Large-scale structure in $f(T)$ gravity

In this work we study the cosmology of the general $f(T)$ gravity theory. We express the modified Einstein equations using covariant quantities, and derive the gauge-invariant perturbation equations in covariant form. We consider a specific choice of $f(T)$, designed to explain the observed late-time accelerating cosmic expansion without including an exotic dark energy component. Our numerical solution shows that the extra degree of freedom of such $f(T)$ gravity models generally decays as one goes to smaller scales, and consequently its effects on scales such as galaxies and galaxies clusters are small. But on large scales, this degree of freedom can produce large deviations from the standard $\Lambda \mathrm{CDM}$ scenario, leading to severe constraints on the $f(T)$ gravity models as an explanation to the cosmic acceleration.

Figure 1

The background evolution for the $f(T)$ gravity model with $f(T)=T-{\mu }^{2(1+n)}/(-T{)}^n$. Upper-left Panel: the fractional energy densities for matter (${\Omega }_m$), radiation (${\Omega }_r$) and the effective dark energy (${\Omega }_{\mathrm{DE}}=1-{\Omega }_m-{\Omega }_r$), as functions of the cosmic scale factor $a$, which is normalized to 1 today. Upper-right Panel: the total effective equation of state $w_{\mathrm{eff}}=-1-\frac{2\dot{H}}{3H^2}$, as a function of $a$. Lower-left Panel: the ratio between the Hubble expansion rates for the $f(T)$ gravity model and for the $\Lambda \mathrm{CDM}$ paradigm, as a function of $a$. Lower-right Panel: $f_T-1$ as a function of $a$. Here, results are shown for $n=0$ (black solid curve), 0.1 (green dotted curve), $-0.1$ (cyan dashed curve), 0.2 (purple dash-dotted curve) and $-0.2$ (pink dash-triple-dotted curve). Note that $n=0$ corresponds to the $\Lambda \mathrm{CDM}$ paradigm. The relevant physical parameters are ${\Omega }_m=0.257$, ${\Omega }_r=8.0331\times {10}^{-5}$ and $H_0=71.9\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$.

Figure 2

The time evolution of frame-independent quantity $ϵ\equiv \ae +\sigma$, with the scale factor, $a(t)$, for the model with $f(T)=T-{\mu }^{2(1+n)}/(-T{)}^n$ and $n=0.1$, on different length scales, characterized, respectively, by $k/(h{\mathrm{Mpc}}^{-1})={10}^{-4}$ (solid curve), ${10}^{-3}$ (dotted curve), ${10}^{-2}$ (dashed curve) and ${10}^{-1}$ (dash-dotted curve).

Figure 3

The power spectra for the large-scale structure of the $f(T)$ gravity model with $f(T)=T-{\mu }^{2(1+n)}/(-T{)}^n$. Upper-left Panel: the CMB spectrum for different values of $n-0$ (black solid curve), 0.1 (green dotted curve), $-0.1$ (cyan dashed curve), 0.2 (purple dash-dotted curve) and $-0.2$ (pink dash-triple-dotted curve). Upper-right Panel: the same as the upper-left panel, but for the matter power spectrum at redshift 0 (today). Lower-left Panel: the late-time evolution of the dark matter density contrast ${\Delta }_{\mathrm{CDM}}$ on different scales (as indicated besides the curves); three values of $n$ have been considered $-n=0.0$ (solid curves), 0.1 (dotted curves) and $-0.1$ (dashed curves). Lower-right Panel: the same as the lower-left panel, but for the late-time evolution of the gravitational potential $\varphi$ on different scales. The physical parameters are the same as listed in the caption of Fig. 1, and three species of massless neutrinos are used.