# Temperature of the CMB when the Earth formed and the faint young Sun paradox

The cosmic microwave background (CMB) has a modern temperature of about 2.7 K. At the time of the origin of the CMB, about 13.6 billion years ago, it had a temperature of about 3000 K.

There is a well-known problem that the Sun should have been too faint at the time the Earth formed, about 4.6 billion years ago, to allow the Earth to avoid becoming a permanent Snowball Earth. This problem is called the faint young Sun paradox.

I was thinking about that, and it occurred to me that those two things may not be completely unrelated.

What was the temperature of the CMB at the time the Earth formed, and what effect would that temperature have on the equilibrium temperature of the Earth?

It seems to me that there should be two effects in play.

1. The equations describing the equilibrium temperature of the Sun then needs to be adjusted for the fact that rather than being a blackbody radiating to a near zero background, it would be radiating to a background with a significant temperature of its own. I would think that would raise the required temperature for the Sun to achieve radiative equilibrium there-by making the Sun brighter.

2. The equilibrium temperature of the Earth should also be boosted by the increased temperature of the CMB for similar reasons of having a less efficient heat 'sink'.

Could those two effects together have shifted the equilibrium temperature of the Earth enough to solve the faint young Sun paradox?

Edit: Having done some more googling on it, the CMB would only have been about a degree or so warmer than it is now because 4.6 billion years only corresponds to a Z of about 0.4. This isn't anywhere near enough to solve the faint young Sun paradox. I'm going to leave this question in place just so if someone else has the same idea this will explain why it can't solve the paradox.

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