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Well, it's not like people pulled it out of thin air. Both inflation and the lambda-CDM models are solutions to the GR equations, so in that sense it's perfectly justifiable to see if general relativity can explain the data. I don't think it's fair to say that it "makes no logical sense".

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I assume that OP was talking about the cosmic inflation theory that claims there was a rapid expansion immediately after the big bang. I don't see how that's a solution to the GR equation, could you maybe explain/give a link?

There is an addition of a inflation field. Observations show that such an inflation happened, and there's several ideas of inflation fields that are compatible with the standard model and match observations. The equations that make the observed expansion happen in the presence of such a field the friedmann equations, which are GR solutions.

The field isn't just something that magically expands things, it just effects energy density. The rapid expansion is then a consequence of GR.

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