Gravity in the Early Universe

Abstract

According to scientific evidence, the universe began after the “Big Bang” with a very high temperature and energy density. At early times, the temperature was certainly so high that all particles such as quarks, leptons, gluons, and so forth were strongly relativistic. Even strongly interacting quark and gluon particles interact relatively weakly due to asymptotic freedom. Therefore, it was only a system of hot and weak particles that can be called Quark-Gluon Plasma. As the universe cooled during the next phase of expansion, quarks, antiquarks, and gluons combined to form hadrons, leading to the formation of baryonic matter, the phase transition of quantum color dynamics (QCD), during which quarks and gluons are confined. In the first-order cosmic phase transition with the formation of bubbles, the transition to a true vacuum occurs. The bubbles expand and collide with each other, and part of the energy stored in the walls of the bubbles turns into gravitational waves.

1. Introduction

Discovering the mystery of the universe from the beginning and how it evolved and expanded is still interesting to people. The Big Bang model provides a general framework for describing the evolution of the universe and has been able to provide several successful fundamental predictions, including the existence of cosmic microwave background radiation and the abundance of light elements in the universe. It is also possible to obtain other indirect information from the early universe through the phase transition that occurred in it and their effects.

In the evolution of the early universe, there were at least two phase transitions. The electroweak theory predicts that at about 100 GeV, a transition from the symmetric high-temperature phase to the broken symmetry phase has occurred, in which the SU(2) × U(1) dimensional symmetry is spontaneously broken and the particle mass, which is proportional to the vacuum expectation of the Higgs, becomes nonzero. Also, quantum color dynamics (QCD) predicts that at an energy of 100 MeV, there is a phase transition from quark-gluon plasma to a confinement phase without free quarks and gluons. At approximately the same energy, we expect that the global chiral symmetry of the QCD with massless fermions is spontaneously broken by the formation of dense quark pairs. Cosmic phase transitions, especially QCD and electroweak phase transitions, may be first-order phase transitions under some conditions [1].

First-order phase transitions in the early universe can be powerful sources of gravitational wave radiation that can suggest a new way to explore the early universe. In a first-order phase transition, the universe starts in a quasi-stable high-temperature phase (symmetric phase) and turns into a stable low-temperature phase (broken symmetry phase). This transition continues through formation of low-temperature phase bubbles in the high-temperature phase. These bubbles then expand and merge. Finally, the universe remains in the phase of broken symmetry.

In a cosmological first-order phase transition that occurs in a thermal bath, bubbles form and expand due to the released vacuum energy. In the hydrodynamic description of the bubble evolution, the bubble velocity, v_b, is an important parameter that affects the production of gravity waves caused by this process. The realistic background is actually a thermal plasma full of relativistic particles that collide with the bubble wall and produce friction. The main problem of bubble expansion in plasma is to understand the relationship between the following quantities, by which the characteristics of the resulting gravitational waves can be known: The velocity of the bubble wall v_b, the friction on the wall from the plasma side; the phase transition intensity α, which measures the energy density of the released vacuum compared to the radiation energy density; and the factors κ and κ_v, which calculate the ability to transfer the energy of the released vacuum to bubble wall expansion, and the fraction of energy converted to the plasma motion, respectively. All parameters are measured at transition temperature. These quantities are used as an interface between theoretical models of particle physics and bubble collision simulations. The bubble collision is one of the sources of gravitational waves [2].

Gravitational waves from first-order phase transitions propagate gravitational field fluctuations and move at the speed of light. Any object in the path of the wave receives the tidal force that is applied perpendicular to the direction of wave propagation. Gravitational waves, after being produced, propagate without any restrictions. It has been proven that these waves are harder to stop than neutrinos. An important change that occurs as a result of their propagation is the decrease in amplitude when moving away from the source. These waves can be redshifted like electromagnetic waves.

In the QCD phase transition, leptonic asymmetry may lead to the generation of a finite bariochemical potential. In the strong baryon generation mechanism where a high baryochemical potential is produced, a short period of supercooling may occur during the QCD phase transition and lead to a dilution of the baryon density. Because in these conditions, the vacuum energy may be much greater than the thermal energy of the universe, and in this case, the plasma becomes very diluted. In this scenario, the phase transition occurs through the expansion and cooling of the universe, which results in the generation of gravitational waves. The phase transition proceeds through the formation of stable and low-energy phase bubbles inside the supercooled phase. These bubbles eventually expand and collide, causing gravitational waves [3].

2. Big bang and the early universe

In 1926, a Belgian priest named Georges Lemaitre proposed the theory that the universe started from a primordial atom. In 1929, the astronomer Edwin Hubble, by analyzing the redshift of the light of the galaxies, observed that they are moving away from each other in all directions. The expansion of the universe reinforces the idea that galaxies must have been closer together in the past when they are increasing in distance today. This theory of the expanding universe eventually led to what is now called the Big Bang model. The framework of this model is based on Einstein’s theory of general relativity and has been able to successfully explain many phenomena of the universe [4].

The evolution of the world is explained based on the Friedmann-Robertson-Walker (FRW) cosmological model or the Big Bang model, which includes the expansion of the world and homogeneous and isotropic approximations of the distribution of matter and energy on a large scale. This model has been so successful that it is called the Standard Model of Cosmology.

Assume a uniform distribution of matter. An observer who looks at the world from all sides and in all directions will find the same world. These two characteristics are called homogeneity and isotropy. According to Einstein’s equations of general relativity, the space-time that includes the mentioned features is given by the Friedman-Robertson-Walker metric:

$$\mathrm{d}{\mathrm{s}}^2={\mathrm{dt}}^2-{\mathrm{a}}^2\left(\mathrm{t}\right)\left[\frac{{\mathrm{dr}}^2}{1-\mathrm{k}{\mathrm{r}}^2}+{\mathrm{r}}^2{\mathrm{d\theta }}^2+{\mathrm{r}}^2{sin}^2\mathrm{\theta }{\mathrm{d\phi }}^2\right]$$E1

where k = 0 is space with zero or Euclidean curvature, k = 1 is closed space with positive curvature, and k = −1 is open space with negative curvature. At k = 0, the Robertson-Walker time space is not flat, but only its t = const subspaces are flat. The variable a(t), which is called cosmic scale factor or expansion factor, is dimensionless.

The general relativity field equations published in 1915 were as follows:

$${\mathrm{G}}_{\mathrm{\mu \nu }}\equiv {\mathcal{R}}_{\mathrm{\mu \nu }}-\frac{1}{2}{\mathrm{g}}_{\mathrm{\mu \nu }}\mathcal{R}=-\mathrm{\kappa }{\mathrm{T}}_{\mathrm{\mu \nu }}$$E2

where ${\mathrm{G}}_{\mathrm{\mu \nu }}$ is the Einstein tensor, ${\mathcal{R}}_{\mathrm{\mu \nu }}$ is the Ricci curvature tensor, ${\mathrm{g}}_{\mathrm{\mu \nu }}$ is the metric tensor and $\mathcal{R}$ is the scalar curvature, $\mathrm{\kappa }$ is the Einstein gravitational constant, and ${\mathrm{T}}_{\mathrm{\mu \nu }}$ is the stress-energy tensor that describes the matter-energy distribution on a large scale. Stress-energy tensor is a diagonal tensor whose elements are relativistic mass density and isotropic pressure. From Eqs. (1) and (2), the relation p = ωρ is obtained, where w is the state parameter.

When ω = −1, the energy of the universe was made up of vacuum energy. ω = 1/3 corresponds to the condition that the energy of the universe is determined by relativistic particles. It means that radiation dominates the universe. In this case, the factor of 1/3 is due to curving in all directions. ω = 0 describes the condition where nonrelativistic particles or matter is dominant in the universe. In this case, the matter of the universe behaves as dust and the pressure is zero. This condition has been true for recent era.

From the combination of Einstein and FRW equations, the following relations can be reached:

$$\frac{\mathrm{d}}{\mathrm{da}}\left(\mathrm{\rho }{\mathrm{a}}^3\right)+3\mathrm{p}{\mathrm{a}}^2=0⟹\mathrm{\rho }\propto {\mathrm{a}}^{-3\left(1+\mathrm{\omega }\right)}$$E3

The Big Bang era starts with a = 0. At this point the cosmic time is zero. Since the singularity means that some terms become infinite and others become zero. As a result, the concept of space-time geometry is violated. Therefore, physicists cannot explain what happened at that moment or before. In other words, the theory of general relativity is not complete, and maybe there is a more complete theory that provides a better explanation, such as hypotheses based on string theory and avoid this space-time singularity [5].

Some theories and hypotheses that try to avoid space-time singularity are:

A theoretical framework of ring quantum gravity that quantizes space and time, leading to the idea that spacetime is granular at the smallest scales, potentially avoiding singularities.

String theory proposes that the fundamental building blocks of the universe are not particles, but tiny strings that can potentially resolve singularities.

Modified theories of gravity that modify general relativity on large scales and aim to avoid singularities without invoking quantum effects.

About 10⁻⁴³ seconds after the Big Bang, the dynamics of the universe is described by the theory of quantum gravity, where the quantum effects of gravity are important. Experimental studies have shown that with increasing energy, the intensity of electromagnetic and weak interactions becomes comparable and the electroweak theory becomes dominant. This prediction was made for energies around 10¹⁵ GeV, approximately 10⁻³⁶ seconds after the Big Bang, which is called the Grand Unification Epoch. In this era, electroweak and strong interactions are unified.

Quarks and leptons become members of a common irreducible group; for example, quarks are converted into a lepton by boson exchange. If this theory is correct, the difference between these three interactions is due to the fact that we are working at low energies, in which case the unification of forces is lost.

Today’s observational evidence confirms the exponential growth at this stage of the world’s expansion. As the temperature decreases around 100 GeV, approximately 10⁻¹⁰ s after the Big Bang, the weak and electromagnetic forces are separated. At times around 10⁻⁵ after the big bang, when the temperature was around 100 MeV, quarks could exist in their bound states, i.e. baryons and mesons, and then protons and neutrons were able to come together and in energy around 1 MeV formed light nuclei [4].

3. Quark-gluon plasma

The universe began with extremely high temperature and energy density. At early times, the temperature was about 100 GeV. At such a temperature, all particles are relativistic so that quarks and gluons, which are strongly interacting particles, interact relatively weakly due to their asymptotic freedom. Thus, a system of weakly interacting hot particles called quark-gluon plasma (QGP) was in equilibrium with other species.

The interaction between quarks becomes vanishingly weakened as they get closer to each other at high density; these quarks are no longer confined inside the hadrons and become free. This process is called asymptotic freedom.

At high temperature, due to the asymptotic freedom, the quark-gluon plasma can be described as the simplest strongly interacting particle system in the context of QCD. As the universe cooled during the next phase of expansion, quarks, antiquarks, and gluons combined to form hadrons, the result of which is baryonic matter. The cooling scenario of the universe presents first- or second-order phase transitions associated with spontaneous symmetries breaking [6, 7, 8].

Phase transition at temperatures of more than 100 GeV causes the spontaneous breaking of the electroweak symmetry, providing masses to elementary particles. It is also related to the electroweak baryon-number-violating processes, which had a major influence on the observed baryon asymmetry of the universe [9].

The next phase transition is the transition from QGP to hadronic matter, which occurs at T < 200 MeV. This transition is related to the spontaneous breaking of the chiral symmetry strong interactions [8]. In a strong first-order phase transition, the QGP supercools before bubbles of hadron gas are formed. Since the hadronic phase is the initial condition for nucleosynthesis, the inhomogeneities in this phase could have a strong effect on the nucleosynthesis epoch [8]. Figure 1 shows in schematic diagram the history of the universe, beginning with the Big Bang and cooling as the universe expanded. The increase in time is accompanied by a decrease in temperature, which is given by the relation $\mathrm{t}=\left(\frac{3}{16\mathrm{\pi G}{\mathrm{g}}_{\ast }\mathrm{A}}\right){\mathrm{T}}^{-2}$, where A is the radiation constant and g_* is the total number of relativistic degrees of freedom. But in the quark-hadron epoch, the equation of state of matter becomes complicated.

Figure 1.

Schematic diagram the history of the universe [10].

4. Cosmic first-order phase transition

The transition between physical states of matter is called phase transition. If the phase transition is accompanied by sudden changes in entropy and definite volume, then it is a first-order phase transition, which usually occurs in the change of state of matter and fundamental alteration in the crystal structure of solids. The first-order phase transition equations are given in Eq. (4):

$$\mathrm{T}1=\text{T2},\mathrm{P}1=\text{P2},\text{G1}\left(\mathrm{P},\mathrm{T}\right)=\text{G2}\left(\mathrm{P},\mathrm{T}\right)$$E4

where T, P, and G are temperature, pressure, and Gibbs thermodynamic potential, respectively. Using Eq. (4), one can obtain the Clausius-Clapeyron equation, which determines the slope of the phase equilibrium curves:

$$\frac{∆\mathrm{P}}{∆\mathrm{T}}=\frac{∆\mathrm{S}}{∆\mathrm{V}}$$E5

In this equation, ∆S and ∆V are entropy changes and volume changes in phase transition, respectively. It is possible to consider the phase transition as a function of volume using the Helmholtz free energy.

Standard cosmology predicts several phase transitions for the evolution of the universe. One of the results of these transitions is the production of gravitational waves. But only if the phase transition is of the first-order type, gravitational waves will be produced. In the cosmic first-order phase transition, tunneling to the new phase is done by creating bubbles. The expansion of the bubbles and their collision with each other produce gravitational waves [11].

If the system starts from the highest Helmholtz free energy level at high temperatures and cools rapidly to below T_c, it is trapped in the high free energy phase. In this case, the system is in a quasi-stable phase. The transition to the lowest free energy phase is discontinuous and occurs through the formation of low free energy phase bubbles in the quasi-stable phase, which is an example of a discontinuous or first-order phase transition. The bubble structure shows a coherent oscillation of the field, which is induced by thermal or quantum effects.

If σ is the surface density and ∆F is the free energy difference between the two phases, the critical radius is R_c = 2σ/∆F, which is suitable for configuration or bubble growth. If R_c < R, surface tension prevails and the bubble shrinks. If ∆F = 0, R_c tends to infinity. R_c also determines the free energy barrier for critical bubble formation. If bubbles form with R_c > R, they will grow and coalesce.

The expansion rate of the universe compared to the time scale of particle interaction is the factor that determines the rate of temperature drop. The expansion rate of the universe for a period of dominant radiation can be obtained from Friedman’s equations, and the curvature constant is considered k = 0:

$$\frac{\dot{\mathrm{R}}}{\mathrm{R}}=\mathrm{H}\equiv 1.66{{\mathrm{g}}_{\ast }}^{1/2}\frac{{\mathrm{T}}^2}{{\mathrm{m}}_{\mathrm{Pl}}},$$E6

In cosmic expansion, all types of particles are in thermal equilibrium until the rate of interaction exceeds the rate of expansion, Γ_int > H. At very high energies, until k_BT ≤ 10^16–17, GeV thermal equilibrium is established.

As the universe expands and cools, the scalar field, which is the cause of symmetry breaking, is trapped in a quasi-steady state. In this case, the energy density of matter at high temperatures includes two terms, one of which comes from relativistic radiation and the other from vacuum energy:

$$\mathrm{\rho }\left(\mathrm{T}\right)=\frac{{\mathrm{\pi }}^{\mathrm{2}}}{\mathrm{30}}{\mathrm{g}}_{\ast }{\mathrm{T}}^{\mathrm{4}}+{\mathrm{V}}_{\mathrm{0}}$$E7

V₀ is the vacuum energy density that is trapped in the quasi-stable minimum. If the temperature is T < (30 V₀/g_* π²)^1/4, the constant vacuum energy dominates the energy density and the scale factor increases exponentially and rapidly. That is why it is called inflation. The universe is expanding at a speed close to the speed of light, and supercooling occurs. A sudden drop in temperature will make quantum effects dominate over thermal effects. Finally, the quantum bubble formation mechanism collapses to the lowest energy state, and the phase transition is complete by merging the bubbles [12].

5. Cosmic phase transition QCD

Quantum chromodynamics (QCD) is a suitable theoretical framework for describing strong interactions. This theory is a part of the standard model of particle physics that describes the physics of quarks and gluons. It also explains mechanisms such as confinement, chiral symmetry breaking, and asymptotic freedom [13].

QCD phase transition usually occurs in time $\mathrm{t}\approx {10}^{-5}$s and temperature T ≈ 150 MeV. This transition results from the temperature evolution of the strong coupling constant gs. At a temperature higher than the transition temperature, the coupling constant is so small that the behavior of the system is disordered and the system is in the phase of quark-gluon plasma.

As the universe cools, the strong coupling constant grows and quarks and gluons confined into colorless hadrons. All multicolors are enclosed in monochromatic baryons. The QCD phase diagram in terms of temperature and baryonic chemical potential is shown in Figure 2.

Figure 2.

A schematic QCD phase diagram in the temperature (T) and chemical potential (μ). The solid lines show the phase boundaries for the indicated phases. The solid circle depicts the critical point [14].

The QCD phase diagram in Figure 2 is predicted for different phases of nuclear matter, which shows the temperature value for different barochemical potential. The bariochemical potential describes the energy required to add or remove a baryon at constant pressure and temperature. At small chemical potential and high temperatures, matter is in the quark-gluon plasma phase. The early universe went through this phase in the first few microseconds after the Big Bbang. At low temperatures and high baryon densities, such as in neutron star cores, color superconducting phases are predicted. The phase transition between quark-gluon plasma and ordinary hadron gas appears continuous for small chemical potentials (dashed line). However, a critical point appears at higher potential values [15].

Numerical calculations show that the first-order QCD phase transition occurs at finite temperature for very small and large quark masses [16]. However, for the average mass of quarks with zero chemical potential, especially for three light quarks and small chemical potential, the first-order phase transition does not occur [17]. QCD has approximate chiral and central symmetries for very small and large quark masses, respectively. The remainder of these symmetries are spontaneously broken, and the corresponding order parameters become nonzero at all temperatures.

The confinement-deconfinement phase transition at the critical temperature T_c is a first order phase transition in pure Yang–Mills theory. While at μ = 0 above the temperature T_c, suggest a crossover in full quantum chromodynamics. At temperatures below T_c, quarks and gluons are confined into hadronic bound states, which are the effective degrees of freedom of low-energy QCD.

In the QCD phase diagram, a little inflation starts when the baryon to photon ratio has a larger initial value. For massless particles, this value is proportional to the ratio of barochemical potential to temperature so that μ/T ∼ 1. Such a large quantity is produced by some primary disequilibrium processes. Then, in the early universe, the conditions are the same as the first-order phase transition line in QCD large barochemical potentials, where the universe is trapped in a pseudo-vacuum state. The constant and nonzero vacuum energy density leads to inflationary expansion, which is accompanied by exponential supercooling and dilution of matter.

During this evolution, the universe is not in equilibrium, while the temperature and barochemical potential decay exponentially with μ/T = const. At low temperatures, the true vacuum barrier becomes so low that the universe spins into a true vacuum. The latent heat released is used to produce particles and eventually causes the world to reheat to a temperature of T ∼ Tc.

During the reheating, the baryon number is constant so that the net baryon density still decreases compared to the initial state before inflation, Today, the observed value of μ/T is approximately 10⁻⁹. After that, standard cosmological evolution continues the Big Bang nucleation process [18].

According to Figure 2, parts of the phase diagram have been investigated by several accelerator-based experimental programs. Experience has shown that T and μ change as a function of the energy of the center of mass [19]. This plan will be followed by experimental programs such as LHC, RHIC, FAIR, and NICA. Among these experiments, the RHIC results provide evidence of QGP formation.

6. Bubble nucleation rate

The bubble nucleation rate per unit volume, Γ, becomes zero at temperatures T ≥ T_c. In other words, the phase transition does not occur exactly at T = T_c. At temperatures T < T_c, the nucleation rate grows continuously from Γ = 0 and may reach a value of $\mathrm{\Gamma }\sim {{\mathrm{T}}_{\mathrm{c}}}^4$. As a result, the entire universe becomes a stable phase. The peak bubble nucleation rate determines the nucleation temperature T_n, T_n < T_c. At this temperature, the nucleation rate is extremely large.

For weak first-order phase transition, although T_n < T_c, the temperature T_n is very close to T_c. However, for a stronger phase transition, we have a lower temperature or even in T = 0. In such a case, there is no temperature T_n and the phase transition can take longer.

The time taken to complete the transition can be expressed as the transition rate β. The value of β is usually positive. Hence, the nucleation rate grows monotonously with time. The parameter β determines the time scale of the phase transition with the approximation $\mathrm{\Gamma }={\mathrm{\Gamma }}_0{\mathrm{e}}^{\mathrm{\beta }\left(\mathrm{t}-{\mathrm{t}}_0\right)}$. Therefore, the duration is given by $∆\mathrm{t}\sim {\mathrm{\beta }}^{-1}$, and the typical bubble size is given by ${\mathrm{R}}_{\mathrm{b}}\sim {\mathrm{v}}_{\mathrm{w}}{\mathrm{\beta }}^{-1}$, where ${\mathrm{v}}_{\mathrm{w}}$ is the bubble wall velocity. If the duration of the phase transition is short enough, the bubble radius can be assumed to be constant.

In fact, the progress of the phase transition depends on the relationship between Γ and the expansion rate H. In other words, bubble nucleation becomes important when Γ is comparable to the Hubble rate [11], since the number of bubbles, N, nucleated in the volume $\mathrm{V}\sim {\mathrm{H}}^{-3}$ in a cosmic time $\mathrm{t}\sim {\mathrm{H}}^{-1}$ will be $\mathrm{N}\sim {\mathrm{\Gamma H}}^{-4}$ and H is approximately given by $\mathrm{H}\sim {\mathrm{T}}^2/{\mathrm{M}}_{\mathrm{Pl}}$, where ${\mathrm{M}}_{\mathrm{Pl}}$ is the Planck mass. Therefore, for T close enough to T_c, the bubble nucleation will be very slow, that is $\mathrm{\Gamma }\ll {\mathrm{H}}^4$. The phase transition generally occurs at an intermediate temperature between T_n and T_c so that ${\mathrm{\Gamma H}}^{-4}\sim 1$, and due to the rapid growth of Γ, ends at a time $∆\mathrm{t}\ll \mathrm{t}$ [20].

The nucleation process transfers part of the vacuum energy to the bubble wall and leads to an increase in the wall velocity [21]. For a very strong phase transition, the wall velocity is ${\mathrm{v}}_{\mathrm{w}}\simeq 1$, but for a weak phase transition, the wall velocity is obtained as a function of T_n. Of course, these calculations may be different in different models and regimes. For example, in Jouguet detonation regime, the wall speed is calculated from Eq. (8) [22]:

$${\mathrm{v}}_{\mathrm{w}}=\frac{\sqrt{{\mathrm{\alpha }}^2+\frac{2\mathrm{\alpha }}{3}}+\sqrt{\frac{1}{3}}}{1+\mathrm{\alpha }}$$E8

In Eq. (8), the quantity α is the ratio of the vacuum energy density to the radiation energy density, which is calculated from relations Eq. (9):

$$\mathrm{\alpha }=\frac{\mathrm{\epsilon }}{{\mathrm{\rho }}_{\mathrm{R}}\left({\mathrm{T}}_{\mathrm{n}}\right)},{\mathrm{\rho }}_{\mathrm{R}}\left({\mathrm{T}}_{\mathrm{n}}\right)=\frac{{\mathrm{\pi }}^2}{30}{\mathrm{g}}_{\ast }{{\mathrm{T}}_{\mathrm{n}}}^4$$E9

The quantity ε represents the vacuum energy density, which is calculated using Eq. (10) for ∆F, where ∆F is the free energy difference.

$$\mathrm{\epsilon }={\left(∆\mathrm{F}\left(\mathrm{T}\right)-\mathrm{T}\frac{\mathrm{d}∆\mathrm{F}\left(\mathrm{T}\right)}{\mathrm{dT}}\right)}_{\mathrm{T}={\mathrm{T}}_{\mathrm{n}}}$$E10

Since gravitational waves are generated during the first-order phase transition, it is very important to calculate the values that play a key role in calculating the gravitational wave spectrum of the early universe. These values are more than the difference of free energy in two phases, vacuum energy density, ratio of vacuum energy to radiant energy, core temperature and wall velocity. In the next section, we will discuss the spectrum of gravitational waves.

7. Gravitational wave spectrum

The discovery of gravitational waves is a promising way to investigate the events of the early universe. One of the sources of the production of these gravitational waves is the cosmic first-order phase transition. When a first-order phase transition occurs in a thermal bath, bubbles are formed. Then, due to the release of the initial phase vacuum energy, the bubbles in the plasma expand and collide with each other. During the collision of the bubbles, their spherical symmetry is broken and part of the stored energy leads to the generation of gravitational waves [13]. For the transition of the first-order phases, during the formation of the bubble, three gravitational wave sources have been proposed: bubble collision, sound waves, and turbulent motion of the fluid. To calculate the energy density spectrum of gravitational waves, the sum of these three sources can be considered:

$${\mathrm{h}}^2\mathrm{\Omega }\left(\mathrm{f}\right)\simeq {\mathrm{h}}^2{\mathrm{\Omega }}_{\mathrm{Col}}\left(\mathrm{f}\right)+{\mathrm{h}}^2{\mathrm{\Omega }}_{\mathrm{Sw}}\left(\mathrm{f}\right)+{\mathrm{h}}^2{\mathrm{\Omega }}_{\mathrm{Tu}}\left(\mathrm{f}\right)$$E11

where the coefficient h is the current Hubble parameter in unit of 100 km/(s MPc). In the following, the contribution of three sources in the energy density of gravitational waves produced during a first-order phase transition is expressed.

The contribution of the collision of bubbles in the frequency spectrum of gravity waves using the envelope approximation has been calculated by numerical simulations as follows [23]:

$${\mathrm{h}}^2{\mathrm{\Omega }}_{\mathrm{Col}}\left(\mathrm{f}\right)=1.67\times {10}^{-5}\left(\frac{0.11{{\mathrm{v}}_{\mathrm{w}}}^3}{0.42+{{\mathrm{v}}_{\mathrm{w}}}^2}\right){\left(\frac{{\mathrm{H}}_{\ast }}{\mathrm{\beta }}\right)}^2{\left(\frac{\mathrm{\kappa \alpha }}{1+\mathrm{\alpha }}\right)}^2{\left(\frac{100}{{\mathrm{g}}_{\ast }}\right)}^{\frac{1}{3}}{\mathrm{S}}_{\mathrm{en}}\left(\mathrm{f}\right)$$E12

The κ parameter is a fraction of the vacuum energy that has been converted into the kinetic energy of the bubbles. In other words, for runaway bubbles, the efficiency factor of the bubble collision source is given by $\mathrm{\kappa }=1-\frac{{\mathrm{\alpha }}_{\infty }}{\mathrm{\alpha }}$, ${\mathrm{\alpha }}_{\infty }$ that is a critical value of α. The Hubble parameter at ${\mathrm{T}}_{\ast }$ is donated by ${\mathrm{H}}_{\ast }=1.66\sqrt{{\mathrm{g}}_{\ast }}{{\mathrm{T}}_{\ast }}^2/{\mathrm{M}}_{\mathrm{Pl}}$where ${\mathrm{T}}_{\ast }$ is the temperature at which gravitational waves are produced. The spectral shape of the gravitational wave is analytically fitted as:

$${\mathrm{S}}_{\mathrm{en}}\left(\mathrm{f}\right)=\frac{3.8{\left(\frac{\mathrm{f}}{{\mathrm{f}}_{\mathrm{en}}}\right)}^{2.8}}{1+2.8{\left(\frac{\mathrm{f}}{{\mathrm{f}}_{\mathrm{en}}}\right)}^{3.8}}$$E13

The peak frequency of redshift is given by Eq. (14):

$${\mathrm{f}}_{\mathrm{en}}=16.5\times {10}^{-6}\left(\mathrm{Hz}\right)\left(\frac{0.62}{1.8-0.1{\mathrm{v}}_{\mathrm{w}}+{{\mathrm{v}}_{\mathrm{w}}}^2}\right)\left(\frac{\mathrm{\beta }}{{\mathrm{H}}_{\ast }}\right)\left(\frac{{\mathrm{T}}_{\ast }}{100\mathrm{GeV}}\right){\left(\frac{{\mathrm{g}}_{\ast }}{100}\right)}^{\frac{1}{6}}$$E14

After the collision of the bubbles and before the expansion in the plasma, sound waves are formed and receive the kinetic energy in the plasma. Due to their distinct peak frequency distribution and the fact that they are potentially a long-period gravitational wave source, they have attracted special attention. In general, these sound waves can lead to a significant gravitational wave signal [24]. Gravitational wave spectrum resulting from sound waves has been calculated based on numerical methods as follows:

$${\mathrm{h}}^2{\mathrm{\Omega }}_{\mathrm{Sw}}\left(\mathrm{f}\right)=2.65\times {10}^{-6}\left(\frac{{\mathrm{H}}_{\ast }}{\mathrm{\beta }}\right){\left(\frac{{\mathrm{\kappa }}_{\mathrm{sw}}\mathrm{\alpha }}{1+\mathrm{\alpha }}\right)}^{100}{\left(\frac{100}{{\mathrm{g}}_{\ast }}\right)}^{\frac{1}{3}}{\mathrm{v}}_{\mathrm{w}}{\mathrm{S}}_{\mathrm{sw}}\left(\mathrm{f}\right)$$E15

The value of ${\mathrm{\kappa }}_{\mathrm{sw}}=\left(1-\mathrm{\delta }\right){\mathrm{\kappa }}_{\mathrm{v}}$ is a fraction of the latent heat that is converted into the movement of the fluid and depends on the state of expansion of the bubble and δ can be of the order of δ = 0.1. Also, ${\mathrm{\kappa }}_{\mathrm{v}}$ is a part of energy and has been transformed into plasma motion that is given by ${\mathrm{\kappa }}_{\mathrm{v}}=\frac{{\mathrm{\alpha }}_{\infty }}{\mathrm{\alpha }}\frac{{\mathrm{\alpha }}_{\infty }}{0.73+0.083\sqrt{{\mathrm{\alpha }}_{\infty }}+{\mathrm{\alpha }}_{\infty }}$ and for non-runaway bubbles is expressed as ${\mathrm{\kappa }}_{\mathrm{v}}=\frac{\mathrm{\alpha }}{0.73+0.083\sqrt{\mathrm{\alpha }}+\mathrm{\alpha }}$.

The shape of the spectrum S_sw and the peak frequency of the redshift f_sw are given by [25]:

$${\mathrm{S}}_{\mathrm{Sw}}\left(\mathrm{f}\right)={\left(\frac{\mathrm{f}}{{\mathrm{f}}_{\mathrm{sw}}}\right)}^3{\left(\frac{7}{4+3{\left(\frac{\mathrm{f}}{{\mathrm{f}}_{\mathrm{sw}}}\right)}^2}\right)}^{\frac{7}{2}},{\mathrm{f}}_{\mathrm{sw}}=1.9\times {10}^{-5}\left(\mathrm{Hz}\right)\left(\frac{1}{{\mathrm{v}}_{\mathrm{w}}}\right)\left(\frac{\mathrm{\beta }}{{\mathrm{H}}_{\ast }}\right)\left(\frac{{\mathrm{T}}_{\ast }}{100\mathrm{GeV}}\right){\left(\frac{{\mathrm{g}}_{\ast }}{100}\right)}^{\frac{1}{6}}$$E16

Magnetohydrodynamic (MHD) occurs in plasma formation after bubbles collide. In a first-order phase transition, bubbles may generate magnetohydrodynamic turbulence. The resulting seed magnetic fields are converted into magnetic fields, and the created MHD perturbation produces gravitational waves that may be detected by gravitational wave detectors. Gravitational waves resulting from the first-order QCD phase transition are placed in the frequency range of pulsar timing arrays, that is, nanohertz frequencies [26]. The contribution of the turbulent motion of the fluid is given by:

$${\mathrm{h}}^2{\mathrm{\Omega }}_{\mathrm{Tu}}\left(\mathrm{f}\right)=3.35\times {10}^{-4}{\left(\frac{{\mathrm{H}}_{\ast }}{\mathrm{\beta }}\right)}^2{\left(\frac{{\mathrm{\kappa }}_{\mathrm{tu}}\mathrm{\alpha }}{1+\mathrm{\alpha }}\right)}^{3/2}{\left(\frac{100}{{\mathrm{g}}_{\ast }}\right)}^{\frac{1}{3}}{\mathrm{v}}_{\mathrm{w}}{\mathrm{S}}_{\mathrm{tu}}\left(\mathrm{f}\right),\mathrm{\delta }={\mathrm{\kappa }}_{\mathrm{tu}}/{\mathrm{\kappa }}_{\mathrm{v}}$$E17

$${\mathrm{S}}_{\mathrm{tu}}\left(\mathrm{f}\right)=\frac{{\left(\frac{\mathrm{f}}{{\mathrm{f}}_{\mathrm{tu}}}\right)}^3}{{\left(1+\frac{\mathrm{f}}{{\mathrm{f}}_{\mathrm{en}}}\right)}^{11/3}\left(1+\frac{8\mathrm{\pi f}}{{\mathrm{h}}_{\ast }}\right)},{\mathrm{h}}_{\ast }=16.5\times {10}^{-6}\left(\mathrm{Hz}\right)\left(\frac{{\mathrm{T}}_{\ast }}{100\mathrm{GeV}}\right){\left(\frac{{\mathrm{g}}_{\ast }}{100}\right)}^{\frac{1}{6}}$$E18

In Eq. (18), ${\mathrm{h}}_{\ast }$ is the redshifted Hubble parameter. The explicit dependence of the spectral shape on ${\mathrm{h}}_{\ast }$ shows that the disturbance acts as a gravitational wave source, several times the Hubble parameter. The peak frequency in the form of the spectrum is given by the following relationship:

$${\mathrm{f}}_{\mathrm{tu}}=2.7\times {10}^{-5}\left(\mathrm{Hz}\right)\left(\frac{1}{{\mathrm{v}}_{\mathrm{w}}}\right)\left(\frac{\mathrm{\beta }}{{\mathrm{H}}_{\ast }}\right)\left(\frac{{\mathrm{T}}_{\ast }}{100\mathrm{GeV}}\right){\left(\frac{{\mathrm{g}}_{\ast }}{100}\right)}^{\frac{1}{6}}$$E19

8. Summary

This chapter is a brief overview of the effects of gravity in the early universe. Because the Big Bang is regarded as the beginning of our universe. It is predicted that in the first moments, the universe was a hot bath of fundamental particles that, although relativistic, had a weak interaction. In fact, in that era, the universe only consisted of quark-gluon plasma, which underwent several phase transitions and entered the confined phase from the deconfined state. This phase transition was accompanied by the nucleation of bubbles that grew and expanded. As a result, some of these bubbles collided with each other and created sound waves and magnetic turbulences, leading to the production of gravitational waves. In fact, during bubble formation, we may have three sources of gravity waves: bubble collisions, sound waves, and turbulent fluid motion. In order to reveal and study the energy density spectrum of gravitational waves, the total of these three sources should be considered and it can be checked which of these three sources has the most contribution. The study of the early universe and the discovery of gravitational waves resulting from it are very popular among physicists, and the results of these studies are consistent with the frequencies of some detectors and arrays.