The first step is that you can measure the spectrum of distant galaxies, a shift twards the red end (redshift) shows they are all getting further apart as expansion happens. On top of that you can measure how the redshift varies with time as light from more distant galaxies was emitted when the universe was younger. If the universe was going to contract eventually, expansion should at least be slowing down. It turns out according to the latest data that the opposite is true, expansion appears instead to be speeding up. Current theories suggest that this is due to the mysterious “dark energy” although it’s not yet clear what this, and its counterpart “dark matter”, actually are.
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