Concordance and Discordance in Cosmology

Abstract: The success of present and future cosmological studies is tied to the ability to detect discrepancies in complex data sets within the framework of a cosmological model. Tensions caused by the presence of unknown systematic effects need to be isolated and corrected to increase the overall accuracy of parameter constraints, while discrepancies due to new physical phenomena need to be promptly identified. We develop a full set of estimators of internal and mutual agreement and disagreement, whose strengths complement each other. These allow to take into account the effect of prior information and compute the statistical significance of both tensions and confirmatory biases. We apply them to a wide range of state of the art cosmological probes and show that these estimators can be easily used, regardless of model and data complexity. We derive a series of results that show that discrepancies indeed arise within the standard LCDM model. Several of them exceed the probability threshold of 95% and deserve a dedicated effort to understand their origin.

Concordance and Discordance in Cosmology: A Comprehensive Analysis

The paper "Concordance and Discordance in Cosmology" by Marco Raveri and Wayne Hu addresses a critical issue in contemporary cosmology: the identification and interpretation of discrepancies within and between cosmological data sets. In the era of precision cosmology, where data from various sources are used to constrain parameters of the $\Lambda$CDM model, it becomes imperative to distinguish between anomalies due to systematic errors, modeling inaccuracies, and new physics. The authors propose a set of methodological tools designed to evaluate the concordance or discordance between data sets, encompassing both internal consistency measures and cross-comparison metrics.

Methodology: Concordance/Discordance Estimators

The authors introduce a suite of concordance/discordance estimators (CDEs) grounded in the Gaussian linear model (GLM). This framework facilitates the linearization of complex model predictions, enabling tractable statistical computations. The CDEs encompass:

1. Goodness of Fit (GoF) Measures: These include the likelihood at maximum posterior (MAP), which accounts for the impact of prior constraints when performing goodness of fit tests. The MAP estimator, as opposed to the traditional maximum likelihood, provides a refined degree-of-freedom counting that integrates prior information.

2. Evidence Ratio Tests: The authors critique the conventional evidence ratio's reliance on the Jeffreys' scale due to its inherent bias toward agreement. They propose a debiasing method, $\Delta \ln\mathrm{C}$, that recalibrates the test to reflect true statistical significance.

3. Parameter Difference Tests: The study introduces generalized parameter difference estimators, such as the update difference in mean, which work optimally in arbitrary multidimensional parameter spaces. These tools are particularly potent in identifying discrepancies masked by marginalization.

Application to Cosmological Data

The paper applies these methodologies to a comprehensive range of cosmological data sets, including the Planck CMB spectra, supernovae measurements, various BAO and galaxy survey data, and local $H_0$ estimates.

1. Internal Consistency of CMB Data: The analysis reveals significant tension between the Planck CMB temperature and polarization data (TT, EE) and the Planck lensing data, suggesting potential biases in measuring the lensing parameter $A_L$. Such discrepancies necessitate further investigation into systematic effects and modeling assumptions.

2. Cross-Comparison of Cosmological Probes: Notably, the study reaffirms the tension between CMB-derived $H_0$ and local measurements, such as SH0ES and H0LiCOW, at high statistical significance (greater than 3$\sigma$). Weak lensing data, specifically from CFHTLenS and KiDS, also exhibit tensions with CMB results, indicating potential issues in modeling the amplitude of structures on linear scales.

3. Assessment of Galaxy Clustering Data: The SDSS LRG survey data show signs of confirmation bias, potentially skewing results when integrated with other cosmological probes, such as Planck CMB data. This necessitates caution in data combination strategies and highlights the importance of robust statistical tools.

Implications and Future Directions

The findings underscore the necessity for refined data analysis techniques in cosmology, especially as the field moves toward more sensitive and complex surveys like Euclid, LSST, and CMB-S4. The methodologies proposed by Raveri and Hu enhance the robustness of cosmological inferences by providing a framework for systematically identifying and quantifying tensions in data sets.

The study's implications extend beyond immediate data consistency checks; they prompt a reevaluation of model assumptions and inspire the development of new physics scenarios when discrepancies are insurmountable by systematic adjustments. As cosmology continues to evolve, these tools offer researchers a nuanced approach to understanding the concordance of our cosmic models, potentially leading to insights that challenge and refine our understanding of the universe.