Stability of the flat FLRW metric in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity

In this paper, we investigate the stability of the flat Friedmann-Lemaitre-Robertson-Walker metric in $f(T)$ gravity. This is achieved by analyzing the small perturbations, $\ensuremath{\delta}$ about the Hubble parameter and the matter energy density, ${\ensuremath{\delta}}_{\mathrm{m}}$. We find that $\ensuremath{\delta}\ensuremath{\propto}\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{H}/H$ and ${\ensuremath{\delta}}_{\mathrm{m}}\ensuremath{\propto}H$. Since the Hubble parameter depends on the function $f(T)$, two models were considered (A) the power-law model $f(T)=\ensuremath{\alpha}(\ensuremath{-}T{)}^{n}$, and (B) the exponential model $f(T)=\ensuremath{\alpha}{T}_{0}(1\ensuremath{-}\mathrm{exp}[\ensuremath{-}p\sqrt{\frac{T}{{T}_{0}}}])$, where the parameters $n$ and $p$ were chosen to give comparable physical results. For the parameters considered, it was found that the solutions are stable with vanishing $\ensuremath{\delta}$ and decaying then constant ${\ensuremath{\delta}}_{\mathrm{m}}$, meaning that the matter perturbations persist during late times.

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