Assuming that observers located inside the Universe measure a proper time
which is different from the time appearing in the Friedmann-Lemaˆıtre equation,
and determining this proper time such that the Universe always appears flat to
these observers, we derive a simple cosmological model which allows to explain
the velocity dispersions of galaxies in galaxy clusters without introducing dark
matter. It also explains the present acceleration of the expansion without any
resort to dark energy and provides a good fit to the observations of distant
supernovæ. Depending on the present value of the matter-energy density, we
calculate an age of the Universe between 15.4 and 16.5 billion years, significantly
larger than the 13.7 billion years of the standard CDM model. Our model
has a slower expansion rate in the early epochs, thus leaving more time for the
formation of structures such as stars and galaxies