arXiv subjects

J. DeRose

Publications and source records attributed to J. DeRose.

At least 127 records · Page 7Linked to original sources

arXiv2018

Dark Energy Survey Year 1 results: Validation of weak lensing cluster member contamination estimates from P(z) decomposition

Weak lensing source galaxy catalogs used in estimating the masses of galaxy clusters can be heavily contaminated by cluster members, prohibiting accurate mass calibration. In this study we test the performance of an estimator for the extent of cluster member contamination based on decomposing the photometric redshift $P(z)$ of source galaxies into contaminating and background components. We perform a full scale mock analysis on a simulated sky survey approximately mirroring the observational properties of the Dark Energy Survey Year One observations (DES Y1), and find excellent agreement between the true number profile of contaminating cluster member galaxies in the simulation and the estimated one. We further apply the method to estimate the cluster member contamination for the DES Y1 redMaPPer cluster mass calibration analysis, and compare the results to an alternative approach based on the angular correlation of weak lensing source galaxies. We find indications that the correlation based estimates are biased by the selection of the weak lensing sources in the cluster vicinity, which does not strongly impact the $P(z)$ decomposition method. Collectively, these benchmarks demonstrate the strength of the $P(z)$ decomposition method in alleviating membership contamination and enabling highly accurate cluster weak lensing studies without broad exclusion of source galaxies, thereby improving the total constraining power of cluster mass calibration via weak lensing.

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arXiv2018

DES Y1 Results: Validating cosmological parameter estimation using simulated Dark Energy Surveys

We use mock galaxy survey simulations designed to resemble the Dark Energy Survey Year 1 (DES Y1) data to validate and inform cosmological parameter estimation. When similar analysis tools are applied to both simulations and real survey data, they provide powerful validation tests of the DES Y1 cosmological analyses presented in companion papers. We use two suites of galaxy simulations produced using different methods, which therefore provide independent tests of our cosmological parameter inference. The cosmological analysis we aim to validate is presented in DES Collaboration et al. (2017) and uses angular two-point correlation functions of galaxy number counts and weak lensing shear, as well as their cross-correlation, in multiple redshift bins. While our constraints depend on the specific set of simulated realisations available, for both suites of simulations we find that the input cosmology is consistent with the combined constraints from multiple simulated DES Y1 realizations in the $Ω_m-σ_8$ plane. For one of the suites, we are able to show with high confidence that any biases in the inferred $S_8=σ_8(Ω_m/0.3)^{0.5}$ and $Ω_m$ are smaller than the DES Y1 $1-σ$ uncertainties. For the other suite, for which we have fewer realizations, we are unable to be this conclusive; we infer a roughly 70% probability that systematic biases in the recovered $Ω_m$ and $S_8$ are sub-dominant to the DES Y1 uncertainty. As cosmological analyses of this kind become increasingly more precise, validation of parameter inference using survey simulations will be essential to demonstrate robustness.

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arXiv2018

Dark Energy Survey Year 1 Results: Methods for Cluster Cosmology and Application to the SDSS

We perform the first blind analysis of cluster abundance data. Specifically, we derive cosmological constraints from the abundance and weak-lensing signal of \redmapper\ clusters of richness $λ\geq 20$ in the redshift range $z\in[0.1,0.3]$ as measured in the Sloan Digital Sky Survey (SDSS). We simultaneously fit for cosmological parameters and the richness--mass relation of the clusters. For a flat $Λ$CDM cosmological model with massive neutrinos, we find $S_8 \equiv σ_{8}(Ω_m/0.3)^{0.5}=0.79^{+0.05}_{-0.04}$. This value is both consistent and competitive with that derived from cluster catalogues selected in different wavelengths. Our result is also consistent with the combined probes analyses by the Dark Energy Survey (DES) and the Kilo-Degree Survey (KiDS), and with the Cosmic Microwave Background (CMB) anisotropies as measured by \planck. We demonstrate that the cosmological posteriors are robust against variation of the richness--mass relation model and to systematics associated with the calibration of the selection function. In combination with Baryon Acoustic Oscillation (BAO) data and Big-Bang Nucleosynthesis (BBN) data, we constrain the Hubble rate to be $h=0.66\pm 0.02$, independent of the CMB. Future work aimed at improving our understanding of the scatter of the richness--mass relation has the potential to significantly improve the precision of our cosmological posteriors. The methods described in this work were developed for use in the forthcoming analysis of cluster abundances in the DES. Our SDSS analysis constitutes the first part of a staged-unblinding analysis of the full DES data set.

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arXiv2018

Dark Energy Survey Year 1 Results: Joint Analysis of Galaxy Clustering, Galaxy Lensing, and CMB Lensing Two-point Functions

We perform a joint analysis of the auto and cross-correlations between three cosmic fields: the galaxy density field, the galaxy weak lensing shear field, and the cosmic microwave background (CMB) weak lensing convergence field. These three fields are measured using roughly 1300 sq. deg. of overlapping optical imaging data from first year observations of the Dark Energy Survey and millimeter-wave observations of the CMB from both the South Pole Telescope Sunyaev-Zel'dovich survey and Planck. We present cosmological constraints from the joint analysis of the two-point correlation functions between galaxy density and galaxy shear with CMB lensing. We test for consistency between these measurements and the DES-only two-point function measurements, finding no evidence for inconsistency in the context of flat $Λ$CDM cosmological models. Performing a joint analysis of five of the possible correlation functions between these fields (excluding only the CMB lensing autospectrum) yields $S_{8}\equiv σ_8\sqrt{Ω_{\rm m}/0.3} = 0.782^{+0.019}_{-0.025}$ and $Ω_{\rm m}=0.260^{+0.029}_{-0.019}$. We test for consistency between these five correlation function measurements and the Planck-only measurement of the CMB lensing autospectrum, again finding no evidence for inconsistency in the context of flat $Λ$CDM models. Combining constraints from all six two-point functions yields $S_{8}=0.776^{+0.014}_{-0.021}$ and $Ω_{\rm m}= 0.271^{+0.022}_{-0.016}$. These results provide a powerful test and confirmation of the results from the first year DES joint-probes analysis.

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arXiv2018

Dark Energy Survey Year 1 Results: Cross-correlation between DES Y1 galaxy weak lensing and SPT+Planck CMB weak lensing

We cross-correlate galaxy weak lensing measurements from the Dark Energy Survey (DES) year-one (Y1) data with a cosmic microwave background (CMB) weak lensing map derived from South Pole Telescope (SPT) and Planck data, with an effective overlapping area of 1289 deg$^{2}$. With the combined measurements from four source galaxy redshift bins, we reject the hypothesis of no lensing with a significance of $10.8σ$. When employing angular scale cuts, this significance is reduced to $6.8σ$, which remains the highest signal-to-noise measurement of its kind to date. We fit the amplitude of the correlation functions while fixing the cosmological parameters to a fiducial $Λ$CDM model, finding $A = 0.99 \pm 0.17$. We additionally use the correlation function measurements to constrain shear calibration bias, obtaining constraints that are consistent with previous DES analyses. Finally, when performing a cosmological analysis under the $Λ$CDM model, we obtain the marginalized constraints of $Ω_{\rm m}=0.261^{+0.070}_{-0.051}$ and $S_{8}\equiv σ_{8}\sqrt{Ω_{\rm m}/0.3} = 0.660^{+0.085}_{-0.100}$. These measurements are used in a companion work that presents cosmological constraints from the joint analysis of two-point functions among galaxies, galaxy shears, and CMB lensing using DES, SPT and Planck data.

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arXiv2018

Dark Energy Survey Year 1 Results: Calibration of redMaGiC Redshift Distributions in DES and SDSS from Cross-Correlations

We present calibrations of the redshift distributions of redMaGiC galaxies in the Dark Energy Survey Year 1 (DES Y1) and Sloan Digital Sky Survey (SDSS) DR8 data. These results determine the priors of the redshift distribution of redMaGiC galaxies, which were used for galaxy clustering measurements and as lenses for galaxy-galaxy lensing measurements in DES Y1 cosmological analyses. We empirically determine the bias in redMaGiC photometric redshift estimates using angular cross-correlations with Baryon Oscillation Spectroscopic Survey (BOSS) galaxies. For DES, we calibrate a single parameter redshift bias in three photometric redshift bins: $z \in[0.15,0.3]$, [0.3,0.45], and [0.45,0.6]. Our best fit results in each bin give photometric redshift biases of $|Δz|<0.01$. To further test the redMaGiC algorithm, we apply our calibration procedure to SDSS redMaGiC galaxies, where the statistical precision of the cross-correlation measurement is much higher due to a greater overlap with BOSS galaxies. For SDSS, we also find best fit results of $|Δz|<0.01$. We compare our results to other analyses of redMaGiC photometric redshifts.

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arXiv2018

Dark Energy Survey Year 1 Results: Weak Lensing Mass Calibration of redMaPPer Galaxy Clusters

We constrain the mass--richness scaling relation of redMaPPer galaxy clusters identified in the Dark Energy Survey Year 1 data using weak gravitational lensing. We split clusters into $4\times3$ bins of richness $λ$ and redshift $z$ for $λ\geq20$ and $0.2 \leq z \leq 0.65$ and measure the mean masses of these bins using their stacked weak lensing signal. By modeling the scaling relation as $\langle M_{\rm 200m}|λ,z\rangle = M_0 (λ/40)^F ((1+z)/1.35)^G$, we constrain the normalization of the scaling relation at the 5.0 per cent level as $M_0 = [3.081 \pm 0.075 ({\rm stat}) \pm 0.133 ({\rm sys})] \cdot 10^{14}\ {\rm M}_\odot$ at $λ=40$ and $z=0.35$. The richness scaling index is constrained to be $F=1.356 \pm 0.051\ ({\rm stat})\pm 0.008\ ({\rm sys})$ and the redshift scaling index $G=-0.30\pm 0.30\ ({\rm stat})\pm 0.06\ ({\rm sys})$. These are the tightest measurements of the normalization and richness scaling index made to date. We use a semi-analytic covariance matrix to characterize the statistical errors in the recovered weak lensing profiles. Our analysis accounts for the following sources of systematic error: shear and photometric redshift errors, cluster miscentering, cluster member dilution of the source sample, systematic uncertainties in the modeling of the halo--mass correlation function, halo triaxiality, and projection effects. We discuss prospects for reducing this systematic error budget, which dominates the uncertainty on $M_0$. Our result is in excellent agreement with, but has significantly smaller uncertainties than, previous measurements in the literature, and augurs well for the power of the DES cluster survey as a tool for precision cosmology and upcoming galaxy surveys such as LSST, Euclid and WFIRST.

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arXiv2018

Survey geometry and the internal consistency of recent cosmic shear measurements

We explore the impact of an update to the typical approximation for the shape noise term in the analytic covariance matrix for cosmic shear experiments that assumes the absence of survey boundary and mask effects. We present an exact expression for the number of galaxy pairs in this term based on the the survey mask, which leads to more than a factor of three increase in the shape noise on the largest measured scales for the Kilo-Degree Survey (KIDS-450) real-space cosmic shear data. We compare the result of this analytic expression to several alternative methods for measuring the shape noise from the data and find excellent agreement. This update to the covariance resolves any internal model tension evidenced by the previously large cosmological best-fit $χ^2$ for the KiDS-450 cosmic shear data. The best-fit $χ^2$ is reduced from 161 to 121 for 118 degrees of freedom. We also apply a correction to how the multiplicative shear calibration uncertainty is included in the covariance. This change, along with a previously known update to the reported effective angular values of the data vector, jointly shift the inferred amplitude of the correlation function to higher values. We find that this improves agreement of the KiDS-450 cosmic shear results with Dark Energy Survey Year 1 and Planck results.

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arXiv2018

Density split statistics: Cosmological constraints from counts and lensing in cells in DES Y1 and SDSS data

We derive cosmological constraints from the probability distribution function (PDF) of evolved large-scale matter density fluctuations. We do this by splitting lines of sight by density based on their count of tracer galaxies, and by measuring both gravitational shear around and counts-in-cells in overdense and underdense lines of sight, in Dark Energy Survey (DES) First Year and Sloan Digital Sky Survey (SDSS) data. Our analysis uses a perturbation theory model (see companion paper Friedrich at al.) and is validated using N-body simulation realizations and log-normal mocks. It allows us to constrain cosmology, bias and stochasticity of galaxies w.r.t. matter density and, in addition, the skewness of the matter density field. From a Bayesian model comparison, we find that the data weakly prefer a connection of galaxies and matter that is stochastic beyond Poisson fluctuations on <=20 arcmin angular smoothing scale. The two stochasticity models we fit yield DES constraints on the matter density $Ω_m=0.26^{+0.04}_{-0.03}$ and $Ω_m=0.28^{+0.05}_{-0.04}$ that are consistent with each other. These values also agree with the DES analysis of galaxy and shear two-point functions (3x2pt) that only uses second moments of the PDF. Constraints on $σ_8$ are model dependent ($σ_8=0.97^{+0.07}_{-0.06}$ and $0.80^{+0.06}_{-0.07}$ for the two stochasticity models), but consistent with each other and with the 3x2pt results if stochasticity is at the low end of the posterior range. As an additional test of gravity, counts and lensing in cells allow to compare the skewness $S_3$ of the matter density PDF to its LCDM prediction. We find no evidence of excess skewness in any model or data set, with better than 25 per cent relative precision in the skewness estimate from DES alone.

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arXiv2018

Modeling projection effects in optically-selected cluster catalogues

The cosmological utility of galaxy cluster catalogues is primarily limited by our ability to calibrate the relation between halo mass and observable mass proxies such as cluster richness, X-ray luminosity or the Sunyaev-Zeldovich signal. Projection effects are a particularly pernicious systematic effect that can impact observable mass proxies; structure along the line of sight can both bias and increase the scatter of the observable mass proxies used in cluster abundance studies. In this work, we develop an empirical method to characterize the impact of projection effects on redMaPPer cluster catalogues. We use numerical simulations to validate our method and illustrate its robustness. We demonstrate that modeling of projection effects is a necessary component for cluster abundance studies capable of reaching $\approx 5\%$ mass calibration uncertainties (e.g. the Dark Energy Survey Year 1 sample). Specifically, ignoring the impact of projection effects in the observable--mass relation --- i.e. marginalizing over a log-normal model only --- biases the posterior of the cluster normalization condition $S_8 \equiv σ_8 (Ω_{\rm m}/0.3)^{1/2}$ by $ΔS_8 =0.05$, more than twice the uncertainty in the posterior for such an analysis.

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arXiv2018

Dark Energy Survey Year 1 Results: Redshift distributions of the weak lensing source galaxies

We describe the derivation and validation of redshift distribution estimates and their uncertainties for the galaxies used as weak lensing sources in the Dark Energy Survey (DES) Year 1 cosmological analyses. The Bayesian Photometric Redshift (BPZ) code is used to assign galaxies to four redshift bins between z=0.2 and 1.3, and to produce initial estimates of the lensing-weighted redshift distributions $n^i_{PZ}(z)$ for bin i. Accurate determination of cosmological parameters depends critically on knowledge of $n^i$ but is insensitive to bin assignments or redshift errors for individual galaxies. The cosmological analyses allow for shifts $n^i(z)=n^i_{PZ}(z-Δz^i)$ to correct the mean redshift of $n^i(z)$ for biases in $n^i_{\rm PZ}$. The $Δz^i$ are constrained by comparison of independently estimated 30-band photometric redshifts of galaxies in the COSMOS field to BPZ estimates made from the DES griz fluxes, for a sample matched in fluxes, pre-seeing size, and lensing weight to the DES weak-lensing sources. In companion papers, the $Δz^i$ are further constrained by the angular clustering of the source galaxies around red galaxies with secure photometric redshifts at 0.15<z<0.9. This paper details the BPZ and COSMOS procedures, and demonstrates that the cosmological inference is insensitive to details of the $n^i(z)$ beyond the choice of $Δz^i$. The clustering and COSMOS validation methods produce consistent estimates of $Δz^i$, with combined uncertainties of $σ_{Δz^i}=$0.015, 0.013, 0.011, and 0.022 in the four bins. We marginalize over these in all analyses to follow, which does not diminish the constraining power significantly. Repeating the photo-z procedure using the Directional Neighborhood Fitting (DNF) algorithm instead of BPZ, or using the $n^i(z)$ directly estimated from COSMOS, yields no discernible difference in cosmological inferences.

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arXiv2018

Dark Energy Survey Year 1 Results: Cosmological Constraints from Cosmic Shear

We use 26 million galaxies from the Dark Energy Survey (DES) Year 1 shape catalogs over 1321 deg$^2$ of the sky to produce the most significant measurement of cosmic shear in a galaxy survey to date. We constrain cosmological parameters in both the flat $Λ$CDM and $w$CDM models, while also varying the neutrino mass density. These results are shown to be robust using two independent shape catalogs, two independent \photoz\ calibration methods, and two independent analysis pipelines in a blind analysis. We find a 3.5\% fractional uncertainty on $σ_8(Ω_m/0.3)^{0.5} = 0.782^{+0.027}_{-0.027}$ at 68\% CL, which is a factor of 2.5 improvement over the fractional constraining power of our DES Science Verification results. In $w$CDM, we find a 4.8\% fractional uncertainty on $σ_8(Ω_m/0.3)^{0.5} = 0.777^{+0.036}_{-0.038}$ and a dark energy equation-of-state $w=-0.95^{+0.33}_{-0.39}$. We find results that are consistent with previous cosmic shear constraints in $σ_8$ -- $Ω_m$, and see no evidence for disagreement of our weak lensing data with data from the CMB. Finally, we find no evidence preferring a $w$CDM model allowing $w\ne -1$. We expect further significant improvements with subsequent years of DES data, which will more than triple the sky coverage of our shape catalogs and double the effective integrated exposure time per galaxy.

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arXiv2017

Dark Energy Survey Year 1 Results: Curved-Sky Weak Lensing Mass Map

We construct the largest curved-sky galaxy weak lensing mass map to date from the DES first-year (DES Y1) data. The map, about 10 times larger than previous work, is constructed over a contiguous $\approx1,500 $deg$^2$, covering a comoving volume of $\approx10 $Gpc$^3$. The effects of masking, sampling, and noise are tested using simulations. We generate weak lensing maps from two DES Y1 shear catalogs, Metacalibration and Im3shape, with sources at redshift $0.2<z<1.3,$ and in each of four bins in this range. In the highest signal-to-noise map, the ratio between the mean signal-to-noise in the E-mode and the B-mode map is $\sim$1.5 ($\sim$2) when smoothed with a Gaussian filter of $σ_{G}=30$ (80) arcminutes. The second and third moments of the convergence $κ$ in the maps are in agreement with simulations. We also find no significant correlation of $κ$ with maps of potential systematic contaminants. Finally, we demonstrate two applications of the mass maps: (1) cross-correlation with different foreground tracers of mass and (2) exploration of the largest peaks and voids in the maps.

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arXiv2017

Dark Energy Survey Year 1 Results: Cross-Correlation Redshifts in the DES -- Calibration of the Weak Lensing Source Redshift Distributions

We present the calibration of the Dark Energy Survey Year 1 (DES Y1) weak lensing source galaxy redshift distributions from clustering measurements. By cross-correlating the positions of source galaxies with luminous red galaxies selected by the redMaGiC algorithm we measure the redshift distributions of the source galaxies as placed into different tomographic bins. These measurements constrain any such shifts to an accuracy of $\sim0.02$ and can be computed even when the clustering measurements do not span the full redshift range. The highest-redshift source bin is not constrained by the clustering measurements because of the minimal redshift overlap with the redMaGiC galaxies. We compare our constraints with those obtained from $\texttt{COSMOS}$ 30-band photometry and find that our two very different methods produce consistent constraints.

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arXiv2017

Dark Energy Survey Year 1 Results: Cross-Correlation Redshifts - Methods and Systematics Characterization

We use numerical simulations to characterize the performance of a clustering-based method to calibrate photometric redshift biases. In particular, we cross-correlate the weak lensing (WL) source galaxies from the Dark Energy Survey Year 1 (DES Y1) sample with redMaGiC galaxies (luminous red galaxies with secure photometric redshifts) to estimate the redshift distribution of the former sample. The recovered redshift distributions are used to calibrate the photometric redshift bias of standard photo-$z$ methods applied to the same source galaxy sample. We apply the method to three photo-$z$ codes run in our simulated data: Bayesian Photometric Redshift (BPZ), Directional Neighborhood Fitting (DNF), and Random Forest-based photo-$z$ (RF). We characterize the systematic uncertainties of our calibration procedure, and find that these systematic uncertainties dominate our error budget. The dominant systematics are due to our assumption of unevolving bias and clustering across each redshift bin, and to differences between the shapes of the redshift distributions derived by clustering vs photo-$z$'s. The systematic uncertainty in the mean redshift bias of the source galaxy sample is $Δz \lesssim 0.02$, though the precise value depends on the redshift bin under consideration. We discuss possible ways to mitigate the impact of our dominant systematics in future analyses.

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arXiv2017

Dark Energy Survey Year 1 Results: Multi-Probe Methodology and Simulated Likelihood Analyses

We present the methodology for and detail the implementation of the Dark Energy Survey (DES) 3x2pt DES Year 1 (Y1) analysis, which combines configuration-space two-point statistics from three different cosmological probes: cosmic shear, galaxy-galaxy lensing, and galaxy clustering, using data from the first year of DES observations. We have developed two independent modeling pipelines and describe the code validation process. We derive expressions for analytical real-space multi-probe covariances, and describe their validation with numerical simulations. We stress-test the inference pipelines in simulated likelihood analyses that vary 6-7 cosmology parameters plus 20 nuisance parameters and precisely resemble the analysis to be presented in the DES 3x2pt analysis paper, using a variety of simulated input data vectors with varying assumptions. We find that any disagreement between pipelines leads to changes in assigned likelihood $Δχ^2 \le 0.045$ with respect to the statistical error of the DES Y1 data vector. We also find that angular binning and survey mask do not impact our analytic covariance at a significant level. We determine lower bounds on scales used for analysis of galaxy clustering (8 Mpc$~h^{-1}$) and galaxy-galaxy lensing (12 Mpc$~h^{-1}$) such that the impact of modeling uncertainties in the non-linear regime is well below statistical errors, and show that our analysis choices are robust against a variety of systematics. These tests demonstrate that we have a robust analysis pipeline that yields unbiased cosmological parameter inferences for the flagship 3x2pt DES Y1 analysis. We emphasize that the level of independent code development and subsequent code comparison as demonstrated in this paper is necessary to produce credible constraints from increasingly complex multi-probe analyses of current data.

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arXiv2017

Real-Time Recovery Efficiencies and Performance of the Palomar Transient Factory's Transient Discovery Pipeline

We present the transient source detection efficiencies of the Palomar Transient Factory (PTF), parameterizing the number of transients that PTF found, versus the number of similar transients that occurred over the same period in the survey search area but that were missed. PTF was an optical sky survey carried out with the Palomar 48-inch telescope over 2009-2012, observing more than 8000 square degrees of sky with cadences of between 1 and 5 days, locating around 50,000 non-moving transient sources, and spectroscopically confirming around 1900 supernovae. We assess the effectiveness with which PTF detected transient sources, by inserting ~7 million artificial point sources into real PTF data. We then study the efficiency with which the PTF real-time pipeline recovered these sources as a function of the source magnitude, host galaxy surface brightness, and various observing conditions (using proxies for seeing, sky brightness, and transparency). The product of this study is a multi-dimensional recovery efficiency grid appropriate for the range of observing conditions that PTF experienced, and that can then be used for studies of the rates, environments, and luminosity functions of different transient types using detailed Monte Carlo simulations. We illustrate the technique using the observationally well-understood class of type Ia supernovae.

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arXiv2016

Imprint of DES super-structures on the Cosmic Microwave Background

Small temperature anisotropies in the Cosmic Microwave Background can be sourced by density perturbations via the late-time integrated Sachs-Wolfe effect. Large voids and superclusters are excellent environments to make a localized measurement of this tiny imprint. In some cases excess signals have been reported. We probed these claims with an independent data set, using the first year data of the Dark Energy Survey in a different footprint, and using a different super-structure finding strategy. We identified 52 large voids and 102 superclusters at redshifts $0.2 < z < 0.65$. We used the Jubilee simulation to a priori evaluate the optimal ISW measurement configuration for our compensated top-hat filtering technique, and then performed a stacking measurement of the CMB temperature field based on the DES data. For optimal configurations, we detected a cumulative cold imprint of voids with $ΔT_{f} \approx -5.0\pm3.7~μK$ and a hot imprint of superclusters $ΔT_{f} \approx 5.1\pm3.2~μK$ ; this is $\sim1.2σ$ higher than the expected $|ΔT_{f}| \approx 0.6~μK$ imprint of such super-structures in $Λ$CDM. If we instead use an a posteriori selected filter size ($R/R_{v}=0.6$), we can find a temperature decrement as large as $ΔT_{f} \approx -9.8\pm4.7~μK$ for voids, which is $\sim2σ$ above $Λ$CDM expectations and is comparable to previous measurements made using SDSS super-structure data.

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