Applications of renormalization group methods to cosmological problems

Abstract

Despite the remarkable progress in understanding the large-scale structure ofthe universe, the nature of its very early moments and its accelerated expansion are still among the biggest puzzles in modern physics. The inflation theory improved our understanding of the early universe by discussing solutions of some relevant problems. The inflation scenario has to be considered with the standard Friemann- Lemaitre-Robertson-Walker cosmological model. One could imagine a conden- sate of a scalar field, called the inflaton field, which moves toward the minimum of a scalar potential during the inflation period. The scenario of inflation proposed by Alan Guth is based on the existence of a false vacuum state before the Big Bang. The inflaton field was trapped in this metastable vacuum state, then via the quantum tunneling the inflaton field freed itself and the inflation period began. This provided a amount of negative energy that forced the cosmic space to expand faster than light immediately before the Big Bang. Some problems in this scenario are related to the proper explanation of the reheating mechanism and the production of radiation. Several models have been proposed over the last decades to explain the in- flation phase of the early universe. Some of these models, such as the quadratic large field inflation, are excluded by the recent Planck observations of the cosmic microwave background. A reasonable candidate model is the slow-roll model. This model replaces the quantum tunneling mechanism of the inflaton field with a slowly rolling field. This requires the inflationary potential to be almost flat in the beginning of inflation. One can use this model to analyze which inflationary po- tential is viable and comparable with the observations. But this can leave us to ask why inflationary field should start at this particular flat situation. It seems to be present a fine tuning problem, referred to as the “initial conditions problem”. One of the proposed solutions to fix this problem is provided by the study of Renor- i malization Group (RG) running of the inflationary potential changing the energy scale (or equivalently with the cosmic time scale) in the era before the inflation. This can provide the smooth exit of the inflaton field from the false vacuum to the true vacuum and get the slow inflation scenario. Another big puzzle of modern cosmology is the cosmological constant prob- lem. This constant was invented in the beginning by Einstein himself to get the static universe case which was believed to be true at that time. After discovering the accelerated expansion of the universe, the cosmological constant is the candi- date to represent the vacuum energy with negative pressure. The problem is the great discrepancy between the theoretically predicted value from quantum theory for this vacuum energy and its measurable value. The disagreement is notoriusly very huge, which implies that there is something missing in the theory. One of the possible solutions to this issue is again related to the RG running of the inflation potential. The inflaton scalar field potential can be usually expanded in terms of the field around its minimum. This produces a field-independent term of the po- tential. The cosmological constant can be represented by this field-independent constant term. In the context of non-perturbative quantum gravity theory, the RG scaling of this constant term can be then investigated. The Functional Renormal- ization Group (FRG) method can be applied to provide a connection between the ultraviolet (UV) and the infrared (IR) values of this constant term. The recently observed cosmological constant (or vacuum energy) value can be considered to be equivalent to the IR value of the constant field-independent term of the inflation potential. This not only can provide a viable solution to the cosmological con- stant problem but also relate the phase of the accelerated expansion of the recent universe with its early time inflation phase. The running scale that can be consid- ered here is the usual momentum (or energy) scale k, which can correspond in this scenario to the cosmic time or the temperature of the universe.