Blueshifts in the Lema\^{\i}tre -- Tolman models

In the Lema\^{\i}tre -- Tolman (L--T) models that have nonconstant bang-time function $t_B(r)$, light emitted close to those points of the Big Bang where $\dril {t_B} r \neq 0$ is blueshifted at the receiver position. The blueshifted rays are expected to perturb the temperature of the cosmic microwave background (CMB) radiation along the lines of sight of the present central observer. It is shown here that, in a general L--T model, the $t_B(r)$ can be chosen so that the blueshift-generating region is hidden before the recombination time, where the L--T model does not apply. The rest of the paper is devoted to investigating blueshifts in one specific L--T model, called L--T$(t_B)$ -- the one that duplicates the luminosity distance vs. redshift relation of the $\Lambda$CDM model using nonconstant $t_B(r)$ alone. The location of the blueshift-generating region in the L--T$(t_B)$ spacetime is determined. Profiles of redshift/blueshift along several rays intersecting the past light cone of the present central observer are calculated. The L--T$(t_B)$ model matched to Friedmann is considered, and profiles of redshift/blueshift in such a composite model are calculated. The requirement of invisibility of blueshifts makes the L--T$(t_B)$ model astrophysically unacceptable if it should apply back to the recombination time, but does not "rule out" a general L--T model -- it only puts limits on $\dril {t_B} r$.