Cosmological Challenges in Theories with Extra Dimensions and Remarks on the Horizon Problem

We consider the cosmology that results if our observable universe is a 3-brane in a higher dimensional universe. In particular, we focus on the case where our 3-brane is located at the $Z_2$ symmetry fixed plane of a $Z_2$ symmetric five-dimensional spacetime, as in the Ho\v{r}ava-Witten model compactified on a Calabi-Yau manifold. As our first result, we find that there can be substantial modifications to the standard Friedmann-Robertson-Walker (FRW) cosmology; as a consequence, a large class of such models is observationally inconsistent. In particular, any relationship between the Hubble constant and the energy density on our brane is possible, including (but not only) FRW. Generically, due to the existence of the bulk and the boundary conditions on the orbifold fixed plane, the relationship is not FRW, and hence cosmological constraints coming from big bang nucleosynthesis, structure formation, and the age of the universe difficult to satisfy. We do wish to point out, however, that some specific choices for the bulk stress-energy tensor components do reproduce normal FRW cosmology on our brane, and we have constructed an explicit example. As our second result, for a broad class of models, we find a somewhat surprising fact: the stabilization of the radius of the extra dimension and hence the four dimensional Planck mass requires unrealistic fine-tuning of the equation of state on our 3-brane. In the last third of the paper, we make remarks about causality and the horizon problem that apply to {\it any} theory in which the volume of the extra dimension determines the four-dimensional gravitational coupling. We point out that some of the assumptions that lead to the usual inflationary requirements are modified.Comment: 15 page REVTeX file; to appear in Phys. Rev. D; clarified the statement of being able to obtain any power dependence of the Hubble expansion rate on the energy density; added reference

The stability for the Cauchy problem for elliptic equations

We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations. We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions. As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality.Comment: 57 pages, review articl

Gravitational and Yang-Mills instantons in holographic RG flows

We study various holographic RG flow solutions involving warped asymptotically locally Euclidean (ALE) spaces of $A_{N-1}$ type. A two-dimensional RG flow from a UV (2,0) CFT to a (4,0) CFT in the IR is found in the context of (1,0) six dimensional supergravity, interpolating between $AdS_3\times S^3/\mathbb{Z}_N$ and $AdS_3\times S^3$ geometries. We also find solutions involving non trivial gauge fields in the form of SU(2) Yang-Mills instantons on ALE spaces. Both flows are of vev type, driven by a vacuum expectation value of a marginal operator. RG flows in four dimensional field theories are studied in the type IIB and type I$'$ context. In type IIB theory, the flow interpolates between $AdS_5\times S^5/\mathbb{Z}_N$ and $AdS_5\times S^5$ geometries. The field theory interpretation is that of an N=2 $SU(n)^N$ quiver gauge theory flowing to N=4 SU(n) gauge theory. In type I$'$ theory the solution describes an RG flow from N=2 quiver gauge theory with a product gauge group to N=2 gauge theory in the IR, with gauge group $USp(n)$. The corresponding geometries are $AdS_5\times S^5/(\mathbb{Z}_N\times \mathbb{Z}_2)$ and $AdS_5\times S^5/\mathbb{Z}_2$, respectively. We also explore more general RG flows, in which both the UV and IR CFTs are N=2 quiver gauge theories and the corresponding geometries are $AdS_5\times S^5/(\mathbb{Z}_N\times \mathbb{Z}_2)$ and $AdS_5\times S^5/(\mathbb{Z}_M\times \mathbb{Z}_2)$. Finally, we discuss the matching between the geometric and field theoretic pictures of the flows.Comment: 32 pages, 3 figures, typoe corrected and a reference adde

Broad-line region in NGC 4151 monitored by two decades of reverberation mapping campaigns. I. Evolution of structure and kinematics

We report the results of long-term reverberation mapping (RM) campaigns of the nearby active galactic nuclei (AGN) NGC 4151, spanning from 1994 to 2022, based on archived observations of the FAST Spectrograph Publicly Archived Programs and our new observations with the 2.3m telescope at the Wyoming Infrared Observatory. We reduce and calibrate all the spectra in a consistent way, and derive light curves of the broad H$\beta$ line and 5100\,{\AA} continuum. Continuum light curves are also constructed using public archival photometric data to increase sampling cadences. We subtract the host galaxy contamination using {\it HST} imaging to correct fluxes of the calibrated light curves. Utilizing the long-term archival photometric data, we complete the absolute flux-calibration of the AGN continuum. We find that the H$\beta$ time delays are correlated with the 5100\,{\AA} luminosities as $\tau_{\rm H\beta}\propto L_{5100}^{0.46\pm0.16}$. This is remarkably consistent with Bentz et al. (2013)'s global size-luminosity relationship of AGNs. Moreover, the data sets for five of the seasons allow us to obtain the velocity-resolved delays of the H$\beta$ line, showing diverse structures (outflows, inflows and disks). Combining our results with previous independent measurements, we find the measured dynamics of the H$\beta$ broad-line region (BLR) are possibly related to the long-term trend of the luminosity. There is also a possible additional $\sim$1.86 years time lag between the variation in BLR radius and luminosity. These results suggest that dynamical changes in the BLR may be driven by the effects of radiation pressure.Comment: Accepted for publication in MNRAS; comments welcome

Assessing the Effectiveness of Tradable Landuse Rights for Biodiversity Conservation: An Application to Canada's Boreal Mixedwood Forest

Ecological reserve networks are an important strategy for conserving biodiversity. One approach to selecting reserves is to use optimization algorithms that maximize an ecological objective function subject to a total reserve area constraint. Under this approach, economic factors such as potential land values and tenure arrangements are often ignored. Tradable landuse rights are proposed as an alternative economic mechanism for selecting reserves. Under this approach economic considerations determine the spatial distribution of development and reserves are allocated to sites with the lowest development value, minimizing the cost of the reserve network. The configuration of the reserve network as well as the biodiversity outcome is determined as a residual. However cost savings can be used to increase the total amount of area in reserve and improve biodiversity outcomes. The appropriateness of this approach for regional planning is discussed in light of key uncertainties associated with biodiversity protection. A comparison of biodiversity outcomes and costs under ecological versus economic approaches is undertaken for the Boreal Forest Natural Region of Alberta, Canada. We find a significant increase in total area protected and an increase in species representation under the TLR approach