A Class Of Scalar-tensor Theory And Its Applications In Cosmological Inflation

General relativity（GR）has been validated by lots of experiments.However,it suffers from problems such as quantization,dark matters and dark energy,etc.A simple way to mod-ify GR is to introduce an extra scalar field to propagate the gravitational interaction,i.e,the scalar-tensor theory of gravity.Horndeski theory is the most general high-order derivative4D scalar-tensor theory,which predicts the second order field equation.This thesis focuses on the extension of Horndeski theory and the applications of Horndeski theory in cosmolog-ical inflation.We study the extension of Horndeski theory from aspect of spatially covariant grav-ity（SCG）.To break spacetime diffeomorphism,the Lagrangian could nonlinearly depend on lapse function.This type of SCG includes Horndeski theory.A way to extend Horn-deski theory is to make lapse function dynamic.In the framework of SCG with dynamic lapse function,we study the minimally modified gravity（MMG）.We derive sufficient and necessary conditions for eliminating the scalar physical Do F.In order to compare with GR,we then construct a class of SCG quadratic actions with only two Do F and dynamic lapse function.In the case that the coupling functions depend on N only,we find that the spatial curvature term cannot enter the Lagrangian and thus this theory possesses no wave solution and cannot recover GR.In the case that the coupling functions depend on the spatial deriva-tives of N,we perform a spatially conformal transformation on a class of quadratic actions with nondynamical lapse function to obtain a class of quadratic actions with˙N.We confirm this theory has two Do Fs by confirming the two sufficient and necessary conditions.With the quadratic action in hand,we then study the correspondence between GR and MMG and find that a class of SCG quadratic action with two Do Fs can be transformed from GR by disformal transformation.An application of Horndeski theory is in cosmological inflation.We study the applica-tions of Horndeski theory on supressing tensor-to-scalar ratio at large scales and enhancing the curvature perturbation at small scales.In order to obtain small tensor-to-scalar ratio,Horndeski theory provides us the frame-work by introducing the nonminimal coupling,derivative coupling,non-canonical kinetic term or canonical inflation with a flat potential.We study two classes of inflation models,helical phase canonical inflation and tachyon inflation,which could predict small tensor-to-scalar ratio.We study the attractor of helical phase inflation and then explore more generic potentials which predictα-attractor by means of potential reconstruction in the framework of tachyon inflation.Choosing helical phase potential as canonical inflationary potential,we study the ob-servation constriants on the model in GR and braneworld inflation and find theαattractor for both GR case and braneworld case.In particular,due to the high energy correction,theαattractor for braneworld case is reached more easily than GR case.Besides,we investi-gate the reheating phase where the inflaton decays into Higgsino and find that the reheating temperature for braneworld case is always lower than GR case due to the high energy cor-rection to the amplitude of the curvature perturbation in braneworld case.In order to find a class of more generic potential that predict the attractor behavior,we then study the potential reconstruction of tachyon inflation.By parameterizing n_s,r respectively and reconstructing the potential,we find that the power-law potential could also predict the attractor behavior.Besides,we study the reheating phase of tachyon inflation.Assuming that the equation of state parameter w_reduring the reheating phase is a constant,we find that for three models,as n_sbecomes larger,the allowed reheating epoch becomes shorter for w_re=-1/3,0 and1/6 while the allowed reheating epoch becomes longer for w_re=2/3.CMB has no obsevation constraints on small scales.If there are large curvature per-turbations at small scales,then primordial black holes（PBHs）and secondary gravitational waves（SIGWs）could be induced at radiation-domination era.We combine G₂,G₃term of Horndeski theory to propose a novel mechanism—K/G inflation,which could enhance curvature perturbation at small scales.Choosing the coupling function as a peak function,the power spectrum is enhanced to the order O(10^-2),which in turn accounts for PBHs and SIGWs.However,the peak function will contribute up to about 20 e-folds so that the potential is restrcited.To solve this problem,we then conbime K/G mechanism with the nonminal coupling and now the Higgs-field-driving inflation is compatible with CMB ob-servation while simultaneously enhancing the curvature perturbations to the order O(10^-2)at small scales.Besides,we introduce double-peaked K/G inflation to simultaneously ex-plain PBHs with different mass ranges.PBHs from K/G inflation can explain LIGO-Virgo events and dark matters.Milihertz SIGWs from K/G inflation can be tested by future space-based detectors like LISA,Tai Ji,and Tian Qin.Besides,models with broad peak in the power spectrum can explain the NANOGrav signal.Due to the violation of the slow-roll condition,the non-Gaussianity may have a significant effect on PBH abundance and SIGWs.For K/G inflation,we find that the non-Gaussianity correction has a significant enhancement on PBH abundance while SIGWs remains the same as the power spectrum receives very tiny correc-tions.