Over at Cosmic Variance, Sean Carroll has a post entitled Do You Think Inflation Probably Happened?. In it, he talks about several speculative ideas in cosmology and particle physics that are there to explain certain unexplained things, but for which there's not direct experimental evidence. Being a bit of a narcissist, I decided to blather on a blog post about this rather than just comment on the comment thread there.
One thing I do want to note before I dive in: all of these theories are different from things like the theory of General Relativity or the theory of biological evolution. Those two theories have tremendous direct support for them, so we believe that they are true. These theories, though, often aren't even theories, but are "paradigms", "ideas", "hypotheses", or things like that. For all of them there are good reasons to think that they might be true, but they're lacking direct solid evidence.
This is the idea that 13.7 billion years ago, about 10 or so minutes before the epoch of Big-Bang nucleosynthesis, there was a period of at least a tiny fraction of a second (but potentially as long as "forever") the Universe was expanding exponentially. During this time, the Universe expanded by at least a factor of 1024 or so (Agullo et al. 2009). Inflation was invented to explain problems such as why the Cosmic Microwave Background is at the same temperature across the sky, even though things more than a degree or so apart on the sky hadn't had time since the beginning of the Universe to send light signals to each other and be able to equilibrate with each other.
I would estimate the probability that inflation really happened at about 80% or so, which makes me feel very non-conservative. However, the paradigm does explain a lot, and there are even details in recent cosmic microwave background results that fit with the idea.
I'm gonna go 50% here, mostly because I am ignorant. I know the basic idea of supersymmetry. However, I have to admit that I'm not conversant enough with Quantum Field Theory to understand the reasons why it's compelling. I do know that people who think a lot about this think that it is compelling, and think that it ought to be out there, or at least that it's reasonable that it's out there, which is why I rate this as high as 50%. It's a matter of trusting their judgment, but not having a great basis myself to form an opinion.
3. String Theory
String Theory remains our leading candidate for unifying gravity with the standard model of particle physics. Physics is in this weird situation where it has two extremely successful theories that have withstood every test they've been put to. One is General Relativity (GR), the other is the Standard Model of Particle Physics (SMPP)— which is built on top of Quantum Field Theory, the special relativistic version of Quantum Mechanics. However, the two don't play nice together. Most of the time, this doesn't matter. In realms where GR is important, that is where gravity is strong, quantum effects are negligible. On tiny scales where you have to worry about Quantum Mechanics, gravity is negligible. However, when densities get high enough (such as at the center of a black hole or in the very early Universe), you need to use both, and when you try you get nonsensical answers.
We haven't experimentally been able to probe these regions, however. It's very weird to have two theories that stand up to every experiment or observation we can make, but which we know, for theoretical reasons, can't both be right.
String theory is our leading candidate for unifying the two. I don't understand string theory beyond the Elegant Universe level. And, indeed, you can find some high-end physicists who have come out with an "anti-String Theory" stance, arguing that a few decades of effort haven't paid off and that too much intellectual capital is now wasted on it.
I'm enough of an iconoclast to be dubious of String Theory, although without really good reason. Part of that is that I find General Relativity much more aesthetically pleasing than Quantum Field Theory, which includes some ugly bits in it (like renormalization). However, if I'm to be honest, I'll have to say that a decent fraction for that reason is that I understand GR better than QFT, so this isn't a fair judgment. Still, part of me wants GR to be "more right" than QFT, so that the former won't have to be as modified to make them play together. So, for these unscientific and poor reasons, I'm only going to rate the probability that String Theory is right at about 15%.
4. Some Form of Higgs Boson
My understanding of the Higgs Mechanism is weak. Somehow, the Higgs Boson is the particle that "gives things mass". Basically, all massive particles couple with the Higgs Field, and the energy of that coupling is what gives them their inertial mass. (It would be more fun if we called the Higgs Field the Molasses Field, but what can you do.) I think (but am not sure) that the need for a Higgs field is the result of electroweak unification, and that's a rather successful theory. Indeed, I think it can be said that the W+, W-, and Z particles are three components of a four (at least?) component Higgs field, so in a sense we already have seen part of the Higgs field.
Again, I'm speaking from not knowing as much as I ought to here, but I have the sense that this whole Higgs business is fairly well established, so I'm going to give you a 75% probability that they're out there, and that we'll find them soon. It would be cool if it's not out there, because it means we've finally found something that doesn't work with the Standard Model.... However, it would be uncool if it's not out there and the LHC doesn't find anything bold and new beyond the Standard Model, because that would make it very difficult to maintain public interest in particle physics.
5. Large Extra Dimensions
Damn, I think it would be really cool if these existed. However, I am going to have to go with a low value of 10% for these. Sure, it's possible, but I have yet to be convinced that there's really any good reason beyond "boy, that would be cool" to think that they exist.
6. WIMP Dark Matter
In answering this question, I have to weasel a bit because I'm not sure what is meant by WIMP. How specific is that? Would (say) a sterile neutrino count as a WIMP? A sterile neutrino is one that doesn't interact via the weak force because it's got the wrong chirality, so technically, it's not a WIMP (since the WI means "Weakly Interacting"). But, it is nonbaryonic mass.
My confidence in nonbaryonic dark matter is extremely high -- I'd place that at something like 98%. We've seen it. Not directly, but we've seen it indirectly just as assuredly as humanity had seen the neutrino indirectly before its direct experimental confirmation in 1956 (Cowan et al. 1956).
But is it something that's Weakly Interacting? I guess I'd have to place that at more like 60%. I'll only be mildly surprised if it turns out that the Dark Matter we know exists doesn't interact via the Weak Force.
7. Any non-cosmological-constant explanation for cosmic acceleration
It makes me sad to say this, but I'd place this at a mere 25%. It would be so much more interesting if it's something other than just a cosmological constant (which is another word for vacuum energy). Especially if it's "phantom dark energy", whose density increases with time, as that would mean that eventually we'll reach a period where the Universe is expanding at the same rate it was during inflation, giving us a nice bookend sort of symmetry for the beginning and end of the Universe. But, this is a case where my gut instinct and my aesthetic desires are at odds. I've thought for five or six years that experiments that try to nail down the equation of state parameter for Dark Energy will just keep shrinking the error bars around the vacuum energy value, never able to either prove or rule out that its vacuum energy. (If the error bars shrink enough and the value is off of the vacuum energy value, then we'd rule it out. But, if they just shrink around that value, we can't rule it out, but we can't be sure that it's just little enough different that we aren't able to yet see the difference.)