We investigate the cosmological perturbations in f(T) gravity. Examining the pure gravitational perturbations in the scalar sector using a diagonal vierbein, we extract the corresponding dispersion relation, which provides a constraint on the f(T) Ans\"atze that lead to a theory free of instabilities. Additionally, upon inclusion of the matter perturbations, we derive the fully perturbed equations of motion, and we study the growth of matter overdensities. We show that f(T) gravity with f(T) constant coincides with General Relativity, both at the background as well as at the first-order perturbation level. Applying our formalism to the power-law model we find that on large subhorizon scales (O(100\,\,Mpc) or larger), the evolution of matter overdensity will differ from $\Lambda${}CDM cosmology. Finally, examining the linear perturbations of the vector and tensor sectors, we find that (for the standard choice of vierbein) f(T) gravity is free of massive gravitons.

We examine dark-energy models in which a quintessence or a phantom field, $\varphi$, rolls near the vicinity of a local minimum or maximum, respectively, of its potential $V(\varphi )$. Under the approximation that $(1/V)(dV/d\varphi )\ll 1$, [although $(1/V)(d^{2}V/d{\varphi }^2)$ can be large], we derive a general expression for the equation-of-state parameter $w$ as a function of the scale factor for these models. The dynamics of the field depends on the value of $(1/V)(d^{2}V/d{\varphi }^2)$ near the extremum, which describes the potential curvature. For quintessence models, when $(1/V)(d^{2}V/d{\varphi }^2)<3/4$ at the potential minimum, the equation-of-state parameter $w(a)$ evolves monotonically, while for $(1/V)(d^{2}V/d{\varphi }^2)>3/4$, $w(a)$ has oscillatory behavior. For phantom fields, the dividing line between these two types of behavior is at $(1/V)(d^{2}V/d{\varphi }^2)=-3/4$. Our analytical expressions agree within 1% with the exact (numerically derived) behavior, for all of the particular cases examined, for both quintessence and phantom fields. We present observational constraints on these models.

We reexamine the evolution of thermal relic particle abundances for the case where the interaction rate depends on the particle velocities. For the case of Sommerfeld enhancement, we show that the standard analytic approximation, modified in a straightforward way, provides an estimate of the relic particle abundance that is accurate to within 10% (in comparison to less than 1% error for the non-Sommerfeld-enhanced case). We examine the effect of kinetic decoupling on relic particle abundances when the interaction rate depends on the velocity. For the case of pure p-wave annihilation, the effect of kinetic decoupling is an increase in the relic abundance, but the effect is negligible when the kinetic decoupling temperature is much less than the chemical decoupling temperature. For the case of Sommerfeld-enhanced s-wave annihilations, after kinetic decoupling occurs, annihilations continue to change the particle abundance down to arbitrarily low temperatures, until either matter domination begins or the Sommerfeld effect cuts off. We derive analytic approximations to give the final relic particle abundances for both of these cases.

We derive slow-roll conditions for thawing k-essence with a separable Lagrangian $p(X,\varphi )=F(X)V(\varphi )$. We examine the evolution of the equation of state parameter, $w$, as a function of the scale factor $a$, for the case where $w$ is close to $-1$. We find two distinct cases, corresponding to $X\approx 0$ and $F_X\approx 0$, respectively. For the case where $X\approx 0$ the evolution of $\varphi$ and hence $w$ is described by only two parameters, and $w(a)$ is model independent and coincides with similar behavior seen in thawing quintessence models. This result also extends to nonseparable Lagrangians where $X\approx 0$. For the case $F_X\approx 0$, an expression is derived for $w(a)$, but this expression depends on the potential $V(\varphi )$, so there is no model-independent limiting behavior. For the $X\approx 0$ case, we derive observational constraints on the two parameters of the model, $w_0$ (the present-day value of $w$), and the $K$, which parametrizes the curvature of the potential. We find that the observations sharply constrain $w_0$ to be close to $-1$, but provide very poor constraints on $K$.

We examine dark energy models in which a phantom field ϕ is rolling near a local minimum of its potential $V(\varphi )$. We require that $(1/V)(dV/d\varphi )\ll 1$, but $(1/V)(d^{2}V/d{\varphi }^2)$ can be large. Using techniques developed in the context of hilltop quintessence, we derive a general expression for w as a function of the scale factor, and as in the hilltop case, we find that the dynamics of the field depend on the value of $(1/V)(d^{2}V/d{\varphi }^2)$ near the minimum. Our general result gives a value for w that is within 1% of the true (numerically-derived) value for all of the particular cases examined. Our expression for $w(a)$ reduces to the previously-derived phantom slow-roll result of Sen and Scherrer in the limit where the potential is flat, $(1/V)(dV/d\varphi )\ll 1$.

Oscillating scalar fields, with an oscillation frequency much greater than the expansion rate, have been proposed as models for dark energy. We examine these models, with particular emphasis on the evolution of the ratio of the oscillation frequency to the expansion rate. We show that this ratio always increases with time if the dark energy density declines less rapidly than the background energy density. This allows us to classify oscillating dark energy models in terms of the epoch at which the oscillation frequency exceeds the expansion rate, which is effectively the time at which rapid oscillations begin. There are three basic types of behavior: early oscillation models, in which oscillations begin during the matter-dominated era; late oscillation models, in which oscillations begin after scalar-field domination; and nonoscillating models. We examine a representative set of models (those with power-law potentials) and determine the parameter range giving acceptable agreement with the supernova observations. We show that a subset of all three classes of models can be consistent with the observational data.