ASG-Corrected Wormholes in Dark Matter Halos

• ASG-corrected wormholes are traversable solutions that incorporate a scale-dependent gravitational coupling, altering the classical spacetime structure near the throat.
• The integration of a dark matter halo with ASG corrections mitigates destabilizing forces and enhances curvature, ensuring a stable, realistic wormhole configuration.
• Observable signatures such as an increased shadow radius offer a direct probe of quantum gravity effects in strong-field astrophysical environments.

An ASG-corrected wormhole is a traversable wormhole solution constructed within the framework of Asymptotically Safe Gravity, wherein the gravitational dynamics are governed by a scale-dependent (running) Newton coupling $G(k)$ as determined by the renormalization group flow, often in the presence of astrophysically realistic matter such as a dark matter halo. The quantum corrections from ASG, parameterized by a characteristic scale $\xi$, are incorporated directly into the field equations, fundamentally altering the geometric, energy-condition, and stability properties of the resulting spacetime. In the presence of a dark matter halo, these corrections can yield solutions that are physically viable, astrophysically relevant, and potentially distinguishable via observables such as the shadow radius in strong-field imaging.

1. Renormalization Group Improved Gravity and the Role of the Running Coupling

In Asymptotically Safe Gravity, the gravitational interaction strength is not static but runs with an energy or length scale, introducing quantum gravitational corrections at all scales. The renormalization group (RG) flow for the Newton coupling in the infrared (IR) regime is typically summarized by

$G(k) = \frac{G_0}{1 + \omega G_0 k^2}$

where $G_0$ is the classical Newton constant and $\omega$ encodes the running induced by quantum corrections. For astrophysical (low energy) applications, the RG scale $k$ is customarily identified as an inverse radial scale, $k(r) = \xi/r$, with $\xi$ as a phenomenological quantum gravity parameter that sets the characteristic length scale of the corrections. This identification leads to a position-dependent gravitational strength: $G(r) = \frac{G_0 r^2}{r^2 + \xi^2}$ Thus, in the deep infrared ($r\gg\xi$), $G(r)\to G_0$, while in the strong-field regime near the wormhole throat ($r\lesssim\xi$), $G(r)$ is suppressed, modifying the classical spacetime structure.

2. Wormhole Solutions in a Realistic Halo: Field Equations, Geometry, and Energy Conditions

The classical spherically symmetric traversable wormhole metric,

$$ds^2 = -e^{2\Phi(r)}dt^2 + \frac{dr^2}{1-S(r)/r} + r^2 (d\theta^2 + \sin^2\theta d\phi^2)$$

is sourced by a halo, here modeled specifically by the astrophysically motivated Dekel--Zhao dark matter density profile. ASG corrections enter directly both through the field equations (by the replacement $G_0\to G(r)$) and via their effect on the energy density and curvature.

The tt-component of the Einstein equations becomes

$S'(r)/r^2 = 8\pi G(r) \rho(r)$

with $S(r)$ determined by integrating $G(r)\rho(r)$. The redshift function is chosen to ensure traversability and regularity, e.g.,

$\Phi(r) = 2\ln\frac{r}{r + r_0}$

for a throat at $r_0$. The flare-out condition, $S'(r_0)\leq 1$, required for a traversable throat, is enforced by appropriate choice of the halo profile and the running parameter $\xi$.

ASG corrections necessarily enhance the curvature near the throat due to the spatial dependence of $G(r)$. This enhancement manifests as a sharper and deeper negative peak in the Ricci scalar in the throat region as $\xi$ increases, a direct reflection of quantum effects becoming dominant in high-curvature domains.

Despite these corrections, the Null Energy Condition (NEC) is always violated at the throat in these models, as required for traversability. The energy-momentum tensor is determined by the dark matter halo; in quantum-improved models, the violation occurs due to the combined effect of the running coupling and the matter source.

3. Impact of Quantum Corrections: The Role and Effects of $\xi$

The parameter $\xi$ serves as a control over the strength and spatial extent of the ASG corrections:

• For $r\gg\xi$, the solution reverts to the classical (Newtonian) regime; quantum corrections are negligible here.
• For $r \sim \xi$, the suppression of $G(r)$ leads to a concentrated region of enhanced curvature around the throat and modifies the shape function such that the wormhole remains traversable even in the presence of realistic matter.

Larger values of $\xi$ correspond to stronger quantum corrections. The modification to $G(r)$ influences both the geometry and the stress-energy tensor, effectively “diluting” gravity near the quantum throat and leading to the observed enhancement in curvature scalars. This “repulsive” quantum gravity behavior acts to partially counterbalance the gravitational attraction of the dark matter halo and thus supports the existence of a stable throat.

4. Wormhole Stability: Adiabatic Sound Speed and Quantum Forces in the TOV Equation

Stability of the wormhole configuration is analyzed both via the average adiabatic sound speed and a modified Tolman–Oppenheimer–Volkoff (TOV) equation adapted to a variable gravitational coupling.

The average squared adiabatic sound speed,

$$\langle v_s^2 \rangle = \frac{d\langle P \rangle}{d\rho} = \frac{d\langle P \rangle / dr}{d\rho / dr}, \quad \langle P \rangle = \frac{P_r + 2P_t}{3}$$

must satisfy $0 \leq \langle v_s^2 \rangle < 1$ for dynamical stability. Quantum corrections (increasing $\xi$) raise $\langle v_s^2 \rangle$, expanding the region of stability even as the presence of dark matter would otherwise tend to destabilize the solution (by lowering $\langle v_s^2 \rangle$).

The modified TOV equation in the presence of $G(r)$ reads: $$- G'(r) P_r + G(r) \left[ -(\rho + P_r)\Phi' - P_r' + \frac{2}{r}(P_t - P_r) \right]= 0$$ The new term $\mathcal{V}_Q = -G'(r) P_r$ functions as a quantum pressure gradient that counteracts destabilizing forces from the matter sector. Increasing $\xi$ amplifies $\mathcal{V}_Q$ near the throat, further stabilizing the configuration.

5. Observational Signatures: Shadow Radius and Astrophysical Implications

A notable phenomenological prediction concerns the size of the wormhole shadow, a quantity directly accessible to black hole imaging observations (e.g., Event Horizon Telescope for Sgr A$^*$). In the ASG-corrected wormhole, the critical photon orbit (photon sphere) coincides with the throat. The corresponding shadow radius $R_{\mathrm{sh}}$ is found to increase nearly linearly with $\xi$, a direct reflection of the quantum corrections' effect on spacetime geometry.

For fiducial values of the dark matter parameters (as in the NFW limit), requiring the shadow radius to lie within the EHT range $4.55M < R_{\mathrm{sh}} < 5.22M$ for Sgr A$^*$ implies that $\xi/M \simeq 0.8$–$0.9$; this compatibility suggests that observational constraints on $\xi$ are in principle accessible with current or near-future data.

Such a linear relationship between $\xi$ and the shadow radius thus presents a potential observational test of strong–field quantum gravity effects in the Galactic Center and similar environments.

6. Synthesis and Theoretical Implications

ASG-corrected wormholes embedded in a realistic dark matter halo exhibit a suite of distinctive features:

• The running gravitational coupling $G(r)$ modifies both the throat geometry and global spacetime structure.
• Enhanced curvature near the throat, regulated by $\xi$, acts as a quantum-gravitational “stabilizer.”
• Although the NEC remains violated at the throat (as required), quantum corrections mitigate the destabilizing effects of dark matter and can, via the quantum force term in the TOV equation, support the maintenance of traversability.
• The shadow radius encodes direct information about $\xi$ and is therefore, in principle, a probe of Planck-suppressed quantum gravity physics at astrophysical scales.

These models establish a direct bridge between the renormalization group formulation of quantum gravity and empirical observables, demonstrating a pathway for direct phenomenological investigation of ASG-corrected spacetimes.

Table: Key Dependencies in ASG-Corrected Wormhole Phenomenology

Feature	Dependency	Quantum Correction Impact (via $\xi$)
Gravitational Coupling $G(r)$	$G_0 r^2 / (r^2 + \xi^2)$	Suppressed at $r \lesssim \xi$
Throat Curvature	$S(r)$, $G(r)$, halo density	Enhanced for larger $\xi$
NEC Violation	Matter profile, quantum-corrected geometry	Always required at the throat, but mitigated
Stability	$\langle v_s^2 \rangle$, modified TOV eqn	Improved for increased $\xi$
Shadow Radius $R_{\mathrm{sh}}$	$\xi$ and halo profile	Grows $\propto \xi$, observable in EHT

This structure encapsulates the multifaceted impact of ASG corrections in traversable wormhole solutions embedded in realistic astrophysical environments.