arXiv subjects

Tim Eifler

Publications and source records attributed to Tim Eifler.

At least 37 records · Page 2Linked to original sources

arXiv2024

Impact of cosmology dependence of baryonic feedback in weak lensing

Robust modeling of non-linear scales is critical for accurate cosmological inference in Stage IV surveys. For weak lensing analyses in particular, a key challenge arises from the incomplete understanding of how non-gravitational processes, such as supernovae and active galactic nuclei - collectively known as baryonic feedback - affect the matter distribution. Several existing methods for modeling baryonic feedback treat it independently from the underlying cosmology, an assumption which has been found to be inaccurate by hydrodynamical simulations. In this work, we examine the impact of this coupling between baryonic feedback and cosmology on parameter inference at LSST Y1 precision. We build mock 3$\times$2pt data vectors using the Magneticum suite of hydrodynamical simulations, which span a wide range of cosmologies while keeping subgrid parameters fixed. We perform simulated likelihood analyses for two baryon mitigation techniques: (i) the Principal Component Analysis (PCA) method which identifies eigenmodes for capturing the effect baryonic feedback on the data vector and (ii) HMCode2020 (Mead et al. 2021) which analytically models the modification in the matter distribution using a halo model approach. Our results show that the PCA method is robust to the coupling between cosmology and baryonic feedback, whereas, when using HMCode2020 there can be up to $0.5σ$ bias in $Ω_\text{m}$-$S_8$. For HMCode2020, the bias also correlates with the input cosmology while for PCA we find no such correlation.

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arXiv2024

Kinematic Lensing Inference II: Cluster Lensing with $\mathcal{O}$(1) Galaxies

We present the first detection of a cluster lensing signal with `Kinematic Lensing' (KL), a novel weak lensing method that combines photometry, spectroscopy, and the Tully-Fisher relation to enable shear measurements with individual source galaxies. This is the second paper in a two-part series aimed at measuring a KL signal from data. The first paper, arXiv:2209.11811, describes the inference pipeline, which jointly forward models galaxy imaging and spectroscopy, and demonstrates unbiased shear inference with simulated data. This paper presents measurements of the lensing signal from the galaxy cluster Abell 2261. We obtain spectroscopic observations of background disk galaxies in the cluster field selected from the CLASH Subaru catalog. The final sample consists of three source galaxies while the remaining are rejected due to insufficient signal-to-noise, spectroscopic failures, and inadequately sampled rotation curves. We apply the KL inference pipeline to the three sources and find the shear estimates to be in broad agreement with traditional weak lensing measurements. The typical shear measurement uncertainty for our sources is $σ(g_+)\approx 0.026$, which represents approximately a ten-fold improvement over the weak lensing shape noise. We identify target selection and observing strategy as the key avenues of improvement for future KL programs.

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arXiv2024

Kinematic Lensing with the Dark Energy Spectroscopic Instrument -- Probing structure formation at very low redshift

We explore the science prospects of a 14,000 deg$^2$ Kinematic Lensing (KL) survey with the Dark Energy Spectroscopic Instrument (DESI) and overlapping imaging surveys. KL infers the cosmic shear signal by jointly forward modeling the observed photometric image and velocity field of a disk galaxy. The latter can be constrained by placing multiple DESI fibers along the galaxy's major and minor axis, a concept similar to the DESI Peculiar Velocity Survey. We study multiple subset galaxy samples of the DESI Legacy Survey Data Release 9 catalog and quantify the residual shape noise, $σ_ε$, of each sample as a function of cuts in $r$-band magnitude using mock observations. We conduct simulated likelihood analyses for these galaxy samples and find that a DESI-KL program can place highly interesting constraints on structure formation at very low redshifts, i.e. $σ_8(z<0.15)$. We conclude that if the $S_8$ tension consolidates as a phenomenon, a DESI-KL survey can provide unique insights into this phenomenon in the very late-time Universe. Given the different footprints of DESI and Rubin Observatory's Legacy Survey of Space and Time (LSST), lensing results from both surveys are highly complementary and can be combined into a joint lensing survey. We further note that DESI-KL benefits multiple additional science cases, e.g. studies of modified gravity models when combined with peculiar velocity surveys, and dark matter studies that are based on galaxy-galaxy lensing of dwarf galaxies.

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arXiv2024

Neural network based emulation of galaxy power spectrum covariances -- A reanalysis of BOSS DR12 data

We train neural networks to quickly generate redshift-space galaxy power spectrum covariances from a given parameter set (cosmology and galaxy bias). This covariance emulator utilizes a combination of traditional fully-connected network layers and transformer architecture to accurately predict covariance matrices for the high redshift, north galactic cap sample of the BOSS DR12 galaxy catalog. We run simulated likelihood analyses with emulated and brute-force computed covariances, and we quantify the network's performance via two different metrics: 1) difference in $χ^2$ and 2) likelihood contours for simulated BOSS DR 12 analyses. We find that the emulator returns excellent results over a large parameter range. We then use our emulator to perform a re-analysis of the BOSS HighZ NGC galaxy power spectrum, and find that varying covariance with cosmology along with the model vector produces $Ω_m = 0.276^{+0.013}_{-0.015}$, $H_0 = 70.2\pm 1.9$ km/s/Mpc, and $σ_8 = 0.674^{+0.058}_{-0.077}$. These constraints represent an average $0.46σ$ shift in best-fit values and a $5\%$ increase in constraining power compared to fixing the covariance matrix ($Ω_m = 0.293\pm 0.017$, $H_0 = 70.3\pm 2.0$ km/s/Mpc, $σ_8 = 0.702^{+0.063}_{-0.075}$). This work demonstrates that emulators for more complex cosmological quantities than second-order statistics can be trained over a wide parameter range at sufficiently high accuracy to be implemented in realistic likelihood analyses.

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arXiv2024

SPHEREx: NASA's Near-Infrared Spectrophotmetric All-Sky Survey

SPHEREx, the Spectro-Photometer for the History of the Universe, Epoch of Reionization, and ices Explorer, is a NASA MIDEX mission planned for launch in 2024. SPHEREx will carry out the first all-sky spectral survey at wavelengths between 0.75 micron and 5 micron with spectral resolving power ~40 between 0.75 and 3.8 micron and ~120 between 3.8 and 5 micron At the end of its two-year mission, SPHEREx will provide 0.75-to-5 micron spectra of each 6.2"x6.2" pixel on the sky - 14 billion spectra in all. This paper updates an earlier description of SPHEREx presenting changes made during the mission's Preliminary Design Phase, including a discussion of instrument integration and test and a summary of the data processing, analysis, and distribution plans.

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arXiv2024

Attention-Based Neural Network Emulators for Multi-Probe Data Vectors Part II: Assessing Tension Metrics

The next generation of cosmological surveys is expected to generate unprecedented high-quality data, consequently increasing the already substantial computational costs of Bayesian statistical methods. This will pose a significant challenge to analyzing theoretical models of cosmology. Additionally, new mitigation techniques of baryonic effects, intrinsic alignment, and other systematic effects will inevitably introduce more parameters, slowing down the convergence of Bayesian analyses. In this scenario, machine-learning-based accelerators are a promising solution, capable of reducing the computational costs and execution time of such tools by order of thousands. Yet, they have not been able to provide accurate predictions over the wide prior ranges in parameter space adopted by Stage III/IV collaborations in studies employing real-space two-point correlation functions. This paper offers a leap in this direction by carefully investigating the modern transformer-based neural network (NN) architectures in realistic simulated Rubin Observatory year one cosmic shear $Λ$CDM inferences. Building on the framework introduced in Part I, we generalize the transformer block and incorporate additional layer types to develop a more versatile architecture. We present a scalable method to efficiently generate an extensive training dataset that significantly exceeds the scope of prior volumes considered in Part I, while still meeting strict accuracy standards. Through our meticulous architecture comparison and comprehensive hyperparameter optimization, we establish that the attention-based architecture performs an order of magnitude better in accuracy than widely adopted NN designs. Finally, we test and apply our emulators to calibrate tension metrics.

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arXiv2024

Machine Learning LSST 3x2pt analyses -- forecasting the impact of systematics on cosmological constraints using neural networks

Validating modeling choices through simulated analyses and quantifying the impact of different systematic effects will form a major computational bottleneck in the preparation for 3$\times$2 analysis with Stage-IV surveys such as Vera Rubin Observatory's Legacy Survey of Space and Time (LSST). We can significantly reduce the computational requirements by using machine learning based emulators, which allow us to run fast inference while maintaining the full realism of the data analysis pipeline. In this paper, we use such an emulator to run simulated 3$\times$2 (cosmic shear, galaxy-galaxy lensing, and galaxy clustering) analyses for mock LSST-Y1/Y3/Y6/Y10 surveys and study the impact of various systematic effects (galaxy bias, intrinsic alignment, baryonic physics, shear calibration and photo-$z$ uncertainties). Closely following the DESC Science Requirement Document (with several updates) our main findings are: {\it a)} The largest contribution to the `systematic error budget' of LSST 3$\times$2 analysis comes from galaxy bias uncertainties, while the contribution of baryonic and shear calibration uncertainties are significantly less important. {\it b)} Tighter constraints on intrinsic alignment and photo-$z$ parameters can improve cosmological constraints noticeably, which illustrates synergies of LSST and spectroscopic surveys. {\it c)} The scale cuts adopted in the DESC SRD may be too conservative and pushing to smaller scales can increase cosmological information significantly. {\it d)} We investigate the impact of photo-$z$ outliers on 3$\times$2 pt analysis and find that we need to determine the outlier fraction to within $5-10\%$ accuracy to ensure robust cosmological analysis. We caution that these findings depend on analysis choices (parameterizations, priors, scale cuts) and can change for different settings.

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arXiv2024

Early dark energy constraints with late-time expansion marginalization

Early dark energy (EDE) is an extension to the $Λ$CDM model, proposed to reduce the tension between the measurements of the Hubble constant $H_0$ from the cosmic microwave background (CMB) and from the local cosmic distance ladder. However, this model increases the $S_8$ tension between CMB and large scale structure measurements. Analyses of galaxy clustering and lensing correlation functions report a decreased preference for EDE and its effect on the Hubble tension. Smooth dark energy models affect growth of structure through the background expansion. In this work, we study the inclusion of a general, smooth late-time dark energy modification in combination with EDE and obtain constraints on EDE marginalized over the late-time expansion. We assess the impact on the $S_8$ and Hubble tensions. In order to generalize the late expansion, we use a late dark energy fluid model with a piecewise constant equation of state $w(z)$ over 3, 5 and 10 redshift bins in the window $z \in [0,3]$. We show that, when analyzing ACT and Planck CMB data combined with Pantheon supernovae, BAO from 6dF, SDSS and BOSS, Planck 2018 CMB lensing and Dark Energy Survey cosmic shear and clustering data, the inclusion of a general smooth dark energy modification at late times has no significant effect on $S_8$ and EDE parameter constraints. Using the aforementioned datasets, the EDE fraction constraint with late-time expansion marginalization is $f_\mathrm{EDE} = 0.067^{+0.019}_{-0.027}$ using 3 redshift bins, with similar results for 5 and 10 redshift bins. This work shows that in order to solve simultaneously the Hubble and $S_8$ tensions, one needs a mechanism for increasing the clustering of matter at late times different from a simple change in the background evolution of late dark energy. [Abridged]

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arXiv2024

Attention-based Neural Network Emulators for Multi-Probe Data Vectors Part I: Forecasting the Growth-Geometry split

We present a new class of machine-learning emulators that accurately model the cosmic shear, galaxy-galaxy lensing, and galaxy clustering real space correlation functions in the context of Rubin Observatory year one simulated data. To illustrate its capabilities in forecasting models beyond the standard $Λ$CDM, we forecast how well LSST Year 1 data will be able to probe the consistency between geometry $Ω^{\rm geo}_\mathrm{m}$ and growth $Ω^{\rm growth}_\mathrm{m}$ dark matter densities in the so-called split $Λ$CDM parameterization. When trained with a few million samples, our emulator shows uniform accuracy across a wide range in an 18-dimensional parameter space. We provide a detailed comparison of three neural network designs, illustrating the importance of adopting state-of-the-art Transformer blocks. Our study also details their performance when computing Bayesian evidence for cosmic shear on three fiducial cosmologies. The transformers-based emulator is always accurate within PolyChord's precision. As an application, we use our emulator to study the degeneracies between dark energy models and growth geometry split parameterizations. We find that the growth-geometry split remains to be a meaningful test of the smooth dark energy assumption.

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arXiv2023

Cosmology from weak lensing, galaxy clustering, CMB lensing and tSZ: I. 10x2pt Modelling Methodology

The overlap of galaxy surveys and CMB experiments presents an ideal opportunity for joint cosmological dataset analyses. In this paper we develop a halo-model-based method for the first joint analysis combining these two experiments using 10 correlated two-point functions (10x2pt) derived from galaxy position, galaxy shear, CMB lensing convergence, and Compton-y fields. We explore this method using the Vera Rubin Observatory Legacy Survey of Space and Time (LSST) and the Simons Observatory (SO) as examples. We find such LSSxCMB joint analyses lead to significant improvement in Figure-of-Merit of $Ω_m$ and $S_8$ over the constraints from using LSS-only probes within $Λ$CDM. We identify that the shear-$y$ and $y$-$y$ correlations are the most valuable additions when tSZ is included. We further identify the dominant sources of halo model uncertainties in the small-scale modelling, and investigate the impact of halo self-calibration due to the inclusion of small-scale tSZ information.

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arXiv2023

Constraining Baryonic Physics with DES Y1 and Planck data -- Combining Galaxy Clustering, Weak Lensing, and CMB Lensing

We constrain cosmology and baryonic feedback scenarios with a joint analysis of weak lensing, galaxy clustering, cosmic microwave background (CMB) lensing, and their cross-correlations (so-called 6$\times$2) using data from the Dark Energy Survey (DES) Y1 and the Planck satellite mission. Noteworthy features of our 6$\times$2 pipeline are: We extend CMB lensing cross-correlation measurements to a band surrounding the DES Y1 footprint (a $\sim 25\%$ gain in pairs), and we develop analytic covariance capabilities that account for different footprints and all cross-terms in the 6$\times$2 analysis. We also measure the DES Y1 cosmic shear two-point correlation function (2PCF) down to $0.^\prime 25$, but find that going below $2.^\prime 5$ does not increase cosmological information due to shape noise. We model baryonic physics uncertainties via the amplitude of Principal Components (PCs) derived from a set of hydro-simulations. Given our statistical uncertainties, varying the first PC amplitude $Q_1$ is sufficient to model small-scale cosmic shear 2PCF. For DES Y1+Planck 6$\times$2 we find $S_8=0.799\pm0.016$, comparable to the 5$\times$2 result of DES Y3+SPT/Planck $S_8=0.773\pm0.016$. Combined with our most informative cosmology priors -- baryon acoustic oscillation (BAO), big bang nucleosynthesis (BBN), type Ia supernovae (SNe Ia), and Planck 2018 EE+lowE, we measure $S_8=0.817\pm 0.011$. Regarding baryonic physics constraints, our 6$\times$2 analysis finds $Q_1=2.8\pm1.8$. Combined with the aforementioned priors, it improves the constraint to $Q_1=3.5\pm1.3$. For comparison, the strongest feedback scenario considered in this paper, the cosmo-OWLS AGN ($ΔT_\mathrm{heat}=10^{8.7}$ K), corresponds to $Q_1=5.84$.

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arXiv2023

NANCY: Next-generation All-sky Near-infrared Community surveY

The Nancy Grace Roman Space Telescope is capable of delivering an unprecedented all-sky, high-spatial resolution, multi-epoch infrared map to the astronomical community. This opportunity arises in the midst of numerous ground- and space-based surveys that will provide extensive spectroscopy and imaging together covering the entire sky (such as Rubin/LSST, Euclid, UNIONS, SPHEREx, DESI, SDSS-V, GALAH, 4MOST, WEAVE, MOONS, PFS, UVEX, NEO Surveyor, etc.). Roman can uniquely provide uniform high-spatial-resolution (~0.1 arcsec) imaging over the entire sky, vastly expanding the science reach and precision of all of these near-term and future surveys. This imaging will not only enhance other surveys, but also facilitate completely new science. By imaging the full sky over two epochs, Roman can measure the proper motions for stars across the entire Milky Way, probing 100 times fainter than Gaia out to the very edge of the Galaxy. Here, we propose NANCY: a completely public, all-sky survey that will create a high-value legacy dataset benefiting innumerable ongoing and forthcoming studies of the universe. NANCY is a pure expression of Roman's potential: it images the entire sky, at high spatial resolution, in a broad infrared bandpass that collects as many photons as possible. The majority of all ongoing astronomical surveys would benefit from incorporating observations of NANCY into their analyses, whether these surveys focus on nearby stars, the Milky Way, near-field cosmology, or the broader universe.

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arXiv2023

Growth and Geometry Split in Light of the DES-Y3 Survey

We test the smooth dark energy paradigm using Dark Energy Survey (DES) Year 1 and Year 3 weak lensing and galaxy clustering data. Within the $Λ$CDM and $w$CDM model we separate the expansion and structure growth history by splitting $Ω_\mathrm{m}$ (and $w$) into two meta-parameters that allow for different evolution of growth and geometry in the Universe. We consider three different combinations of priors on geometry from CMB, SNIa, BAO, BBN that differ in constraining power but have been designed such that the growth information comes solely from the DES weak lensing and galaxy clustering. For the DES-Y1 data we find no detectable tension between growth and geometry meta-parameters in both the $Λ$CDM and $w$CDM parameter space. This statement also holds for DES-Y3 cosmic shear and 3x2pt analyses. For the combination of DES-Y3 galaxy-galaxy lensing and galaxy clustering (2x2pt) we measure a tension between our growth and geometry meta-parameters of 2.6$σ$ in the $Λ$CDM and 4.48$σ$ in the $w$CDM model space, respectively. We attribute this tension to residual systematics in the DES-Y3 RedMagic galaxy sample rather than to new physics. We plan to investigate our findings further using alternative lens samples in DES-Y3 and future weak lensing and galaxy clustering datasets.

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arXiv2023

Primordial non-Gaussianity with Angular correlation function: Integral constraint and validation for DES

Local primordial non-Gaussianity (PNG) is a promising observable of the underlying physics of inflation, characterised by $f_{\rm NL}^{\rm loc}$. We present the methodology to measure $f_{\rm NL}^{\rm loc}$ from the Dark Energy Survey (DES) data using the 2-point angular correlation function (ACF) with scale-dependent bias. One of the focuses of the work is the integral constraint. This condition appears when estimating the mean number density of galaxies from the data and is key in obtaining unbiased $f_{\rm NL}^{\rm loc}$ constraints. The methods are analysed for two types of simulations: $\sim 246$ GOLIAT-PNG N-body small area simulations with $f_{\rm NL}$ equal to -100 and 100, and 1952 Gaussian ICE-COLA mocks with $f_{\rm NL}=0$ that follow the DES angular and redshift distribution. We use the ensemble of GOLIAT-PNG mocks to show the importance of the integral constraint when measuring PNG, where we recover the fiducial values of $f_{\rm NL}$ within the $1σ$ when including the integral constraint. In contrast, we found a bias of $Δf_{\rm NL}\sim 100$ when not including it. For a DES-like scenario, we forecast a bias of $Δf_{\rm NL} \sim 23$, equivalent to $1.8σ$, when not using the IC for a fiducial value of $f_{\rm NL}=100$. We use the ICE-COLA mocks to validate our analysis in a realistic DES-like setup finding it robust to different analysis choices: best-fit estimator, the effect of IC, BAO damping, covariance, and scale choices. We forecast a measurement of $f_{\rm NL}$ within $σ(f_{\rm NL})=31$ when using the DES-Y3 BAO sample, with the ACF in the $1\ {\rm deg}<θ<20\ {\rm deg}$ range.

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arXiv2022

Kinematic Lensing with the Roman Space Telescope

Kinematic lensing (KL) is a new cosmological measurement technique that combines traditional weak lensing (WL) shape measurements of disc galaxies with their kinematic information. Using the Tully-Fisher relation KL breaks the degeneracy between intrinsic and observed ellipticity and significantly reduces the impact of multiple systematics that are present in traditional WL. We explore the performance of KL given the instrument capabilities of the $\textit{Roman Space Telescope}$, assuming overlap of the High Latitude Imaging Survey (HLIS), the High Latitude Spectroscopy Survey (HLSS) over 2,000 deg$^2$. Our KL suitable galaxy sample has a number density of $n_{\mathrm{gal}}=4~\mathrm{arcmin}^{-1}$ with an estimated shape noise level of $σ_ε=0.035$. We quantify the cosmological constraining power on $Ω_{\mathrm{m}}$-$S_8$, $w_p$-$w_a$ by running simulated likelihood analyses that account for redshift and shear calibration uncertainties, intrinsic alignment and baryonic feedback. Compared to a traditional WL survey we find that KL significantly improves the constraining power on $Ω_{\mathrm{m}}$-$S_8$ (FoM$_{\mathrm{KL}}$=1.70FoM$_{\mathrm{WL}}$) and $w_p$-$w_a$ (FoM$_{\mathrm{KL}}$=3.65FoM$_{\mathrm{WL}}$). We also explore a "narrow tomography KL survey" using 30 instead of the default 10 tomographic bins, however we find no meaningful enhancement to the FoM even when assuming a significant time-dependence in our fiducial dark energy input scenarios.

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arXiv2022

Kinematic Lensing Inference I: Characterizing Shape Noise with Simulated Analyses

The unknown intrinsic shape of source galaxies is one of the largest uncertainties of weak gravitational lensing (WL). It results in the so-called shape noise at the level of $σ_ε^{\mathrm{WL}} \approx 0.26$, whereas the shear effect of interest is of order percent. Kinematic lensing (KL) is a new technique that combines photometric shape measurements with resolved spectroscopic observations to infer the intrinsic galaxy shape and directly estimate the gravitational shear. This paper presents a KL inference pipeline that jointly forward-models galaxy imaging and slit spectroscopy to extract the shear signal. We build a set of realistic mock observations and show that the KL inference pipeline can robustly recover the input shear. To quantify the shear measurement uncertainty for KL, we average the shape noise over a population of randomly oriented disc galaxies and estimate it to be $σ_ε^{\mathrm{KL}}\approx 0.022-0.041$ depending on emission line signal-to-noise. This order of magnitude improvement over traditional WL makes a KL observational program feasible with existing spectroscopic instruments. To this end, we characterize the dependence of KL shape noise on observational factors and discuss implications for the survey strategy of future KL observations. In particular, we find that prioritizing quality spectra of low inclination galaxies is more advantageous than maximizing the overall number density.

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arXiv2022

Weak lensing in the blue: a counter-intuitive strategy for stratospheric observations

The statistical power of weak lensing measurements is principally driven by the number of high redshift galaxies whose shapes are resolved. Conventional wisdom and physical intuition suggest this is optimised by deep imaging at long (red or near IR) wavelengths, to avoid losing redshifted Balmer break and Lyman break galaxies. We use the synthetic Emission Line EL-COSMOS catalogue to simulate lensing observations using different filters, from various altitudes. Here were predict the number of exposures to achieve a target z > 0.3 source density, using off-the-shelf and custom filters. Ground-based observations are easily better at red wavelengths, as (more narrowly) are space-based observations. However, we find that SuperBIT, a diffraction-limited observatory operating in the stratosphere, should instead perform its lensing-quality observations at blue wavelengths.

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arXiv2022

Constraints on dark matter to dark radiation conversion in the late universe with DES-Y1 and external data

We study a phenomenological class of models where dark matter converts to dark radiation in the low redshift epoch. This class of models, dubbed DMDR, characterizes the evolution of comoving dark matter density with two extra parameters, and may be able to help alleviate the observed discrepancies between early- and late-time probes of the universe. We investigate how the conversion affects key cosmological observables such as the CMB temperature and matter power spectra. Combining 3x2pt data from Year 1 of the Dark Energy Survey, {\it Planck}-2018 CMB temperature and polarization data, supernovae (SN) Type Ia data from Pantheon, and baryon acoustic oscillation (BAO) data from BOSS DR12, MGS and 6dFGS, we place new constraints on the amount of dark matter that has converted to dark radiation and the rate of this conversion. The fraction of the dark matter that has converted since the beginning of the universe in units of the current amount of dark matter, $ζ$, is constrained at 68\% confidence level to be $<0.32$ for DES-Y1 3x2pt data, $<0.030$ for CMB+SN+BAO data, and $<0.037$ for the combined dataset. The probability that the DES and CMB+SN+BAO datasets are concordant increases from 4\% for the $Λ$CDM model to 8\% (less tension) for DMDR. The tension in $S_8 = σ_8 \sqrt{Ω_{\rm m}/0.3}$ between DES-Y1 3x2pt and CMB+SN+BAO is slightly reduced from $2.3σ$ to $1.9σ$. We find no reduction in the Hubble tension when the combined data is compared to distance-ladder measurements in the DMDR model. The maximum-posterior goodness-of-fit statistics of DMDR and $Λ$CDM model are comparable, indicating no preference for the DMDR cosmology over $Λ$CDM.

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