I bet that among the "vocabulary" of string theory the two that more broadly have expanded in the layman are "Calabi-Yau" and "warped (or brane world) universes". I have already made (in Spanish) a brie introduction to compactifications and Calabi-Yaus. Now I am going to talk about the warped universes and related questions.
I dón´t know the details of the history, but the "heuristic" aspects are clear. After the second string revolution there were two mayor new ingredients in string theory, D-Branes and M-Theory. D-Branes allows the picture of vectorial guage particles trapped in a brane while gravity (.i.e. gravitons) being able to escape the brane and exploring the extra dimensions. This wouldn´t be so important without M-theory. M-Theory lives in 11 dimensions, instead of the superstrings who live in 10 and the nature of the 11th dimensions is somewhat different to that of the 6 extra dimensions of superstrings. In particular it is reasonable to expect that it´s size could be greater than the (Calabi-Yau, or similar) compactified dimensions of the superstrings.
The "natural" scenario seems clear. The universe could be a D3-Brane with the standard model attached to it. The particles of the standard model would be related to a "Calabi-Yau" compactification of 6 of the additional dimensions to a planckanian size and we still have an aditional dimensiosn whose size is not constrained to be planckanian. How great could actually be?.
Well, in the late nineties Arkanni-Hammed, Savas Dimopulus and Dvalli analyzed the question in an a very simple and general framework. They proposed that we would live in a D3-brane and that all other dimensions could be macroscopic and only accessible to gravity. This trivially implied that to short distances Newtonian gravity should be modified so that it would have an 1/R2 + n behaviour, being n the number of additional macroscopic dimensions, instead of the usual, with n=0. The implications of this new behaviour would be that at short distances gravity would be more intense. Surprisingly at that time gravity hadn´t been measured at short distances and the bounds for the distances where the extra dimensions would appear were as great as a millimetre. Soon a few experiments were realized and new bounds arrived. I am not totally sure, but nowadays the allowed size is on the order of at most a decimal of a millimetre. Also it has been discarded (or at least is very unlikely) that more than an additional dimension would be macroscopic (in the sense of non Planckanian).
This ADD scenario was certainly very naive. But the idea of an extra macroscopic deserved further attention. Two different ways to approach it appeared. A phenomenological one the Randall-Sundrum brane world models, and purely string theoretic one, the Horava-Witten theory which realised the idea of heterotic M-theory. I will begin discussing the first ones.
Lisa Randall and Cundrum Sumdrum proposed an scenario where gravity was prevented from "leaking" in the extra dimension was a curvature effect (a warped compactification). They proposed a metric of the form:
Here x represent the coordinates of the usual 4 dimensions and y is the additional dimension. It is assumed that the four dimensional brane is the boundary of a bulk which is a portion of an AdS5 geometry. (this also allow to relate the Randall-Sundrum sceneries to the AdS/CFT correspondence, which, incidentally, has been proved in a very recent, yesterday when this is being written, paper, available here). The parameter l in the metric of the previous equation would be the curvature of that AdS space.
The exponential factor of the metric is responsible of confining gravity. The reason behind the y being in modulus is that we are in an scenario with two branes. One of this correspond to the visible world, the other is a "hidden" brane (usually called also the planck brane). The branes are postioned at y=0 and y=L and there is a Z2-symmetry identification y <-> -y, y+L <->L-y. This relates, at least conceptually, this scenario to the Horava-Witten, as it will be shown later.
From this departure there are two models. The RS-2 model keeps the distance between branes finite. The RS-1 model sends the hidden brane to infinity (i.e. L->infinity) and it effectively behaves as if it would only have one brane. It is the bes suited for being related to AdS/CFT correspondence and an string realization based on flux (warped) compactifications of Type II B superstrings (ore Type II A related to them by mirror symmetry). There are however a few characteristics that this models share. One of them is that they solve the "hierarchy problem". Loosely speaking this problem consists in that the large difference of energy between the electroweak scale and the planck scale requires a very fine tunning of many constants. Supersimmetry could be a solution for this problem, but the Randall-sundrum models solve it in a different way. It can be shown that the energy of the particles in the planck brane is seen in the visible sector damped by the exponential factor of the metric. This automatically addresses the problem. This was, seemingly, the major goal of the first papers in the subject. This explains that the branes are plane Minkowsky space. This means that model is not a cosmologicla model. But there are possible additions which allows to convert it into a cosmological model, let´s say a few about it.
One thing that one would care about going into cosmological considerations is the cosmological constant. It can be seems that the visible brane has a positive tension (understood as self-gravity; in an "stringy" viewpoint it would be the brane tension, i.e. density of energy) while the hidden brane has a negative tension. The bulk, being a AdS, has a negative cosmological constant. The tensions of the branes compensate so the visible brane has a 0 cosmological constant. Introducing matter in the bulk it can get an small positive cosmological constant, but that requires a lot of fine tuning.
The inclusion of matter opens new perspectives. If we must go to a cosmological model we must have matter, and get an FRW scenario (to begin with). The inclusion of matter in this phenomenological model has a curious feature The standard model matter is confined to the branes by a delta function (which certainly is not a "first principles" way to proceed, but, ey it is a phenomenological model). I´ll not dwell into the details (you can find them, for example, in the article that Roy Marteens wrote in the subject for living reviews in relativity, available in the links section of this blog).I´ll simple state that it is actually possible to get an FRW metric.
My main interest here is to present the interplay between this "warped worlds) and string theory, but I can´t avoid to say a few things about two more characteristics of them. The two things are related to black holes.
The first, and most famous, is that the increase of gravity intensity for small distances own to the additional macroscopic dimension. This means that the threesold for the production of a black hole in a collision of particles is seriously reduced. In particular there is an small possibility that in the energy available to the LHC microblack holes could formate (and later evaporate by Hawking process). In fact in the Strings 2007 conferences Lisa-Randall was actually pessimistic about this possibility, but still there is an small window for it.
The other black hole issue is that the rate of evaporation of primordial black holes is modified, actually slowed down. by the extra dimension. In particular that means that most of the primordial black holes wouldn’t have evaporated (that explain why the gamma ray bust corresponding to the last moments of their existence have not been observed) and that the density of that black holes would be high (the probability of being one in the solar system being reasonably high). I must say that the mathematical description of a black hole in the brane worlds is a delicate issue and that it requires numerical calculations.
After heaving done a description of some aspects of the brane world universes let’s go to the Horava Witten theory. It consists of two steps. First one compatifies the theory in a S1/Z(2) orbifold. Such and orbifold is a circle with the upper and inferior half identified that is, a segment. The extremes of the segment are fixed points under the action of the Z(2) group. This means that we have two ten dimensional plains and a 11 dimensional bulk. Requirements of cancellation of gravitational and gauge anomalies in the two orbifold plains require that they must implement an E(8)xE(8)symmetry, the same that most phenomenologically promising heterotic superstring theories. The ten dimensional planes must, actually, compactified into planck sized spaces (usually Calabi-Yaus). The details of the calculations involving anomalies predict a relation between the gauge coupling and the size of the eleventh dimension of the form:
Restrictions from cosmology imply that the size of the orbifold would be of the oorder of 10 times the planck size. This is a mayor departure form the RS models. Another thing is that the bulk, would, at least in the simplest models, supersymmetric, i.e. not AdS. Going beyond the "heuristic" description and getting actual effective equations for the model is not a trivial task, but it has been realized. From those equations it is possible to calculate which matter is actually available in the model. And one should try to get a FRW model from them. Well, one can get a hubble era with relativistic matter (i.e. matter at relativistic velocities) but it is now allowed to get non relativistic matter. In fact if one allows some modifications of the model, for example including M5 branes in the bulk, or similar nonperturbative effects to the low energy equations of motion, it could, perhaps, be possible to actually get non relativistic matter in the branes (I am not totally sure of which is the "last day" status of the question because most of the reviews that I have read in this topic are at most from the 2005).
I didn´t present all the details (it would be impossible in a blog sized post) but it is clear that although similar the RS models and the Horava Witten don´t fit totally. This has lead to different lines of research.. On one side there are the Stiendard-Turok Ekpyroctic model. This realizes, at least it is what they claims, very closely the Horava Witten model, but with a very significant diference. Instead of purchasing a FRW model they propose a ciclic model. They propose that form tiem to time (actually thousands mof millions of years) an M5 (or something similar) brane forms in the bulkd and is atracted towards the visible brane, where it desintegrtes resulting in a great increment of the energy which resembles similarities with an expanding universe but actually consist of separation of the particles (and creation of new ones own to the available energy). I didn´t read too much of this model so I can´t say much ore than this.
The other approach is to try to "enginer" RS scenaries form other tools of string theoy. For example one can do wrapped compactifications, as I mentioned at the begining. In that constrcuctioins the visible brane could be an stack of coincident branes. But not necessarilly D3-branes. They could be, for exampled, Dm-branes wraped around n dimesnional "supersymmetric" homological cicles of the compactifeid dimension (which result in a flux) or another constructions.The AdS/CFT correspondence that I have mentiones ocassionaly rquieres the existence (betwen the planck and the standard model branes) of a region many AdS radii in size.The RS sndrum scenario would be the strong coupling version of an older idea for solving the hierarchy problem. One startswith some ultraviolete fixed point CFT around the UV scale (planck scale) and perturbs it by a marginally relevant operator (whose is dimension is close to, say, 4 - ε) then one can naturally generate scales much lower than Mpl. The RG (renormalization group) runningof the couplings in the perturbed field theory is logaritmic, and therefore the relevant coupling will have sgnificant dynamical effects only afther a vast amount of RG running. The translation of this scenario to the RS models is via the AdS/CFT dictionary.
Describing, even "bloglike" the detaills would require another post. But I hope that with what I have told here the reader could get an overview of how goes the interplay betwen phenomenological models and more microscopic stringy considerations.
As a finall comment to sy that not all the string theorist are to conveiced that these secenaries are the most probably realized by nature. Neither it is probable that they would shide light into more fundamental problems of string theory. But what is clear that they have many intriguing posibilties, some of them which could connect to measurable effects and that it is reasonable that people work on them.