Inflation is no doubt a leading theory of the very early Universe. It provides a satisfactory explanation to such diverse observations as that of flatness of the Universe or a scale-invariance of the density fluctuations. Its predictions have been found consistent with a range of observables probing the Universe from the largest, e.g., Cosmic Microwave Background anisotropies, to the very small ones, say, Lyman-alpha forest. However, when looked upon in more detail, inflation is not what one might have wished for, i.e., a consistent well-defined model, but rather a bundle of diverse, specific implementation of the inflation's generic ideas. This has led some to refer to inflation more as a paradigm, i.e., a framework within which to think of the early Universe, rather than a scientific model.
Paradigms can be changed, in an instance of Kuhnian paradigm shift, but can not be easily falsified, as some tweaking, think of Ptolemaic epicycles, is usually available on hand to help to address new observational constraints, or even if not what other choice we have than to stick to it ?! From this perspective what is left for most of us is to look for a successful, most likely tentative, but necessarily inflationary model, which while minimal in the 'Ockham's razor' sense, fulfills all current, observational constraints and produces meaningful and testable predictions. If any of those are later found to be false, the specific model is patched up within the limits of the general paradigm, and the game - a.k.a, 'the standard science' - starts over. It does so at least as long as somebody does not come up with a better or simpler way of looking at the Universe - think Keplerian ellipses - what can lead to a paradigm change.
However all the inflationary models unavoidably share some generic features. They are all part of the same paradigm after all ! It is therefore invariably tempting to try to test those directly and thus, in a dream straight from the Wild West, try to "shoot'em all" with one shot, i.e., with help of a single, well-designed observational test. This is how, in an essence, a goal of this recent paper could be construed, though an outcome of which, turns out at the end to be quite more modest. As at least I would argue for.
The paper singles out the presence of a rapidly accelerating expansion phase in the early Universe followed by its deceleration as a 'smoking gun' of inflation as such and discuss a method for its observational verification. The period of acceleration is the inflationary epoch itself, while the deceleration marks the fact that it ended. To test this sequence of events the authors propose a simple relation, which they nickname a closure relation. It relates a number of so-called e folds, i.e., a number of times the Universe's size has increased by a factor e=2.71..., computed during and after the inflation, essentially tracking the changes in a physical size of some selected co-moving length size from some specific moment during inflation (so called horizon exit) and the present day. Simple as the relation is, not all the quantities appearing in it are directly measurable. In fact what one needs to know is the Hubble parameter, i.e., Universe's expansion rate, at the time of the inflation. Sounds like a lot to ask for. In fact, turning somewhat the tables around, if we knew the Hubble parameter history over the time, there would have been no question if inflation really took place. We would have just known it ! We can of course try to reconstruct the missing piece from present day observations. However to do so we need usually to restrict ourselves to some specific model of inflation and thus losing all the generality we have been after. So a vicious circle ?!
For some it could be, however the authors dare to differ. What they propose is to get around the issue by representing the Hubble constant via a Taylor series (in powers of the inflation-driving field value), terms of which are straightforwardly related to some observables. Of higher an order a term is, more observations needs to be done to pin it down. Thus the derived relation is essentially a constraint on an infinite sum of an infinite number of observables (or some functions thereof), which uniquely, or nearly so, determines its value. The fact that may seem rather daunting indeed.
Not so for the authors ! Rather than to deal with a single test, which however comes with infinitely many terms, and observables, they propose a sequence of tests, each assuming a truncated, finite number of terms. The authors' strategy is to start with the lowest order term and then progressively include more terms, one-by-one. These are calculated using the observational constraints, until, hopefully, a truncated closure relation of some order is satisfied. In general there is no apriori reason why that would happen, but if it does, inflation is, the authors posit, the correct paradigm. This could be made even more striking if, as the authors propose, one then uses the closure relation, now successfully fulfilled at some order, to make predictions about values of the subsequent observables corresponding to the next order. If the predictions are confirmed it will be even more difficult to argue against inflation. So we seem to have a way out of the problem. As it may look, or at least feel, a bit like a Baron Munchausen feat of dragging himself out of the swap, the authors propose to call the approach the bootstrap tests.
However, as the authors indeed point out, within the inflationary paradigm the fact that the closure relation can be satisfied when using one of the truncated series expansions, does tell us something more about inflation than what they defined at the onset of the paper as the inflation "smoking gun". In fact, barring any fancy fine-tunings, this seems to point towards some simplest models of inflation at the peril of some more complex ones, such as, e.g., hybrid or string-theory motivated proposals. In fact if any of the latter is indeed a correct model, the proposed sequence of tests would never successfully converged.
The question seems therefore be now what we are really testing: the paradigm, or maybe one family of the inflationary models - the "simple" ones - as opposed to another one more complex ?
Considered as a test of the paradigm the proposed approach does not look fully satisfactory at least on a methodological level. This is because it can only confirm it, while have no power to reject, from the very start posing no threat to inflation. Its failure still may mean that some "fancy" inflation may be plausible. Moreover, the closure relation in its original unabridged form is a smoking gun, or even say a definition of inflation. However, is that so for a truncated one ? In fact we do not have any immediate answer to the last question, even if non-inflation giving predictions so closed to inflation may look to us as a rather weird a possibility. Therefore, for time being, the only statement we can make is that if the test works, then we have confirmed some class of specific models within the inflationary paradigm.
Leaving this kind of discussions on a side, and putting our pragmatic hat on, we can follow the authors and ask how the test fares giving the data as they are at this time.
One should first note that the observables needed to calculate the subsequent expansion terms are indeed part of the standard cosmological package - tensor-to-scalar ratio, power spectrum tilt, running of the power spectrum - plus their extensions - "running of the running" etc - and at least two observables, tensor-to-scalar ration, r, and power spectrum index, ns, which are necessary and sufficient to calculate the first two lowest order terms, are already constrained by some data and indeed seem to fulfill the 'closure constraint' reasonable well. Given that the authors make a rather specific prediction concerning the running of the spectral index. This is constrained much more poorly than the other two at this time so only time will show how good the prediction is.
And the last but not least ... it seems worth pointing out that whatever its implementation the CMB B-mode experiments will feature prominently in it as they provide the best and most direct way to constraint the tensor-to-scalar ratio, which is indeed the first observable of the considered expansions.
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