Algebraic fiber spaces and curvature of higher direct images

In this article we are interested in the differential geometric properties of certain higher direct images of exterior powers of the sheaf of relative differentials twisted with a line bundle. We obtain explicit curvature formulas, especially in case where the said line bundle satisfies a natural curvature assumption. Several applications are obtained, including a proof of a result by Viehweg-Zuo in the context of a canonically polarized family of maximal variation.Comment: A few results and details adde

A measurement of the Hubble constant using galaxy redshift surveys

We perform a measurement of the Hubble constant, $H_0$, using the latest baryonic acoustic oscillations (BAO) measurements from galaxy surveys of 6dFGS, SDSS DR7 Main Galaxy Sample, BOSS DR12 sample, and eBOSS DR14 quasar sample, in the framework of a flat $\Lambda$CDM model. Based on the Kullback-Leibler (KL) divergence, we examine the consistency of $H_0$ values derived from various data sets. We find that our measurement is consistent with that derived from Planck and with the local measurement of $H_0$ using the Cepheids and type Ia supernovae. We perform forecasts on $H_0$ from future BAO measurements, and find that the uncertainty of $H_0$ determined by future BAO data alone, including complete eBOSS, DESI and Euclid-like, is comparable with that from local measurements.Comment: 5 pages, 2 figures, 2 tables; accepted for publication in Ap

Post-PlanckPlanck constraints on interacting vacuum energy

We present improved constraints on an interacting vacuum model using updated astronomical observations including the first data release from Planck. We consider a model with one dimensionless parameter, $\alpha$, describing the interaction between dark matter and vacuum energy (with fixed equation of state $w=-1$). The background dynamics correspond to a generalised Chaplygin gas cosmology, but the perturbations have a zero sound speed. The tension between the value of the Hubble constant, $H_0$, determined by Planck data plus WMAP polarisation (Planck+WP) and that determined by the Hubble Space Telescope (HST) can be alleviated by energy transfer from dark matter to vacuum ($\alpha>0$). A positive $\alpha$ increases the allowed values of $H_0$ due to parameter degeneracy within the model using only CMB data. Combining with additional datasets of including supernova type Ia (SN Ia) and baryon acoustic oscillation (BAO), we can significantly tighten the bounds on $\alpha$. Redshift-space distortions (RSD), which constrain the linear growth of structure, provide the tightest constraints on vacuum interaction when combined with Planck+WP, and prefer energy transfer from vacuum to dark matter ($\alpha<0$) which suppresses the growth of structure. Using the combined datasets of Planck+WP+Union2.1+BAO+RSD, we obtain the constraint on $\alpha$ to be $-0.083<\alpha<-0.006$ (95% C.L.), allowing low $H_0$ consistent with the measurement from 6dF Galaxy survey. This interacting vacuum model can alleviate the tension between RSD and Planck+WP in the $\Lambda$CDM model for $\alpha<0$, or between HST measurements of $H_0$ and Planck+WP for $\alpha>0$, but not both at the same time.Comment: 15 pages, 12 figures, 3 tables; accepted for publication in Phys. Rev.