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We present an analysis of an $f(T, \mathcal{T})$ extension of theTeleparallel Equivalent of General Relativity, where $T$ denotes the torsionand $\mathcal{T}$ the trace of the energy-momentum tensor. This extensionincludes non--minimal couplings between torsion and matter. In particular, weconstruct two specific models that recover the usual continuity equation,namely, $f(T, \mathcal{T})=T+g(\mathcal{T})$ and $f(T, \mathcal{T})=T\timesg(\mathcal{T})$. We then constrain the parameters of each model by fitting thepredicted distance modulus to that measured from type Ia supernovae, and findthat both models can reproduce the late--time cosmic acceleration. We alsoobserve that one of the models satisfies well the observational constraints andyields a goodness--of--fit similar to the $\Lambda$CDM model, thusdemonstrating that $f(T,\mathcal{T})$ gravity theory encompasses viable modelsthat can be an alternative to $\Lambda$CDM.

Constraining $f(T, \mathcal{T})$ gravity models using type Ia supernovae