Stable soliton dark matter wormhole in non-minimally coupled $f({\cal Q},{\cal T})$ gravity

TL;DR

This work demonstrates that non-minimally coupled f({\cal Q},{\cal T}) gravity can yield stable, traversable wormholes that do not require exotic matter, by exploiting a linear coupling {\beta}{\cal T} and realistic DM density profiles. By deriving two DM-based shape functions (soliton core + NFW halo) under the linear theory {f({\cal Q},{\cal T})={\cal Q}+{\beta}{\cal T}}, the authors show that the flaring-out condition can be satisfied while matter obeys standard energy conditions for suitably chosen {\beta} (often {\beta}<-{1\over2}), with the NEC violation appearing effectively in the total (effective) description. A modified Tolman–Oppenheimer–Volkoff equation reveals a new coupling force that, together with hydrostatic and anisotropic forces, can dynamically stabilize the wormhole. The study also discusses observational probes—gravitational lensing, shadows, and gravitational-wave echoes—that could distinguish these wormholes from black holes, and provides explicit parameter ranges for {\beta} in two DM scenarios. Overall, the paper offers a concrete route to healthy wormholes in a physically motivated gravity theory, tied to DM halo structure and testable via astrophysical observations.

Abstract

We show that non-minimal coupling between matter and geometry can indeed help in constructing stable, traversable, wormholes (WHs) without requiring exotic matter under certain conditions. In models like $f({\cal Q},{\cal T})={\cal Q}+β{\cal T}$ gravity, where ${\cal Q}$ is the non-metricity scalar, and ${\cal T}$ is the trace of the energy-momentum tensor, the coupling between matter and geometry introduces additional degrees of freedom in terms of the parameter $β$. These can mimic the effects of exotic matter or even replace it entirely under specific parameter choice. The analysis involves deriving WH shape functions based on two dark matter (DM) density profiles: a solitonic core at the center of DM halos, and the outer halo follows the universal Navarro-Frenk-White (NFW) density profile of cold DM (CDM). The wormhole solutions derived in these models satisfy important geometric conditions like: Flaring-out condition (necessary for traversability) and asymptotic flatness condition. For large positive coupling parameter, the null energy condition (NEC) can be satisfied at the wormhole throat, meaning exotic matter is not needed, while the wormhole is no longer Lorentzian and the flaring-out condition is broken. However, for large negative coupling parameter, the NEC can be satisfied, allowing for healthy wormholes without exotic matter, provided the coupling strength remains within certain bounds. In the latter case, the NEC is broken only effectively. We investigate the stability of the obtained wormhole solutions by virtue of a modified version of Tolman-Oppenheimer-Volkoff (TOV) equation, which includes a new force due to matter-geometry non-minimal, showing that these wormholes can be dynamically stable.

Figures (12)

• Figure 1: Model I: \ref{['fig:space1']} The parameter space $\{r_0,\delta\}$ of the soliton DM core WH model of galaxy NGC 2366 ($\sigma_c=15 \times 10^{-3}~M_\odot$/pc$^{3}$ and $r_c=3$ kpc Banares-Hernandez:2023axy). We use the faring out constraint on the shape function at the WH throat $h'(r_0)<1$ to evaluate critical values of the non-minimal coupling parameter $\beta$, namely \ref{['eq:model1_beta1']} and \ref{['eq:model1_beta2']}, at arbitrary throat size $0<r_0<r_c$ where $r_c$ is the soliton core radius of galaxy NGC 2366. Given that $\beta\neq -3/8$, we show both cases when $\beta<-3/8$ ($\delta<0$) and $\beta>-3/8$ ($\delta>0$). The positive (negative) $\delta$ curve is multiplied by a factor $10^{-9}$ ($10^{9}$) to fit the curve into the scale of the graph. \ref{['fig:flaring1']} The flaring-out condition at the WH throat is satisfied for different selected values of the parameter $\beta$ where the WH throat is arbitrarily chosen $0<r_0<r_c$, it shows that $h'(r_0)<1$ is fulfilled.
• Figure 2: Model I: WH embedding diagrams: \ref{['fig:shapefn1']} The plots show that $h(r)<r$ and $h'(r)<1$ at $r>r_0$, while $h/r\to 0$ and $h'\to 0$ as $r\to \infty$. \ref{['fig:2demedding1']} The embedding surface integral, Eq. \ref{['eq:surface_int']}, shows that $z'(r_0)\to \infty$, $z(r>r_0)$ is finite and $z(r)\to \infty$ as $r\to \infty$. \ref{['fig:3demedding1']} Since $z'(r_0)\to \infty$, then $z(r)$ is vertical in the 3D embedding diagram. We set $r_0=1$ pc, for the galaxy NGC 2366, where $\sigma_c=15 \times 10^{-3}~M_\odot$/pc$^{3}$ and $r_c=3$ kpc Banares-Hernandez:2023axy.
• Figure 3: Model II: \ref{['fig:space1']} The parameter space $\{r_0,\delta\}$ of the NFW CDM WH model of galaxy NGC 2366 ($\sigma_s=3.11 \times 10^{-3}~\text{M}_{\odot}/\text{pc}^3$ and $r_s=1.447$ kpc). We use the faring out constraint on the shape function at the WH throat $h'(r_0)<1$ to evaluate critical values of the non-minimal coupling parameter $\beta$, namely \ref{['eq:model2_beta1']} and \ref{['eq:model2_beta2']}, at arbitrary throat size $0<r_0<r_c$ where $r_c$ is the soliton core radius of galaxy NGC 2366. Given that $\beta\neq -3/8$, we show both cases when $\beta<-3/8$ ($\delta<0$) and $\beta>-3/8$ ($\delta>0$). The positive (negative) $\delta$ curve is multiplied by a factor $10^{-10}$ ($10^{10}$) to fit the curve into the scale of the graph. \ref{['fig:flaring2']} The flaring-out condition at the WH throat is satisfied for different selected values of the parameter $\beta$ where the WH throat is arbitrarily chosen $0<r_0<r_s$, it shows that $h'(r_0)<1$ is fulfilled.
• Figure 4: Model II: WH embedding diagrams: \ref{['fig:shapefn2']} The plots show that $h(r)<r$ and $h'(r)<1$ at $r>r_0$, while $h/r\to 0$ and $h'\to 0$ as $r\to \infty$. \ref{['fig:2demedding2']} The embedding surface integral, Eq. \ref{['eq:surface_int']}, shows that $z'(r_0)\to \infty$, $z(r>r_0)$ is finite and $z(r)\to \infty$ as $r\to \infty$. \ref{['fig:3demedding2']} Since $z'(r_0)\to \infty$, then $z(r)$ is vertical in the 3D embedding diagram. We set $r_0=1$ pc, for the galaxy NGC 2366, $\sigma_s=3.11 \times 10^{-3}~\text{M}_{\odot}/\text{pc}^3$ and ${r_s} = 1.447$ kpc Banares-Hernandez:2023axy.
• Figure 5: Model I with $\beta=-0.6$: \ref{['fig:model1_fluid']} The matter fluid, the density has a flat profile at the core as suggested to solve the core-cusp problem with $p_r>0$ and $p_\theta<0$. \ref{['fig:model1_MEC']} The matter energy conditions, where the NEC $\sigma c^2+p_r>0$ is satisfied and $\sigma c^2+p_\theta<0$ at $r>r_0$. \ref{['fig:model1_EEC']} The effective energy conditions, where the NEC $\tilde{\sigma} c^2+\tilde{p}_r>0$ is broken, but $\tilde{\sigma} c^2+\tilde{p}_\theta>0$ is satisfied at $r>r_0$. The alternative behavior of the NEC in the matter and the effective sector is understood since $\beta=-0.6<-1/2$, see Eqs. \ref{['eq:linearEC1']} and \ref{['eq:linearEC2']}. Other energy conditions, namely SEC and DEC, are broken in both sectors. We set $r_0=1$ pc, for the galaxy NGC 2366, where $\sigma_c=15 \times 10^{-3}~M_\odot$/pc$^{3}$ and $r_c=3$ kpc Banares-Hernandez:2023axy.