The Evolution of Dark Energy in a Model of a Gas of Thin Tubes of a Massless Scalar Field ^†

Abstract

We have shown the possibility of realizing the phase of phantom dark energy in a gas of thin tubes of a massless scalar field, and we have also shown that in the process of evolution, the phantom component of dark energy in such a gas can become equal to zero. In this case, starting from a certain point in time, the total value of the parameter of the equation of state for a gas of thin tubes of a massless scalar field will satisfy the inequality $w>-1$.

1. Introduction

The large body of observational data accumulated to date records significant discrepancies between early and late cosmological measurements. Among the most discussed cosmological parameters with existing deviations in measurements, at early and late times, for example, are the Hubble constant ($H_0$), and the parameter $S_8$, which characterizes the measure of growth of cosmic structures [1].

Attempts to soften the existing tensions between early and late cosmological measurements have led to the emergence of a large number of models with variable dynamics of the expansion of the Universe. In particular, such models include dynamical dark energy (DDE) models. In the DDE models, the equation of state parameter w depends on time [2,3,4]. The gas of thin tubes of a massless scalar field (TToMSF) can also be classified as a dynamical dark energy model, since the parameter of its equation of state w changes over time and depends on the density of the gas. For the rarefied TToMSF gas, $w\to -1$, and for the compressed gas, $w\to 1$ [5,6,7]. The realization of such asymptotics of the parameter w in dynamical dark energy models is not unique. For example, in models of modified gravity $f\left(R\right)$, at short times, under conditions of ultra-high curvature of the early Universe, the equations of modified gravity generate an effective equation of state of the medium with $w_{eff}\to 1$. At the same time, at large times, when the Universe expands and the curvature drops to extremely small values, the geometry of space begins to behave like Einstein’s classical cosmological constant with $p=-\rho$ [8,9]. The possibility of realizing the evolution of the equation of state parameter in the range $-1\le w\le 1$ is also demonstrated by the modified Chaplygin gas models [10,11], the k-essence models [12,13], and the viscous fluid models [14,15]. The advantage of the TToMSF gas model, compared to other models, is related to the possibility of influencing both the global and local evolution of cosmological parameters. At the same time, in this work we showed that taking into account the features of gravitational interaction in the TToMSF gas can expand the region of variation of the parameter w to the phase of phantom dark energy.

The global evolution of cosmological parameters in the TToMSF model is associated with a global decrease in gas density during the expansion of the Universe, and local evolution is associated with local changes in gas density, the source of which, for example, can be the gravitational interaction of the TToMSF gas with non-uniformly distributed baryonic matter [7]. At the same time, local inhomogeneities of the TToMSF gas density are the source of local equations of state with different values and different laws of change of the parameter w, which will further lead to local inhomogeneities in the distribution of matter, which are in tension with the standard model of cosmology (SMoC). Observed “local” inhomogeneities that are in tension with the SMoC include, for example, galaxies with masses ${10}^9-{10}^{11}M\odot$, discovered at redshifts $z>10$ [16,17], as well as regions with reduced matter density, for example, the Keenan–Barger–Cowie void [18,19,20] or Ho’oleilana [21].

The concept of a thin tube of massless scalar field is closely related to the concepts of a classical cosmic string and a null string. Classical cosmic strings are “one-dimensional” topological defects that could have formed in the early Universe as a result of phase transitions. The null string is a massless limit of the classical cosmic string, moving at the speed of light and having finite energy [22,23]. The one-dimensional model often used to describe null strings can be physically justified only on large scales, where the gravitational influence of a null string on its environment is not significant. At the same time, when studying the gravitational influence of a null string on its local environment, it is physically more justified to move from the one-dimensional model to the TToMSF model.

The application of strings in cosmology is not limited to classical cosmic strings; for example, within the framework of superstring theory, a field of research called string gas cosmology (SGC) is successfully developing [24,25,26].

In this work, we investigated how the trajectories of a test null string change when moving in the evolving gravitational field of a thin tube of a massless scalar field, and how these trajectories can influence the accelerated expansion of the TToMSF gas.

2. Distribution Function and Gravitational Field

It is known that the Nambu–Goto action cannot be directly applied to describe cosmic strings that move at the speed of light (null strings). At the same time, the action functional for the null string (one-dimensional model) can be obtained from the action, which is classically equivalent to the Nambu–Goto action [27]

$$S=\int d\tau d\sigma \left({\displaystyle \frac{h}{E}}-{\rho }_m^{2}E\right),$$

(1)

where ${\rho }_m$ is the linear mass density, $\tau$ and $\sigma$ are the parameters on the world surface of the classical cosmic string, the function $E=E(\tau ,\sigma )$ is the two-dimensional sheet density on the world surface of the string, $h=|h_{ln}|$, and $h_{ln}={\partial }_l{x}^i{g}_{ij}\left(x\right){\partial }_n{x}^j$ and $g_{ij}\left(x\right)$ are, respectively, the metric tensor on the string worldsheet and the metric tensor of the background spacetime, $l;n=0,1$; $i;j=0,\dots ,D-1$. By varying (1) over the field E, we find the equation of motion for the field E

$$E={\rho }_m^{-1}\sqrt{-h}.$$

(2)

For (2), the action (1) coincides with the Nambu–Goto action. For (1), in calibration

$$E=\xi g_{ij}{x}_{,\sigma }^i{x}_{,\sigma }^j,g_{ij}{x}_{,\tau }^i{x}_{,\sigma }^j=0,$$

the components of the energy-momentum tensor of a null string (condition ${\rho }_m=0$) have the form

$$T^{ij}={\displaystyle \frac{\varrho }{\sqrt{-g}}}\int d\tau d\sigma x_{,\tau }^i{x}_{,\tau }^j{\delta }^4\left(x^k-x^k\left(\tau ,\sigma \right)\right),$$

(3)

where $g=|g_{ij}|$, $x_{,\sigma }^i=\partial x^i/\partial \sigma$, $x_{,\tau }^i=\partial x^i/\partial \tau$, $\varrho$ and $\xi$ are constants.

For a real massless scalar field $\phi$, the components of the energy-momentum tensor are determined by the equalities [28]

$$T_{ij}={\phi }_{,i}{\phi }_{,j}-{\displaystyle \frac{1}{2}}{g}_{ij}{g}^{mn}{\phi }_{,m}{\phi }_{,n},$$

(4)

where ${\phi }_{,i}=\partial \phi /\partial x^i$ and indices $m,n,i,j$ take the values $0,1,2,3$.

If a closed TToMSF has the shape of a torus and, over a certain time interval, does not change its shape and size while moving, then, in a cylindrical coordinate system ($x^0=t$, $x^1=r$, $x^2=\theta$, $x^3=z$), the distribution function of the scalar field $\phi$, which will model such a TToMSF, can be represented as [6]

$$\phi =-\gamma ln\left(\alpha \left(q\right)+\lambda \left(q\right)f\left(r\right)\right),$$

(5)

where $q=t+z$, $\gamma$ is a positive constant ($\gamma >0$), the function $\lambda \left(q\right)$ is related to the function $\alpha \left(q\right)$ by the equality

$$\alpha \left(q\right)=1-f_0\lambda \left(q\right),f_0=const,$$

(6)

and the functions $\lambda \left(q\right)$ and $f\left(r\right)$ are bounded

$${\left.\lambda \left(q\right)\right|}_{q\in (-\infty ,-\Delta q)\cup (+\Delta q,+\infty )}\to 0,{\left.\lambda \left(q\right)\right|}_{q\to 0}\to 1/f_0,$$

(7)

$${\left.f\left(r\right)\right|}_{r\in \left(0,R-\Delta r\right)\cup (R+\Delta r,+\infty )}\to f_0,{\left.f\left(r\right)\right|}_{r\to R}\to 0,$$

(8)

where $R=const.$ ($R>0$), $\Delta q$ and $\Delta r$ are small positive constants that determine the “thickness” of the TToMSF (core radius).

The constant $\gamma$ in the function (5) can be considered an evolutionary parameter. If the functions $\alpha \left(q\right)$ and $f\left(r\right)$ are given, then larger values of the constant $\gamma$ will correspond to larger values of the scalar field $\phi$, and, accordingly, larger values of the constant $\gamma$ can be associated with earlier moments in the evolution of the Universe. In [6], by integrating Einstein’s equations for the function (5), restrictions on the possible values of the constant $\gamma$ were found, namely,

$$0<\gamma \le \sqrt{2/\chi },$$

(9)

where $\chi =8\pi G$ (in the system of units $c=1$, where c is the speed of light), G is the gravitational constant. For $\gamma \in \left(0;\sqrt{1/\chi }\right)\cup \left(\sqrt{1/\chi };\sqrt{2/\chi }\right)$, the solution to Einstein’s equations has the form [6]

$$d{S}^2=e^{2\nu }\left({\left(dt\right)}^2-{\left(dz\right)}^2\right)-A{\left(dr\right)}^2-B{\left(d\theta \right)}^2,$$

(10)

where

$$e^{2\nu }={\alpha }_0\left|{\lambda }_{,q}\right|{\displaystyle \frac{{\left(\alpha \left(q\right)+\lambda \left(q\right)f\left(r\right)\right)}^{2-\sqrt{4-2\chi {\gamma }^2}}}{{({\lambda }_0\lambda \left(q\right)+{\beta }_0)}^k}},$$

(11)

$$B={\displaystyle \frac{{\left(\alpha \left(q\right)+\lambda \left(q\right)f\left(r\right)\right)}^{\sqrt{4-2\chi {\gamma }^2}}}{{({\lambda }_0\lambda \left(q\right)+{\beta }_0)}^{\frac{\sqrt{4-2\chi {\gamma }^2}}{1-\chi {\gamma }^2}}}},$$

(12)

$$A={\displaystyle \frac{{\gamma }^2{\left(\lambda \left(q\right)f_{,r}\right)}^2}{c_0}}{\displaystyle \frac{{\left(\alpha \left(q\right)+\lambda \left(q\right)f\left(r\right)\right)}^{2-\sqrt{4-2\chi {\gamma }^2}}}{{({\lambda }_0\lambda \left(q\right)+{\beta }_0)}^k}},$$

(13)

$$k=\left(2+{\displaystyle \frac{2-\sqrt{4-2\chi {\gamma }^2}}{1-\chi {\gamma }^2}}\right),$$

(14)

$${\lambda }_0=-d{\displaystyle \frac{1-\chi {\gamma }^2}{\sqrt{4-2\chi {\gamma }^2}}},$$

(15)

${\alpha }_0$, ${\lambda }_0$, ${\beta }_0$, $c_0$, and d are constants. Solutions of Einstein’s equations for the values $\gamma =\sqrt{2/\chi }$ and $\gamma =\sqrt{1/\chi }$ can be found in [6].

The conditions under which a thin tube of a scalar field can be considered as a null string can be found by comparing the corresponding components of the energy-momentum tensors (3) and (4). For the quadratic form (10) such conditions have the form [6]

$${\displaystyle \frac{{\left({\phi }_{,r}\right)}^2}{A}}\to 0,{\displaystyle \frac{{\phi }_{,q}{\phi }_{,r}}{e^{2\nu }}}\to 0,{\displaystyle \frac{{\phi }_{,q}{\phi }_{,r}}{A}}\to 0,$$

(16)

$${\left.{\displaystyle \frac{{\left({\phi }_{,q}\right)}^2}{e^{2\nu }}}\right|}_{q\to 0,\rho \to R}\to \infty ,{\left.{\displaystyle \frac{{\left({\phi }_{,q}\right)}^2}{e^{2\nu }}}\right|}_{q\ne 0,\rho \ne R}\to 0.$$

(17)

3. Motion of a Test Null String

By the term test null string we mean a one-dimensional massless object whose gravitational influence is neglected. The motion of a test null string in an external gravitational field with tensor $g_{ij}$ is described by the following equations [29]:

$$x_{,\tau \tau }^{\alpha }+{\mathsf{\Gamma }}_{pq}^{\alpha }{x}_{,\tau }^p{x}_{,\tau }^q=0,$$

(18)

$$g_{\alpha \beta }{x}_{,\tau }^{\alpha }{x}_{,\tau }^{\beta }=0,g_{\alpha \beta }{x}_{,\tau }^{\alpha }{x}_{,\sigma }^{\beta }=0,$$

(19)

where ${\mathsf{\Gamma }}_{pq}^{\alpha }$ denotes the Christoffel symbols, $\tau$ and $\sigma$ are the parameters on the world surface of the null string, $x_{,\tau }^{\alpha }=\partial x^{\alpha }/\partial \tau$, the indices $\alpha$, $\beta$, p, q, i, j take the values 0, 1, 2, and 3, and the function $x^{\alpha }$ determines the trajectory of the null string (the world surface).

By integrating Equations (18) and (19) for (10), we find a relation between the variables t, z, r, and $\theta$, with the parameters $\tau$ and $\sigma$. In particular, if the value of the variable $\theta$ for each point of the test null string does not change over time (does not depend on the parameter $\tau$), then the relation of the variables t, z, and r, with the parameters $\tau$ and $\sigma$, is determined by the following equalities:

$$f_L^i\left(r\right)=U_L^i+h_L^i{\left(\lambda \left(q\right)\right)}^{-1},$$

(20)

$${|\lambda \left(q\right)}_{,\tau }|{\displaystyle \frac{{\left(1+h_L^i-\lambda \left(q\right)\left(f_0-U_L^i\right)\right)}^{2-\sqrt{4-2\chi {\gamma }^2}}}{{({\lambda }_0\lambda \left(q\right)+{\beta }_0)}^k}}={\displaystyle \frac{P_q}{{\alpha }_0}},$$

(21)

$$s_L=s_0+n_L\left(f_0-{\left(\lambda \left(q\right)\right)}^{-1}\right),$$

(22)

where $s=t-z$ and index L takes the values I–$IV$ and denotes the number of the region to which the solution corresponds (for I: $q<0$ and $r>R$; for $II$: $q<0$ and $r<R$; for $III$: $q>0$ and $r>R$; for $IV$: $q>0$ and $r<R$), index i takes the values 0 and 1 ($i=0$ corresponds to the case $r_{,\tau }>0$; $i=1$ corresponds to the case $r_{,\tau }<0$):

$$U_I^0=U_{II}^1=U_{III}^1=U_{IV}^0=f_0\left(1+{\alpha }_o{\displaystyle \frac{P_r}{P_q}}{\displaystyle \frac{c_0}{{\gamma }^2}}\right),$$

(23)

$$U_I^1=U_{II}^0=U_{III}^0=U_{IV}^1=-f_0{\alpha }_o{\displaystyle \frac{P_r}{P_q}}{\displaystyle \frac{c_0}{{\gamma }^2}},$$

(24)

$$h_I^0=h_{II}^1=h_{III}^1=h_{IV}^0=-{\alpha }_o{\displaystyle \frac{P_r}{P_q}}{\displaystyle \frac{c_0}{{\gamma }^2}},$$

(25)

$$h_I^1=h_{II}^0=h_{III}^0=h_{IV}^1={\alpha }_o{\displaystyle \frac{P_r}{P_q}}{\displaystyle \frac{c_0}{{\gamma }^2}},$$

(26)

$$n_I=n_{II}=-n_{III}=-n_{IV}={\alpha }_o{\left({\displaystyle \frac{P_r}{P_q}}\right)}^2{\displaystyle \frac{c_0}{{\gamma }^2}},$$

(27)

$P_r$, $P_q$, and $s_0$ are constants.

From the given equalities, it is clear that the evolutionary parameter $\gamma$ is included both in the metric functions that determine the gravitational field TToMSF (see equalities (10)–(15)) and in equalities (20)–(22), which describe the motion of the test null string in the gravitational field of TToMSF.

4. Evolution of Dark Energy in the TToMSF Gas Model

It can be noted that for $q\to \pm \infty$, taking into account (7), the right side of Equation (20) takes on unlimited values. At the same time, the left side of Equation (20) can take values only in a limited interval $(0,f_0)$ (see (8)). In this case, Equation (20) defines the limits of change of the variable q and therefore limits the possible values of the parameter $\tau$ (the dependence of the variable q on the parameter $\tau$ is determined by the equality (21)), as well as the possible values of the variable s (see Equation (22)). As a consequence of the above, the motion of the test null string will be determined only within a limited region of space–time. This limited region of space–time was called the “interaction zone”.

The boundaries of the interaction zone are determined by the maximum possible values of the left side of Equation (20). In regions I and $II$, the left boundary of the interaction zone is realized ($f_I^1=f_{II}^0\to f_0$), and in regions $III$ and $IV$, respectively, the right boundary of the interaction zone is realized ($f_{III}^0=f_{IV}^1\to f_0$). These limits are reached when

$$\lambda \left(q\right)\to {\displaystyle \frac{1}{f_0}}{\left(1+{\displaystyle \frac{{\gamma }^2}{{\alpha }_o{c}_0}}{\displaystyle \frac{P_q}{P_r}}\right)}^{-1}.$$

(28)

From Equation (28) it is directly evident that

$$\underset{\gamma \to 0}{lim}\lambda \left(q\right)\to {\displaystyle \frac{1}{f_0}}.$$

(29)

For (9), the earliest possible moment of formation of TToMSF corresponds to the value $\gamma =\sqrt{2/\chi }$, and the values $\gamma \to 0$ correspond to the latest moments of the evolution of the Universe. From Equation (28), it follows that as the value of the constant $\gamma$ decreases, the “width” (size) of the interaction zone also decreases. Comparing the asymptotic equalities (7) and (29), one can notice that for values $\gamma \to 0$, the size of the interaction zone also tends to zero.

The studies conducted in [5,6,7,30] allow us to identify a number of quantities (processes) that influence the evolution of TToMSF gas:

$P1.$ TToMSF gas density (degree of rarefaction). For the rarefied TToMSF gas (if we do not take into account the features of gravitational interaction in the gas), the parameter of the equation of state is $w\to -1$, and for the compressed gas, it is $w\to 1$.

$P2.$ Thickness (core radius) of thin tubes of a massless scalar field. The thinner the tubes that form the gas, the faster, upon expansion, this gas can reach the equation of state with the parameter $w\to -1$ (if we do not take into account the features of gravitational interaction in the gas).

$P3.$ The moment of formation and evolution of a scalar field (the value of the constant $\gamma$ in the distribution function (5)). In [6], it is shown that the smaller the value of the constant $\gamma$ in (5), the faster the TToMSF gas will reach the equation of state with the parameter $w\to -1$ (if we do not take into account the features of gravitational interaction in the gas).

$P4.$ Features of gravitational interaction in TToMSF gas. This feature is related to the fact that two TToMSFs, having different spatial forms, can exist in the gas only outside the zone of interaction between each other [30]. Depending on the initial conditions, the size of the interaction zone for each TToMSF in gas can be different. At the same time, there are always some critical values of the initial conditions for which the interaction zone for an individual TToMSF in the gas can occupy the entire space. In this case, the impermeability and size (“width”) of the interaction zones, for TToMSFs having different spatial forms, may be the reason for the realization of “strong” gravitational repulsion in such a gas and, as a consequence, be the source of long-term accelerated expansion of the TToMSF gas [30]. The phase of “strong” gravitational repulsion is realized when the total volume of impenetrable interaction zones for TToMSFs in the gas significantly exceeds the size of the region that contains the gas.

Taking into account the above, the total value of the equation of state parameter for TToMSF gas can be represented as

$$w=w_{P1;P2;P3}+w_{P4},w_{P1;P2;P3}>-1,w_{P4}<0,$$

(30)

where $w_{P1;P2;P3}$ is a parameter of the equation of state of the TToMSF gas that does not take into account the features of gravitational interaction (taking into account the influence of the parameters $P1,P2,$ and $P3$), and $w_{P4}$ is a parameter that takes into account the features of gravitational interaction in the TToMSF gas.

5. Discussion

It can be noted that Equation (30) allows for the realization of the phantom dark energy phase in the TToMSF gas. At the same time, in the process of evolution, the contribution of the parameter $w_{P4}$ to the total value of the parameter of the equation of state (30) can only decrease. Reasons leading to the decrease in $|w_{P4}|$ include the following:

• When expanding, the volume of the region that contains the gas increases. In this case, the total volume of impermeable interaction zones for TToMSFs in gas will gradually approach the increasing volume of the region that contains the gas. As a consequence, starting from some point in time, the volume of the region that contains the gas may exceed the total volume of impermeable interaction zones in this gas, which will lead to $w_{P4}=0$.
• In the process of evolution, the value of the constant $\gamma$ in function (5) decreases, i.e., the size (“width”) of the interaction zones for TToMSFs in gas decreases, which accelerates the approach of the moment when $w_{P4}=0$.

As a consequence, during evolution, the phantom component of dark energy in the TToMSF gas can only decrease and, starting from a certain point in time, the total value of the parameter of the equation of state for the TToMSF gas can satisfy the inequality $w>-1$. The obtained result is consistent with recently published observational data from DESI (Dark Energy Spectroscopic Instrument), according to which the preferred $w\left(z\right)$ shows a phase $w>-1$ at low redshifts and a phantom crossing with $w<-1$ above redshifts of $z\simeq 0.4$ [31].