Cosmological redshift doesn't take expansion into account.
The Big Bang Theory assumes that redshift can only be caused by relative velocity, and since all photons coming from a distance are redshifted, the Universe must be expanding. Some of those photons are showing a redshift that indicates an age of 13 billion years, or only 700 million years after the Big Bang (i.e., when the Universe was 5% of its present age). But there is a disconnect in that reasoning. Photons generated when the Universe was only 5% of its present age did not have 13 billion light years to traverse in order to get to us, because at the time, the Universe was only 5% of its present volume. So the photons had 700 million light years to traverse at most. That light should have arrived a long time ago, and it should be showing the redshift of something only 700 million light years away.
To hold onto their interpretation of redshift within the Big Bang framework, cosmologists have concluded that space itself must be expanding. That way, photons generated when the Universe was only 5% of its present age could take 13 billion years to get to us, because they had the same amount of distance to traverse 13 billion years ago. In other words, the yard stick was shorter back then, but the speed of light, when measured in yards, was the same.
But this begs the question of how the Universe can be said to be expanding. The Big Bang Theory, plus the metric expansion of space, equals the Steady State Theory. It cannot be said that the Universe was more dense when it was younger, because density is mass over volume, and with a shorter yard stick, the volume works out to be the same number as it is now. And without a change in density, there was no Big Bang.
Analogously, suppose we were to cut football fields in half, making them only 50 yards from goal to goal. That would change the game of football. But not if we redefined a yard as being 18 inches instead of 36, and not if the game could only be played by midgets — a shorter field, using a shorter yard, and with shorter players, amounts to the same game, just at a different scale, while we'd never know the difference if we were all midgets as well.
All in all, Big Bang Theory, like a lot of the elements of modern astronomy, is just sophistry — it's a fancy way of saying something that just isn't correct, and the twist is just in the words, not in the physics.

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