# Is pulsar timing a hopeful tool for detection of relic gravitational waves by using GW150914 data?

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## Abstract

The inflation stage has a behaviour as power law expansion like \(S(\eta )\propto \eta ^{1+\beta }\) where \(\beta \) constrained on the \(1+\beta <0\). If the inflation were preceded by a radiation era, then there would be thermal spectrum of relic gravitational waves at the time of inflation. Based on this idea we find new upper bound on \(\beta \) by comparison the thermal spectrum with strain sensitivity of single pulsar timing. Also we show that sensitivity curve of single pulsar timing may be hopeful tool for detection of the spectrum in usual and thermal case by using the GW150914 data.

## 1 Introduction

*S*and \(\eta \) are scale factor and conformal time respectively and \(\beta \) constrained on the \(1+\beta <0\)

^{1}[1, 2]. If the inflation were preceded by a radiation era, then there would be thermal spectrum of gravitational waves at the time of inflation [3, 4, 5]. Based on this idea we have shown in [6] that there is some more chances for the detection of relic gravitational waves (RGWs) in usual and thermal spectrum by using the data of GW150914. On the other hand, there is an important procedure for direct detection of RGWs that called pulsar timing (PT). This procedure is based on the investigation of fluctuation of the pulses of the pulsars due to RGWs. The frequency range of PT is \(10^{-9}\)–\(10^{-6}\) Hz [7]. The authors in [7] have found an upper bound on \(\beta \) and shown that there is so low chance for the detection of RGWs by using the mentioned method. But we think that the upper bound on \(\beta \) will modify if we use the thermal spectrum. Also the chance of detection of RGWs based on the strain sensitivity of PT will increase by using the obtained \(\beta \) for GW150914 [6] in usual and thermal case. Therefore the main purpose of this work is investigation of this detection. In the present work, we use the unit \(c=\hbar = k_{B} =1\).

## 2 The spectrum of gravitational waves in usual and thermal case

*a*) operators then,

*A*without scalar running as follows [14]

## 3 The analysis of the spectrum

Therefore based on the our results, the PT may be more hopeful tool for the detection of RGWs.

## 4 Discussion and conclusion

The thermal spectrum of RGWs causes an enhanced amplitude that called ‘modified amplitude’. The obtained results based on the thermal spectrum cause new upper bound on \(\beta \) which can give us more information about the inflation. Also using the obtained amounts of \(\beta \) based on the GW150914 data tell us that PT may be more hopeful tool for detection of RGWs in usual and thermal case.

## Footnotes

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