![\"Open](\"https://colab.research.google.com/assets/colab-badge.svg\"){kind=icon}
"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "aeUxtTpEfzbn"
},
"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\""
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "Dc8zj3wngiYu"
},
"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`."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true,
"id": "Tsv8JYinDZv0"
},
"outputs": [],
"source": [
"!pip install \"gwpopulation>=1.3\" unxt --quiet --progress-bar off"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "TdiqeGZtHwRw"
},
"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."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/"
},
"collapsed": true,
"id": "s8bqvo26HABd",
"outputId": "824319a4-56d0-4258-e4b1-c16009dd6a14"
},
"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, 32.8MB/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 [00:01<00:00, 33.5MB/s]\n"
]
}
],
"source": [
"!gdown https://drive.google.com/uc?id=16gStLIjt65gWBkw-gNOVUqNbZ89q8CLF\n",
"!gdown https://drive.google.com/uc?id=10pevUCM3V2-D-bROFEMAcTJsX_9RzeM6"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "I1MQIoc_gvR-"
},
"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."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"id": "mKC8rhSUDmtg"
},
"outputs": [],
"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"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "6MIE6_uMg4vw"
},
"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."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"id": "1pAakjJ1DwrK"
},
"outputs": [],
"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))"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "82GMaSlJhKm1"
},
"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."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"id": "IuHqDXFQE9Mw"
},
"outputs": [],
"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))"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "I5FzBbnniQfl"
},
"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."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"id": "7IrVTnK5GbaZ"
},
"outputs": [],
"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",
")"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "lQW_m3bNioih"
},
"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)."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"id": "IRf7FFQDGda-"
},
"outputs": [],
"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(minimum=1, maximum=6, latex_label=\"$\\\\alpha_{\\\\chi}$\")\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=\"$\\\\lambda_{z}$\")"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "ncci_peFisbm"
},
"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.)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/"
},
"id": "fgkofuRrGfX7",
"outputId": "9885a108-f977-48c8-854d-b9fe03292e09"
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"-262.40427928666327\n",
"CPU times: user 449 ms, sys: 171 ms, total: 620 ms\n",
"Wall time: 451 ms\n",
"-262.40427928666327\n",
"CPU times: user 12.1 s, sys: 502 ms, total: 12.6 s\n",
"Wall time: 12.1 s\n",
"-262.40427928666327\n",
"CPU times: user 20.6 ms, sys: 114 µs, total: 20.7 ms\n",
"Wall time: 21 ms\n"
]
}
],
"source": [
"parameters = priors.sample()\n",
"likelihood.log_likelihood_ratio(parameters)\n",
"%time print(likelihood.log_likelihood_ratio(parameters))\n",
"jit_likelihood = JittedLikelihood(likelihood)\n",
"%time print(jit_likelihood.log_likelihood_ratio(parameters))\n",
"%time print(jit_likelihood.log_likelihood_ratio(parameters))"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "fX0FoqX0ixo0"
},
"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."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 729,
"referenced_widgets": [
"93d868c809304d0cad98490ede26fa0f",
"32d0417bc4464e7497fd88e6c9a96dd0",
"ad6a13798eea4bc0b5e5a88315e86011",
"93ae286cfff5473fbafd302be8e369bc",
"b20b77e48a2b4a599e68b0c6634084d3",
"c0f3c4f667904b6081d85edd69e3d3cb",
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},
"id": "lHB2loW6Ghx-",
"outputId": "f1f72f88-1d9d-4e91-dc6b-fb2194bab0a8"
},
"outputs": [
{
"output_type": "stream",
"name": "stderr",
"text": [
"20:18 bilby INFO : Running for label 'cosmo', output will be saved to 'outdir'\n",
"20:18 bilby INFO : Analysis priors:\n",
"20:18 bilby INFO : alpha=Uniform(minimum=-2, maximum=4, name=None, latex_label='$\\\\alpha$', unit=None, boundary=None)\n",
"20:18 bilby INFO : beta=Uniform(minimum=-4, maximum=12, name=None, latex_label='$\\\\beta$', unit=None, boundary=None)\n",
"20:18 bilby INFO : mmin=Uniform(minimum=2, maximum=2.5, name=None, latex_label='$m_{\\\\min}$', unit=None, boundary=None)\n",
"20:18 bilby INFO : mmax=Uniform(minimum=80, maximum=100, name=None, latex_label='$m_{\\\\max}$', unit=None, boundary=None)\n",
"20:18 bilby INFO : lam=Uniform(minimum=0, maximum=1, name=None, latex_label='$\\\\lambda_{m}$', unit=None, boundary=None)\n",
"20:18 bilby INFO : mpp=Uniform(minimum=10, maximum=50, name=None, latex_label='$\\\\mu_{m}$', unit=None, boundary=None)\n",
"20:18 bilby INFO : sigpp=Uniform(minimum=1, maximum=10, name=None, latex_label='$\\\\sigma_{m}$', unit=None, boundary=None)\n",
"20:18 bilby INFO : alpha_chi=Uniform(minimum=1, maximum=6, name=None, latex_label='$\\\\alpha_{\\\\chi}$', unit=None, boundary=None)\n",
"20:18 bilby INFO : beta_chi=Uniform(minimum=1, maximum=6, name=None, latex_label='$\\\\beta_{\\\\chi}$', unit=None, boundary=None)\n",
"20:18 bilby INFO : xi_spin=Uniform(minimum=0, maximum=1, name=None, latex_label='$\\\\xi$', unit=None, boundary=None)\n",
"20:18 bilby INFO : sigma_spin=Uniform(minimum=0.3, maximum=4, name=None, latex_label='$\\\\sigma$', unit=None, boundary=None)\n",
"20:18 bilby INFO : H0=Uniform(minimum=20, maximum=200, name=None, latex_label='$H_0$', unit=None, boundary=None)\n",
"20:18 bilby INFO : Om0=Uniform(minimum=0, maximum=1, name=None, latex_label='$\\\\Omega_{m,0}$', unit=None, boundary=None)\n",
"20:18 bilby INFO : w0=Uniform(minimum=-1.5, maximum=-0.5, name=None, latex_label='$w_{0}$', unit=None, boundary=None)\n",
"20:18 bilby INFO : lamb=Uniform(minimum=-1, maximum=10, name=None, latex_label='$\\\\lambda_{z}$', unit=None, boundary=None)\n",
"20:18 bilby INFO : gaussian_mass_maximum=100\n",
"20:18 bilby INFO : amax=1\n",
"20:18 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", "743.000000 | \n", "743.000000 | \n", "743.000000 | \n", "743.000000 | \n", "743.0 | \n", "743.000000 | \n", "743.000000 | \n", "743.0 | \n", "743.000000 | \n", "743.000000 | \n", "... | \n", "743.000000 | \n", "743.000000 | \n", "743.000000 | \n", "743.000000 | \n", "743.000000 | \n", "743.000000 | \n", "743.000000 | \n", "743.000000 | \n", "743.000000 | \n", "743.000000 | \n", "
| mean | \n", "92.360953 | \n", "0.486399 | \n", "2.770012 | \n", "1.746442 | \n", "1.0 | \n", "1.762433 | \n", "4.718197 | \n", "100.0 | \n", "0.112395 | \n", "2.402824 | \n", "... | \n", "0.000807 | \n", "0.001125 | \n", "0.000445 | \n", "0.000319 | \n", "0.000676 | \n", "0.000758 | \n", "0.000865 | \n", "1.320810 | \n", "-0.996708 | \n", "0.612861 | \n", "
| std | \n", "41.752044 | \n", "0.255232 | \n", "0.325525 | \n", "0.431440 | \n", "0.0 | \n", "0.960041 | \n", "0.868764 | \n", "0.0 | \n", "0.181103 | \n", "0.902655 | \n", "... | \n", "0.000371 | \n", "0.000574 | \n", "0.000180 | \n", "0.000121 | \n", "0.000265 | \n", "0.000303 | \n", "0.000341 | \n", "0.581550 | \n", "0.277397 | \n", "0.258564 | \n", "
| min | \n", "27.361213 | \n", "0.015903 | \n", "0.073127 | \n", "1.005281 | \n", "1.0 | \n", "-0.623073 | \n", "2.140122 | \n", "100.0 | \n", "0.007416 | \n", "-0.804947 | \n", "... | \n", "0.000178 | \n", "0.000276 | \n", "0.000143 | \n", "0.000104 | \n", "0.000157 | \n", "0.000206 | \n", "0.000215 | \n", "0.364017 | \n", "-1.497507 | \n", "0.000294 | \n", "
| 25% | \n", "60.450052 | \n", "0.289400 | \n", "2.606277 | \n", "1.405255 | \n", "1.0 | \n", "1.089937 | \n", "4.137047 | \n", "100.0 | \n", "0.025116 | \n", "1.876491 | \n", "... | \n", "0.000560 | \n", "0.000718 | \n", "0.000332 | \n", "0.000241 | \n", "0.000493 | \n", "0.000562 | \n", "0.000633 | \n", "0.938326 | \n", "-1.228801 | \n", "0.425478 | \n", "
| 50% | \n", "81.385499 | \n", "0.482400 | \n", "2.794633 | \n", "1.720532 | \n", "1.0 | \n", "1.708881 | \n", "4.881393 | \n", "100.0 | \n", "0.036918 | \n", "2.425547 | \n", "... | \n", "0.000740 | \n", "0.001022 | \n", "0.000406 | \n", "0.000295 | \n", "0.000637 | \n", "0.000700 | \n", "0.000802 | \n", "1.166464 | \n", "-0.990194 | \n", "0.652782 | \n", "
| 75% | \n", "117.471092 | \n", "0.682141 | \n", "2.948939 | \n", "2.048675 | \n", "1.0 | \n", "2.362898 | \n", "5.420889 | \n", "100.0 | \n", "0.084015 | \n", "2.993242 | \n", "... | \n", "0.000966 | \n", "0.001403 | \n", "0.000522 | \n", "0.000370 | \n", "0.000815 | \n", "0.000911 | \n", "0.001031 | \n", "1.545677 | \n", "-0.763237 | \n", "0.833905 | \n", "
| max | \n", "199.657560 | \n", "0.998753 | \n", "3.664634 | \n", "3.058372 | \n", "1.0 | \n", "6.947037 | \n", "5.992614 | \n", "100.0 | \n", "0.984410 | \n", "5.258572 | \n", "... | \n", "0.003369 | \n", "0.005578 | \n", "0.001749 | \n", "0.001094 | \n", "0.002131 | \n", "0.002914 | \n", "0.003131 | \n", "5.849521 | \n", "-0.501485 | \n", "0.998934 | \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", "0.104769 | \n", "0.057727 | \n", "0.031850 | \n", "-0.515273 | \n", "0.119441 | \n", "-0.100635 | \n", "-0.003047 | \n", "0.076560 | \n", "-0.021058 | \n", "0.088277 | \n", "-0.064720 | \n", "-0.072049 | \n", "0.075218 | \n", "0.403147 | \n", "0.185284 | \n", "0.121324 | \n", "
| beta | \n", "0.104769 | \n", "1.000000 | \n", "0.027970 | \n", "0.061236 | \n", "-0.138127 | \n", "0.077205 | \n", "-0.060520 | \n", "-0.075358 | \n", "-0.040275 | \n", "-0.048708 | \n", "-0.003847 | \n", "-0.145900 | \n", "0.060501 | \n", "-0.015096 | \n", "-0.156298 | \n", "0.036758 | \n", "0.108856 | \n", "
| mmin | \n", "0.057727 | \n", "0.027970 | \n", "1.000000 | \n", "0.064558 | \n", "-0.011677 | \n", "0.019201 | \n", "-0.074863 | \n", "-0.005770 | \n", "0.051799 | \n", "-0.027292 | \n", "-0.043057 | \n", "0.044674 | \n", "0.017065 | \n", "0.010939 | \n", "-0.053468 | \n", "0.135694 | \n", "0.101285 | \n", "
| mmax | \n", "0.031850 | \n", "0.061236 | \n", "0.064558 | \n", "1.000000 | \n", "0.018959 | \n", "-0.038920 | \n", "0.040200 | \n", "0.010222 | \n", "0.039795 | \n", "0.018212 | \n", "-0.050326 | \n", "0.039923 | \n", "0.078974 | \n", "-0.018565 | \n", "0.037521 | \n", "-0.047827 | \n", "-0.046889 | \n", "
| lam | \n", "-0.515273 | \n", "-0.138127 | \n", "-0.011677 | \n", "0.018959 | \n", "1.000000 | \n", "-0.769588 | \n", "0.581677 | \n", "0.032600 | \n", "-0.012777 | \n", "0.056083 | \n", "-0.081695 | \n", "0.619254 | \n", "-0.018853 | \n", "-0.041354 | \n", "0.046305 | \n", "-0.454545 | \n", "-0.147537 | \n", "
| mpp | \n", "0.119441 | \n", "0.077205 | \n", "0.019201 | \n", "-0.038920 | \n", "-0.769588 | \n", "1.000000 | \n", "-0.755787 | \n", "-0.025355 | \n", "0.002958 | \n", "-0.000804 | \n", "-0.008536 | \n", "-0.826711 | \n", "0.073687 | \n", "0.043599 | \n", "-0.216987 | \n", "0.599396 | \n", "0.245498 | \n", "
| sigpp | \n", "-0.100635 | \n", "-0.060520 | \n", "-0.074863 | \n", "0.040200 | \n", "0.581677 | \n", "-0.755787 | \n", "1.000000 | \n", "-0.027702 | \n", "-0.079613 | \n", "-0.056792 | \n", "-0.067881 | \n", "0.415370 | \n", "-0.080341 | \n", "-0.080442 | \n", "0.165945 | \n", "-0.719298 | \n", "-0.477722 | \n", "
| alpha_chi | \n", "-0.003047 | \n", "-0.075358 | \n", "-0.005770 | \n", "0.010222 | \n", "0.032600 | \n", "-0.025355 | \n", "-0.027702 | \n", "1.000000 | \n", "0.663680 | \n", "-0.025392 | \n", "0.278962 | \n", "-0.059012 | \n", "0.032951 | \n", "0.078771 | \n", "0.105113 | \n", "-0.059531 | \n", "-0.103207 | \n", "
| beta_chi | \n", "0.076560 | \n", "-0.040275 | \n", "0.051799 | \n", "0.039795 | \n", "-0.012777 | \n", "0.002958 | \n", "-0.079613 | \n", "0.663680 | \n", "1.000000 | \n", "0.025947 | \n", "0.070095 | \n", "-0.056962 | \n", "0.071960 | \n", "0.025606 | \n", "0.138051 | \n", "0.110210 | \n", "0.362311 | \n", "
| xi_spin | \n", "-0.021058 | \n", "-0.048708 | \n", "-0.027292 | \n", "0.018212 | \n", "0.056083 | \n", "-0.000804 | \n", "-0.056792 | \n", "-0.025392 | \n", "0.025947 | \n", "1.000000 | \n", "-0.063362 | \n", "0.036526 | \n", "0.053075 | \n", "-0.079162 | \n", "-0.036799 | \n", "0.241529 | \n", "0.177782 | \n", "
| sigma_spin | \n", "0.088277 | \n", "-0.003847 | \n", "-0.043057 | \n", "-0.050326 | \n", "-0.081695 | \n", "-0.008536 | \n", "-0.067881 | \n", "0.278962 | \n", "0.070095 | \n", "-0.063362 | \n", "1.000000 | \n", "-0.032461 | \n", "0.042798 | \n", "0.039152 | \n", "0.194012 | \n", "-0.202366 | \n", "-0.259614 | \n", "
| H0 | \n", "-0.064720 | \n", "-0.145900 | \n", "0.044674 | \n", "0.039923 | \n", "0.619254 | \n", "-0.826711 | \n", "0.415370 | \n", "-0.059012 | \n", "-0.056962 | \n", "0.036526 | \n", "-0.032461 | \n", "1.000000 | \n", "-0.244578 | \n", "-0.066930 | \n", "0.065973 | \n", "-0.387321 | \n", "-0.088668 | \n", "
| Om0 | \n", "-0.072049 | \n", "0.060501 | \n", "0.017065 | \n", "0.078974 | \n", "-0.018853 | \n", "0.073687 | \n", "-0.080341 | \n", "0.032951 | \n", "0.071960 | \n", "0.053075 | \n", "0.042798 | \n", "-0.244578 | \n", "1.000000 | \n", "-0.062555 | \n", "0.027233 | \n", "0.043844 | \n", "0.061437 | \n", "
| w0 | \n", "0.075218 | \n", "-0.015096 | \n", "0.010939 | \n", "-0.018565 | \n", "-0.041354 | \n", "0.043599 | \n", "-0.080442 | \n", "0.078771 | \n", "0.025606 | \n", "-0.079162 | \n", "0.039152 | \n", "-0.066930 | \n", "-0.062555 | \n", "1.000000 | \n", "0.049391 | \n", "0.089234 | \n", "0.004628 | \n", "
| lamb | \n", "0.403147 | \n", "-0.156298 | \n", "-0.053468 | \n", "0.037521 | \n", "0.046305 | \n", "-0.216987 | \n", "0.165945 | \n", "0.105113 | \n", "0.138051 | \n", "-0.036799 | \n", "0.194012 | \n", "0.065973 | \n", "0.027233 | \n", "0.049391 | \n", "1.000000 | \n", "-0.082165 | \n", "0.021441 | \n", "
| log_likelihood | \n", "0.185284 | \n", "0.036758 | \n", "0.135694 | \n", "-0.047827 | \n", "-0.454545 | \n", "0.599396 | \n", "-0.719298 | \n", "-0.059531 | \n", "0.110210 | \n", "0.241529 | \n", "-0.202366 | \n", "-0.387321 | \n", "0.043844 | \n", "0.089234 | \n", "-0.082165 | \n", "1.000000 | \n", "0.433768 | \n", "
| variance | \n", "0.121324 | \n", "0.108856 | \n", "0.101285 | \n", "-0.046889 | \n", "-0.147537 | \n", "0.245498 | \n", "-0.477722 | \n", "-0.103207 | \n", "0.362311 | \n", "0.177782 | \n", "-0.259614 | \n", "-0.088668 | \n", "0.061437 | \n", "0.004628 | \n", "0.021441 | \n", "0.433768 | \n", "1.000000 | \n", "