It is usually believed that “observable primordial gravitational waves prove inflation”, which I consider as a misconception. In fact, observing primordial gravitational waves (PGW) is just a one-sided test between the Ekpyrotic scenario and inflation.

A few clarifications

Before explanations, let me first clarify a few things (that you may like to skip):

  • This article is not attacking PGW. The current experimental efforts on searching for PGW is absolutely valuable. Seeing PGW tells us a lot about inflation (or alternative-to-inflation scenarios). In fact I was extremely excited in the BICEP2 days.
  • This article is not attacking inflation. I think inflation is the leading paradigm and most likely to become real.
  • This article is not attacking alternative-to-inflation. I indeed think all current alternative-to-inflation scenarios have their difficulties. But the study and improvements are still valuable.
  • This article is not a philosophical argument about a physical theory cannot be “proven”. By “proof” I just mean a crucial experiment, after which the current alternatives to inflation are considered ruled out without unnatural modifications, and it becomes much harder to come up with new alternatives.
  • Just to mention another related common misconception, PGW is not the only prediction of inflation which has not yet been verified. Primordial non-Gaussianity of slow roll inflation is the other not-yet-verified prediction of inflation. (Non-Gaussianities of slow roll inflation is hard to see. But PGW could also be hard.)
  • I indeed have a personal motivation to write this article: instead, I consider the quantum primordial standard clock to be more reliable to distinguish inflation between alternatives. But I try my best to be unbiased and you are encouraged to tell from the text below.
  • Some of the arguments of this article comes from paper 1, paper 2, paper 3 by Xingang Chen, Mohammad Hossein Namjoo and me. I also thank Xingang Chen for discussions while writing this article.

Assumptions of the well-known formula

Why PGW was considered as a proof of inflation? It is because of the below power spectrum formula for PGW: $$ P_\mathrm{PGW} = \frac{2H^2}{\pi^2 M_p^2}~. $$ So it seems that, once PGW is seen over a range of scales, the Hubble parameter is thus seen over this range of scales, specified by the above formula. So we observe the scale factor of the universe and prove inflation.

However, note that this formula is derived assuming inflation. In alternative-to-inflation, some assumptions are false and then the formula is different. The two outstanding assumptions are:

  1. Constant mode dominates. As a dynamical field, PGW obeys a second order differential equation and has two modes. The above formula assumes that, on super-Hubble scales, the constant mode dominates. This is true for an expanding universe: the other mode is decaying. But to tell between inflation and alternative, one cannot assume an expanding universe. In fact, in the matter bounce scenario, the growing mode of PGW gives a nearly scale-invariant spectrum. Naturally the size is $r\sim 1$. But one can use the multi-field trick to suppress $r$ to a consistent level.
  2. Vacuum initial condition. This is natural for inflation, but may not for all alternatives. For example, in string gas cosmology, stringy holographic thermal initial condition is considered and as a result, nearly scale invariant PGW arises with possibly observable tensor-to-scalar ratio $r$. Indeed, inflation tends to have red tilt of PGW spectrum (with exceptions such as Galileons), string gas tends to have blue tilt. But seeing the tilt is a different challenge and is two orders of magnitude harder than seeing PGW.

These two assumptions can be thought about in a more abstract way. Assumption 1 indicates that, with all other assumptions except expansion (which cannot be assumed when the evolution history of the universe is what to be tested), PGW does not prove inflation. Assumption 2 indicates that, with an almost flat power spectrum, PGW provides too little information and thus is easy to be obtained once additional assumptions are applied (such like the non-vacuum initial condition which is the defining feature of string gas cosmology. The Ekpyrotic scenario will also be commented below for a similar reason).

Conclusions

So observing PGW cannot distinguish inflation between string gas scenario, matter bounce, or any other alternative-to-inflation which does not share the same assumptions of the above formula and predicts an observable and nearly scale invariant spectrum of PGW.

Instead, observing PGW only tells between inflation and the Ekpyrotic scenario. (This is because the Ekpyrotic scenario predicts vanishingly small PGW at cosmological scales. Indeed, if one patches the Ekpyrotic scenario, scale invariant PGW is also possible. But smart people can always patch anything…) Even between these two, PGW is not a “fair” test because observing PGW “falsifies” Ekpyrotic scenario, but not observing PGW does not falsify inflation in any sense.

The situation between inflation and Ekpyrotic scenario lets me recall a bet between me and my classmate C when we were undergraduates. The bet is: If C get the Nobel Prize in the future, he buys me a dinner. If C does not get the Nobel Prize, I buy him a dinner. The truth is, I will never buy him a dinner because he will never be sure not to get the Nobel Prize throughout his life.

Of course, PGW is still an important prediction of inflation (and it tells the energy scale of inflation, and much more). But seeing a prediction does not prove a theory (otherwise inflation has been proven in 1990s by the primordial density fluctuations). Instead, observing a clear advantage compared to alternatives is more important for proving inflation.