If the neutrino flux is variable with time that could mean that the fusion reaction rate is variable with time (there is no good reason to assume the fusion rate is rigidly constant in any case), or it could indicate new neutrino physics. A crucial study along these lines appears to be
Sturrock, et al., 2005. They analyze the Super-Kamiokande dataset, and compare their analysis & results with several previous studies, some of which report evidence for a variable solar neutrino flux, and some of report the contrary. They conclude, in part, "
It appears that current evidence for variability of the solar neutrino flux from the Homestake (99.9% CL), GALLEX (99.9% CL) and Super-Kamiokande (98%-99% CL) experiments, when taken together, comprise a strong case for the existence of a sterile neutrino which, together with the needed large transition magnetic moment, would be harbingers of important new physics." (where CL = Confidence Level). However,
Aharmim, et al., 2010 did not find any evidence for variability of the
8B neutrinos at SNO, whereas
Ranucci & Rovere, et al., 2007 did find evidence for a variability of the
8B neutrinos at SNO, but were able to identify the source of the variability as the Earth's elliptical orbit around the Sun. This shows that the variability, if real, is very weak and/or complicated (maybe some neutrinos vary in flux while others so not). So whatever side one wants to take in the argument, definitive claims that there definitely is or is not a variability in solar neutrino flux are at best premature.
( ... )
Once again, the
bottom line is this: There is no way that a variable solar neutrino flux can reasonably be interpreted as in any way falsifying, or even weakening, the theory that the source of solar energy is nuclear fusion in the core.