arXiv:2007.06691v1 [gr-qc] 13 Jul 2020 The Power Spectrum Of Gravitational Waves in the Different Hubble rates Taimur Mohammadi, * Behrooz Malekolkalami, † and Khabat Ghamari ‡ July 15, 2020 Abstract The power spectrum of the gravitational waves (as spacetime perturbations) in the flat back- ground FRW universe are presented when the evolution equation of perturbations are described in terms of scale factor. It is a straightforward method to calculate the power spectrum (diagram) different from the ones used in [3, 7, 9], namely, method dependent on the transfer function. The presence of the Hubble parameter in the evolution equations allows to calculate power spectrum for different evolution stages of the universe. Specially, it is shown that, the incorporation of dark energy has a reinforcing effect on the power spectrum. Keywords: Gravitational waves; Power Spectrum; Hubble Parameter. 1 Introduction Gravitational waves (GW) are propagating perturbations in spacetime which can be produced by rapidly accelerating material bodies. In the other words, the ripples in the spacetime due to the accelerated object would make disorderliness and this perturbations on linear order would produce GW which was predicated by Einstein in 1916 [1]. In recent decays, study of the GW is a topic interesting the theoretical and experimental physics, for example GW due to the early universe or the compact binary system [2, 3]. The first direct detections of GW from the merging of black hole and neutron stars recently measured by the LIGO/VIRGO [4, 5, 6]. Studying the GW can be important from the standpoint of cosmological paradigms as inflation or LSS of the universe. In the latter, it can play a potentially role to infer information about the LSS, what is known as cosmography. Another important aspect of this study comes from the fact that, the GWs are tensorial perturbations and this can provide the data to extract relevant information which cannot be derived from the scalar perturbations, specially for the relic gravitational waves from the early evolution of the universe. In this work, we are interested in studying and analysing the GW originating in the early universe which to understand and describe of its physical conditions, the inflationary paradigm is a very successful theory. The inflationary cosmology was welcomed due to solving the major cosmological problems. The inflation theory is well–supported by observational data such as LSS of the present universe which is well–described by the scalar perturbations. On the other hand, the tensorial perturbations predict the existence of a uniform background of gravitational radiation in our universe. It is true that despite the very low intensity of GW, observational evidences and technological tools have succeeded in detecting them, but the hope is that by improved technologies, one should be able to test this prediction from inflation which would be a huge success in confirming the theory. * Department of Physics, University of Kurdistan, P.O.Box 66177-15175, Sanandaj, Iran. Email: t.mohammadi@uok.ac.ir † Department of Physics, University of Kurdistan, P.O.Box 66177-15175, Sanandaj, Iran. Email: b.malakolkalami@uok.ac.ir ‡ Kurdistan Meteorological Office. Email: xabatq@gmail.com 1