Inflationary Perturbation Spectra At Next-to-leading Slow-roll Order In Effective Field Theory Of Inflation

Inflation stands as the leading paradigm of modern cosmology.In order to solve several conceptual problems,such as the flatness,horizon and monopoles problems of standard Big-Bang cosmology,we advance this theory.As the same time,it also predicts the nearly scale-invariant primordial.These cosmological predictions of the inflation have been tested by lots of high-precision cosmological observations.There are many models describing the inflation,according to the latest observational data of cosmic microwave background by Plank group,it supports the standard single-field inflationary models.The EFT provides a general framework for describing the most generic single scalar field theory.In general,the inflation occurred at the early universe with a very high energy standard,and it was very sensitive to the new physics under this condition.In the framework of the effective field theory,we can understand the possible effects of different new physical mechanisms in the early universe by studying the effects of various higher-order terms that dominate the high energy period of the inflation.On the other hand,with the development of science and observation methods,we get more accurate observations about the power spectra,spectral indices,and tensor-to-scalar ratio.In the era of precise cosmology,in order to detect different inflationary models by using accurate observations,it also requires that we can calculate the various theoretical predictions of inflationary models with high accuracy.There are a lot of methods can achieve this goal,like WKB approximation,improved-WKB approximation,Green’s function method,Bessel function approximation and the phase integral method.In this paper,we use the uniform asymptotic approximation method to obtain the most accurate analytical results on various observational predictions of the EFT of inflation.We organize the rest of the paper as follows.In Sec.1,we briefly introduce the inflation,the effective field theory of inflation,and several methods to calculate the primordial perturbation power spectra in the current literature.In Sec.2,we present the action of the EFT of inflation,the motion equations of the cosmological scalar and tensor perturbations,etc.and introduce four new slow-roll parameters.In Sec.3,we introduced some general formulas form the uniform asymptotic approximation method and its application.In Sec.4,we calculate the power spectra,spectral indices,and running of the spectral indices of both scalar mode and tensor mode,which is the most accurate calculation at present,and expressions at the next-to-leading order,new slow-roll parameters can affect the primordial perturbation spectra at the first-order and the corresponding spectral indices at the second-order.At the same time,we also compare our results with other calculations in the literature.In Sec.5,we present our conclusion and discussions.