We revisit the possibility and detectability of a stochastic gravitational wave background (SGWB) produced by a cosmological population of newborn neutron stars (NSs) with r-mode instabilities. We show that the resultant SGWB is insensitive to the ch
oice of CSFR models, but depends strongly on the evolving behavior of CSFR at low redshifts. Our results show that the dimensionless energy density $Omega_{rm{GW}}$ could have a peak amplitude of $simeq (1-3.5) times10^{-8}$ in the frequency range $(200-1000)$~Hz. However, such a high mode amplitude is unrealistic as it is known that the maximum value is much smaller and at most $10^{-2}$. A realistic estimate of $Omega_{rm{GW}}$ should be at least 4 orders of magnitude lower ($sim 10^{-12}$), which leads to a pessimistic outlook for the detection of r-mode background. We consider different pairs of terrestrial interferometers (IFOs) and compare two approaches to combine multiple IFOs in order to evaluate the detectability of this GW background. Constraints on the total emitted GW energy associated with this mechanism to produce a detectable stochastic background are $sim 10^{-3} M_{odot} c^2$ for two co-located advanced LIGO detectors, and $2 times 10^{-5} M_{odot} c^2$ for two Einstein Telescopes. These constraints may also be applicable to alternative GW emission mechanisms related to oscillations or instabilities in NSs depending on the frequency band where most GWs are emitted.
