  We consider an alternative cold dark matter candidate, ultralight bosons
($m>10^{-22}$eV) described by a complex scalar field (SFDM) with global U(1)
symmetry, with comoving particle number density conserved after particle
production during standard reheating. We allow for repulsive self-interaction.
In a Lambda-SFDM universe, SFDM starts relativistic, evolving from stiff (w=1)
to radiation-like (w=1/3), becoming nonrelativistic (w=0) at late times. Thus,
a stiff-SFDM-dominated era precedes the familiar radiation-dominated era. SFDM
particle mass $m$ and quartic self-interaction strength \lambda, are therefore
constrained by cosmological observables, N_{eff}, the effective number of
neutrino species during BBN, and z_{eq}, the matter-radiation equality
redshift. Since the stochastic gravitational wave background (SGWB) from
inflation is amplified during the stiff-SFDM-dominated era, it can also
contribute a radiationlike component large enough to affect these observables.
Remarkably, this amplification makes this SGWB detectable by current GW
experiments, e.g., aLIGO/Virgo and LISA, for Lambda-SFDM models satisfying
cosmological constraints, for a range of reheat temperatures T_{re} and
currently allowed values of tensor-to-scalar ratio $r$. For given r and
$\lambda/(mc^2)^2$, the marginally-allowed Lambda-SFDM model for each T_{re}
has the smallest m that satisfies cosmological constraints. For example, for
marginally-allowed models with r=0.01 and
$\lambda/(mc^2)^2=10^{-18}$eV$^{-1}$cm$^3$, null detection by the aLIGO O1 run
excludes 8.75*10^3<T_{re} (GeV)<1.7*10^5 at 95% confidence, demonstrating that
GW experiments already place a new kind of cosmological constraint on SFDM. A
wider parameter range should be accessible to aLIGO/Virgo O5, with potential to
detect this signature of Lambda-SFDM. For this same illustrative family,
3-sigma detection is predicted for 600<T_{re} (GeV)<10^7.
