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Like dark matter and dark energy, there is no candidate of inflaton in the Standard Model (SM) of the particle physics but its various extension does have candidates of dark matter. Does the minimal supersymmetric SM or minimal supergravity theories have any candidate(s) that could behave like inflaton(s)? What about other non-SUSY scenarios such as extra dimensions?

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    $\begingroup$ It's not true that there is no inflaton candidate in the Standard Model. Some people theorize that the Higgs could be the inflaton if you assume non minimal coupling to gravity. See arxiv.org/abs/0710.3755 for example $\endgroup$ – FrodCube May 5 '18 at 17:28
  • $\begingroup$ There are two things called the "Standard Model" in physics. One is the Standard Model of Particle Physics; the other a.ka. the LambdaCDM model is the Standard Model of Cosmology. Which one do you mean? $\endgroup$ – ohwilleke May 23 '18 at 19:26
  • $\begingroup$ @ohwilleke Edited! $\endgroup$ – SRS May 23 '18 at 19:27
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"SUSY Higgs Inflation" could be something.

More generally, there's a lot of work on inflatons in gauge-mediated SUSY breaking. These are generally beyond minimal SUSY. For example, Baumann wrote:

we explore the novel possibility that the inflaton responsible for cosmological inflation is a gauge non-singlet matter field in supersymmetric (SUSY) Grand Unified Theories (GUTs). We consider hybrid-like inflation models in SUSY (more specifically tribrid inflation models 2), where we show that the scalar components of gauge non-singlet superfields, together with fields in conjugate representations, may form a D-flat direction suitable for inflation.

We apply these ideas to SUSY models with an Abelian gauge group, a Pati-Salam gauge group and finally Grand Unified Theories based on SO(10). Here, the scalar components of the matter superfields in the 16s may combine with a single $\bar{16}$ to form the inflaton. Thus in some sense this “unified particle” can act as the “mother of the universe”, as it contains the inflaton responsible for inflation as well as all the matter present today.

Almost 20 years of exploratory theoretical work has brought forward some ideas and killed others, but there's clearly still a lot of work to do.

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