I've looked in and out the forum, and found no precise definition of the meaning of fine-tuning in physics.
QUESTION
Is it possible to give a precise definition of fine-tuning?
Of course, I guess most of us understand the empirical meaning of the phrase... but it seem so ethereal, that's the reason behind my question.
All we can do precisely is give a probability for some physical quantity to have its observed value. For example (subject to various assumptions!) the probability of the cosmological constant having it's observed value is around 1 in $10^{120}$. Since this is absurdly low we say it's fine tuned.
But where you draw the line between fine tuned and not fined tuned is a matter of debate. Most of us wouldn't consider a 10% probability fine tuned, but what about 1% or 0.1%? Particle physics required a $5\sigma$ probability to be considered proof, and this is about 1 in 3.5 million and this is about 0.00003%, so that seems like a reasonable lower bound for not fine tuned. However I'd guess most people would consider considerably higher probabilities than this as evidence of fine tuning.
The point is that the probability of an observed value can be calculated precisely, but whether this corresponds to fine tuning is a matter of personal opinion.
John Rennie's answer describes the term "fine tuning" as used in high-energy physics, but the term is often used in a very different way in the study of critical phenomena. In that context, it often has a much sharper definition: a Hamiltonian is "fine-tuned" if it lies on a particular lower-dimensional submanifold of Hamiltonian parameter space (typically a critical hypersurface). In this case we can even say that multicritical points are more fine-tuned than singly critical points, because they lie on a submanifold of higher codimension (and are corresponding more difficult to achieve in an experiment).
