![\"Open](\"https://colab.research.google.com/assets/colab-badge.svg\"){kind=icon}
"
]
},
{
"cell_type": "markdown",
"source": [
"# Spectral Siren Population Inference on GWTC-3\n",
"\n",
"The thrid gravitational-wave transient catalog [GWTC-3](https://arxiv.org/abs/1811.12907) includes all compact binary coalescences observed during Advanced LIGO/Virgo's first three oberving runs.\n",
"\n",
"\"Spectral siren\" cosmology (see, e.g., [Ezquiaga & Holz](https://arxiv.org/abs/2202.08240)) uses features in the black hole mass spectrum to constrain the distance-redshift relation.\n",
"\n",
"`GWPopulation` builds upon [Bilby](https://git.ligo.org/lscsoft/bilby) ([arXiv:1811.02042](https://arxiv.org/abs/1811.02042)) to provide simple, modular, user-friendly, population inference.\n",
"`wcosmo` provides optimized cosmology functionality for `GWPopulation`.\n",
"\n",
"There are many [implemented models](https://colmtalbot.github.io/gwpopulation/_autosummary/gwpopulation.models.html#module-gwpopulation.models) and an example of defining custom models is included below.\n",
"In this example we use:\n",
"\n",
"- A mass distribution in primary mass and mass ratio from Talbot & Thrane (2018) ([arXiv:1801:02699](https://arxiv.org/abs/1801.02699)). This is equivalent to the `PowerLaw + Peak` model used in LVK analyses without the low-mass smoothing for computational efficiency.\n",
"- Half-Gaussian + isotropic spin tilt distribution from Talbot & Thrane (2017) ([arXiv:1704.08370](https://arxiv.org/abs/1704.08370)).\n",
"- Beta spin magnitude distribution from Wysocki+ (2018) ([arXiv:1805:06442](https://arxiv.org/abs/1805.06442)).\n",
"- Each of these are also available with independent but identically distributed spins.\n",
"- Redshift evolution model as in Fishbach+ (2018) ([arXiv:1805.10270](https://arxiv.org/abs/1805.10270)).\n",
"- A variable Flat Lambda CDM cosmology using a modified version of the approximation in [arXiv:1111.6396](https://arxiv.org/abs/1111.6396) as implemented in `wcosmo`.\n",
"\n",
"For more information on `GWPopulation` see the [git repository](https://github.com/ColmTalbot/gwpopulation), [documentation](https://colmtalbot.github.io/gwpopulation/).\n",
"\n",
"\n",
"## Setup\n",
"\n",
"If you're using colab.research.google.com you will want to choose a GPU-accelerated runtime (I'm going to use a T4 GPU).\n",
"\n",
"\"runtime\"->\"change runtime type\"->\"Hardware accelerator = GPU\""
],
"metadata": {
"id": "aeUxtTpEfzbn"
}
},
{
"cell_type": "markdown",
"source": [
"## Install some needed packages\n",
"\n",
"Almost all of the dependencies for this are integrated into `GWPopulation`.\n",
"These include `wcosmo` for cosmology, and `Bilby` and `dynesty` for sampling.\n",
"\n",
"The one exception is [`unxt`](https://unxt.readthedocs.io/en/latest/) which provides `astropy`-like units compatible with `jax`."
],
"metadata": {
"id": "Dc8zj3wngiYu"
}
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"id": "Tsv8JYinDZv0",
"collapsed": true
},
"outputs": [],
"source": [
"!pip install gwpopulation unxt --quiet --progress-bar off"
]
},
{
"cell_type": "markdown",
"source": [
"## Download data\n",
"\n",
"We need to download the data for the events and simmulated \"injections\" used to characterize the detection sensitivity.\n",
"\n",
"### Event posteriors\n",
"\n",
"We're using the posteriors from the GWTC-3 data release in a pre-processed format.\n",
"\n",
"The file was produced by [gwpopulation-pipe](https://docs.ligo.org/ratesAndPopulations/gwpopulation_pipe) to reduce the many GB of posterior sample files to a single ~30Mb file.\n",
"\n",
"The choice of events in this file was not very careful and should only be considered qualitatively correct.\n",
"\n",
"The data file can be found [here](https://drive.google.com/drive/folders/1wyfR6sYvYVdBefF9_vrVTp0Btu03OlzL?usp=drive_link).\n",
"The original data can be found at [zenodo:5546663](https://zenodo.org/records/5546663) and [zenodo:6513631](https://zenodo.org/records/6513631) along with citation information.\n",
"\n",
"### Sensitivity injections\n",
"\n",
"Again I have pre-processed the full injection set using `gwpopulation-pipe` to reduce the filesize.\n",
"The original data is available at [zenodo:7890398](https://zenodo.org/records/7890398) along with citation information."
],
"metadata": {
"id": "TdiqeGZtHwRw"
}
},
{
"cell_type": "code",
"source": [
"!gdown https://drive.google.com/uc?id=16gStLIjt65gWBkw-gNOVUqNbZ89q8CLF\n",
"!gdown https://drive.google.com/uc?id=10pevUCM3V2-D-bROFEMAcTJsX_9RzeM6"
],
"metadata": {
"collapsed": true,
"id": "s8bqvo26HABd",
"outputId": "6c4674b2-44d6-4612-d950-cd2ae48b2c3c",
"colab": {
"base_uri": "https://localhost:8080/"
}
},
"execution_count": 2,
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Downloading...\n",
"From: https://drive.google.com/uc?id=16gStLIjt65gWBkw-gNOVUqNbZ89q8CLF\n",
"To: /content/gwtc-3-injections.pkl\n",
"100% 2.69M/2.69M [00:00<00:00, 19.0MB/s]\n",
"Downloading...\n",
"From: https://drive.google.com/uc?id=10pevUCM3V2-D-bROFEMAcTJsX_9RzeM6\n",
"To: /content/gwtc-3-samples.pkl\n",
"100% 36.4M/36.4M [00:00<00:00, 85.6MB/s]\n"
]
}
]
},
{
"cell_type": "markdown",
"source": [
"## Imports\n",
"\n",
"Import the packages required for the script.\n",
"We also set the backend for array operations to `jax` which allows us to take advantage of just-in-time (jit) compilation in addition to GPU-parallelisation when available."
],
"metadata": {
"id": "I1MQIoc_gvR-"
}
},
{
"cell_type": "code",
"source": [
"import bilby as bb\n",
"import gwpopulation as gwpop\n",
"import jax\n",
"import matplotlib.pyplot as plt\n",
"import numpy as np\n",
"import pandas as pd\n",
"from bilby.core.prior import PriorDict, Uniform\n",
"from gwpopulation.experimental.cosmo_models import CosmoModel\n",
"from gwpopulation.experimental.jax import JittedLikelihood\n",
"from wcosmo.astropy import Planck15\n",
"from wcosmo.utils import disable_units\n",
"\n",
"disable_units()\n",
"gwpop.set_backend(\"jax\")\n",
"\n",
"xp = gwpop.utils.xp\n",
"\n",
"%matplotlib inline"
],
"metadata": {
"id": "mKC8rhSUDmtg"
},
"execution_count": 3,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"## Load posteriors\n",
"\n",
"We remove two events from the file that shouldn't be there that have NS-like secondaries as we are just interested in BBHs for this demonstration.\n",
"\n",
"We also need to modify some of the stored parameters as we are going to include a cosmological fit so we will fit detector frame masses and luminosity distance.\n",
"\n",
"When using the `JAX` backend, this could probably be accelerated by sprinkling some `jax.jit`s."
],
"metadata": {
"id": "6MIE6_uMg4vw"
}
},
{
"cell_type": "code",
"source": [
"posteriors = pd.read_pickle(\"gwtc-3-samples.pkl\")\n",
"del posteriors[15]\n",
"del posteriors[38]\n",
"\n",
"for post in posteriors:\n",
" zs = post.pop(\"redshift\").values\n",
" post[\"mass_1_detector\"] = post.pop(\"mass_1\") * (1 + zs)\n",
" post[\"luminosity_distance\"] = np.asarray(Planck15.luminosity_distance(zs))\n",
" post[\"prior\"] /= np.asarray(Planck15.dDLdz(zs) * (1 + zs))"
],
"metadata": {
"id": "1pAakjJ1DwrK"
},
"execution_count": 4,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"## Load injections\n",
"\n",
"Load the injections used to characterize the sensitivity of the gravitaitonal-wave survey.\n",
"\n",
"Again, we need to make some modifications for fitting in detector-frame quantities."
],
"metadata": {
"id": "82GMaSlJhKm1"
}
},
{
"cell_type": "code",
"source": [
"import dill\n",
"\n",
"with open(\"gwtc-3-injections.pkl\", \"rb\") as ff:\n",
" injections = dill.load(ff)\n",
"\n",
"zs = np.asarray(injections.pop(\"redshift\"))\n",
"injections[\"mass_1_detector\"] = injections.pop(\"mass_1\") * (1 + zs)\n",
"injections[\"luminosity_distance\"] = np.asarray(Planck15.luminosity_distance(zs))\n",
"injections[\"prior\"] /= np.asarray(Planck15.dDLdz(zs) * (1 + zs))"
],
"metadata": {
"id": "IuHqDXFQE9Mw"
},
"execution_count": 5,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"## Define some models and the likelihood\n",
"\n",
"We need to define `Bilby` `Model` objects for the numerator and denominator independently as these cache some computations interally.\n",
"\n",
"Note that we are using a `CosmoModel`, this in an experimental feature and so the specific API may change in future, but the basic funcionality should be stable.\n",
"We create a model that can infer the three parameters of `wCDM`.\n",
"\n",
"The `HyperparameterLikelihood` marginalises over the local merger rate, with a uniform-in-log prior.\n",
"The posterior for the merger rate can be recovered in post-processing.\n",
"\n",
"We provide:\n",
"- `posteriors`: a list of `pandas` DataFrames.\n",
"- `hyper_prior`: our population model, as defined above.\n",
"- `selection_function`: anything which evaluates the selection function.\n",
"\n",
"We can also provide:\n",
"- `conversion_function`: this converts between the parameters we sample in and those needed by the model, e.g., for sampling in the mean and variance of the beta distribution.\n",
"- `max_samples`: the maximum number of samples to use from each posterior, this defaults to the length of the shortest posterior."
],
"metadata": {
"id": "I5FzBbnniQfl"
}
},
{
"cell_type": "code",
"source": [
"model = CosmoModel(\n",
" model_functions=[\n",
" gwpop.models.mass.two_component_primary_mass_ratio,\n",
" gwpop.models.spin.iid_spin,\n",
" gwpop.models.redshift.PowerLawRedshift(cosmo_model=\"FlatwCDM\"),\n",
" ],\n",
" cosmo_model=\"FlatwCDM\",\n",
")\n",
"\n",
"vt = gwpop.vt.ResamplingVT(model=model, data=injections, n_events=len(posteriors))\n",
"\n",
"likelihood = gwpop.hyperpe.HyperparameterLikelihood(\n",
" posteriors=posteriors,\n",
" hyper_prior=model,\n",
" selection_function=vt,\n",
")"
],
"metadata": {
"id": "7IrVTnK5GbaZ"
},
"execution_count": 6,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"## Define our prior\n",
"\n",
"The mass model has eight parameters that we vary that are described in arXiv:1801:02699. This model is sometimes referred to as \"PowerLaw+Peak\"\n",
"\n",
"The spin magnitude model is a `Beta` distribution with the usual parameterization, and the spin orientation model is a mixure of a uniform component and a truncated Gaussian that peaks at aligned spin. This combination is sometimes referred to as \"Default\".\n",
"\n",
"For redshift we use a model that looks like\n",
"\n",
"$$p(z) \\propto \\frac{d V_{c}}{dz} (1 + z)^{λ - 1}$$\n",
"\n",
"Finally, we set priors on the three parameters of the `wCDM` model, $H_0$ (the Hubble constant), $\\Omega_{m,0}$ (the matter fraction of the Universe at current time), $w_0$ (the constant dark energy equation of state)."
],
"metadata": {
"id": "lQW_m3bNioih"
}
},
{
"cell_type": "code",
"source": [
"priors = PriorDict()\n",
"\n",
"# mass\n",
"priors[\"alpha\"] = Uniform(minimum=-2, maximum=4, latex_label=\"$\\\\alpha$\")\n",
"priors[\"beta\"] = Uniform(minimum=-4, maximum=12, latex_label=\"$\\\\beta$\")\n",
"priors[\"mmin\"] = Uniform(minimum=2, maximum=2.5, latex_label=\"$m_{\\\\min}$\")\n",
"priors[\"mmax\"] = Uniform(minimum=80, maximum=100, latex_label=\"$m_{\\\\max}$\")\n",
"priors[\"lam\"] = Uniform(minimum=0, maximum=1, latex_label=\"$\\\\lambda_{m}$\")\n",
"priors[\"mpp\"] = Uniform(minimum=10, maximum=50, latex_label=\"$\\\\mu_{m}$\")\n",
"priors[\"sigpp\"] = Uniform(minimum=1, maximum=10, latex_label=\"$\\\\sigma_{m}$\")\n",
"priors[\"gaussian_mass_maximum\"] = 100\n",
"# spin\n",
"priors[\"amax\"] = 1\n",
"priors[\"alpha_chi\"] = Uniform(\n",
" minimum=1, maximum=6, latex_label=\"$\\\\alpha_{\\\\chi}$\"\n",
")\n",
"priors[\"beta_chi\"] = Uniform(minimum=1, maximum=6, latex_label=\"$\\\\beta_{\\\\chi}$\")\n",
"priors[\"xi_spin\"] = Uniform(minimum=0, maximum=1, latex_label=\"$\\\\xi$\")\n",
"priors[\"sigma_spin\"] = Uniform(minimum=0.3, maximum=4, latex_label=\"$\\\\sigma$\")\n",
"\n",
"priors[\"H0\"] = Uniform(minimum=20, maximum=200, latex_label=\"$H_0$\")\n",
"priors[\"Om0\"] = Uniform(minimum=0, maximum=1, latex_label=\"$\\\\Omega_{m,0}$\")\n",
"priors[\"w0\"] = Uniform(minimum=-1.5, maximum=-0.5, latex_label=\"$w_{0}$\")\n",
"priors[\"lamb\"] = Uniform(minimum=-1, maximum=10, latex_label=\"$\\\\lambda_{z}$\")"
],
"metadata": {
"id": "IRf7FFQDGda-"
},
"execution_count": 7,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"## Just-in-time compile\n",
"\n",
"We JIT compile the likelihood object before starting the sampler.\n",
"This is done using the `gwpopulation.experimental.jax.JittedLikelihood` class.\n",
"\n",
"We then time the original likelihood object and the JIT-ed version.\n",
"Note that we do two evaluations for each object as the first evaluation must compile the likelihood and so takes longer. (In addition to the JIT compilation, `JAX` compiles GPU functionality at the first evaluation, but this is less extreme than the full JIT compilation.)"
],
"metadata": {
"id": "ncci_peFisbm"
}
},
{
"cell_type": "code",
"source": [
"parameters = priors.sample()\n",
"likelihood.parameters.update(parameters)\n",
"likelihood.log_likelihood_ratio()\n",
"%time print(likelihood.log_likelihood_ratio())\n",
"jit_likelihood = JittedLikelihood(likelihood)\n",
"jit_likelihood.parameters.update(parameters)\n",
"%time print(jit_likelihood.log_likelihood_ratio())\n",
"%time print(jit_likelihood.log_likelihood_ratio())"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/"
},
"id": "fgkofuRrGfX7",
"outputId": "5638c2f2-9256-45ed-a88b-c8c37575c5af"
},
"execution_count": 8,
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"-234.23005649570018\n",
"CPU times: user 2.25 s, sys: 242 ms, total: 2.49 s\n",
"Wall time: 3.11 s\n",
"-234.23005649570018\n",
"CPU times: user 6.57 s, sys: 536 ms, total: 7.11 s\n",
"Wall time: 7.12 s\n",
"-234.23005649570018\n",
"CPU times: user 10.2 ms, sys: 0 ns, total: 10.2 ms\n",
"Wall time: 23.1 ms\n"
]
}
]
},
{
"cell_type": "markdown",
"source": [
"## Run the sampler\n",
"\n",
"We'll use the sampler `dynesty` and use a small number of live points to reduce the runtime (total runtime should be approximately 5 minutes on T4 GPUs via Google colab).\n",
"The settings here may not give publication quality results, a convergence test should be performed before making strong quantitative statements.\n",
"\n",
"`bilby` times a single likelihood evaluation before beginning the run, however, this isn't well defined with JAX.\n",
"\n",
"**Note:** sometimes this finds a high likelihood mode, likely due to [breakdowns in the approximation](https://arxiv.org/abs/2304.06138) used to estimate the likelihood. If you see `dlogz > -80`, you should interrupt the execution and restart."
],
"metadata": {
"id": "fX0FoqX0ixo0"
}
},
{
"cell_type": "code",
"source": [
"result = bb.run_sampler(\n",
" likelihood=jit_likelihood,\n",
" priors=priors,\n",
" sampler=\"dynesty\",\n",
" nlive=100,\n",
" label=\"cosmo\",\n",
" sample=\"acceptance-walk\",\n",
" naccept=5,\n",
" save=\"hdf5\",\n",
" resume=False,\n",
")"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 781,
"referenced_widgets": [
"72d055b18aa940a6b0b7631150dccd60",
"8782ae2a54d547acba5516c1b8afa1a5",
"d0d3c27fdce9454d96ae4c3bc5e720d9",
"333d64bb31a9488f965d35829d1653ec",
"d70bed921fee457d9d99114c32dbca2e",
"27e703e67d07495d8b9876ca4cc180b3",
"410a3e9d0153406ca218c5bb27e5d0d6",
"17ce52a4bdb94ac898c258491f0fbe83",
"d7d88647a7704491bb1b8a45b518e6d3",
"2f5856f1fe214e74bdcc0acbb62fff91",
"cf3481e3b7d5429db2d8462474cbe3ef"
]
},
"id": "lHB2loW6Ghx-",
"outputId": "b6fe482e-3b6b-472b-e662-a2c93bc59dc6"
},
"execution_count": 9,
"outputs": [
{
"output_type": "stream",
"name": "stderr",
"text": [
"15:26 bilby INFO : Running for label 'cosmo', output will be saved to 'outdir'\n",
"/usr/local/lib/python3.10/dist-packages/bilby/core/utils/log.py:73: UserWarning: The '__version__' attribute is deprecated and will be removed in MarkupSafe 3.1. Use feature detection, or `importlib.metadata.version(\"markupsafe\")`, instead.\n",
" vdict[key] = str(getattr(sys.modules[key], \"__version__\", \"N/A\"))\n",
"/usr/local/lib/python3.10/dist-packages/_distutils_hack/__init__.py:32: UserWarning: Setuptools is replacing distutils. Support for replacing an already imported distutils is deprecated. In the future, this condition will fail. Register concerns at https://github.com/pypa/setuptools/issues/new?template=distutils-deprecation.yml\n",
" warnings.warn(\n",
"15:26 bilby INFO : Analysis priors:\n",
"15:26 bilby INFO : alpha=Uniform(minimum=-2, maximum=4, name=None, latex_label='$\\\\alpha$', unit=None, boundary=None)\n",
"15:26 bilby INFO : beta=Uniform(minimum=-4, maximum=12, name=None, latex_label='$\\\\beta$', unit=None, boundary=None)\n",
"15:26 bilby INFO : mmin=Uniform(minimum=2, maximum=2.5, name=None, latex_label='$m_{\\\\min}$', unit=None, boundary=None)\n",
"15:26 bilby INFO : mmax=Uniform(minimum=80, maximum=100, name=None, latex_label='$m_{\\\\max}$', unit=None, boundary=None)\n",
"15:26 bilby INFO : lam=Uniform(minimum=0, maximum=1, name=None, latex_label='$\\\\lambda_{m}$', unit=None, boundary=None)\n",
"15:26 bilby INFO : mpp=Uniform(minimum=10, maximum=50, name=None, latex_label='$\\\\mu_{m}$', unit=None, boundary=None)\n",
"15:26 bilby INFO : sigpp=Uniform(minimum=1, maximum=10, name=None, latex_label='$\\\\sigma_{m}$', unit=None, boundary=None)\n",
"15:26 bilby INFO : alpha_chi=Uniform(minimum=1, maximum=6, name=None, latex_label='$\\\\alpha_{\\\\chi}$', unit=None, boundary=None)\n",
"15:26 bilby INFO : beta_chi=Uniform(minimum=1, maximum=6, name=None, latex_label='$\\\\beta_{\\\\chi}$', unit=None, boundary=None)\n",
"15:26 bilby INFO : xi_spin=Uniform(minimum=0, maximum=1, name=None, latex_label='$\\\\xi$', unit=None, boundary=None)\n",
"15:26 bilby INFO : sigma_spin=Uniform(minimum=0.3, maximum=4, name=None, latex_label='$\\\\sigma$', unit=None, boundary=None)\n",
"15:26 bilby INFO : H0=Uniform(minimum=20, maximum=200, name=None, latex_label='$H_0$', unit=None, boundary=None)\n",
"15:26 bilby INFO : Om0=Uniform(minimum=0, maximum=1, name=None, latex_label='$\\\\Omega_{m,0}$', unit=None, boundary=None)\n",
"15:26 bilby INFO : w0=Uniform(minimum=-1.5, maximum=-0.5, name=None, latex_label='$w_{0}$', unit=None, boundary=None)\n",
"15:26 bilby INFO : lamb=Uniform(minimum=-1, maximum=10, name=None, latex_label='$\\\\lambda_{z}$', unit=None, boundary=None)\n",
"15:26 bilby INFO : gaussian_mass_maximum=100\n",
"15:26 bilby INFO : amax=1\n",
"15:26 bilby INFO : Analysis likelihood class: | \n", " | H0 | \n", "Om0 | \n", "alpha | \n", "alpha_chi | \n", "amax | \n", "beta | \n", "beta_chi | \n", "gaussian_mass_maximum | \n", "lam | \n", "lamb | \n", "... | \n", "var_67 | \n", "var_68 | \n", "var_69 | \n", "var_7 | \n", "var_70 | \n", "var_8 | \n", "var_9 | \n", "variance | \n", "w0 | \n", "xi_spin | \n", "
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| count | \n", "687.000000 | \n", "687.000000 | \n", "687.000000 | \n", "687.000000 | \n", "687.0 | \n", "687.000000 | \n", "687.000000 | \n", "687.0 | \n", "687.000000 | \n", "687.000000 | \n", "... | \n", "687.000000 | \n", "687.000000 | \n", "687.000000 | \n", "687.000000 | \n", "687.000000 | \n", "687.000000 | \n", "687.000000 | \n", "687.000000 | \n", "687.000000 | \n", "687.000000 | \n", "
| mean | \n", "86.192321 | \n", "0.463870 | \n", "2.785490 | \n", "1.699249 | \n", "1.0 | \n", "2.029660 | \n", "4.672059 | \n", "100.0 | \n", "0.046022 | \n", "2.281156 | \n", "... | \n", "0.000902 | \n", "0.001298 | \n", "0.000506 | \n", "0.000339 | \n", "0.000781 | \n", "0.000857 | \n", "0.001010 | \n", "1.578096 | \n", "-1.037032 | \n", "0.656418 | \n", "
| std | \n", "33.315514 | \n", "0.264367 | \n", "0.227932 | \n", "0.425861 | \n", "0.0 | \n", "1.084555 | \n", "0.828649 | \n", "0.0 | \n", "0.067595 | \n", "0.923766 | \n", "... | \n", "0.000432 | \n", "0.000799 | \n", "0.000242 | \n", "0.000153 | \n", "0.000300 | \n", "0.000415 | \n", "0.000481 | \n", "0.820307 | \n", "0.278118 | \n", "0.244211 | \n", "
| min | \n", "28.461434 | \n", "0.012781 | \n", "2.014633 | \n", "1.001600 | \n", "1.0 | \n", "-0.575890 | \n", "2.176011 | \n", "100.0 | \n", "0.007235 | \n", "-0.860138 | \n", "... | \n", "0.000166 | \n", "0.000260 | \n", "0.000140 | \n", "0.000107 | \n", "0.000179 | \n", "0.000216 | \n", "0.000239 | \n", "0.495687 | \n", "-1.499852 | \n", "0.000738 | \n", "
| 25% | \n", "62.020596 | \n", "0.241354 | \n", "2.629028 | \n", "1.362416 | \n", "1.0 | \n", "1.278567 | \n", "4.072469 | \n", "100.0 | \n", "0.022109 | \n", "1.709052 | \n", "... | \n", "0.000615 | \n", "0.000760 | \n", "0.000346 | \n", "0.000236 | \n", "0.000595 | \n", "0.000579 | \n", "0.000679 | \n", "1.025316 | \n", "-1.275062 | \n", "0.501259 | \n", "
| 50% | \n", "80.248760 | \n", "0.457905 | \n", "2.793779 | \n", "1.664151 | \n", "1.0 | \n", "1.894215 | \n", "4.742172 | \n", "100.0 | \n", "0.031899 | \n", "2.228540 | \n", "... | \n", "0.000811 | \n", "0.001100 | \n", "0.000444 | \n", "0.000304 | \n", "0.000738 | \n", "0.000762 | \n", "0.000894 | \n", "1.336759 | \n", "-1.044505 | \n", "0.693256 | \n", "
| 75% | \n", "105.724361 | \n", "0.667858 | \n", "2.940473 | \n", "1.940837 | \n", "1.0 | \n", "2.606386 | \n", "5.371636 | \n", "100.0 | \n", "0.045664 | \n", "2.897086 | \n", "... | \n", "0.001078 | \n", "0.001568 | \n", "0.000607 | \n", "0.000405 | \n", "0.000906 | \n", "0.001025 | \n", "0.001201 | \n", "1.858051 | \n", "-0.810291 | \n", "0.861111 | \n", "
| max | \n", "193.002066 | \n", "0.995524 | \n", "3.430792 | \n", "3.125901 | \n", "1.0 | \n", "6.657481 | \n", "5.998233 | \n", "100.0 | \n", "0.704479 | \n", "6.065530 | \n", "... | \n", "0.003160 | \n", "0.005728 | \n", "0.001807 | \n", "0.001424 | \n", "0.002516 | \n", "0.003410 | \n", "0.003835 | \n", "6.507338 | \n", "-0.500004 | \n", "0.999019 | \n", "
8 rows × 164 columns
\n", "| \n", " | alpha | \n", "beta | \n", "mmin | \n", "mmax | \n", "lam | \n", "mpp | \n", "sigpp | \n", "alpha_chi | \n", "beta_chi | \n", "xi_spin | \n", "sigma_spin | \n", "H0 | \n", "Om0 | \n", "w0 | \n", "lamb | \n", "log_likelihood | \n", "variance | \n", "
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| alpha | \n", "1.000000 | \n", "0.141622 | \n", "0.090023 | \n", "0.080669 | \n", "-0.357669 | \n", "-0.056450 | \n", "-0.038718 | \n", "0.110190 | \n", "0.136683 | \n", "0.035176 | \n", "0.131746 | \n", "0.092676 | \n", "-0.059821 | \n", "-0.011571 | \n", "0.441030 | \n", "0.191771 | \n", "0.166605 | \n", "
| beta | \n", "0.141622 | \n", "1.000000 | \n", "-0.033839 | \n", "0.067055 | \n", "-0.009699 | \n", "-0.052296 | \n", "0.017018 | \n", "-0.069778 | \n", "-0.055648 | \n", "0.054644 | \n", "-0.096338 | \n", "-0.059046 | \n", "0.068654 | \n", "0.147231 | \n", "-0.149448 | \n", "-0.114000 | \n", "0.183815 | \n", "
| mmin | \n", "0.090023 | \n", "-0.033839 | \n", "1.000000 | \n", "0.054066 | \n", "-0.068222 | \n", "0.109581 | \n", "-0.007212 | \n", "-0.033472 | \n", "-0.006678 | \n", "-0.117315 | \n", "0.027034 | \n", "-0.106144 | \n", "0.133593 | \n", "-0.133617 | \n", "0.037905 | \n", "0.005377 | \n", "-0.043536 | \n", "
| mmax | \n", "0.080669 | \n", "0.067055 | \n", "0.054066 | \n", "1.000000 | \n", "0.110677 | \n", "-0.094240 | \n", "-0.006212 | \n", "0.090794 | \n", "0.154900 | \n", "-0.005070 | \n", "0.056983 | \n", "0.131370 | \n", "-0.042789 | \n", "0.027122 | \n", "-0.006661 | \n", "-0.139059 | \n", "0.087109 | \n", "
| lam | \n", "-0.357669 | \n", "-0.009699 | \n", "-0.068222 | \n", "0.110677 | \n", "1.000000 | \n", "-0.691310 | \n", "0.413978 | \n", "0.069220 | \n", "0.004225 | \n", "-0.064605 | \n", "-0.055335 | \n", "0.300825 | \n", "-0.041868 | \n", "0.001544 | \n", "-0.011827 | \n", "-0.409244 | \n", "-0.079433 | \n", "
| mpp | \n", "-0.056450 | \n", "-0.052296 | \n", "0.109581 | \n", "-0.094240 | \n", "-0.691310 | \n", "1.000000 | \n", "-0.460154 | \n", "-0.094476 | \n", "-0.080426 | \n", "0.037703 | \n", "-0.006029 | \n", "-0.613080 | \n", "0.137499 | \n", "-0.105793 | \n", "-0.165381 | \n", "0.396366 | \n", "0.056222 | \n", "
| sigpp | \n", "-0.038718 | \n", "0.017018 | \n", "-0.007212 | \n", "-0.006212 | \n", "0.413978 | \n", "-0.460154 | \n", "1.000000 | \n", "0.022190 | \n", "-0.106280 | \n", "-0.145172 | \n", "0.073256 | \n", "-0.205229 | \n", "-0.024965 | \n", "-0.026839 | \n", "0.111204 | \n", "-0.593371 | \n", "-0.485489 | \n", "
| alpha_chi | \n", "0.110190 | \n", "-0.069778 | \n", "-0.033472 | \n", "0.090794 | \n", "0.069220 | \n", "-0.094476 | \n", "0.022190 | \n", "1.000000 | \n", "0.592209 | \n", "-0.071230 | \n", "0.318560 | \n", "0.068430 | \n", "-0.133062 | \n", "0.008132 | \n", "0.181618 | \n", "-0.165641 | \n", "-0.149992 | \n", "
| beta_chi | \n", "0.136683 | \n", "-0.055648 | \n", "-0.006678 | \n", "0.154900 | \n", "0.004225 | \n", "-0.080426 | \n", "-0.106280 | \n", "0.592209 | \n", "1.000000 | \n", "-0.040551 | \n", "0.002835 | \n", "0.162779 | \n", "-0.094758 | \n", "-0.029511 | \n", "0.191065 | \n", "0.164174 | \n", "0.350465 | \n", "
| xi_spin | \n", "0.035176 | \n", "0.054644 | \n", "-0.117315 | \n", "-0.005070 | \n", "-0.064605 | \n", "0.037703 | \n", "-0.145172 | \n", "-0.071230 | \n", "-0.040551 | \n", "1.000000 | \n", "0.093104 | \n", "0.059087 | \n", "-0.058887 | \n", "0.078590 | \n", "0.026096 | \n", "0.254043 | \n", "0.187677 | \n", "
| sigma_spin | \n", "0.131746 | \n", "-0.096338 | \n", "0.027034 | \n", "0.056983 | \n", "-0.055335 | \n", "-0.006029 | \n", "0.073256 | \n", "0.318560 | \n", "0.002835 | \n", "0.093104 | \n", "1.000000 | \n", "-0.071397 | \n", "-0.016116 | \n", "-0.030013 | \n", "0.102910 | \n", "-0.221670 | \n", "-0.333567 | \n", "
| H0 | \n", "0.092676 | \n", "-0.059046 | \n", "-0.106144 | \n", "0.131370 | \n", "0.300825 | \n", "-0.613080 | \n", "-0.205229 | \n", "0.068430 | \n", "0.162779 | \n", "0.059087 | \n", "-0.071397 | \n", "1.000000 | \n", "-0.396252 | \n", "0.028811 | \n", "0.001228 | \n", "-0.009208 | \n", "0.219151 | \n", "
| Om0 | \n", "-0.059821 | \n", "0.068654 | \n", "0.133593 | \n", "-0.042789 | \n", "-0.041868 | \n", "0.137499 | \n", "-0.024965 | \n", "-0.133062 | \n", "-0.094758 | \n", "-0.058887 | \n", "-0.016116 | \n", "-0.396252 | \n", "1.000000 | \n", "0.010338 | \n", "-0.022531 | \n", "0.033099 | \n", "0.030473 | \n", "
| w0 | \n", "-0.011571 | \n", "0.147231 | \n", "-0.133617 | \n", "0.027122 | \n", "0.001544 | \n", "-0.105793 | \n", "-0.026839 | \n", "0.008132 | \n", "-0.029511 | \n", "0.078590 | \n", "-0.030013 | \n", "0.028811 | \n", "0.010338 | \n", "1.000000 | \n", "-0.088142 | \n", "-0.037276 | \n", "-0.034192 | \n", "
| lamb | \n", "0.441030 | \n", "-0.149448 | \n", "0.037905 | \n", "-0.006661 | \n", "-0.011827 | \n", "-0.165381 | \n", "0.111204 | \n", "0.181618 | \n", "0.191065 | \n", "0.026096 | \n", "0.102910 | \n", "0.001228 | \n", "-0.022531 | \n", "-0.088142 | \n", "1.000000 | \n", "0.067673 | \n", "0.007794 | \n", "
| log_likelihood | \n", "0.191771 | \n", "-0.114000 | \n", "0.005377 | \n", "-0.139059 | \n", "-0.409244 | \n", "0.396366 | \n", "-0.593371 | \n", "-0.165641 | \n", "0.164174 | \n", "0.254043 | \n", "-0.221670 | \n", "-0.009208 | \n", "0.033099 | \n", "-0.037276 | \n", "0.067673 | \n", "1.000000 | \n", "0.344516 | \n", "
| variance | \n", "0.166605 | \n", "0.183815 | \n", "-0.043536 | \n", "0.087109 | \n", "-0.079433 | \n", "0.056222 | \n", "-0.485489 | \n", "-0.149992 | \n", "0.350465 | \n", "0.187677 | \n", "-0.333567 | \n", "0.219151 | \n", "0.030473 | \n", "-0.034192 | \n", "0.007794 | \n", "0.344516 | \n", "1.000000 | \n", "