  In the baseline design for advanced LIGO interferometers, the most serious
noise source is tiny, dynamically fluctuating bumps and valleys on the faces of
the arm-cavity mirrors, caused by random flow of heat in the mirrors' sapphire
substrates: so-called "thermoelastic noise". We propose replacing the
interferometers' baseline arm-cavity light beams, which have Gaussian-shaped
intensity profiles, by beams with mesa-shaped profiles that are flat in their
central ~7 cm of radius, and that then fall toward zero as quickly as is
allowed by diffraction in LIGO's 4 km arms. The mesa beams average the bumps
and valleys much more effectively than the Gaussian beams. As a result, if the
beam radii are adjusted so diffraction losses per bounce are about 10 ppm,
replacing Gaussian beams by mesa beams reduces the thermoelastic noise power by
about a factor 3. If other thermal noises are kept negligible, this will permit
advanced LIGO to beat the Standard Quantum Limit by about a factor 1.5 over a
bandwidth about equal to frequency, and the event rate for inspiraling neutron
star binaries will increase by about a factor 2.5. The desired mesa beams can
be produced from input, Gaussian-profile laser light, by changing the shapes of
the arm cavities' mirror faces from their baseline spherical shapes (with radii
of curvature of order 60 km) to Mexican-hat (MH) shapes that have a shallow
bump in the center but are otherwise much flatter in the central 10 cm than the
spherical mirrors, and then flare upward strongly in the outer 6 cm, like a
sombrero. We describe mesa beams and MH mirrors mathematically and we report
the results of extensive modeling calculations, which show that mesa-beam
interferometers are not much more sensitive than the Gaussian-beam
interferometers to errors in mirror figures, positions, and orientations.
