  The conventional $\Lambda$CDM cosmological model supplemented by the
inflation concept describes the Universe very well. However, there are still a
few concerns: new Planck data impose constraints on the shape of the inflaton
potential, which exclude a lot of inflationary models; dark matter is not
detected directly, and dark energy is not understood theoretically on a
satisfactory level. In this brief sketch we investigate an alternative
cosmological model with spherical spatial geometry and an additional perfect
fluid with the constant parameter $\omega=-1/3$ in the linear equation of
state. It is demonstrated explicitly that in the framework of such a model it
is possible to satisfy the supernovae data at the same level of accuracy as
within the $\Lambda$CDM model and at the same time suppose that the observed
cosmic microwave background (CMB) radiation originates from a very limited
space region. This is ensured by introducing an additional condition of light
propagation between the antipodal points during the age of the Universe.
Consequently, the CMB uniformity can be explained without the inflation
scenario. The corresponding drawbacks of the model with respect to its
comparison with the CMB data are also discussed.
