  ATLAS has measured two-particle correlations as a function of relative
azimuthal-angle, $\Delta \phi$, and pseudorapidity, $\Delta \eta$, in
$\sqrt{s}$=13 and 2.76 TeV $pp$ collisions at the LHC using charged particles
measured in the pseudorapidity interval $|\eta|$<2.5. The correlation functions
evaluated in different intervals of measured charged-particle multiplicity show
a multiplicity-dependent enhancement at $\Delta \phi \sim 0$ that extends over
a wide range of $\Delta\eta$, which has been referred to as the "ridge".
Per-trigger-particle yields, $Y(\Delta \phi)$, are measured over
2<$|\Delta\eta|$<5. For both collision energies, the $Y(\Delta \phi)$
distribution in all multiplicity intervals is found to be consistent with a
linear combination of the per-trigger-particle yields measured in collisions
with less than 20 reconstructed tracks, and a constant combinatoric
contribution modulated by $\cos{(2\Delta \phi)}$. The fitted Fourier
coefficient, $v_{2,2}$, exhibits factorization, suggesting that the ridge
results from per-event $\cos{(2\phi)}$ modulation of the single-particle
distribution with Fourier coefficients $v_2$. The $v_2$ values are presented as
a function of multiplicity and transverse momentum. They are found to be
approximately constant as a function of multiplicity and to have a
$p_{\mathrm{T}}$ dependence similar to that measured in $p$+Pb and Pb+Pb
collisions. The $v_2$ values in the 13 and 2.76 TeV data are consistent within
uncertainties. These results suggest that the ridge in $pp$ collisions arises
from the same or similar underlying physics as observed in $p$+Pb collisions,
and that the dynamics responsible for the ridge has no strong $\sqrt{s}$
dependence.
