  The main aim of this paper is to provide cosmological constraints on the
Multi Scalar Field Dark Matter model (MSFDM), in which we assume the dark
matter is made up of different ultra-light scalar fields. As a first
approximation, we consider they are real and do not interact with each other.
We study the equations for both the background and perturbations for $N$-fields
and present the evolution of the density parameters, the mass power spectrum
and the CMB spectrum. In particular, we focus on two scalar fields with several
combinations for the potentials $V(\phi) = 1/2 m_{\phi}^2 \phi^2$, $V(\phi) =
m_{\phi}^2f^2\left[1+\cos(\phi/f)\right]$ and $V(\phi) =
m_{\phi}^2f^2\left[\cosh(\phi/f)-1\right]$, however the work, along with the
code, could be easily extended to more fields. We use the data from BAO, Big
Bang Nucleosynthesis, Lyman-$\alpha$ forest and Supernovae to find constraints
on the sampling parameters for the cases of a single field and double field,
along with the Bayesian evidence. We found that some combinations of the
potentials get penalized through the evidence, however for others there is a
preference as good as for the cold dark matter.
