| At small scales,primordial perturbations could be large enough to induce the second-order GWs（scalar-induced GWs,SIGWs）.Such GWs are likely to be observed by space-borne GW observatories like Tian Qin,Taiji and LISA,which could give a constraint on small-scale perturbations.Therefore,theoretical studies on SIGWs provide guidance on fu-ture observations of GWs.Cosmological perturbations are usually conserved outside the horizon and start their evolution after horizon reentry,which is also true for SIGWs.In some literature,SIGWs are assumed to be generated well outside the horizon to simplify calculation.However,this assumption brings deviation from the actual situation.So we study this deviation.We com-pared the results in several typical spectra,and we derived the semi-analytic expression for SIGWs generated after horizon reentry as a by-product.Significant deviation occurs only in the low-frequency band of SIGWs induced by monochromatic spectra and the Lognormal spectra with small variance.Normally,the deviation can be ignored,and we can use the assumption to simplify calculation.Observables are gauge invariants.But SIGWs are gauge dependent,and this is referred to as the gauge issue of SIGWs.To study this issue,we derived the gauge transformation of second-order tensor perturbation induced by scalar perturbations,as well as its compat-ible equation of motion.Based on this,we gave the solutions of scalar-induced 2nd-order tensor perturbation in six gauges,including synchronous gauge.We also discussed how to eliminate the effect from redundant gauge modes.Even when the effect from gauge modes is eliminated,SIGWs are still gauge dependent.In fact,GWs propagate with a speed of light in general relativity,and it is interesting to notice that the terms in the solutions of scalar-induced2nd-order tensor perturbation that propagate with the speed of light are gauge independent.We point out that only they are genuine SIGWs that are observable.Our proposal provides a novel solution to the gauge issue.Besides SIGWs,primordial black holes（PBHs）can originate from large perturbations as well.Those PBHs can be used to explain dark matters.On the other hand,PBHs could be the source of some GW events,which attracts many interests.So far,there is no observational constraint on small-scale perturbations,but the constraint on PBHs abundance can lay a limit on primordial spectrum,which can be used to further constrain SIGWs.We parameterized the constraints from PBHs on primordial spectrum with a broken power law.This spectrum gives an upper bound on the energy density of SIGWs,which locates above the sensitivities of space-borne GW observatories.So the future GW observations could test the primordial perturbations at small scales from inflations.Meanwhile,the observed SIGWs and PBHs can provide strong evidence of the mechanism of PBH formation.The observations of cosmic microwave background constrain the large-scale spectrum to be O(10-9).To produce a significant amount of PBHs and SIGWs,the small-scale spec-trum should reach O(10-2).However,it is hard to realize the enhancement of primordial spectrum in inflation models.We proposed a novel mechanism that uses a non-canonical kinetic term with a coupling function（1+G（?））to enhance primordial spectrum.If there is a peak at small scales in coupling function G（?）,then the spectrum will be enhanced ac-cordingly.At large scales,G（?）1 and inflation meets the constraint from CMB.This mechanism overcomes the fine-tuning problem of inflection-point inflations,as well as the strong requirements on the choice of inflaton potential.Our models predict PBHs that make up whole dark matter,and the induced GWs can be verified by future observations,so as to verify the inflation model.Moreover,our models predict non-trivial non-Gaussianity with f NL O（10）at ultra-slow-roll phases.Taking non-Gaussianity into account,there are much more PBHs to form than we estimate with Gaussian perturbation.This fact may relax the requirement on perturbations. |
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