Several Applications Of Scalar Field In Gravitational Theories

Gravitational theories with a scalar field include quintessence,k-essence,Brans-Dicke theory and so on,and the most general scalar-tensor theory with second-order equation of motion in 4 dimensional spacetime is Horndeski theory.As a gravitational theory,we will discuss the role of the newly introduced scalar field.It can not only act as the inflaton driving inflation,but also describe the behavior of dark matter and dark energy,and even change the properties of our black holes by introducing scalar hair.The nontrivial coupling between scalar field and gravity will also bring interesting effects on the primordial perturbations of the universe.This thesis studies the dark matter problem and several related problems of the early universe through the gravitational theories with a scalar field.The formation mechanism of primordial black holes as a candidate of dark matter has been widely studied.How to form enough primordial black holes to act as dark matter and how to form primordial black holes with different mass ranges are two important problems.We answer this question through the inflation with multi-bumpy potentials in the canonical scalar field(quintessence).It shows that the inflation with multi-bumpy potential can explain the primordial black hole in the mass ranges of several tons,asteroid mass,planetary mass.At the same time,the gravitational wave signals induced by our model can explain the NANOGrav results.However,our numerical results show that our model is not enough to prove that all dark matter is composed of primordial black holes in the early universe.In another way,we further study a subclass of Horndeski theory possessing shift-symmetry,so as to explain the dark-matter-like behavior in late universe and explain the dark matter halo distribution accounting for galaxy rotation curve.Finally we can well understand the interpretation of dark matter in the gravitational theories with a scalar field.The gravitational theories with a scalar field can be used to study the problems of the early universe,which mainly focuses on dealing with the primordial perturbations in the early universe.The important observable variables we investigate are the scalar power spectrum Ps and its spectral index ns,tensor power spectrum PT and its spectral index nT。First,we try to study the duality of different evolution scenarios of the early universe,which reflects the degeneracy of the evolution models of the early universe.This cosmic duality reflects that different cosmic evolution scenarios can lead to the same observational results,in which the physical quantities investigated are PS、nS、PT、nT.Here we use the theory with a scalar field called k-essence,which can give the varying sound speed of scalar perturbations.Our results show that constant sound speed can only give the duality relationship about the scalar mode,but not the tensor one.Therefore,through the varying sound speed in k-essence theory,we can obtain the duality relationship about PS、nS and uT,but it can not include the tensor power spectrum PT.Therefore,cosmic duality is violated by tensor power spectrum,that is,tensor-to-scalar ratio r=PT/PS.The early universe models can be distinguished by future detection of primordial gravitational waves.Second,we study the nontrivial coupling between gravity and scalar field.We investigate two different couplings:nonminimal coupling and disformal coupling.We find that nonminimal coupling can greatly alleviate the tight bound by " Trans-Planckian Censorship Conjecture"(TCC)on the tensor-to-scalar ratio r,so as to meet the requirement of future detection sensitivity of O(10-3).The key is that we propose a conformal invariant TCC formula,so as to adjust r through the coupling coefficient f.For disformal coupling,we consider a single parameter M affecting ns-r,we find that it has different degrees of influence on different inflation models.In particular,it can make the monodromy inflation model closer to the central region of the allowed observational constraints,so as to improve the matching degree between this model and the observations.