142. Chaotic inflation on the landscape?

By a chance, do you remember the paper by Nemanja Kaloper and Lorenzo Sorbo that we have recently discussed? There, the authors were modeling quintessence by axion-like fields that dynamically mix with 4-form fields. The mixing introduced mass terms for the axions approximately preserving the shift symmetry. (By the way, I was surprised to see how many people are looking for “Caloper Sorbo quintessence” in Google – it seems that the topic is going to become hot soon.)

What if instead of a very small mass we would take a mass of the order GeV? In this case, of course, we would have high energy chaotic inflation driven by the axion field instead of the axion-quintessence, and this simple observation is the topic of another paper by Kaloper and Sorbo.

So, let me try to understand the Kaloper-Sorbo setup a bit better this time. Axions (or what seems to be axions from the point of view of a 4-dimensional observer) appear as components of higher rank forms in the internal 6-dimensional space. Typically, it is hard to make a nice model with axion-driven inflation based on a string theory-inspired model. Let us see why.

Since the internal space is compact, we can say that the axion potential is periodic and write

. (1)

Since we want long high energy inflation, has to be much larger than regime (otherwise the potential is not flat enough), but it is hard to obtain large decay constants for the axion in this regime – expand the cosine and note that the higher order terms will be suppressed by higher powers of inverse (this terms are responsible for the decay of the coherent axion condensate in the end of inflation). Thus. of large automatically means overclosed Universe after the end of axion-driven inflation.

On the other hand, if  is relatively small, constants acquire large instanton corrections, and we loose analytic control over the form of the potential (1).

Kaloper and Sorbo propose athen interesting way out. Suppose, they say, the axion is mixed with 4-form field through a term

(2)

in the Lagrangian, to which we also want to add the term

(3)

where is the Lagrangian multiplier (the constraint (3) basically says that is the field strength of the 4-form).

Fixing the background 4-form field freezes the value of . Integrating over the fluctuations near this background, we therefore get terms linear sand quadratic w.r.t. – so, the overall effective potential for the axion now is

. (4)

Without the 4-form background the shift symmetry is preserved (shifting simultaneously shifts since the latter is the Lagrangian multiplier). As long as is fixed, the shift symmetry is broken (slightly – for small backgrounds), and we can have chaotic inflation in the potential (4).

So, why the Kaloper-Sorbo setup is interesting and important,  why so much struggle for the shift symmetry? The reason is that when it is preserved, we do not need to care about radiative corrections to the axion potential – they are automatically zero. If shift symmetry is broken slightly, it is automatically guaranteed that radiative corrections also remain small (they are suppressed by powers of the symmetry breaking potential). We therefore keep corrections to the axion (inflaton) potential under our analytic control.