**4. Generalized exponential power-law inflation**

The scheme for constructing models of the early universe based on the evolution of the scalar field in the context of the inflationary paradigm can be represented as follows:


To build cosmological models corresponding to observational data, we propose the principle of constructing the inflationary models with generalized exponential power-law expansion. For this aim we consider any exact solutions f g *ϕ; H;V* of Eqs. (3)–(5) for which the substitution of the slow-roll parameters (8)–(10) into Eqs. (13)–(17) doesn't correspond to observational constraints (17)–(18).

After the following transformations

$$
\overline{H} = \mathfrak{n}H + \lambda \tag{21}
$$

$$\overline{a}(t) = \mathbf{C}a^n(t)e^{\lambda t}, \mathbf{C} = \overline{a}\_0/a\_0^n \tag{22}$$

$$
\varphi = \sqrt{n}\phi\tag{23}
$$

$$
\overline{V}(\phi) = 3n^2H^2 + 6\lambda nH - nH\_{\phi}^{\prime 2} + 3\lambda^2,\\
\overline{V}(\phi) = \overline{V}(\phi(\varphi))\tag{24}
$$

one has new exact solutions *φ; H;V* � � with new slow-roll parameters

$$\overline{\epsilon} = n\epsilon \left( n + \frac{\lambda}{H(\epsilon)} \right)^{-2} \tag{25}$$

$$\overline{\delta} = \delta \left( n + \frac{\lambda}{H(\epsilon)} \right)^{-1} \tag{26}$$

and with the conformity to observational constraints which can be achieved by choosing the values of free constant parameters *n* and *λ*.

The proposed approach has two limitations:

a. The original scale factor *a t*ð Þ doesn't violate the law of accelerated expansion.

*The Early Universe as a Source of Gravitational Waves DOI: http://dx.doi.org/10.5772/intechopen.87946*

b.The potential *V*ð Þ *φ* corresponding to the scale factor (20) implies the evolution of the scalar field *φ*, according to the inflationary paradigm.

Transformations (21)–(24) define a class of models with the generalized exponential power-law dynamics, and the original scale factor *a t*ð Þ may not correspond to the condition of accelerated expansion *a*€>0; however, the resulting scale factor *a t*ð Þ implies a combination of the de Sitter solution (for *n* ¼ 0) and the power-law expansion (for *λ* ¼ 0), which corresponds to the basic feature of the inflationary paradigm implying a graceful exit from the stage of accelerated expansion to the power-law non-accelerated expansion.

#### **5. Relic gravitational waves**

Also, we note that the relic gravitational waves were not directly observed, which leads to a large number of theoretical models of cosmological inflation, which provide an explanation of the origin and evolution of the large-scale structure

The scheme for constructing models of the early universe based on the evolution of the scalar field in the context of the inflationary paradigm can be represented

a. The generating solutions of background dynamic equations (excluding quantum fluctuations of the scalar field) for a given potential, the law of accelerated expansion of the early universe, or the evolution of a scalar field.

b.Analysis of the quantum fluctuations of a scalar field and the corresponding metric perturbations on the basis of the theory of cosmological perturbation for the previously obtained background solutions. The result of this analysis is the values of the spectral parameters of cosmological perturbations

c. Comparison of the obtained spectral parameters of cosmological perturbations

To build cosmological models corresponding to observational data, we propose the principle of constructing the inflationary models with generalized exponential power-law expansion. For this aim we consider any exact solutions f g *ϕ; H;V* of Eqs. (3)–(5) for which the substitution of the slow-roll parameters (8)–(10) into Eqs. (13)–(17) doesn't correspond to observational constraints (17)–(18).

*λt*

*<sup>φ</sup>* <sup>¼</sup> ffiffiffi

*, C* <sup>¼</sup> *<sup>a</sup>*0*=an*

*<sup>ϕ</sup>* <sup>þ</sup> <sup>3</sup>*λ*<sup>2</sup>

*λ H*ð Þϵ � ��<sup>2</sup>

*λ H*ð Þϵ � ��<sup>1</sup>

and with the conformity to observational constraints which can be achieved by

a. The original scale factor *a t*ð Þ doesn't violate the law of accelerated expansion.

*H* ¼ *nH* þ *λ* (21)

*<sup>n</sup>* <sup>p</sup> *<sup>ϕ</sup>* (23)

<sup>0</sup> (22)

*,V*ð Þ¼ *φ V*ð Þ *ϕ φ*ð Þ (24)

(25)

(26)

of the universe and correspond to the observational constraints.

**4. Generalized exponential power-law inflation**

which can be calculated from Eqs. (13)–(17).

After the following transformations

*<sup>V</sup>*ð Þ¼ *<sup>ϕ</sup>* <sup>3</sup>*n*<sup>2</sup>

with the corresponding observational data (18)–(20).

*a t*ðÞ¼ *Ca<sup>n</sup>*ð Þ*<sup>t</sup> <sup>e</sup>*

*<sup>H</sup>*<sup>2</sup> <sup>þ</sup> <sup>6</sup>*λnH* � *nH*0<sup>2</sup>

one has new exact solutions *φ; H;V* � � with new slow-roll parameters

ϵ ¼ *n*ϵ *n* þ

*δ* ¼ *δ n* þ

choosing the values of free constant parameters *n* and *λ*. The proposed approach has two limitations:

as follows:

*Progress in Relativity*

**228**

As an additional verification tool for cosmological models, we consider the possibility of direct detection of the relic gravitational waves. The detection of relic gravitational waves is extremely important for determining the parameters of the models of early universe. Additionally, such a detection enhances the position of the inflationary paradigm compared to alternative scenarios, for example, the models with a rebound from singularity in which cosmological gravitational waves are absent [19].

As the main observational characteristic of relic gravitational waves, we consider the energy density, which is usually determined by the dimensionless quantity [20]:

$$
\Omega\_{GW}(f) = \frac{1}{\rho\_c} \frac{d\rho\_{GW}}{d\ln f} \tag{27}
$$

where *<sup>f</sup>* is the linear frequency, *<sup>ρ</sup><sup>c</sup>* <sup>¼</sup> <sup>3</sup>*H*<sup>2</sup> <sup>0</sup> is the critical energy density, *H*<sup>0</sup> is the value of the Hubble parameter in the modern era, and *ρGW* is the energy density of gravitational waves.

Also, the energy density of relic gravitational waves can be represented in terms of the power spectrum:

$$
\Omega\_{GW}(k) = \frac{k^2}{12H\_0^2} P\_T(k) \tag{28}
$$

The frequency and energy density of relic gravitational waves are limited by the following conditions [20]:

a. The energy density of relic gravitational waves should not exceed

$$\int\_{f\_0}^{\infty} \Omega\_{GW} d\ln f \le \mathbf{1.1} \times \mathbf{10}^{-5} \tag{29}$$

where *f* <sup>0</sup>≈10�<sup>9</sup> Hz.

a. The temperature of the scalar field *T*<sup>∗</sup> and the frequency of gravitational waves *f* at the end of the inflation stage are

$$T\_\* = 5.85 \times 10^6 \left(\frac{f}{Hz}\right) \left(\frac{g\_\*}{106.75}\right)^{1/6} \text{GeV} \tag{30}$$

$$f = 1.71 \times 10^{-7} \left(\frac{T\_\*}{\text{GeV}}\right) \left(\frac{\text{g}\_\*}{106.75}\right)^{-1/6} Hz \tag{31}$$

**6. Conclusion**

observational constraints.

**Author details**

**231**

Vladimir Gladyshev\* and Igor Fomin

provided the original work is properly cited.

Bauman Moscow State Technical University, Moscow, Russia

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: vgladyshev@mail.ru

exponential power-law dynamics are proposed.

*The Early Universe as a Source of Gravitational Waves DOI: http://dx.doi.org/10.5772/intechopen.87946*

low-frequency optical resonance proposed in [23, 24].

We considered the basis of building and verifying of the inflationary models of early universe. As the method for constructing the exact cosmological solutions corresponding to observational constraints, the models with generalized

The verification of the relevance of such models is related to the estimation of the contribution of relic gravitational waves to the anisotropy and polarization of the cosmic microwave background radiation. Therefore, there are a lot of infla-

The most obvious way to significantly reduce the number of theoretical models

The most promising methods in this area of experimental research are using the

interferometers as detectors. The interesting direction of the observation is the detection of high-frequency relic gravitational waves using the effect of

tionary models with different scalar field potentials that will satisfy the

of cosmological inflation is direct detection of relic gravitational waves.

where *g* <sup>∗</sup> is the effective number of the degrees of freedom (in standard model of elementary particles *g* <sup>∗</sup> ¼ 106*:*75).

Therefore, conditions (18) and (21)–(23) impose restrictions on the parameters of relic gravitational waves.

The application to the analysis of the models of the early universe only of the slow-roll approximation implies a low-frequency spectrum of relic gravitational waves in the range of 10�<sup>18</sup> � <sup>10</sup>�<sup>16</sup> Hz [20]. However, the predominance of the kinetic energy of the scalar field during the evolution of the early universe provides a theoretical justification for the existence of high-frequency relic gravitational waves in models with one scalar field in the range of 10<sup>2</sup> � <sup>10</sup><sup>4</sup> Hz [21] which can be used as affordable means of verification of models of the early universe in the presence of physical effects that increase the sensitivity of the detector to the required level.

Currently, the most productive method of direct detection of gravitational waves is the use of interferometers as detectors, which was proposed in the article by Gertsenshtein and Pustovoit [22]. This principle is widely used in modern laser interference gravitational antennas, the main element of which is the Fabry-Perot interferometer. These are broadband gravitational antennas, which offer a lot of opportunities as to the methods of recording of gravitational waves and extracting signals, as well as the use of quantum non-perturbative measurements and the inclusion of gravitational antennas in the network. The main element of laser interference gravitational antennas, as a rule, is Fabry-Perot multipath free-mass resonator, on whose properties the sensitivity and noise immunity of the entire gravitational antenna largely depend [4, 23, 24].

After creating the first laser interferometer for detecting gravitational waves, systematic work began on the creation and improvement of such devices in various laboratories around the world. The experience of gravitational antenna projects by VIRGO (Italy, France), LIGO (USA), TAMA (Japan), CLIO (Japan), GEO-600 (Germany), and OGRAN (Russia) will certainly be used to create more compact and highly sensitive antennas of new generation [4]. Also, as the most promising project for the direct detection of gravitational waves, work on the creation of a space interferometer in a helio-stationary orbit should be noted, in which the distance between the mirrors will be about 1 million kilometers. This project is called Laser Interferometer Space Antenna (LISA) [25]. The implementation of the LISA project is scheduled for 2029.

One of the promising methods for increasing the sensitivity of gravitational antennas in the high-frequency region of the spectrum is to use the phenomenon of low-frequency optical resonance, which distinguishes this approach from other projects on the detection of gravitational waves. The presence of this effect in Fabry-Perot interferometers was first considered in [23, 24]. At the moment, there is a high-frequency gravitational wave detector, which was built at the University of Birmingham, United Kingdom [26]. Also, it is planned to build the high-frequency gravitational wave detectors in Japan [27].

Thus, at the moment there are a large number of promising methods for direct observation of gravitational waves, which correspond to the ability to measure the characteristics of relic gravitational waves for a better understanding of the physical processes occurring in the early universe.
