Modified gravity theories have received increased attention lately due to combined motivation coming from high-energy physics, cosmology, and astrophysics. Among numerous alternatives to Einstein's theory of gravity, theories that include higher-order curvature invariants, and specifically the particular class of $f(R)$ theories, have a long history. In the last five years there has been a new stimulus for their study, leading to a number of interesting results. Here $f(R)$ theories of gravity are reviewed in an attempt to comprehensively present their most important aspects and cover the largest possible portion of the relevant literature. All known formalisms are presented---metric, Palatini, and metric affine---and the following topics are discussed: motivation; actions, field equations, and theoretical aspects; equivalence with other theories; cosmological aspects and constraints; viability criteria; and astrophysical applications.

f ( R ) theories of gravity

Modified Gravity and Cosmology

The status of experimental tests of general relativity and of theoretical frameworks for analyzing them is reviewed and updated. Einstein’s equivalence principle (EEP) is well supported by experiments such as the Eotvos experiment, tests of local Lorentz invariance and clock experiments. Ongoing tests of EEP and of the inverse square law are searching for new interactions arising from unification or quantum gravity. Tests of general relativity at the post-Newtonian level have reached high precision, including the light deflection, the Shapiro time delay, the perihelion advance of Mercury, the Nordtvedt effect in lunar motion, and frame-dragging. Gravitational wave damping has been detected in an amount that agrees with general relativity to better than half a percent using the Hulse-Taylor binary pulsar, and a growing family of other binary pulsar systems is yielding new tests, especially of strong-field effects. Current and future tests of relativity will center on strong gravity and gravitational waves.

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The Confrontation Between General Relativity and Experiment

2. GRAVITY 7. MICROSCOPIC PHYSICS 13. TOPOLOGY OF DEFECTS 18. ANOMALOUS NON-CONSERVATION OF FERMIONIC CHARGE 22. EDGE STATES AND FERMION ZERO MODES ON SOLITON 26. LANDAU CRITICAL VELOCITY 29. CASIMIR EFFECT AND VACUUM ENERGY

The Universe in a Helium Droplet

Cosmology and the fate of dilatation symmetry

We conducted three torsion-balance experiments to test the gravitational inverse-square law at separations between 9.53 mm and $55\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$, probing distances less than the dark-energy length scale ${\ensuremath{\lambda}}_{d}=\sqrt[4]{\ensuremath{\hbar}c/{\ensuremath{\rho}}_{d}}\ensuremath{\approx}85\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$. We find with 95% confidence that the inverse-square law holds ($|\ensuremath{\alpha}|\ensuremath{\le}1$) down to a length scale $\ensuremath{\lambda}=56\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$ and that an extra dimension must have a size $R\ensuremath{\le}44\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$.

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Tests of the gravitational inverse-square law below the dark-energy length scale.

Motivated by higher-dimensional theories that predict new effects, we tested the gravitational 1/r(2) law at separations ranging down to 218 microm using a 10-fold symmetric torsion pendulum and a rotating 10-fold symmetric attractor. We improved previous short-range constraints by up to a factor of 1000 and find no deviations from Newtonian physics.

Submillimeter test of the gravitational inverse-square law: a search for "large" extra dimensions.

Motivated by a variety of theories that predict new effects, we tested the gravitational ${1/r}^{2}$ law at separations between 10.77 mm and $137\ensuremath{\mu}\mathrm{m}$ using two different 10-fold azimuthally symmetric torsion pendulums and rotating 10-fold symmetric attractors. Our work improves upon other experiments by up to a factor of about 100. We found no deviation from Newtonian physics at the 95% confidence level and interpret these results as constraints on extensions of the standard model that predict Yukawa or power-law forces. We set a constraint on the largest single extra dimension (assuming toroidal compactification and that one extra dimension is significantly larger than all the others) of ${R}_{*}l~160\ensuremath{\mu}\mathrm{m},$ and on two equal-sized large extra dimensions of ${R}_{*}l~130\ensuremath{\mu}\mathrm{m}.$ Yukawa interactions with $|\ensuremath{\alpha}|g~1$ are ruled out at 95% confidence for $\ensuremath{\lambda}g~197\ensuremath{\mu}\mathrm{m}.$ Extra-dimensions scenarios stabilized by radions are restricted to unification masses ${M}_{*}g~3.0\mathrm{TeV}{/c}^{2},$ regardless of the number of large extra dimensions. We also provide new constraints on power-law potentials $V(r)\ensuremath{\propto}{r}^{\ensuremath{-}k}$ with k between 2 and 5 and on the ${\ensuremath{\gamma}}_{5}$ couplings of pseudoscalars with $ml~10\mathrm{meV}{/c}^{2}.$

Sub-millimeter tests of the gravitational inverse-square law

We use a continuously rotating torsion balance to make new tests of the universality of free fall (UFF). We study differential accelerations of Be-Cu and Be-Al test-body pairs in the fields of Earth, the Sun, our Galaxy, and in the direction of the cosmic microwave dipole. We also compare the acceleration towards the Sun and our galactic center of Cu and single-crystal Si in an Al shell (this pair of bodies approximates the elemental compositions of Earth's core and the Moon or Earth's crust, respectively). In terms of the classic UFF parameter \ensuremath{\eta}, our Earth-source results are \ensuremath{\eta}(Be,Cu)=(-1.9\ifmmode\pm\else\textpm\fi{}2.5)\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}12}$ and \ensuremath{\eta}(Be,Al)=(-0.2\ifmmode\pm\else\textpm\fi{}2.8)\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}12}$ where all errors are 1\ensuremath{\sigma}. Thus our limit on UFF violation for Be and a composite Al/Cu body is \ensuremath{\eta}=(-1.1\ifmmode\pm\else\textpm\fi{}1.9)\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}12}$. Our solar-source results are \ensuremath{\Delta}a(Be,Cu) =(-3.0\ifmmode\pm\else\textpm\fi{}3.6)\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}12}$ cm/${\mathrm{s}}^{2}$, \ensuremath{\Delta}a(Be,Al)=(+2.4\ifmmode\pm\else\textpm\fi{}5.8)\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}12}$ cm/${\mathrm{s}}^{2}$, and \ensuremath{\Delta}a(Si/Al,Cu)=(+3.0 \ifmmode\pm\else\textpm\fi{}4.0)\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}12}$ cm/${\mathrm{s}}^{2}$. This latter result, when added to the lunar laser-ranging result that senses both composition-dependent forces and gravitational binding-energy anomalies, yields a nearly model-independent test of the UFF for gravitational binding energy at the 1% level.

New tests of the universality of free fall.

The lunar-ranging test of the equivalence principle for gravitational self-energy is ambiguous. Although the Earth has more gravitational self-energy than the Moon, its sizable Fe/Ni core also gives it a different composition than the Moon. We removed this ambiguity by comparing, in effect, the accelerations of {open_quotes}miniature{close_quotes} earths and moons toward the Sun. Our composition-dependent Earth-Moon acceleration, {Delta}a{sub CD}/a{sub s}= (+0.1{plus_minus}2.7{plus_minus}1.7){times}10{sup {minus}13} , and lunar-ranging data provide an unambiguous test at the 1.3{times}10{sup {minus}3} level. {copyright} {ital 1999} {ital The American Physical Society}

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H. E. Swanson

Papers

Tests of the gravitational inverse-square law below the dark-energy length scale.

Submillimeter test of the gravitational inverse-square law: a search for "large" extra dimensions.

Sub-millimeter tests of the gravitational inverse-square law

New tests of the universality of free fall.

Improved test of the equivalence principle for gravitational self-energy