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Primordial density perturbations in the radiation-dominated era of the earlyUniverse are expected to generate stochastic gravitational waves (GWs) due tononlinear mode coupling. In this \emph{Letter}, we report on a search for sucha stochastic GW background in the data of the two LIGO detectors during theirsecond observing run (O2). We focus on the primordial perturbations in therange of comoving wavenumbers $10^{16}-10^{18}~{\rm Mpc}^{-1}$ for which thestochastic background falls within the detectors' sensitivity band. We do notfind any conclusive evidence of this stochastic signal in the data, and thusplace the very first GW-based constraints on the amplitude of the powerspectrum at these scales. We assume a lognormal shape for the power spectrumand Gaussian statistics for the primordial perturbations, and vary the width ofthe power spectrum to cover both narrow and broad spectra. Derived upper limits($95\%$) on the amplitude of the power spectrum are $0.01-0.1$. As a byproduct,we are able to infer upper limits on the fraction of the Universe's mass inultralight primordial black holes ($M_\mathrm{PBH} \simeq10^{-20}-10^{-19}M_{\odot}$) at their formation time to be $\lesssim 10^{-25}$.

Search for the stochastic gravitational-wave background induced by primordial curvature perturbations in LIGO's second observing run