Analyzing 9-year NANOGrav data¶

In this tutorial we will use enterprise to analyze the NANOGrav 9-year data release for a stochastic GW background. We will reproduce the power-law GWB limit from this paper.

Function to convert PAL2 noise parameters to enterprise parameter dict¶

def get_noise_from_pal2(noisefile):
    psrname = noisefile.split('/')[-1].split('_noise.txt')[0]
    fin = open(noisefile, 'r')
    lines = fin.readlines()
    params = {}
    for line in lines:
        ln = line.split()
        if 'efac' in line:
            par = 'efac'
            flag = ln[0].split('efac-')[-1]
        elif 'equad' in line:
            par = 'log10_equad'
            flag = ln[0].split('equad-')[-1]
        elif 'jitter_q' in line:
            par = 'log10_ecorr'
            flag = ln[0].split('jitter_q-')[-1]
        elif 'RN-Amplitude' in line:
            par = 'log10_A'
            flag = ''
        elif 'RN-spectral-index' in line:
            par = 'gamma'
            flag = ''
        else:
            break
        if flag:
            name = [psrname, flag, par]
        else:
            name = [psrname, par]
        pname = '_'.join(name)
        params.update({pname: float(ln[1])})
    return params

Get par, tim, and noise files¶

Here we collect the tim and par files as well as noise files made from the PAL2 code. These are the same par, tim, and noise files used in the 9-year analysis papers. We use the convienience function above to convert from PAL2 noise files to enterprise parameter dictionaries.

parfiles = sorted(glob.glob(datadir + '/*.par'))
timfiles = sorted(glob.glob(datadir + '/*.tim'))
noisefiles = sorted(glob.glob(datadir + '/*noise.txt'))

# 18 pulsars used in 9 year analysis
p9 = np.loadtxt(datadir+'/9yr_pulsars.txt', dtype='S42')

# filter
parfiles = [x for x in parfiles if x.split('/')[-1].split('_')[0] in p9]
timfiles = [x for x in timfiles if x.split('/')[-1].split('_')[0] in p9]
noisefiles = [x for x in noisefiles if x.split('/')[-1].split('_')[0] in p9]

Load into Pulsar class list¶

psrs = []
for p, t in zip(parfiles, timfiles):
    psr = Pulsar(p, t, ephem='DE421')
    psrs.append(psr)

Get parameter dict from noisefiles¶

params = {}
for nfile in noisefiles:
    params.update(get_noise_from_pal2(nfile))

Set up model¶

When setting up the model for our upper limit run we fix all of the white noise (EFAC, EQUAD, and ECORR) parameters to the values obtained from the noise files. This is done by using Constant parameters. In this case we do not specify a default value for all instances of that parameter but instead will set them, based on their initialized pulsar and backend specific name, later via the set_default_params method of PTA.

Speaking of white noise parameters here, we also use the Selection object.

Another feature to notice is that we do not use a uniform prior on the log of the red noise or GWB amplitude. Instead we use a LinearExp prior (short for linear-exponent prior), that is a prior of the form \(p(x)\propto 10^x\). This is how we can still use the log of the parameter to sample but place a uniform prior on the parameter itself. We do this for both the red noise and GWB amplitude parameters.

Next, in order to save on computing time we do not include spatial correlations here. Instead we model the GWB as a common red process across all pulsars. In enterprise we can do this with a simple trick. We pre-initialize the parameters before passing them to the Signal model. In this way the same parameter instance is used for all pulsars. Lastly, we fixt the spectral index of the GWB to be 13/3 (4.33) using the Constant parameter.

# find the maximum time span to set GW frequency sampling
tmin = [p.toas.min() for p in psrs]
tmax = [p.toas.max() for p in psrs]
Tspan = np.max(tmax) - np.min(tmin)

# selection class to break white noise by backend
selection = selections.Selection(selections.by_backend)

##### parameters and priors #####

# white noise parameters
# since we are fixing these to values from the noise file we set
# them as constant parameters
efac = parameter.Constant()
equad = parameter.Constant()
ecorr = parameter.Constant()

# red noise parameters
log10_A = parameter.LinearExp(-20,-12)
gamma = parameter.Uniform(0,7)

# GW parameters (initialize with names here to use parameters in common across pulsars)
log10_A_gw = parameter.LinearExp(-18,-12)('log10_A_gw')
gamma_gw = parameter.Constant(4.33)('gamma_gw')

##### Set up signals #####

# white noise
ef = white_signals.MeasurementNoise(efac=efac, selection=selection)
eq = white_signals.EquadNoise(log10_equad=equad, selection=selection)
ec = white_signals.EcorrKernelNoise(log10_ecorr=ecorr, selection=selection)

# red noise (powerlaw with 30 frequencies)
pl = utils.powerlaw(log10_A=log10_A, gamma=gamma)
rn = gp_signals.FourierBasisGP(spectrum=pl, components=30, Tspan=Tspan)

# gwb (no spatial correlations)
cpl = utils.powerlaw(log10_A=log10_A_gw, gamma=gamma_gw)
gw = gp_signals.FourierBasisGP(spectrum=cpl, components=30, Tspan=Tspan)

# for spatial correltions you can do...
#orf = utils.hd_orf()
#crn = gp_signals.FourierBasisCommonGP(cpl, orf, components=30, name='gw', Tspan=Tspan)

# timing model
tm = gp_signals.TimingModel()

# to add solar system ephemeris modeling...
#eph = deterministic_signals.PhysicalEphemerisSignal(use_epoch_toas=True)

# full model is sum of components
model = ef + eq + ec + rn + tm + gw

# intialize PTA
pta = signal_base.PTA([model(psr) for psr in psrs])

Set white noise parameters¶

pta.set_default_params(params)

Set initial parameters drawn from prior and evaluate likelihood to fill caches¶

Evaluating the likelihood is not necessary, the caches will be filled the first time it is called within the sampler if not called here.

xs = {par.name: par.sample() for par in pta.params}
print pta.get_lnlikelihood(xs);
print pta.get_lnprior(xs);

1396202.32558
-32.2501201076

Set up sampler¶

# dimension of parameter space
ndim = len(xs)

# initial jump covariance matrix
cov = np.diag(np.ones(ndim) * 0.01**2)

# set up jump groups by red noise groups
ndim = len(xs)
groups  = [range(0, ndim)]
groups.extend(map(list, zip(range(0,ndim,2), range(1,ndim,2))))
groups.extend([[36]])

sampler = ptmcmc(ndim, pta.get_lnlikelihood, pta.get_lnprior, cov, groups=groups,
                 outDir='chains/nano_9_gwb/')

Sample!¶

# sampler for N steps
N = 1000000
x0 = np.hstack(p.sample() for p in pta.params)
sampler.sample(x0, N, SCAMweight=30, AMweight=15, DEweight=50, )

Plot output¶

chain = np.loadtxt('chains/nano_9_gwb/chain_1.txt)
pars = sorted(xs.keys())
burn = int(0.25 * chain.shape[0])

plt.hist(chain[burn:,-5], 50, normed=True, histtype='step', lw=2);
plt.xlabel(pars[-1]);

Upper limit value¶

We see that the upper limit agrees perfectly with the published value.

upper = 10**np.percentile(chain[burn:, -5], q=0.95)
print(upper)

1.49899289556e-15