Suspending test masses in terrestrial millihertz gravitational-wave detectors: a case study with a magnetic assisted torsion pendulum

Current terrestrial gravitational-wave detectors operate at frequencies above 10 Hz. There is strong astrophysical motivation to construct low-frequency gravitational-wave detectors capable of observing 10 mHz - 10Hz signals. While space-based detectors provide one means of achieving this end, one may also consider terretrial detectors. However, there are numerous technological challenges. In particular, it is difficult to isolate test masses so that they are both seismically isolated and freely falling under the influence of gravity at millihertz frequencies. We investigate the challenges of low-frequency suspension in a hypothetical terrestrial detector. As a case study, we consider a Magnetically Assisted Gravitational-wave Pendulum Intorsion (MAGPI) suspension design. We construct a noise budget to estimate some of the required specifications. In doing so, we identify what are likely to be a number of generic limiting noise sources for terrestrial millihertz gravitational-wave suspension systems (as well as some peculiar to the MAGPI design). We highlight significant experimental challenges in order to argue that the development of millihertz suspensions will be a daunting task. Any system that relies on magnets faces even greater challenges. Entirely mechanical designs such as Zollner pendulums may provide the best path forward.Comment: 6 pages, 4 figure

Local elastic response measured near the colloidal glass transition

We examine the response of a dense colloidal suspension to a local force applied by a small magnetic bead. For small forces, we find a linear relationship between the force and the displacement, suggesting the medium is elastic, even though our colloidal samples macroscopically behave as fluids. We interpret this as a measure of the strength of colloidal caging, reflecting the proximity of the samples' volume fractions to the colloidal glass transition. The strain field of the colloidal particles surrounding the magnetic probe appears similar to that of an isotropic homogeneous elastic medium. When the applied force is removed, the strain relaxes as a stretched exponential in time. We introduce a model that suggests this behavior is due to the diffusive relaxation of strain in the colloidal sample.Comment: long-delayed followup paper to Habdas et al., EPL 67, 477-483 (2004

When is price discrimination profitable?

We analyze a model of a quality-constrained monopolist's product line decision that encompasses a variety of important examples of second-degree price discrimination, including intertemporal price discrimination, coupons, advance purchase discounts, versioning of information goods, and damaged goods. We derive necessary and sufficient conditions for price discrimination to be profitable that generalize existing results in the literature. Specifically, we show that when a continuum of product qualities are feasible, price discrimination is profitable if and only if the ratio of the marginal social value from an increase in quality to the total social value of the good is increasing in consumers' willingness to pay. Unlike third-degree price discrimination, we show that second-degree price discrimination may result in a Pareto improvement. However, in general the welfare effects are ambiguous

How Sales Taxes Affect Customer and Firm Behavior: The Role of Search on the Internet

When a multichannel retailer opens its first retail store in a state, the firm is obligated to collect sales taxes on all Internet and catalog orders shipped to that state. This article assesses how opening a store affects Internet and catalog demand. The authors analyze purchase behavior among customers who live far from the retail store but must now pay sales taxes on catalog and Internet purchases. A comparable group of customers in a neighboring state serves as a control. The results show that Internet sales decrease significantly, but catalog sales are unaffected. Further investigation indicates that the difference in these outcomes is partly attributable to the ease with which customers can search for lower prices at competing retailers. The authors extend the analysis to a panel of multichannel firms and show that retailers that earn a large proportion of their revenue from direct channels avoid opening a first store in high-tax states. They conclude that current U.S. sales taxes laws have significant effects on both customer and firm behavior

Future singularities if the universe underwent Starobinsky inflation in the past

The effects which quantum fields and an $\alpha_0 R^2$ term in the gravitational Lagrangian have on future singularities are investigated. While all values of $\alpha_0$ are considered, an emphasis is placed on those values which are compatible with the universe having undergone Starobinsky inflation in the past. These are also values which lead to stable solutions to the semiclassical backreaction equations in the present universe. The dark energy is modeled as a perfect fluid, and the focus is on type I-IV singularities and little rips which result when the classical Einstein equations are solved with various types of dark energy as a source. First, evidence is provided that the energy densities of massive conformally coupled scalar fields approach that of the conformally invariant scalar field as a type III singularity is approached. Then, solutions to the semiclassical backreaction equations are investigated when conformally invariant fields and the $\alpha_0 R^2$ term in the gravitational Lagrangian are present. General proofs regarding the behaviors of the solutions are given. The proofs are illustrated by analytic and numerical calculations in specific cases.Comment: Two references and minor changes added to the PRD version, 45 pages, 5 figure

Semiclassical Gravity in the Far Field Limit of Stars, Black Holes, and Wormholes

Semiclassical gravity is investigated in a large class of asymptotically flat, static, spherically symmetric spacetimes including those containing static stars, black holes, and wormholes. Specifically the stress-energy tensors of massless free spin 0 and spin 1/2 fields are computed to leading order in the asymptotic regions of these spacetimes. This is done for spin 0 fields in Schwarzschild spacetime using a WKB approximation. It is done numerically for the spin 1/2 field in Schwarzschild, extreme Reissner-Nordstrom, and various wormhole spacetimes. And it is done by finding analytic solutions to the leading order mode equations in a large class of asymptotically flat static spherically symmetric spacetimes. Agreement is shown between these various computational methods. It is found that for all of the spacetimes considered, the energy density and pressure in the asymptotic region are proportional to 1/r^5 to leading order. Furthermore, for the spin 1/2 field and the conformally coupled scalar field, the stress-energy tensor depends only on the leading order geometry in the far field limit. This is also true for the minimally coupled scalar field for spacetimes containing either a static star or a black hole, but not for spacetimes containing a wormhole.Comment: 43 pages, 2 figures. Reference added, minor changes, PRD versio

Stress-Energy Tensor for the Massless Spin 1/2 Field in Static Black Hole Spacetimes

The stress-energy tensor for the massless spin 1/2 field is numerically computed outside and on the event horizons of both charged and uncharged static non-rotating black holes, corresponding to the Schwarzschild, Reissner-Nordstrom and extreme Reissner-Nordstr\"om solutions of Einstein's equations. The field is assumed to be in a thermal state at the black hole temperature. Comparison is made between the numerical results and previous analytic approximations for the stress-energy tensor in these spacetimes. For the Schwarzschild (charge zero) solution, it is shown that the stress-energy differs even in sign from the analytic approximation. For the Reissner-Nordstrom and extreme Reissner-Nordstrom solutions, divergences predicted by the analytic approximations are shown not to exist.Comment: 5 pages, 4 figures, additional discussio