180. Cosmology of F-theory GUTs

Posted by Jonathan Heckman

January 14, 2009

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This is a guest blog post by Jonathan Heckman who is working at Harvard with Cumrun Vafa. Dmitry.

Let me first thank Dmitry for the invitation to write a blog entry on my recent work with A. Tavanfar and C. Vafa entitled “Cosmology of F-theory GUTs”, arXiv:0812.3155 [hep-th]. Before proceeding to a discussion of the actual contents of this paper, I will first provide some background and motivation for F-theory GUTs, and what we can hope to learn by studying the cosmology of these models. As a general comment, I will not be particularly fastidious in referring to all of the relevant papers, and instead refer the interested reader to our paper and references therein as an access point to the literature.

At a broad level, we are interested in determining how the Standard Model fits within the framework of string theory, a candidate for the unification of particle physics and gravity. F-theory GUTs aim to study Grand Unified Theories (GUTs) in the context of a strongly coupled formulation of IIB string theory known as F-theory. The operative assumptions which enter into the study of F-theory GUTs are:

Low energy ${\cal N}=1$ supersymmetry will soon be found.
There exists a supersymmetric “Grand Unified Theory” (GUT).
There exists a limit in the parameters of the compactification where gravity can in principle decouple from gauge theory.

While any of these criteria in isolation appear to allow a large amount of flexibility in string based constructions, far fewer vacua appear to satisfy all criteria simultaneously. At a general level, this is perhaps the most exciting feature of F-theory GUTs, because out of the many vacua of the string theory landscape, crude, qualitative considerations allow us to narrow the search for semi-realistic vacua. The last assumption deserves a few comments. By assuming the existence of such a limit, we are implicitly restricting attention to models where the forces of the Standard Model unify at some energy scale which need not be the same scale as which the dynamics of gravity are relevant. Said differently, the primary focus in F-theory GUTs is on the particle physics implications, rather than the gravitational implications of the model.

In keeping with this philosophy, our main interest will be on the cosmology of the F-theory GUT after the Universe exits some period of high scale dynamics dictated by a period of inflation, or some alternative such as string gas cosmology so that the FRW Universe provides a good approximation. While the high scale dynamics is of course an important piece of a complete story and is sensitive to UV details of a given model, for the purposes of particle physics considerations, we can parameterize our ignorance of “what comes earlier” in terms of an initial reheating temperature $T_{RH}^{0}$ .

Below the initial reheating temperature, we would like to investigate whether F-theory GUTs are compatible with constraints from cosmology. For the purposes
of this post, I would like to focus on just two potential constraints, which quite interestingly “cancel each other out” in F-theory GUTs. These are:

Constraints from gravitinos.
Constraints from cosmological moduli.

In models such as F-theory GUTs where the gravitino (the superpartner of the graviton) is stable, there is a potential danger that thermal processes can overproduce these particles. Indeed, when the initial reheating temperature is higher than the freeze out temperature of gravitinos $T_{3/2}^{f}$ , the relic abundance of gravitinos is:

$\Omega_{3/2}^{T}h^{2}\sim\frac{m_{3/2}}{2\text{ keV}}$

where in the above, $m_{3/2}$ is the mass of the gravitino, and $\Omega_{3/2}^{T}$ is the ratio of the energy density stored in thermally produced gravitinos to the critical density. To put this in perspective, in F-theory GUT models, solving the supersymmetric $\mu$ problem requires that the scale of supersymmetry breaking is in the range of $\sim10^{8}\div{}10^{9}$ GeV. This in turn sets the mass of the gravitino in the range of $10\div{}100$ MeV. At first, this sounds very problematic, because observational requirements demand that the total energy density stored in any thermal relic satisfy the bound:

$\Omega_{\text{relic}}h^{2}\lesssim{}0.1$ ,

which is the expected contribution from all dark matter.

It is important to note that here we have made two assumptions. The first is that $T_{RH}^{0}>T_{3/2}^{f}$ . The second is that below the initial reheating temperature, the Universe continues to expand and cool during an epoch where radiation dominates the energy density of the Universe down through the start of big bang nucleosynthesis (BBN), at a temperature of $T_{BBN}\sim{}1$ MeV, and somewhat beyond this temperature as well. A model can bypass this constraint when an appropriate violation of either assumption has occurred.

Indeed, one common way to avoid the “gravitino problem” is to lower the initial reheating temperature $T_{RH}^{0}$ below the freeze out temperature of gravitinos. How low does $T_{RH}^{0}$ need to be to avoid over-production of gravitinos? In the context of F-theory GUTs, the freeze out temperature $T_{3/2}^{f}\sim{}10^{10}$ GeV, so this would appear to require $T_{RH}^{0}<10^{10}$ GeV. In fact, going through the details, satisfying the bound $\Omega_{relic}h^{2}\lesssim0.1$ actually requires $T_{RH}^{0}\lesssim10^{6}$ GeV. The more general formula for the gravitino relic abundance can then be written as:

$\Omega_{3/2}^{T}h^{2}\sim C\cdot\frac{{\rm min}(T_{RH}^{0},T_{3/2}^{f})} {10^{10}\text{ GeV}}$ ,

for some numerical constant. This clashes, however, with the assumption of a limit where gravity and gauge theory questions can decouple. In keeping with the general philosophy espoused earlier, it would be far more natural if the cosmological scenario depended only minimally on the value of $T_{RH}^{0}$ .

Leaving aside for the moment this potential worry, cosmological moduli are another potential source of concern. These correspond to flat directions of the supersymmetric effective potential which only develop a mass due to the effects of supersymmetry breaking. In a string based construction, such moduli can originate for example, from fields which set the shapes of the internal directions of the compactification. Why are moduli so problematic? Because their mass is set by supersymmetry breaking effects, they will generically have masses in the range of $\sim{}1$ TeV. The decay rate for such fields is:

$\Gamma_{\phi}\sim\fra{1}{64\pi}\frac{m_{\phi}^{3}}{\Lambda^{2}}$ ,

where $\Lambda$ is a mass scale which is typically on the order of the GUT scale or Planck scale. The fact that $m_{\phi}/\Lambda$ is so small translates into the fact that such moduli are typically long-lived, decaying after the start of BBN. To make matters worse, the oscillation of these fields which commences at a temperature $T_{osc}^{\phi}$ can come to dominate the energy density of the Universe so that when they decay, they will release a significant amount of entropy into the Universe. As a consequence, some of the most successful features of the standard cosmology, such as BBN are now potentially in jeopardy. In scenarios where the modulus dominates the energy density of the Universe, the amount of dilution expected from the decay of the modulus is given by the ratio of entropy densities before and after the decay of the modulus:

$\frac{s_{before}}{s_{after}}\equiv D_{\phi}\sim\frac{M_{PL}^{2}}{\phi_{0}^{2}}\cdot\frac{T_{RH}^{\phi}}{\min(T_{RH}^{0},T_{osc}^{\phi})}$

where $M_{PL}$ is the Planck mass, and $\phi_{0}$ is the initial amplitude of the modulus, and $T_{RH}^{\phi}$ denotes the reheating temperature at which the modulus decays. If the value of $T_{RH}^{\phi}$ is below the starting temperature of BBN so that $T_{RH}^{\phi}<1$ MeV, the model will disrupt BBN. This is indeed the case for a generic cosmological modulus.

In F-theory GUTs, most metric moduli are assumed to be stabilized by high scale supersymmetric dynamics. This is necessary for the scenario to be consistent with the assumption that there exists a limit where gravity decouples from the GUT. The model is not, however, free from moduli, because it also contains the QCD axion. The existence of this single real degree of freedom allows F-theory GUTs to solve the strong CP problem. The QCD axion has a lifetime greater than that of the present Universe, so as long as the energy density stored in this field is within an acceptable range, it will not pose any problem as a cosmological modulus. In the context of supersymmetric theories, however, all degrees of freedom must be appropriately “complexified”. As such, there is another scalar degree of freedom besides the QCD axion, known as the saxion. This field can decay, for example to axions, or to Standard Model particles. Returning to our general decay rate formula, the primary difference is that as opposed to $\Lambda\sim10^{16}$ GeV or $10^{19}$ GeV, for the saxion $\Lambda\sim10^{12}$ GeV. As a consequence, the saxion is a modulus which typically decays at a temperature $T_{RH}^{sax}\sim1$ GeV, which is above the start of BBN.

This alone is encouraging, because although the saxion is a modulus, it decays early enough to avoid the most serious problems from cosmological moduli. The decay of the saxion can also have beneficial consequences for the over-production of gravitinos. In the most common F-theory GUT models, the saxion eventually comes to dominate the energy density of the Universe. The era of saxion domination terminates when the field decays. The resulting relic abundance of thermally produced gravitinos is then given by:

$\Omega_{3/2}^{actual}h^{2}\sim D_{sax}\cdot\Omega_{3/2}^{T}h^{2}+\Omega_{3/2}^{NT}h^{2}$ .

The first term is the diluted relic abundance from thermally produced gravitinos, and $\Omega_{3/2}^{NT}$ are gravitinos generated from the decay of the saxion itself. This second term is typically a subdominant contribution in F-theory GUTs, so we shall instead focus on the first term, $D_{sax}\cdot\Omega_{3/2}^{T}h^{2}$ :

$D_{sax}\cdot\Omega_{3/2}^{T}h^{2}\sim C\cdot10^{-10}\frac{M_{PL}^{2}}{s_{0}^{2}}\frac{T_{RH}^{sax}}{\rm GeV}\frac{{\rm min}(T_{RH}^{0},T_{3/2}^{f})}{\min(T_{RH}^{0},T_{osc}^{sax})}$ (1)

where $s_{0}$ denotes the initial amplitude of the saxion, which is generically of order the Kaluza-Klein scale for the associated field, and is given by $\sim10^{15}$ GeV.

The main point is that in F-theory GUTs, it turns out that the freeze out temperature of the gravitinos and the axion oscillation temperature are actually nearly identical:

$T_{3/2}^{f}\sim T_{osc}^{sax}\sim10^{10}{\rm GeV}$

which did not have to be the case. Returning to equation (1), the actual relic abundance of gravitinos is independent of $T_{RH}^{0}$ . While the actual relic abundance is sensitive to various numerical factors, at a rough level of approximation, the actual relic abundance of gravitinos can potentially provide a prominent component of the dark matter. There is, of course, still the possibility that other components could be present as well.

This is only one aspect of the full story. One might also ask whether the dilution due to the saxion is compatible with generating a sufficient level of baryon asymmetry in the early Universe. We have checked that once the effects of the dilution factor are taken into account, in F-theory GUTs, the typical range of parameters for thermal leptogenesis yields the expected asymmetry. There are many more details to be found in the paper, but I hope I have given a rough outline of some of the main features of our work.

Cosmology, Journal club, Master Postdoc, Quantum field theory, String theory

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