)*np.sin(d2) + np.cos(d1)*np.cos(d2)*np.cos(r1-r2) )
########################################################################
# Estimate the 3D power spectrum of a density field. #
########################################################################
def getpkspec(datgrid1,datgrid2,nx,ny,nz,lx,ly,lz,kmin,kmax,nkbin):
'''
If auto-correlating then datgrid1 should equal datgrid2
'''
w = np.ones((nx,ny,nz)) #Give all a constant weighting
w = w / np.sum(w) #normalise weighting so they integrate to 1
datgrid1,datgrid2 = datgrid1*w,datgrid2*w
fgrid1 = np.fft.rfftn(datgrid1)
fgrid2 = np.fft.rfftn(datgrid2)
pkspec = np.real( fgrid1 * np.conj(fgrid2) )
vol_box = lx*ly*lz
nc = nx*ny*nz
vol_cell = vol_box / nc
pkspec = vol_cell * pkspec / np.sum(w**2) #L.Wolz method: https://arxiv.org/pdf/1510.05453.pdf
return pkspec
########################################################################
# Compute correction due to alias effect (pixelisation onto grid) #
########################################################################
def W_alias(nx,ny,nz,lx,ly,lz,p):
'''
Following Jing et al compute correction based on mass assignment function
where p=1 (for NGP), p=2 (for CIC) or p=3 (for TSC). Divide measured power
through by W^2 to complete correction
'''
kx = 2*np.pi*np.fft.fftfreq(nx,d=lx/nx)[:,np.newaxis,np.newaxis]
ky = 2*np.pi*np.fft.fftfreq(ny,d=ly/ny)[np.newaxis,:,np.newaxis]
kz = 2*np.pi*np.fft.fftfreq(nz,d=lz/nz)[:int(nz/2)+1][np.newaxis,np.newaxis,:]
kx[kx==0]=1e-30; ky[ky==0]=1e-30; kz[kz==0]=1e-30 #Amend to avoid divide by zeros
return (np.sin(kx*lx/(2*nx))/(kx*lx/(2*nx)) * np.sin(ky*ly/(2*ny))/(ky*ly/(2*ny)) * np.sin(kz*lz/(2*nz))/(kz*lz/(2*nz)))**p
########################################################################
# Bin 3D power spectrum in angle-averaged bins. #
########################################################################
def binpk(pkspec,nx,ny,nz,lx,ly,lz,kmin,kmax,nkbin):
kspec,muspec,indep = getkspec(nx,ny,nz,lx,ly,lz)
pkspec = pkspec[indep == True]
kspec = kspec[indep == True]
ikbin = np.digitize(kspec,np.linspace(kmin,kmax,nkbin+1))
nmodes,pk = np.zeros(nkbin,dtype=int),np.zeros(nkbin)
for ik in range(nkbin):
nmodes[ik] = int(np.sum(np.array([ikbin == ik+1])))
if (nmodes[ik] > 0):
pk[ik] = np.mean(pkspec[ikbin == ik+1])
return pk
########################################################################
# Bin 3D power spectrum in angle-averaged bins, weighting by Legendre #
# polynomials. #
########################################################################
def binpole(pkspec,nx,ny,nz,lx,ly,lz,kmin,kmax,nkbin):
kspec,muspec,indep = getkspec(nx,ny,nz,lx,ly,lz)
pkspec = pkspec[indep == True]
kspec = kspec[indep == True]
muspec = muspec[indep == True]
leg2spec = ((3*(muspec**2))-1)/2
leg4spec = ((35*(muspec**4))-(30*(muspec**2))+3)/8
ikbin = np.digitize(kspec,np.linspace(kmin,kmax,nkbin+1))
nmodes,pk0,pk2,pk4 = np.zeros(nkbin,dtype=int),np.zeros(nkbin),np.zeros(nkbin),np.zeros(nkbin)
for ik in range(nkbin):
nmodes[ik] = int(np.sum(np.array([ikbin == ik+1])))
if (nmodes[ik] > 0):
pk0[ik] = np.mean(pkspec[ikbin == ik+1])
pk2[ik] = 5*np.mean((pkspec*leg2spec)[ikbin == ik+1])
pk4[ik] = 9*np.mean((pkspec*leg4spec)[ikbin == ik+1])
return [pk0,pk2,pk4],nmodes
########################################################################
# Obtain 3D grid of k-modes. #
########################################################################
def getkspec(nx,ny,nz,lx,ly,lz):
kx = 2*np.pi*np.fft.fftfreq(nx,d=lx/nx)
ky = 2*np.pi*np.fft.fftfreq(ny,d=ly/ny)
kz = 2*np.pi*np.fft.fftfreq(nz,d=lz/nz)[:int(nz/2)+1]
indep = getindep(nx,ny,nz)
indep[0,0,0] = False
kspec = np.sqrt(kx[:,np.newaxis,np.newaxis]**2 + ky[np.newaxis,:,np.newaxis]**2 + kz[np.newaxis,np.newaxis,:]**2)
kspec[0,0,0] = 1.
muspec = np.absolute(kz[np.newaxis,np.newaxis,:])/kspec
kspec[0,0,0] = 0
return kspec,muspec,indep
########################################################################
# Obtain array of independent 3D modes. #
########################################################################
def getindep(nx,ny,nz):
indep = np.full((nx,ny,int(nz/2)+1),False,dtype=bool)
indep[:,:,1:int(nz/2)] = True
indep[1:int(nx/2),:,0] = True
indep[1:int(nx/2),:,int(nz/2)] = True
indep[0,1:int(ny/2),0] = True
indep[0,1:int(ny/2),int(nz/2)] = True
indep[int(nx/2),1:int(ny/2),0] = True
indep[int(nx/2),1:int(ny/2),int(nz/2)] = True
indep[int(nx/2),0,0] = True
indep[0,int(ny/2),0] = True
indep[int(nx/2),int(ny/2),0] = True
indep[0,0,int(nz/2)] = True
indep[int(nx/2),0,int(nz/2)] = True
indep[0,int(ny/2),int(nz/2)] = True
indep[int(nx/2),int(ny/2),int(nz/2)] = True
return indep
########################################################################
# Convert power spectrum multipoles P_l(k) to P(k,mu) or P(kpar,kperp).#
########################################################################
def pkpoletopkmu(nmu,pkmults):
nkbin = len(pkmults[0])
dmu = 1/nmu
pkmuobs = np.zeros((nkbin,nmu))
for imu in range(nmu):
mu1 = dmu*imu
mu2 = dmu*(imu+1)
obs = np.zeros(nkbin)
for i in range(3):
coeff = quad(pkmuint,mu1,mu2,args=(i*2))[0]
obs[:] += (coeff/dmu)*pkmults[i]
pkmuobs[:,imu] = obs
return pkmuobs
def pkmuint(mu,l):
return getleg(l,mu)
def getleg(l,mu):
if (l == 2):
leg = (3*(mu**2)-1)/2
elif (l == 4):
leg = (35*(mu**4)-30*(mu**2)+3)/8
else:
leg = 1
return leg
def pkmutopk2(kmin2,kmax2,nk2,kmin,kmax,nk,nmu,pkmu):
# Generate random points across (kperp,kpar) space
nran = 1000000
rkperp = kmin2 + (kmax2-kmin2)*np.random.rand(nran)
rkpar = kmin2 + (kmax2-kmin2)*np.random.rand(nran)
# Convert these points to (k,mu) values
rk = np.sqrt(rkperp**2 + rkpar**2)
rmu = rkpar/rk
# Bin these points in (k,mu) bins
klims = np.concatenate((np.linspace(kmin,kmax,nk+1),np.array([1.])))
ikbin = np.digitize(rk,klims) - 1
mulims = np.linspace(0.,1.,nmu+1)
mulims[0],mulims[nmu] = -0.01,1.01
imubin = np.digitize(rmu,mulims) - 1
# Power spectrum values of these points
pkmuobs1 = np.zeros((nk+1,nmu))
pkmuobs1[:nk,:] = pkmu
rpkobs = pkmuobs1[ikbin,imubin]
# Count points in (k,mu) bins
pkmucount,edges = np.histogramdd(np.vstack([ikbin+0.5,imubin+0.5]).transpose(),bins=(nk,nmu),range=((0,nk),(0,nmu)))
# Count points in (kperp,kpar) bins
pk2count,edges = np.histogramdd(np.vstack([rkperp,rkpar]).transpose(),bins=(nk2,nk2),range=((kmin2,kmax2),(kmin2,kmax2)))
# Bin these points in (kperp,kpar) bins
pk2obs,edges = np.histogramdd(np.vstack([rkperp,rkpar]).transpose(),bins=(nk2,nk2),range=((kmin2,kmax2),(kmin2,kmax2)),normed=False,weights=rpkobs)
pk2obs = pk2obs/pk2count
return pk2obs