def xi(self, rin, **keys):
"""
Name:
xi
Calling Sequence:
xi(r, nplk=)
Purpose:
Calculate the 3d linear correlation function by integrating
over the power spectrum.
Inputs:
r: radius in Mpc
nplk=: The number of points in P(k) per log k. See
pkgen() for the default.
"""
r = numpy.array(rin, dtype='f8', copy=False, ndmin=1)
# first get P(k) on a log spaced grid
log_kmin = -5.0
log_kmax = 6.0
k,Pk = self.pkgen(log_kmin, log_kmax, **keys)
rmax=2000.0
if r.max() > rmax:
raise ValueError("Some rmax greater than %s" % rmax)
xi = numpy.zeros(r.size, dtype='f8')
for i in xrange(r.size):
#xi[i] = self.xi_int_pk(k,Pk,r[i])
xi[i] = self.xi_int_pk_slow(k,Pk,r[i])
w=where1(r < 25.0)
rr,ss = self.xi2sigmavar(r[w], xi[w])
s8=exp(interplin(log(ss),log(rr),log(8.0)))
xi=xi*(self.sigma_8/s8)**2
return xi
def xi2sigmavar(self,r,xi):
"""
Compute sigma variance from xi where xi is the 3d correlation
function. returns rr and sig, rr is not exactly the same as r
You can calculate sigma8 from the results
rr,ss = self.xi2sigmavar(r, xi)
s8=exp(interplin(log(ss),log(rr),log(8.0)))
"""
w = where1( xi <= 0 )
if w.size > 0:
raise ValueError("xi must be positive to perform powerlawe interp")
# choose a new set of points, logarithmically spaced such that a
# whole number of points (nper2) correspond to a factor of two
nper2 = 100
rmax = r.max()
rmin = r.min()
tmp = int32( ceil(log(rmax/rmin)/log(2)) )
num = 1+nper2*tmp
tmp = arange(num,dtype='f8')/nper2
rr = rmin*2.0**tmp
log_r = log(r)
log_xi = log(xi)
log_rr = log(rr)
lxx = interplin(log_xi,log_r,log_rr)
xxi = exp(lxx)
# roll is same as shift in IDL
al=(roll(lxx,-1)-lxx)/(roll(log_rr,-1)-log_rr)
al[num-1]=al[num-2]
A=xxi/rr**(al)
Ra=rr.copy()
Ra[0]=0.0
# first one , integrate to zero
Rb=roll(rr,-1)
# 3 integrals to do
beta=0.0
ex=3.0+al+beta
int0=A*(1.0/ex) *(Rb**ex -Ra**ex)
beta=1.0
ex=3.0+al+beta
int1=A*(1.0/ex) *(Rb**ex -Ra**ex)
beta=3.0
ex=3.0+al+beta
int2=A*(1.0/ex) *(Rb**ex -Ra**ex)
# array of sub integrals, last one to be ignored
# Now add up the sub integrals with 0 to rmin integral
T0=roll(int0.cumsum(),-nper2)
T1=roll(int1.cumsum(),-nper2)
T2=roll(int2.cumsum(),-nper2)
sig2=3.0*T0/(rr**3) + (-9.0/4)*T1/(rr**4) + (3.0/16)*T2/(rr**6)
# now trim off last bunch
rr=rr[0:num-nper2-2]
log_rr=log_rr[0:num-nper2-2]
sig2=sig2[0:num-nper2-2]
sig=sqrt(sig2)
# now interpolate back onto original points
ss=exp(interplin(log(sig),log_rr,log_r))
wcut=where1(r*2 < rmax)
rr=r[wcut]
sig=ss
sig=sig[wcut]
return rr, sig
def xi_int_pk(self, k, Pk, r):
"""
get xi at position r by integrating P(k). r should be a scalar
the integral is done in two parts
"""
import scipy.integrate
# some of this could be pre-computed, but it might
# not matter
NumPerPer = 30.0
# Number of samples per period, should always be > 5 at least
npd = 100
# depends on cosmology but this is ballpark
L_wiggle=170.0
Pref=1.0 /(2.0 * PI**2 * r)
dk1=2.0 *PI/(NumPerPer*L_wiggle)
dk2=2.0 *PI/(NumPerPer*r)
dktry=min([dk1,dk2])
# first section
kmin=0.0
kmax=2.0
numk=int32((kmax-kmin)/dktry)
dk=(kmax-kmin)/(1.0*numk-1)
kk=arange(numk,dtype='f8')*dk+kmin
# Dave used cubic spline interpolation
Pkk=interplin(Pk,k,kk)
tab=Pkk*kk*sin(kk*r)
integ = scipy.integrate.simps(tab,kk)
xi_1=Pref*integ
# next section/method
# now need integral \int_xmax^infinity x P(x/r) sin(x)
Pref2=Pref/(r**2)
xmax=r*kmax
num_dec=5
num_per_dec=npd
numx=num_dec*num_per_dec
x=xmax*10.0**(arange(numx,dtype='f8')/num_per_dec)
Px=interplin(Pk,k,x/r)
y=x*Px
ly=log(y)
n=numx
al=(roll(ly,-1)-ly)/(roll(x,-1)-x)
al[n-1]=al[n-2]
Amp=y/exp(al*x)
# integral boundaries
a=x
b=roll(x,-1)
norm=y/(1+al**2)
Ta=al*sin(a)-cos(a)
Tb=al*sin(b)-cos(b)
dif=exp(al*(b-a))*Tb-Ta
d=norm*dif
d=d[0:n-2]
integ=d.sum()
xi_2=integ*Pref2
xi=xi_1+xi_2
return xi
def xi_int_pk_slow(self, k, Pk, r, debug=False, doplot=False):
"""
Maybe try a slow but more accurate version here
get xi at position r by integrating P(k). r should be a scalar
the integral is done in two parts
"""
import scipy.integrate
# some of this could be pre-computed, but it might
# not matter
NumPerPer = 30.0
# Number of samples per period, should always be > 5 at least
npd = 100
# depends on cosmology but this is ballpark
L_wiggle=170.0
Pref=1.0 /(2.0 * PI**2 * r)
dk1=2.0 *PI/(NumPerPer*L_wiggle)
dk2=2.0 *PI/(NumPerPer*r)
dktry=min([dk1,dk2])
# first section
kmin=0.0
kmax=2.0
numk=int32((kmax-kmin)/dktry)
dk=(kmax-kmin)/(1.0*numk-1)
kk=arange(numk,dtype='f8')*dk+kmin
# Dave used *local* cubic spline interpolation (the /spline)
# keyword for interpol. This seems to give radically different
# results! Using interpsp2 above gets us within about 6%
# of dave's answer. Which is right? I think the right thing
# to do is use more values for Pk,k and do linear interpolation
Pkk=interplin(Pk,k,kk)
#Pkk=interp(kk,k,Pk)
tab=Pkk*kk*sin(kk*r)
integ = scipy.integrate.simps(tab,x=kk)
xi_1=Pref*integ
if debug:
print 'r=',r
if dk1 < dk2:
print 'Baryon wiggle limited'
else:
print 'Sin(k*r) limited'
print 'Numk=',numk
print 'dk=',dk
integ_trap = scipy.integrate.trapz(tab,x=kk)
integ_qg1000 = self.qg1000.integrate(kk,tab)
print 'integ=',integ
print 'integ_trap=',integ_trap
print 'integ_qg1000=',integ_qg1000
nprint=20
if doplot:
wk=where1(k < kmax)
wkk=where1(kk > 1.e-8)
ptab=Table(2,1)
lplt=FramedPlot()
lplt.add( Points(k[wk],Pk[wk],type='filled circle',size=1) )
lplt.add( Points(kk[wkk],Pkk[wkk], color='red',
type='filled circle', size=0.7))
lplt.add( Curve(kk[wkk],Pkk[wkk], color='blue'))
lplt.xlog=True
ptab[0,0] = lplt
splt = FramedPlot()
splt.add(Points(k[wk],Pk[wk]*k[wk]*sin(k[wk]*r),
type='filled circle',size=1))
splt.add(Points(kk[wkk],tab[wkk],
color='red',type='filled circle',size=0.7))
splt.add(Curve(kk[wkk],tab[wkk], color='blue'))
splt.xlog=True
ptab[1,0] = splt
ptab.show()
# next section/method
# now need integral \int_xmax^infinity x P(x/r) sin(x)
Pref2=Pref/(r**2)
xmax=r*kmax
num_dec=5
num_per_dec=npd
numx=num_dec*num_per_dec
x=xmax*10.0**(arange(numx,dtype='f8')/num_per_dec)
Px=interplin(Pk,k,x/r)
y=x*Px
ly=log(y)
n=numx
al=(roll(ly,-1)-ly)/(roll(x,-1)-x)
al[n-1]=al[n-2]
Amp=y/exp(al*x)
# integral boundaries
a=x
b=roll(x,-1)
norm=y/(1+al**2)
Ta=al*sin(a)-cos(a)
Tb=al*sin(b)-cos(b)
dif=exp(al*(b-a))*Tb-Ta
d=norm*dif
d=d[0:n-2]
integ=d.sum()
xi_2=integ*Pref2
xi=xi_1+xi_2
if debug:
print 'xi_1:',xi_1
print 'xi_2:',xi_2
return xi