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, H0_, and parameter S8, which characterizes the measure of growth of cosmic structure. One of the possible modifications of ΛCDM, capable of resolving the tensions H0 and S8, is associated with the development of dark energy (DE) models, in which the equation of state parameter w can influence the growth rate of the cosmological structure. Models with this property include, for example, the gas of thin tubes of a massless scalar field (TToMSF), since the parameter of its equation of state $w$ depends on the density of the gas, namely, for rarefied gas TToMSF w→-1 and for compressed gas w→1. In this case, changes in the density of the TToMSF gas, which may be associated, for example, with its gravitational interaction with non-uniformly distributed baryonic matter, during the period of baryon acoustic oscillations (BAOs), should lead to the implementation of local equations of state with different values and different laws of change of parameter w.

In this work, we investigated how the trajectories of a test null string change as it moves 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.