  We analyze the two-body momentum correlation function for a uniform weakly
interacting one-dimensional Bose gas. We show that the strong positive
correlation between opposite momenta, expected in a Bose-Einstein condensate
with a true long-range order, almost vanishes in a phase-fluctuating
quasicondensate where the long-range order is destroyed. Using the Luttinger
liquid approach, we derive an analytic expression for the momentum correlation
function in the quasicondensate regime, showing (i) the reduction and
broadening of the opposite-momentum correlations (compared to the singular
behavior in a true condensate) and (ii) an emergence of anticorrelations at
small momenta. We also numerically investigate the momentum correlations in the
crossover between the quasicondensate and the ideal Bose-gas regimes using a
classical field approach and show how the anticorrelations gradually disappear
in the ideal-gas limit.
