Cosmological Data Sets Research Articles

Quantifying concordance of correlated cosmological data sets

We develop estimators of agreement and disagreement between correlated cosmological data sets. These account for data correlations when computing the significance of both tensions and excess confirmation while remaining statistically optimal. We discuss and thoroughly characterize different approaches commenting on the ones that have the best behavior in practical applications. We complement the calculation of their statistical distribution within the Gaussian model with one estimator that takes non-Gaussianities fully into account. To illustrate the use of our techniques, we apply these estimators to supernovae measurements of the distance-redshift relation, absolutely calibrated by the local distance ladder. The suite of best estimators that we discuss finds results that are in excellent agreement between estimators and find no indications of significant internal inconsistencies in this data set above the $1\%$ probability threshold. This shows the robustness of local determinations of the Hubble constant to features in the distance-redshift relation.

CosmoHub: Interactive exploration and distribution of astronomical data on Hadoop

We present CosmoHub ( https://cosmohub.pic.es ), a web application based on Hadoop to perform interactive exploration and distribution of massive cosmological datasets. Recent Cosmology seeks to unveil the nature of both dark matter and dark energy mapping the large-scale structure of the Universe, through the analysis of massive amounts of astronomical data, progressively increasing during the last (and future) decades with the digitization and automation of the experimental techniques. CosmoHub, hosted and developed at the Port d’Informació Científica (PIC), provides support to a worldwide community of scientists, without requiring the end user to know any Structured Query Language (SQL). It is serving data of several large international collaborations such as the Euclid space mission, the Dark Energy Survey (DES), the Physics of the Accelerating Universe Survey (PAUS) and the Marenostrum Institut de Ciències de l’Espai (MICE) numerical simulations. While originally developed as a PostgreSQL relational database web frontend, this work describes the current version of CosmoHub, built on top of Apache Hive, which facilitates scalable reading, writing and managing huge datasets. As CosmoHub’s datasets are seldomly modified, Hive it is a better fit. Over 60 TiB of cataloged information and 50 × 1 0 9 astronomical objects can be interactively explored using an integrated visualization tool which includes 1D histogram and 2D heatmap plots. In our current implementation, online exploration of datasets of 1 0 9 objects can be done in a timescale of tens of seconds. Users can also download customized subsets of data in standard formats generated in few minutes.

Forecasting interacting vacuum-energy models using gravitational waves

The physics of the dark sector has remained one of the controversial areas of modern cosmology at present and hence it naturally attracts massive attention to the scientific community. With the developments of the astronomical data, the physics of the dark sector is becoming much more transparent than it was some twenty years back. The detection of gravitational waves (GWs) has now opened a cluster of possibilities in the cosmological regime. Being motivated by the detection of GWs and its possible impact on the physics of dark matter and dark energy, in this work we focus on the interacting dark energy models. Assuming the simplest possibility in which the vacuum energy with equation-of-state wx =−1 is allowed to interact with the pressureless dark matter, we have extracted the constraints of the cosmological parameters. We find that the addition of the GWs data to the CMB measurements significantly improves up to a factor 4 of the parameters space, and up to a factor 2 for the full combination of current cosmological datasets, namely CMB+BAO+Pantheon+RSD+R16+CC+WL. The most affected parameters by the inclusion of the GWs are Ωch2, θMC, ξ, and the derived parameters Ωm, σ8 and H0.

Gravity in the infrared and effective nonlocal models

We provide a systematic and updated discussion of a research line carried out by our group over the last few years, in which gravity is modified at cosmological distances by the introduction of nonlocal terms, assumed to emerge at an effective level from the infrared behavior of the quantum theory. The requirement of producing a viable cosmology turns out to be very stringent and basically selects a unique model, in which the nonlocal term describes an effective mass for the conformal mode. We discuss how such a specific structure could emerge from a fundamental local theory of gravity, and we perform a detailed comparison of this model with the most recent cosmological datasets, confirming that it fits current data at the same level as ΛCDM. Most notably, the model has striking predictions in the sector of tensor perturbations, leading to a very large effect in the propagation of gravitational wave (GWs) over cosmological distances. At the redshifts relevant for the next generation of GW detectors such as Einstein Telescope, Cosmic Explorer and LISA, this leads to deviations from GR that could be as large as 80%, and could be verified with the detection of just a single coalescing binary with electromagnetic counterpart. This would also have potentially important consequences for the search of the counterpart since, for a given luminosity distance to the source, as inferred through the GW signal, the actual source redshift could be significantly different from that predicted by ΛCDM. At the redshifts relevant for advanced LIGO/Virgo/Kagra the effect is smaller, but still potentially observable over a few years of runs at target sensitivity.

Hi_class background evolution, initial conditions and approximation schemes

Cosmological datasets have great potential to elucidate the nature of dark energy and test gravity on the largest scales available to observation. Theoretical predictions can be computed with hi_class (www.hiclass-code.net), an accurate, fast and flexible code for linear cosmology, incorporating a wide range of dark energy theories and modifications to general relativity. We introduce three new functionalities into hi_class: (1) Support for models based on covariant Lagrangians, including a constraint-preserving integration scheme for the background evolution and a series of worked-out examples: Galileon, nKGB, quintessence (monomial, tracker) and Brans-Dicke. (2) Consistent initial conditions for the scalar-field perturbations in the deep radiation era, identifying the conditions under which modified-gravity isocurvature perturbations may grow faster than adiabatic modes leading to a loss of predictivity. (3) An automated quasi-static approximation scheme allowing order-of-magnitude improvement in computing performance without sacrificing accuracy for wide classes of models. These enhancements bring the treatment of dark energy and modified gravity models to the level of detail comparable to software tools restricted to standard ΛCDM cosmologies. The hi_class code is publicly available (https://github.com/miguelzuma/hi_class_public), ready to explore current data and prepare for next-generation experiments.

Analytic Calculation of Covariance between Cosmological Parameters from Correlated Data Sets, with an Application to SPTpol.

Consistency checks of cosmological data sets are an important tool because they may suggest systematic errors or the type of modifications to ΛCDM necessary to resolve current tensions. In this work, we derive an analytic method for calculating the level of correlations between model parameters from two correlated cosmological data sets, which complements more computationally expensive simulations. This method is an extension of the Fisher analysis that assumes a Gaussian likelihood and a known data covariance matrix. We apply this method to the South Pole Telescope Polarimeter (SPTpol) temperature and polarization cosmic microwave background (CMB) spectra (TE and EE). We find weak correlations between ΛCDM parameters with a 9% correlation between the TE-only and EE-only constraints on H 0 and a 25% and 32% correlation for log(A s ) and n s respectively. The TE-EE parameter differences are consistent with zero, with a probability to exceed of 0.53. Using simulations we show that this test is independent of the consistency of the SPTpol TE and EE band powers with the best-fit ΛCDM model spectra. Despite the negative correlations between the TE and EE power spectra, the correlations between TE-only and EE-only ΛCDM parameters are positive. Ignoring correlations in the TT-TE and TE-EE comparisons biases the χ 2 low, artificially making parameters look more consistent. Therefore, we conclude that these correlations need to be accounted for when performing internal consistency checks of the TT versus TE versus EE power spectra for future CMB analyses.

Cosmological discordances. III. More on measure properties, large-scale-structure constraints, the Hubble constant andPlanckdata

Consistency between cosmological data sets is essential for ongoing and future cosmological analyses. We first investigate the questions of stability and applicability of some moment-based inconsistency measures to multiple data sets. We show that the recently introduced index of inconsistency (IOI) is numerically stable while it can be applied to multiple data sets. We use an illustrative construction of constraints as well as an example with real data sets (i.e. WMAP versus Planck) to show some limitations of the application of the Karhunen-Loeve decomposition to discordance measures. Second, we perform various consistency analyzes using IOI between multiple current data sets while \textit{working with the entire common parameter spaces}. We find current Large-Scale-Structure (LSS) data sets (Planck CMB lensing, DES lensing-clustering and SDSS RSD) all to be consistent with one another. This is found to be not the case for Planck temperature (TT) versus polarization (TE,EE) data, where moderate inconsistencies are present. Noteworthy, we find a strong inconsistency between joint LSS probes and Planck with IOI=5.27, and a moderate tension between DES and Planck with IOI=3.14. Next, using the IOI metric, we compare the Hubble constant from five independent probes. We confirm previous strong tensions between local measurement (SH0ES) and Planck as well as between H0LiCOW and Planck, but also find new strong tensions between SH0ES measurement and the joint LSS probes with IOI=6.73 (i.e. 3.7-$\sigma$ in 1D) as well as between joint LSS and combined probes SH0ES+H0LiCOW with IOI=8.59 (i.e. 4.1-$\sigma$ in 1D). Whether due to systematic effects in the data sets or problems with the underlying model, sources of these old and new tensions need to be identified and dealt with.

Testing deviations from GR at cosmological scales including dynamical dark energy, massive neutrinos, functional or binned parametrizations, and spatial curvature

We introduce a new version of the Integrated Software in Testing General Relativity (ISiTGR) which is a patch to the software CAMB and CosmoMC. ISiTGR is intended to test deviations from GR at cosmological scales using cosmological data sets. While doing so, it allows for various extensions to the standard flat $\Lambda$CDM model. In this new release, we have support for the following: 1) dynamical dark energy parametrizations with a constant or time-dependent equation of state; 2) a consistent implementation of anisotropic shear to model massive neutrinos throughout the full formalism; 3) multiple commonly-used parametrizations of modified growth (MG) parameters; 4) functional, binned and hybrid time- and scale-dependencies for all MG parameters; 5) spatially flat or curved backgrounds. ISiTGR is designed to allow cosmological analyses to take full advantage of ongoing and future surveys to test simultaneously or separately various extensions to the standard model. We describe here the formalism and its implementation in the CMB code, the Integrated Sachs-Wolfe (ISW) effect, and the 3x2 point statistics. Next, we apply ISiTGR to current data sets from Planck-2018, Planck-2015, Dark Energy Survey YR1 release, Baryonic Acoustic Oscillations (BAO), Redshift Space Distortions (BAO/RSD) from the BOSS Data Release 12, the 6DF Galaxy Survey and the SDSS Data Release 7 Main Galaxy Sample, and Supernova from the Pantheon compilation, joint SNLS/SDSS data analysis and the Hubble Space Telescope. We derive constraints on MG parameters for various combinations of the five features above and find that GR is consistent with current data sets in all cases. The code is made publicly available at \url{https://github.com/mishakb/ISiTGR}.

Challenging bulk viscous unified scenarios with cosmological observations

In a spatially flat Friedmann-Lema\^{i}tre-Robertson-Walker universe, we investigate a unified cosmic fluid scenario endowed with bulk viscosity in which the coefficient of the bulk viscosity has a power law evolution. The power law in the bulk viscous coefficient is a general case in this study which naturally includes several choices as special cases. Considering such a general bulk viscous scenario, in the present work we have extracted the observational constraints using the latest cosmological datasets and examine their behaviour at the level of both background and perturbations. From the observational analyses, we find that a non-zero bulk viscous coefficient is always favored and some of the models in this series are able to weaken the current tension on $H_0$ for some dataset. However, from the Bayesian evidence analysis, $\Lambda$CDM is favored over the bulk viscous model.

Forecasted constraints on modified gravity from Sunyaev-Zel’dovich tomography

Observational cosmology has become an important laboratory for testing General Relativity, with searches for modified gravity forming a significant portion of the science case for existing and future surveys. In this paper, we illustrate how future measurements of the Cosmic Microwave Background (CMB) temperature and polarization anisotropies can be combined with large galaxy surveys to improve constraints on modified gravity using the technique of Sunyaev Zel'dovich (SZ) tomography. SZ tomography uses the correlations between the kinetic/polarized SZ contributions to the small-angular scale CMB and the distribution of structure measured in a galaxy redshift survey to reconstruct the remote dipole and quadrupole fields, e.g. the CMB dipole and quadrupole observed throughout the Universe. We compute the effect of a class of modifications of gravity on the remote dipole and quadrupole fields, illustrating that these observables combine a number of the desirable features of existing probes. We then perform a fisher forecast of constraints on a two-parameter class of modifications of gravity for next-generation CMB experiments and galaxy surveys. By incorporating information from the reconstructed remote dipole and quadrupole fields, we find that it is possible to improve the constraints on this model by a factor of $\sim 2$ beyond what is possible with a galaxy survey alone. We conclude that SZ tomography is a promising method for testing gravity with future cosmological datasets.

Reconstructing the dark matter and dark energy interaction scenarios from observations

We consider a class of interacting dark energy models in a flat and nonflat FLRW universe where the interaction is characterized by the modified evolution of the pressureless dark matter as $a^{-3+\delta (a)}$, $a$ being the FLRW scale factor and $\delta (a)$ quantifies the interaction rate. By assuming the most natural and nonsingular parametrization for $\delta (a)$ as $\delta \left( a\right) =\sum_{i} \delta_i (1-a)^i $, where $\delta _{i}$'s ($i=0,1,2,3,..$) are constants, we reconstruct the expansion history of the universe for three particular choices of the DE sector using different cosmological datasets. Our analyses show that the non-interacting scenario is consistent with the observations while the interaction is not strictly ruled out. We reconstruct in the following way. We start with the first two terms of $\delta (a)$ above and constrain $\delta_0$, $\delta_1$. Then we consider up to the second order terms in $\delta (a)$ but fix $\delta_0$, $\delta_1$ to their constrained values and constrain $\delta_2$; similarly we constrain $\delta_3$, and finally we constrain $(\delta_0, \delta_1, \delta_1, \delta_3)$ by keeping all of them to be free as a generalized case. Our reconstruction technique shows that the constraints on $\delta_2$ (fixing $\delta_0$ and $\delta_1$) and $\delta_3$ (fixing $\delta_0$, $\delta_1$ and $\delta_2$) are almost zero for any interaction model and thus the effective scenario is well described by the linear parametrization $\delta (a)\simeq \delta_{0}+\delta _{1}(1-a)$. Additionally, a strong negative correlation between $\delta_0$, $\delta_1$ is observed independently of the dark energy fluid and the curvature of our universe.

Improved renormalization group computation of likelihood functions for cosmological data sets

Evaluation of likelihood functions for cosmological large scale structure data sets (including CMB, galaxy redshift surveys, etc.) naturally involves marginalization, i.e., integration, over an unknown underlying random signal field. Recently, I showed how a renormalization group method can be used to carry out this integration efficiently by first integrating out the smallest scale structure, i.e., localized structure on the scale of differences between nearby data cells, then combining adjacent cells in a coarse graining step, then repeating this process over and over until all scales have been integrated. Here I extend the formulation in several ways in order to reduce the prefactor on the method's linear scaling with data set size. The key improvement is showing how to integrate out the difference between specific adjacent cells before summing them in the coarse graining step, compared to the original formulation in which small-scale fluctuations were integrated more generally. I suggest some other improvements in details of the scheme, including showing how to perform the integration around a maximum likelihood estimate for the underlying random field. In the end, an accurate likelihood computation for a million-cell Gaussian test data set runs in two minutes on my laptop, with room for further optimization and straightforward parallelization.

CMB Tensions with Low-Redshift H0 and S8 Measurements: Impact of a Redshift-Dependent Type-Ia Supernovae Intrinsic Luminosity

With the recent increase in precision of our cosmological datasets, measurements of Λ CDM model parameter provided by high- and low-redshift observations started to be in tension, i.e., the obtained values of such parameters were shown to be significantly different in a statistical sense. In this work we tackle the tension on the value of the Hubble parameter, H 0 , and the weighted amplitude of matter fluctuations, S 8 , obtained from local or low-redshift measurements and from cosmic microwave background (CMB) observations. We combine the main approaches previously used in the literature by extending the cosmological model and accounting for extra systematic uncertainties. With such analysis we aim at exploring non standard cosmological models, implying deviation from a cosmological constant driven acceleration of the Universe expansion, in the presence of additional uncertainties in measurements. In more detail, we reconstruct the Dark Energy equation of state as a function of redshift, while we study the impact of type-Ia supernovae (SNIa) redshift-dependent astrophysical systematic effects on these tensions. We consider a SNIa intrinsic luminosity dependence on redshift due to the star formation rate in its environment, or the metallicity of the progenitor. We find that the H 0 and S 8 tensions can be significantly alleviated, or even removed, if we account for varying Dark Energy for SNIa and CMB data. However, the tensions remain when we add baryon acoustic oscillations (BAO) data into the analysis, even after the addition of extra SNIa systematic uncertainties. This points towards the need of either new physics beyond late-time Dark Energy, or other unaccounted systematic effects (particulary in BAO measurements), to fully solve the present tensions.

Constraints on quintessence scalar field models using cosmological observations

We consider a varieties of quintessence scalar field models in a homogeneous and isotropic geometry of the universe with zero spatial curvature aiming to provide stringent constraints using a series of cosmological data sets, namely, the cosmic microwave background observations (CMB), baryon acoustic oscillations (BAO), joint light curve analysis (JLA) from supernovae type Ia, redshift space distortions (RSD), and the cosmic chronometers (CC). From the qualitative evolution of the models, we find all of them are able to execute a fine transition from the past decelerating phase to the presently accelerating expansion where in addition, the equation of state of the scalar field (also the effective equation of state) might be close to that of the $\Lambda$CDM cosmology depending on its free parameters. From the observational analyses, we find that the scalar field parameters are unconstrained irrespective of all the observational datasets. In fact, we find that the quintessence scalar field models are pretty much determined by the CMB observations since any of the external datasets such as BAO, JLA, RSD, CC does not add any constraining power to CMB. Additionally, we observe a strong negative correlation between the parameters $H_0$ (present value of the Hubble parameter), $\Omega_{m0}$ (density parameter for the matter sector, i.e., cold dark matter plus baryons) exists, while no correlation between $H_0$, and $\sigma_8$ (amplitude of the matter fluctuation) are not correlated. We also comment that the present models are unable to reconcile the tension on $H_0$. Finally, we conclude our work with the Bayesian analyses which report that the non-interacting $\Lambda$CDM model is preferred over all the quintessence scalar field models.

Testing modified gravity at cosmological distances with LISA standard sirens

Modifications of General Relativity leave their imprint both on the cosmic expansion history through a non-trivial dark energy equation of state, and on the evolution of cosmological perturbations in the scalar and in the tensor sectors. In particular, the modification in the tensor sector gives rise to a notion of gravitational-wave (GW) luminosity distance, different from the standard electromagnetic luminosity distance, that can be studied with standard sirens at GW detectors such as LISA or third-generation ground based experiments. We discuss the predictions for modified GW propagation from some of the best studied theories of modified gravity, such as Horndeski or the more general degenerate higher order scalar-tensor (DHOST) theories, non-local infrared modifications of gravity, bigravity theories and the corresponding phenomenon of GW oscillation, as well as theories with extra or varying dimensions. We show that modified GW propagation is a completely generic phenomenon in modified gravity. We then use a simple parametrization of the effect in terms of two parameters (Ξ0,n), that is shown to fit well the results from a large class of models, to study the prospects of observing modified GW propagation using supermassive black hole binaries as standard sirens with LISA . We construct mock source catalogs and perform detailed Markov Chain Monte Carlo studies of the likelihood obtained from LISA standard sirens alone, as well as by combining them with CMB, BAO and SNe data to reduce the degeneracies between cosmological parameters. We find that the combination of LISA with the other cosmological datasets allows one to measure the parameter Ξ0 that characterizes modified GW propagation to the percent level accuracy, sufficient to test several modified gravity theories. LISA standard sirens can also improve constraints on GW oscillations induced by extra field content by about three orders of magnitude relative to the current capability of ground detectors. We also update the forecasts on the accuracy on H0 and on the dark-energy equation of state using more recent estimates for the LISA sensitivity.

MGCAMB with massive neutrinos and dynamical dark energy

We present a major upgrade of MGCAMB, a patch for the Einstein-Boltzmann solver CAMB used for phenomenological tests of general relativity against cosmological datasets. This new version is compatible with the latest CosmoMC code and includes a consistent implementation of massive neutrinos and dynamical dark energy. The code has been restructured to make it easier to modify to fit the custom needs of specific users. To illustrate the capabilities of the code, we present joint constraints on the modified growth, massive neutrinos and the dark energy equation of state from the latest cosmological observations, including the recent galaxy counts and weak lensing measurements from the Dark Energy Survey, and find a good consistency with the ΛCDM model.

Cosmology of surviving Horndeski theory: The road ahead

In the context of the effective field theory of dark energy (EFT) we perform agnostic explorations of Horndeski gravity. We choose two parametrizations for the free EFT functions, namely a power law and a dark energy density-like behaviour on a non trivial Chevallier-Polarski-Linder background. We restrict our analysis to those EFT functions which do not modify the speed of propagation of gravitational waves. Among those, we prove that one specific function cannot be constrained by data, since its contribution to the observables is below the cosmic variance, although we show it has a relevant role in defining the viable parameter space. We place constraints on the parameters of these models combining measurements from present day cosmological datasets and we prove that the next generation galaxy surveys can improve such constraints by one order of magnitude. We then verify the validity of the quasi-static limit within the sound horizon of the dark field, by looking at the phenomenological functions $\mu$ and $\Sigma$, associated respectively to clustering and lensing potentials. Furthermore, we notice up to $5\%$ deviations in $\mu, \Sigma$ with respect to General Relativity at scales smaller than the Compton one. For the chosen parametrizations and in the quasi-static limit, future constraints on $\mu$ and $\Sigma$ can reach the $1\%$ level and will allow us to discriminate between certain models at more than $3\sigma$, provided the present best-fit values remain.

Impact of astrophysics on cosmology forecasts for 21 cm surveys

We use the results of previous work building a halo model formalism for the distribution of neutral hydrogen, along with experimental parameters of future radio facilities, to place forecasts on astrophysical and cosmological parameters from next generation surveys. We consider 21 cm intensity mapping surveys conducted using the BINGO, CHIME, FAST, TianLai, MeerKAT and SKA experimental configurations. We work with the 5-parameter cosmological dataset of {$\Omega_m, \sigma_8, h, n_s, \Omega_b$} assuming a flat $\Lambda$CDM model, and the astrophysical parameters {$v_{c,0}, \beta$} which represent the cutoff and slope of the HI- halo mass relation. We explore (i) quantifying the effects of the astrophysics on the recovery of the cosmological parameters, (ii) the dependence of the cosmological forecasts on the details of the astrophysical parametrization, and (iii) the improvement of the constraints on probing smaller scales in the HI power spectrum. For an SKA I MID intensity mapping survey alone, probing scales up to $\ell_{\rm max} = 1000$, we find a factor of $1.1 - 1.3$ broadening in the constraints on $\Omega_b$ and $\Omega_m$, and of $2.4 - 2.6$ on $h$, $n_s$ and $\sigma_8$, if we marginalize over astrophysical parameters without any priors. However, even the prior information coming from the present knowledge of the astrophysics largely alleviates this broadening. These findings do not change significantly on considering an extended HIHM relation, illustrating the robustness of the results to the choice of the astrophysical parametrization. Probing scales up to $\ell_{\rm max} = 2000$ improves the constraints by factors of 1.5-1.8. The forecasts improve on increasing the number of tomographic redshift bins, saturating, in many cases, with 4 - 5 redshift bins. We also forecast constraints for intensity mapping with other experiments, and draw similar conclusions.

Testing the Etherington distance duality relation at higher redshifts: Combined radio quasar and gravitational wave data

In this paper we analyse the implications of the latest cosmological data sets to test the Etherington's distance duality relation (DDR), which connects the luminosity distance $D_L$ and angular diameter distance $D_A$ at the same redshift. For $D_L$ we consider the simulated data of gravitational waves from the third-generation gravitational wave detector (the Einstein Telescope, ET), which can be considered as standard candles (or standard siren), while the angular diameter distances $D_A$ are derived from the newly-compiled sample of compact radio quasars observed by very-long-baseline interferometry (VLBI), which represents a type of new cosmological standard ruler. Alleviating the absorption and scattering effects of dust in the Universe, this will create a valuable opportunity to directly test DDR at much higher precision with the combination of gravitational wave (GW) and electromagnetic (EM) signals. Our results show that, with the combination of the current radio quasar observations, the duality-distance relation can be verified at the precision of $10^{-2}$. Moreover, the Einstein Telescope ET would produce more robust constraints on the validity of such distance duality relation (at the precision of $10^{-3}$), with a larger sample of compact milliarcsecond radio quasars detected in future VLBI surveys.

Renormalization group computation of likelihood functions for cosmological data sets

I show how a renormalization group (RG) method can be used to incrementally integrate the information in cosmological large-scale structure data sets (including CMB, galaxy redshift surveys, etc.). I show numerical tests for Gaussian fields, where the method allows arbitrarily close to exact computation of the likelihood function in order $\sim N$ time, even for problems with no symmetry, compared to $N^3$ for brute force linear algebra (where $N$ is the number of data points -- to be fair, methods already exist to solve the Gaussian problem in at worst $N \log N$ time, and this method will not necessarily be faster in practice). The method requires no sampling or other Monte Carlo (random) element. Non-linearity/non-Gaussianity can be accounted for to the extent that terms generated by integrating out small scale modes can be projected onto a sufficient basis, e.g., at least in the sufficiently perturbative regime. The formulas to evaluate are straightforward and require no understanding of quantum field theory, but this paper may also serve as a pedagogical introduction to Wilsonian RG for astronomers.