Galaxies are gradually being extinguished. Most star formation activity occurs near the start of a galaxy's life, or in response to merger activity with other galaxies.
The star formation rate of the universe peaked at redshifts of around 3, corresponding to a look-back time of around 9 billion years. Since then the star formation rate has declined as the universe expands; mergers are less frequent, gas is driven out of galaxies by supernovae and active galactic nuclei.
However, most of the stars that have been formed are of lower mass (K- and M-dwarfs) than the Sun and will live on for tens or hundreds of billions of years. They are however much fainter than the Sun. So although high-mass, luminous O- and B stars live their short lives and are not replaced at the same rate, the low-mass stars continue to shine. This means that galaxies will get fainter on average as the remaining stellar populations increase in average age and decrease in average mass.
It is a slow process though. High mass stars are still being formed in our galaxy after 12 billion years, and most spiral galaxies have ongoing star formation. However, star formation has more-or-less ceased in gas-poor elliptical galaxies.
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