143. The structure of correlation functions in single-field inflation

This is a guest post by Sarah Shandera from the University of Columbia. Dmitry.

Dmitry has asked me to write a post about my recent paper, arXiv:0812.0818, about correlation functions in single field inflation. The motivation behind this work is the potential of very near-future data (from the Planck satellite and large scale structure surveys) to more significantly test the Gaussianity of primordial fluctuations. In standard single-field slow-roll models, the flatness (and smoothness) of the potential guarantees that the primordial fluctuations are Gaussian to a part in . Current observational bounds only require the primordial fluctuations to be Gaussian to a part in or – still pretty Gaussian, but leaving a surprising amount of room for some interesting physics. Measurements of non-Gaussianity would be a very useful tool for understanding the fundamental picture of inflation, because they probe interactions of the inflaton, and because observations can distinguish between wide classes of models. There are many different statistics that can be used to test Gaussianity. Clearly the -point functions are one class of possibilities, but there are also measures like Minkowski functionals or galaxy cluster number counts that depend on an integrated contribution from a series of correlation functions. Often, this series can be simply truncated to depend on a few lower-order moments (for example, when the non-Gaussianity comes from the non-linear gravitational evolution), and this truncation looks suggestively like a perturbative series for the fundamental interactions.

In single-field models with a standard kinetic term, the small amplitude of the primordial fluctuations combined with the flatness of the potential seems to be enough to trust the expansion in moments (and at any rate we are a long way from being able to test non-Gaussianity at the slow-roll level). But, the story is still interesting because the leading order answer for the curvature correlations depends on not just the scalar field interactions but on the metric perturbations. The result for the three-point function was given by Acquaviva et al (astro-ph/0209156) and Maldacena (astro-ph/0210603), and the four-point was most exhaustively calculated by Arroja and Koyama (arXiv:0802.1167) following on results by Jarnhus and Sloth (arXiv:0709.2708). Beyond the linear order, there is not a clean separation between scalar and tensor fluctuations, so although it seems reasonable that higher order correlation functions are slow-roll suppressed by increasing powers of the slow-roll parameter , it isn’t entirely clear (at least to me) what the exact pattern to all orders is.

However, an interesting case where the curvature correlations are simply related (at first order) to the fundamental interactions is single-field inflation with higher order derivative terms. One reason to consider these terms is that a very flat potential can be explained if the scalar inflaton field has an approximate shift symmetry, so that its interactions are invariant under , where is a constant. The interactions that preserve this symmetry exactly are derivative interactions. In addition, derivative interactions can change the dynamics of the field in interesting ways – the original motivation for DBI inflation (named for the Dirac-Born-Infeld action for a D-brane) was that the kinetic structure alleviated the need for a very flat potential by enforcing a “speed limit” (see Silverstein and Tong, hep-th/0310221). The strength of derivative interactions for models with only single derivative terms (k-inflation) is effectively captured by a single parameter, the sound speed , which is one for a standard kinetic term and decreases toward 0 as the derivative interactions become more important. Roughly when , the derivative interactions give the dominant contribution to the non-Gaussianity. Interestingly, increasing powers of this small number appear in the denominator of higher order correlation functions. So, we might wonder: under what conditions is the expansion in moments sensible?

To answer this question, it is first important to notice that derivative interactions can come with a new scale (in addition to and the Hubble scale, ). Dimensionally, if we define , we will have terms like . If is very close to , we will always have since the equation of state for inflation requires , , and , that allows the curvature correlation functions from the most general single field model to be under control, and including only first derivative terms should usually capture the most important effects. This agrees, of course, with other ways of bounding the sound speed (as in Cheung et al., arXiv:0709.0293), and should allow us to correctly interpret the constraints on single field models in the perturbative regime from all types of future observations.