Will cosmological gravitational waves be weaker or stronger than astrophysical ones? Will gravitational waves of cosmological origin be weaker or stronger (higher amplitude $h \simeq\Delta L/L$) than those created from astrophysical sources?
I'm having a real hard time finding the amplitude of cosmological gravitational waves (from inflation and from cosmic defects (strings etc.)) in terms of $h$, so that I can make a comparison.
Any help on this would be great.
 A: You probably should look here first for some relatively up-to-date predictions of signal-to-noise from cosmological sources.
Astrophysicists are very confident that advanced LIGO will see signals from merging compact objects like neutron star binaries and black hole binaries, on the order of a few tens per year (see first link). It is far less certain whether gravitational waves from the very early universe will be detected by LIGO, eLISA, or even the Big Bang Observer if it ever gets built. The answer you get will be different depending on who you ask and how optimistic they are.
Regarding signals from the end of inflation, the frequency at which the signal peaks  depends on, among other parameters, an unknown inflationary energy scale. I'm afraid I can't provide you with the expected signal in terms of $\Delta L / L$, but you can see some predictions for this sort of signal plotted against the LIGO, LISA, and BBO noise curves in this paper, and how they vary with the energy scale. The chances of detection don't seem very high, and require the inflationary energy scale to be quite low, $< 10^9$ GeV, to be in a favorable frequency range. (Take these results with a grain of salt, though - they are based on highly simplified models and should not be viewed as definitive).
A: there is chance to detect the waves
Observable Spectra of Induced Gravitational Waves from inflation
https://arxiv.org/abs/1203.4663
...A recently discovered observational by-product of an enhanced power spectrum on small scales, induced gravitational waves, have been shown to be within the range of proposed space based gravitational wave detectors; such as NASA's LISA and BBO detectors, and the Japanese DECIGO detector...
