14.2 Modified gravity theories

Motivation and Varieties of Modified Gravity Theories

Modified gravity theories attempt to explain the universe's accelerated expansion and large-scale structure without relying on dark energy or dark matter. Instead of adding mysterious new substances to the cosmic inventory, these theories modify the gravitational equations themselves, changing how gravity behaves on the largest scales.

Motivation for Modified Gravity Theories

The standard ΛCDM model fits observational data remarkably well, but it carries unresolved baggage. The cosmological constant problem asks why the observed value of dark energy is roughly 120 orders of magnitude smaller than quantum field theory predicts. Meanwhile, decades of experiments have failed to directly detect dark matter particles. These gaps motivate a different approach: what if the issue isn't missing matter or energy, but an incomplete theory of gravity?

Modified gravity theories tackle this by altering the Einstein-Hilbert action of general relativity, introducing additional fields, or embedding spacetime in higher dimensions. The goals include:

• Explaining accelerated cosmic expansion without a cosmological constant
• Reproducing the gravitational effects currently attributed to dark matter (flat galaxy rotation curves, gravitational lensing patterns)
• Potentially unifying gravity with the other fundamental forces (electromagnetism, strong nuclear force, weak nuclear force)

Comparison of Modified Gravity Models

Three major families of modified gravity stand out, each with a distinct strategy:

f(R) Gravity replaces the Ricci scalar $R$ in the Einstein-Hilbert action with a general function $f(R)$. By choosing the right form of this function, the theory can mimic dark energy and drive accelerated expansion. The trade-off is that the resulting field equations contain higher-order derivatives, which can introduce Ostrogradsky instabilities that make certain solutions physically unreliable.

Scalar-Tensor Theories add a scalar field that couples directly to the metric tensor. Brans-Dicke theory is the classic example, while the broader Horndeski framework represents the most general scalar-tensor theory with second-order field equations (avoiding those dangerous higher-order instabilities). These theories can reproduce effects similar to both dark energy and dark matter, depending on how the scalar field evolves.

The DGP (Dvali-Gabadadze-Porrati) Model takes a geometric approach, proposing that our 4D spacetime sits on a brane embedded in a 5D bulk. Gravity "leaks" into the extra dimension at very large distances, weakening its pull and producing accelerated expansion. However, the self-accelerating branch of this model suffers from ghost instabilities, where certain field modes carry negative kinetic energy, signaling a fundamental theoretical problem.

Observational Consequences and Evaluation of Modified Gravity Theories

Observational Tests of Modified Gravity

Modified gravity theories earn their keep by making predictions that differ from general relativity on testable scales. The key observational signatures fall into two categories.

Cosmological-scale tests probe how gravity shapes the universe's largest structures:

• Growth rate of structure: Modified gravity changes how fast galaxy clusters and cosmic filaments form. Measuring redshift-space distortions in galaxy surveys reveals this growth rate and can flag deviations from ΛCDM predictions.
• Weak gravitational lensing: The bending of light by large-scale matter distributions depends on the theory of gravity. Modified models predict subtle differences in lensing patterns that surveys like Euclid and the Vera Rubin Observatory are designed to detect.
• CMB power spectrum: Alterations to gravity affect how photons travel through evolving gravitational potentials (the integrated Sachs-Wolfe effect), leaving imprints on the CMB at large angular scales.
• Gravitational slip parameter: In general relativity, the two metric potentials governing space curvature and time dilation are equal. Many modified gravity theories break this equality, and measuring the "slip" between them provides a direct test.

Smaller-scale tests check whether modifications to gravity conflict with well-established physics:

• Deviations in planetary and lunar orbital motion within the solar system
• Anomalous precession of orbiting bodies beyond what general relativity predicts
• Differences in gravitational wave signals from compact binary mergers (neutron star collisions, black hole mergers), where the speed and polarization of gravitational waves can reveal extra degrees of freedom

Strengths vs. Weaknesses in Cosmology

Strengths:

• Provide a path to accelerated expansion without invoking an unexplained cosmological constant
• Some frameworks unify the roles of dark matter and dark energy into a single gravitational modification
• May offer a more natural resolution to the cosmological constant problem by removing the need to fine-tune vacuum energy

Weaknesses:

• Many models harbor instabilities (ghost modes, gradient instabilities) that undermine their physical viability
• Additional free parameters reduce predictive power and make it easier to fit data without genuinely explaining it
• Solar system tests and laboratory gravity experiments place stringent constraints that many models struggle to satisfy simultaneously with their cosmological predictions

Current observational precision cannot yet definitively distinguish most modified gravity theories from ΛCDM. Upcoming surveys and gravitational wave observatories will tighten these constraints considerably, but for now, modified gravity remains a collection of promising alternatives rather than a confirmed replacement for the standard model.