# Tests of inflation theory

So I'm following a course in Early Universe Cosmology right now. For the next lecture we have to solve a few questions about inflation, just as preparation. I read some closely related lecture notes from different sources but did not find much concerning these questions:

1. Explain how CMB observations can be used to test specific inflationary models.
2. Given a slow-roll potential V (φ), how does one relate it to cosmological parameters, and how can these parameters be measured?
3. What is the current observational status of the simple inflationary model with potential V (φ) ∼ $$φ^2$$?

For the first question I found that we can measure tensor perturbations in the CMB, who are closely related on what energy scale inflation took place (So we can distinguish between different inflation theories at different energy scales). For the rest I just find a lot of info about how CMB can tell us the values of different parameters of the $$\Lambda CMB$$ but not specifically inflation. I don't expect a complete answer on all these three questions, but a push in the right direction, or some ideas/examples would be welcome. Or if you really insist, a complete answer would also be welcome! Thanks in advance.

• I've added the homework-and-exercises tag. In the future, please use this tag on this type of question. – Ben Crowell Apr 28 at 21:52
• I believe they measure the CMB then work backwards. And I believe inflation solved the particle horizon problem - also known as the flatness problem. Also the CMB would have extended mere 38 centimeters instead being present on a cosmological scale. – Cinaed Simson Apr 29 at 7:10