308. Dark matter via many copies of the Standard Model

The authors of this post are Alex Vikman and Iggy Sawicki. Alex is an old friend of mine (actually, a long time ago we studied English together in the same group at MIPT and even shared the same table ). After receiving his PhD from Munich U. (his advisor was Slava Mukhanov), Alex joined NYU as a postdoc. There he mostly works with Gia Dvali. Iggy is also postdoc at NYU. On the his path of scientific career he had a chance to work with Sean Carroll and Wayne Hu. Dmitry.

We would like to thank Dmitry for giving us the opportunity to blog about our latest paper (arXiv:0903.0660), “Dark Matter via Many Copies of the Standard Model” that we have written together with Gia Dvali.

In this paper, we propose that large number,  , of copies of the standard model (“SM”) can be naturally responsible for dark matter (“DM”) . The key idea is that all the baryonic matter in the universe is a member of the same SM copy, while DM is made of protons and antiprotons which are members of all the other copies.

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In particular the same number of weakly coupled copies of SM can address both the hierarchy problem and explain the existence and the observed abundance of dark matter.

As is usual, we produce the particle content at the end of inflation: we endow each copy of the SM with its own inflaton. One of the inflatons drives the last stages of inflation and, when it decays, it does so mainly to its own SM copy. The density of matter in that copy is large and it quickly thermalises. The thermal history in this sector proceeds in the usual manner with most of the baryons and antibaryons annihilating, with the remainder forming today’s baryonic matter.

However, this inflaton will also have a suppressed coupling to the other SM copies, allowing them to become populated in much smaller numbers. Given a large enough  , the other sectors are not dense enough to ever thermalise, annihilation is always frozen out and they provide for an effectively collisionless fluid today that behaves as is required of dark matter. The prediction is that dark matter would be made of protons and antiprotons in other copies of the standard model.

In our model, the ratio of DM and baryon abundances is predicted in terms of measured quantities: the amplitude of the power spectrum of inflationary perturbations,  , and the photon-to-baryon-number ratio,  ,

(1)

Our idea is motivated by the recent proofs that quantum field theories which contain a large number of species,  N , have a new energy scale  at which gravity in the theory becomes strongly-coupled. It is  rather than  that serves as the real cut-off of the theory. This allows us to address the hierarchy problem: in the SM, the mass of the Higgs is naturally expected to be at the cut off, i.e. around the Planck mass, since no symmetries exist to protect it from radiative corrections. However, in a theory with  , the cut-off is actually at  and the required weak-scale mass is naturally obtained without fine tuning. Since gravity becomes strong at much lower energies, these large- N models will have an interesting signature at the LHC. Despite this motivation, we leave  N as a free parameter is our discussion.

One of the implications of the existence of this cut off is the fact that the Hubble parameter is also bound,  , which prevents the usual chaotic-inflationary mechanism from producing a sufficiently large amplitude for cosmological perturbations. It is known that having a large number of inflatons does not change the amplitude of perturbations in first order in slow roll approximation. Thus, for theories with  , we need an alternative mechanism for the generation of inflationary perturbations.

A simple solution is provided by modulated reheating. One introduces the modulator  , a new scalar field, light during inflation. The modulator is common to all the species and is present in all the leading-order couplings between the inflaton and matter. Its vacuum expectation value (“VEV”) controls the decay rate of the inflaton and, therefore, the position of the reheating surface. Since it will obtain perturbations during inflation, the reheating surface will also be perturbed, resulting in density perturbations after reheating.

In particular, we assume that reheating be dominated by the following interactions

where  is the inflaton and  are quarks and  and  are dimensionless couplings. In our paper, we show that the amplitude of perturbations after reheating ends is related to the VEV of  as

where  is the Hubble parameter during inflation and therefore the VEV of the modulator is fixed by the observed amplitude. We will simplify our discussion henceforth by assuming that inflation occur around the cut-off of the theory, i.e.  .

Another universal implication of large- theories is the suppression of couplings between different species required by the self-consistency of the theory. For example, if there exists a cross-species coupling  , then we can construct annihilation diagrams for species  into species  mediated by loops of all the other species. Since the species index on each loop is free, such amplitudes will involve summations over all the species, and therefore will be relatively enhanced compared to one-species theories. In particular, diagrams with  consecutive loops are possible, and will scale as  . This implies that  .

We can apply this sort of argument to the values of our couplings  and  : for  , for example, consider a diagram for annihilation of quarks  to  mediated by a virtual  (with an additional two external inflaton legs). We can integrate the modulator out and obtain an effective cross-species six-point coupling proportional to  . We can then apply the loop argumentation presented above to find that the coupling is constrained to be  . A similar argument can be made for  , except here we can take  diagrams, each mediated by a different species of virtual inflaton in the presence of a non-zero VEV of  . This constrains the coupling to be 

We assume that these couplings take their most natural values, the largest for which the theory remains consistent, i.e. ones saturating the above bounds. This then allows us to predict the ratio of densities of matter in the each of the dark species,  , to that in our species,  , immediately after reheating,

As is evident from the above, the total density in the dark sector summer over all the species is independent of  .

The final element of the calculation is provided by the fact that, in our sector, the plasma thermalises and mostly annihilates, with only a fraction  of the baryons surviving. In the paper, we show that given  the annihilation in the other sectors is always frozen out and no depletion of density occurs. This leads to our final result that the ratio of the dark matter to baryonic abundance is given by equation (1) given that the number of species lies between  .

In summary, we have shown that, in models with a large number of species, we can naturally reproduce the observed dark-matter abundance. The parameters of such models are constrained by the requirement for perturbative unitarity: the effective coupling between species must be suppressed in such a way that amplitudes with loops of species do not diverge. Saturating these constraints, coupled with a modulated-reheating mechanism for generating cosmological perturbations during inflation allows the inflaton to decay to our copy of the standard model (which we interpret as baryonic matter) and equally to the other copies. Since interaction between species are required to be suppressed, the resulting fluid is collisionless, replicating the properties of cold dark matter.