Phase-space density is recovered from Gaia data for M4 and disrupted streams by correcting entropy injection and minimizing stream entropy, enabling original mass inference via Liouville's theorem.
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In DHOST theories consistent with GW observations, deviations from GR suppress high-redshift galaxy cluster abundance relative to ΛCDM when using spherical collapse and analytic mass functions matched to eROSITA data.
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Applying Liouville's Theorem to Gaia Data
Phase-space density is recovered from Gaia data for M4 and disrupted streams by correcting entropy injection and minimizing stream entropy, enabling original mass inference via Liouville's theorem.