Advanced LIGO's present baseline design uses arm cavities with Gaussian light beams supported by spherical mirrors. Because Gaussian beams have large intensity gradients in regions of high intensity, they average somewhat poorly over fluctuating bumps and valleys on the mirror surfaces (thermal noise). Flattopped light beams (mesa beams) are being considered as an alternative because they average over thermal noise more effectively. However, the proposed mesa beams are supported by nearly-flat mirrors, which experience a very serious tilt instability. In this paper we propose an alternative configuration in which mesa-shaped beams are supported by nearly-concentric spheres, which experience only a weak tilt instability. The tilt instability is analyzed for these mirrors in a companion paper by Savov and Vyatchanin. We also propose a one-parameter family of light beams and mirrors in which, as the parameter varies continuously from 0 to , the beams and supporting mirrors get deformed continuously from the nearly-flat-mirrored mesa configuration (FM) at 0, to the nearly-concentricmirrored mesa configuration (CM) at . The FM and CM configurations at the endpoints are close to optically unstable, and as moves away from 0 or , the optical stability improves.

SUMMARY The geoid is the true physical figure of the Earth, a particular equipotential surface of the Earth's gravity field that accounts for the effect of all subsurface density variations. Its shape approximates best (in the sense of least squares) the mean level of oceans, but the geoid is more difficult to determine over continents. Satellite missions carry out distance measurements and derive the gravity field to provide geoid maps over the entire globe. However, they require calibration and extensive computations including integration, which is a non‐unique operation. Here we propose a direct method and a new tool that directly measures geopotential differences on continents using atomic clocks. General relativity theory predicts constant clock rate at sea level, and faster (slower) clock rate above (below) sea level. The technology of atomic clocks is on the doorstep of reaching an accuracy level in clock rate (frequency ratio inaccuracy of 10−18), which is equivalent to 1 cm in determining equipotential surface (including geoid) height. We discuss the value and future applicability of such measurements including direct geoid mapping on continents, and joint gravity–geopotential surveying to invert for subsurface density anomalies. Our synthetic calculations show that the geoid perturbation caused by a 1.5 km radius sphere with 20 per cent density anomaly buried at 2 km depth in the Earth's crust is already detectable by atomic clocks of achievable accuracy. Therefore atomic clock geopotential surveys, used together with relative gravity data to benefit from their different depth sensitivities, can become a useful tool in mapping density anomalies within the Earth.

We discuss the thermal evolution and Bose-Einstein condensation of ultra-light dark matter particles at finite, realistic cosmological temperatures. We find that if these particles decouple from regular matter before Standard model particles annihilate, their temperature will be about 0.9 K. This temperature is substantially lower than the temperature of CMB neutrinos and thus Big Bang Nucleosynthesis remains unaffected. In addition the temperature is consistent with WMAP 7-year+BAO+H0 observations without fine-tuning. We focus on particles of mass of m ∼ 10 −23 eV, which have Compton wavelengths of galactic scales. Agglomerations of these particles can form stable halos and naturally prohibit small scale structure. They avoid over-abundance of dwarf galaxies and may be favored by observations of dark matter distributions. We present numerical as well as approximate analytical solutions of the Friedmann-Klein-Gordon equations and study the cosmological evolution of this scalar field dark matter from the early universe to the era of matter domination. Today, the particles in the ground state mimic presureless matter, while the excited state particles are radiation like.

According to general relativity, a clock experiencing a shift in the gravitational potential ∆U will measure a frequency change given by ∆f /f ≈ ∆U/c 2 . The best clocks are optical clocks. After about 7 hours of integration they reach stabilities of ∆f /f ∼ 10 −18 , and can be used to detect changes in the gravitational potential that correspond to vertical displacements of the centimetre level. At this level of performance, ground-based atomic clock networks emerge as a tool that is complementary to existing technology for monitoring a wide range of geophysical processes by directly measuring changes in the gravitational potential. Vertical changes of the clock's position due to magmatic, post-seismic or tidal deformations can result in measurable variations in the clock tick rate. We illustrate the geopotential change arising due to an inflating magma chamber using the Mogi model, and apply it to the Etna volcano. Its effect on an observer on the Earth's surface can be divided into two different terms: one purely due to uplift (free-air gradient) and one due to the redistribution of matter. Thus, with the centimetre-level precision of current clocks it is already possible to monitor volcanoes. The matter redistribution term is estimated to be 3 orders of magnitude smaller than the uplift term. Additionally, clocks can be compared over distances of thousands of kilometres over short periods of time, which improves our ability to monitor periodic effects with long-wavelength like the solid Earth tide.