A comparison between axion-like and power law potentials in cosmological background

In this paper, we compare the scalar field dynamics in axion-like and power law potentials for both positive and negative values of the exponents. We find that, for positive exponents, both the potentials exhibit similar scalar field dynamics and it can be difficult to distinguish them at least at the background level. Even though the potentials are oscillatory in nature for positive exponents scaling solutions can be achieved for larger values of the exponent for which the dynamics can be different during early times. Because of the presence of this scaling nature there is a turnaround in the values of the scalar field equation of state as we increase the values of the exponent in both the potentials. This indicates the deviation from the oscillatory behaviour for the larger values of the exponent. For negative values of the exponent, the dynamics of the scalar field is distinguishable and axion-like potential can give rise to cosmologically viable tracker solutions unlike the power law potentials. For negative values of the exponent, axion-like potential can behave like a cosmological constant around its minima and the dark energy scale can be related to the potential scale. Due to the cosmological constant like behavior of the axion-like potential for negative exponent around its minima the late time dynamics can be similar to $\Lambda$CDM and we get similar observational constraint on the parameters for both $\Lambda$CDM and axion-like potential with negative exponent. So, while for positive exponents we may not distinguish the two potentials for negative exponents the dynamics of the scalar field is distinguishable.