Using the chirally invariant overlap Dirac operator we remove its lowest-lying quasizero modes from the valence quark propagators and study evolution of isovector mesons with J = 1. At the truncation level about 50 MeV SU (2)L × SU (2)R and U (1)A symmetries get restored. However, we observe a degeneracy not only within the chiral and U (1)A multiplets, but also a degeneracy of all possible chiral multiplets, i.e., the observed quantum levels have a symmetry larger than U (2)L ×U (2)R and their energy does not depend on the spin orientation of quarks and their parities. We offer a possible interpretation of these energy levels as the quantum levels of the dynamical QCD string. The structure of the radial J = 1 spectrum is compatible with E = (nr + 1) ω with ω = 900 ± 70 MeV.

In a dynamical lattice simulation with the overlap Dirac operator and N f = 2 mass degenerate quarks we study all possible J = 0 and J = 1 correlators upon exclusion of the low lying "quasizero" modes from the valence quark propagators. After subtraction of a small amount of such Dirac eigenmodes all disconnected contributions vanish and all possible point-to-point J = 0 correlators with different quantum numbers become identical, signaling a restoration of the SU (2)L × SU (2)R × U (1)A. The original ground state of the π meson does not survive this truncation, however. In contrast, in the I = 0 and I = 1 channels for the J = 1 correlators the ground states have a very clean exponential decay. All possible chiral multiplets for the J = 1 mesons become degenerate, indicating a restoration of an SU (4) symmetry of the dynamical QCD-like string.

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Cosmic spatial curvature is a fundamental geometric quantity of the Universe. We investigate a model independent, geometric approach to measure spatial curvature directly from observations, without any derivatives of data. This employs strong lensing time delays and supernova distance measurements to measure the curvature itself, rather than just testing consistency with flatness. We define two curvature estimators, with differing error propagation characteristics, that can crosscheck each other, and also show how they can be used to map the curvature in redshift slices, to test constancy of curvature as required by the Robertson-Walker metric. Simulating realizations of redshift distributions and distance measurements of lenses and sources, we estimate uncertainties on the curvature enabled by next generation measurements. The results indicate that the model independent methods, using only geometry without assuming forms for the energy density constituents, can determine the curvature at the ∼ 6 × 10 −3 level.

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