Modern cosmology is now established on mature theoretical grounds and preci-sion observational probes.Our understanding of the Universe grows as we make new discoveries.On the one hand,some theoretical hypothesis are waiting to be confirmed,or falsified,by experiments:Inflation as a paradigm for the very-early Universe,pri-mordial black holes as candidate for dark matter,and much more.On the other hand,some observations still require a consensual explanation,such as the two dark clouds of dark energy and dark matter.These two hands form one path that one can explore from each sides.One way to test our theoretical assumptions is to design specific phe-nomenological predictions to be confronted to observations.Meanwhile,new physics beyond the existing theoretical framework are to be explored if we want to unravel the yet mysterious facets of the Universe.This thesis is at the intersection of these two directions.Either from the perspective of effective theories,or from the ’bottom-up’,we will scrutiny certain peculiar aspects of our Universe,building a bridge between observations and theory.This thesis consists of two axes.First,for the first-order scalar perturbation dur-ing inflation,a special parameterization cs2=1-2ξ[1-cos(2k*τ)]of its propagation speed is given.The propagation speed of this perturbation has a slight time-dependent oscillatory term.Through further analysis of the equation of motion,it is found that,under the de Sitter approximation,the equation is actually a Mathieu equation,that is,the equation of parametric resonance in mechanics.Under the governing of this equa-tion of motion,the perturbation mode function near the characteristic scale k.*will be exponentially amplified to close to unitary within the Hubble radius,and converted into curvature perturbation outside the Hubble radius.Such a sharp and narrow curva-ture perturbation can trigger the formation of the primordial black hole during radiation domination.We found that the mass spectrum of primordial black holes produced by the perturbations is also narrow and sharp,which can be limited by current observations,which in turn can limit the parameterization of sound speed resonance.Based on this model-independent sound speed resonance mechanism,we calculated the energy spec-trum of induced gravitational waves in Chapter 4.There is a nonlinear coupling term of scalar-scalar-tensor for second-order tensor perturbations and primordial scalar pertur-bations at second-order perturbations theories.Since the scalar perturbations near k*are exponentially amplified by the sound speed resonance mechanism during inflation,we find that both in the radiation domination and during inflation,an induced gravitational wave signal with a sufficiently high energy spectrum is generated.In addition,we have calculated the induced gravitational waves in radiation domination,inside the Hubble radius and outside the Hubble radius during inflation,and found that the spectrum dur-ing inflation is dominated by it produced inside the Hubble radius,and that produced outside the Hubble radius will be suppressed by slow-roll parameter.And the superim-posed energy spectrum can fall within the detection sensitivity range of LISA or PTA.By the signals of induced gravitational waves,we can indirectly constrain the formation of primordial black holes,and we can use multi-messenger astronomical observations that combine primordial black holes and induced gravitational waves to constrain and filter the sound speed resonance mechanism.Second,we also study the realization of an effective model with torsional gravity,and how it alleviates cosmological tensions.First,we constructed an effective action based on the effective field theory of teleparallel torsional gravity and selected a normal parameterization f(T)=-T-2∧/Mp2+αTβ and reconstructed it with observational data.Then it can alleviate H0 tension and σ8 tension simultaneously,hence,offers an additional argument in favor of gravitational modification.