  The hypothesis of the self-induced collapse of the inflaton wave function was
introduced as a candidate for the physical process responsible for the
emergence of inhomogeneity and anisotropy at all scales. In particular, we
consider different proposal for the precise form of the dynamics of the
inflaton wave function: i) the GRW-type collapse schemes proposals based on
spontaneous individual collapses which generate non-vanishing expectation
values of various physical quantities taken as ansatz modifications of the
standard inflationary scenario; ii) the proposal based on a Continuous
Spontaneous Localization (CSL) type modification of the Schr\"odinger evolution
of the inflaton wave function, based on a natural choice of collapse operator.
We perform a systematic analysis within the semi-classical gravity
approximation, of the standing of those models considering a full quasi-de
sitter expansion scenario. We note that the predictions for the Cosmic
Microwave Background (CMB) temperature and polarization spectrum differ
slightly from those of the standard cosmological model. We also analyse these
proposals with a Bayesian model comparison using recent CMB and Baryonic
Acoustic Oscillations (BAO) data. Our results show a moderate preference of the
joint CMB and BAO data for one of the studied collapse schemes model over the
$\Lambda$CDM one, while there is no preference when only CMB data are
considered. Additionally, analysis using CMB data provide the same Bayesian
evidence for both the CSL and standard models, i.e. the data have not
preference between the simplicity of the LCDM model and the complexity of the
collapse scenario.
