Narlikar Gravity Model

• Narlikar Gravity Model is a gravitational theory that supplements the Einstein–Hilbert action with a massless scalar creation field to dynamically exchange energy with matter.
• It incorporates Weyl invariance and non-local modifications to achieve quantum finiteness and resolve classical singularities in black holes and cosmological models.
• Observational diagnostics in FLRW cosmologies show that these models can mimic dark energy and partially reconcile cosmic acceleration and Hubble tension.

Narlikar Gravity Model refers to a class of gravitational theories inspired by or directly developed from the original work of Narlikar and collaborators, encompassing both the classical Hoyle–Narlikar theory—which introduces a matter creation field as a dynamical degree of freedom in cosmology—and subsequent extensions incorporating conformal (Weyl) invariance, non-locality, and quantum finiteness. These theories modify the standard Einstein–Hilbert action through the inclusion of a scalar “creation” field or compensator field, yielding models with novel cosmological dynamics, singularity resolution, and alternative mechanisms for cosmic acceleration.

1. Theoretical Foundations and Action Principles

The canonical Hoyle–Narlikar gravity model supplements the Einstein–Hilbert action with a massless scalar creation field $\mathcal{C}$, interacting non-minimally with ordinary matter. The action in covariant form is: $$S = \int d^4x\,\sqrt{-g}\;\left[\frac{1}{16\pi G}\,R + \mathcal{L}_\mathrm{m}(g_{ij},\Psi) - \frac{s\,\zeta}{2}\,g^{ij}\,\nabla_i\mathcal{C}\,\nabla_j\mathcal{C}\right]$$ where $R$ is the Ricci scalar, $\mathcal{L}_\mathrm{m}$ is the matter Lagrangian, $\zeta$ is the coupling constant, and $s=\pm1$ tracks the sign of the $\mathcal{C}$-field kinetic term. The field equations read: $$G_{ij} = -8\pi G \left[T_{ij}^{(m)} + T_{ij}^{(\mathcal{C})}\right]$$ with the energy–momentum tensors for matter and creation field specified as

$$T_{ij}^{(m)} = (\rho + p)u_i u_j + p\,g_{ij} \quad ; \quad T_{ij}^{(\mathcal{C})} = -s\,\zeta\left(\nabla_i\mathcal{C}\,\nabla_j\mathcal{C} - \frac{1}{2}g_{ij}\nabla^k\mathcal{C}\,\nabla_k\mathcal{C}\right)$$

The equation of motion for $\mathcal{C}$ encodes energy exchange between matter and the creation field: $$S = \int d^4x\,\sqrt{-g}\;\left[\frac{1}{16\pi G}\,R + \mathcal{L}_\mathrm{m}(g_{ij},\Psi) - \frac{s\,\zeta}{2}\,g^{ij}\,\nabla_i\mathcal{C}\,\nabla_j\mathcal{C}\right]$$0 which, under cosmological symmetry, leads to a continuity equation for energy densities with a source (or sink) term controlled by $$S = \int d^4x\,\sqrt{-g}\;\left[\frac{1}{16\pi G}\,R + \mathcal{L}_\mathrm{m}(g_{ij},\Psi) - \frac{s\,\zeta}{2}\,g^{ij}\,\nabla_i\mathcal{C}\,\nabla_j\mathcal{C}\right]$$1 (Singh et al., 13 Oct 2025).

In bulk viscous extensions, the action takes the form (Yadav et al., 18 Sep 2025)

$$S = \int d^4x\,\sqrt{-g}\;\left[\frac{1}{16\pi G}\,R + \mathcal{L}_\mathrm{m}(g_{ij},\Psi) - \frac{s\,\zeta}{2}\,g^{ij}\,\nabla_i\mathcal{C}\,\nabla_j\mathcal{C}\right]$$2

with $$S = \int d^4x\,\sqrt{-g}\;\left[\frac{1}{16\pi G}\,R + \mathcal{L}_\mathrm{m}(g_{ij},\Psi) - \frac{s\,\zeta}{2}\,g^{ij}\,\nabla_i\mathcal{C}\,\nabla_j\mathcal{C}\right]$$3 denoting coupling between the creation field $$S = \int d^4x\,\sqrt{-g}\;\left[\frac{1}{16\pi G}\,R + \mathcal{L}_\mathrm{m}(g_{ij},\Psi) - \frac{s\,\zeta}{2}\,g^{ij}\,\nabla_i\mathcal{C}\,\nabla_j\mathcal{C}\right]$$4 and gravity.

2. Weyl Invariance, Quantum Finiteness, and Singularity Resolution

A significant extension is the conformal (Weyl-invariant) gravity model, as reviewed by Modesto and Rachwał, which realizes anomaly-free, finite quantum gravity in arbitrary dimension (Modesto et al., 2016). The action, recast in terms of the compensator $$S = \int d^4x\,\sqrt{-g}\;\left[\frac{1}{16\pi G}\,R + \mathcal{L}_\mathrm{m}(g_{ij},\Psi) - \frac{s\,\zeta}{2}\,g^{ij}\,\nabla_i\mathcal{C}\,\nabla_j\mathcal{C}\right]$$5 and the conformal metric $$S = \int d^4x\,\sqrt{-g}\;\left[\frac{1}{16\pi G}\,R + \mathcal{L}_\mathrm{m}(g_{ij},\Psi) - \frac{s\,\zeta}{2}\,g^{ij}\,\nabla_i\mathcal{C}\,\nabla_j\mathcal{C}\right]$$6, is

$$S = \int d^4x\,\sqrt{-g}\;\left[\frac{1}{16\pi G}\,R + \mathcal{L}_\mathrm{m}(g_{ij},\Psi) - \frac{s\,\zeta}{2}\,g^{ij}\,\nabla_i\mathcal{C}\,\nabla_j\mathcal{C}\right]$$7

with entire-function form factors $$S = \int d^4x\,\sqrt{-g}\;\left[\frac{1}{16\pi G}\,R + \mathcal{L}_\mathrm{m}(g_{ij},\Psi) - \frac{s\,\zeta}{2}\,g^{ij}\,\nabla_i\mathcal{C}\,\nabla_j\mathcal{C}\right]$$8 ensuring ghost-freedom, and “killer” curvature polynomials tuning all beta functions to vanish.

The spontaneous breaking of Weyl invariance via gauge fixing $$S = \int d^4x\,\sqrt{-g}\;\left[\frac{1}{16\pi G}\,R + \mathcal{L}_\mathrm{m}(g_{ij},\Psi) - \frac{s\,\zeta}{2}\,g^{ij}\,\nabla_i\mathcal{C}\,\nabla_j\mathcal{C}\right]$$9 to a constant reduces the physical degrees of freedom analogously to the Higgs mechanism.

The introduction of explicit, local Weyl symmetry and the compensator field $R$0 allows mapping classical solutions with curvature singularities (Schwarzschild, FRW, BKL cosmologies) into regular, geodesically complete spacetimes via appropriate conformal transformations (Modesto et al., 2016).

3. Cosmological Dynamics and Observational Constraints

In FLRW cosmology, the inclusion of a creation field modifies the Friedmann equations. For the Hoyle–Narlikar model (Singh et al., 13 Oct 2025, Yadav et al., 18 Sep 2025): $R$1

$R$2

In the presence of bulk viscosity and for the ancillary field ansatz $R$3, with $R$4 constant, the resulting Friedmann and continuity equations are further modified by $R$5 and viscous terms.

Parameter fits using Hubble, Pantheon SNe Ia, and BAO data yield

• $R$6 km s$R$7 Mpc$R$8,
• $R$9,
• $\mathcal{L}_\mathrm{m}$0 (deceleration–acceleration transition),
• Universe age $\mathcal{L}_\mathrm{m}$1 Gyr (Yadav et al., 18 Sep 2025).

The creation field effectively behaves as a dark-energy component, with its kinetic term providing late-time cosmic acceleration, and the energy exchange $\mathcal{L}_\mathrm{m}$2 ensuring the field dynamically mimics an asymptotic cosmological constant. Observationally, the best-fit $\mathcal{L}_\mathrm{m}$3 partially alleviates the Hubble tension, outperforming $\mathcal{L}_\mathrm{m}$4CDM in correlating $\mathcal{L}_\mathrm{m}$5, $\mathcal{L}_\mathrm{m}$6, and $\mathcal{L}_\mathrm{m}$7 (Singh et al., 13 Oct 2025, Yadav et al., 18 Sep 2025).

4. Black Hole and Singularity Structure

A central prediction of the finite conformal model is the existence of exact, singularity-free black hole solutions. For example, starting from the Schwarzschild metric $\mathcal{L}_\mathrm{m}$8, a conformal rescaling with

$\mathcal{L}_\mathrm{m}$9

produces a one-parameter family of regular spacetimes with all curvature invariants finite as $\zeta$0: $\zeta$1

$\zeta$2

Radial geodesics take infinite proper time to reach $\zeta$3; the resulting Penrose diagram shares the causal structure of Schwarzschild but lacks a singular boundary. Similarly, all FRW cosmologies are conformally equivalent to flat spacetimes, rendering the big bang singularity a pure gauge artifact in this framework (Modesto et al., 2016).

A key theorem proves that weak non-locality without explicit Weyl symmetry does not remove singularities: Ricci-flat solutions and traceless-matter FRW solutions of the nonlocal action remain singular, underscoring the necessity of the conformal compensator for singularity resolution (Modesto et al., 2016).

5. Comparison with ΛCDM and Observational Diagnostics

Relative to classical $\zeta$4CDM, where $\zeta$5, the creation field provides a dynamical alternative to the cosmological constant:

• The creation sector acts as an effective dark energy, with its equation of state evolving toward $\zeta$6 at late times.
• $\zeta$7–$\zeta$8 phase-space analysis in both the canonical (Singh et al., 13 Oct 2025) and viscous (Yadav et al., 18 Sep 2025) Narlikar models reveals thawing and freezing behavior, with all trajectories converging to the $\zeta$9CDM fixed point $s=\pm1$0.
• The transition redshift, present deceleration parameter, and cosmological parameters returned by these models are consistent with late-universe cosmic acceleration and are viable under current cosmological data, maintaining a stable attractor and satisfying all necessary energy conditions except for SEC (violated as required for acceleration).

The table below summarizes key fit parameters from (Singh et al., 13 Oct 2025, Yadav et al., 18 Sep 2025):

Dataset	$s=\pm1$1 [km s$s=\pm1$2 Mpc$s=\pm1$3]	$s=\pm1$4	$s=\pm1$5 (transition)
H(z)	72.00	0.40	$s=\pm1$60.58
Pantheon$s=\pm1$7 + BAO	72.00	0.398	--
OHD + Pantheon (viscous)	71.2 $s=\pm1$8 2.1	0.41	0.63

These results indicate a partial mitigation of the Hubble tension and robust compatibility with late-time acceleration.

6. Physical Implications and Model Stability

The Narlikar gravity class of models demonstrates several physically significant predictions:

• Generic resolution of classical singularities is achieved through conformal (Weyl) invariance and a dynamical scalar compensator field.
• All black holes are non-singular; infalling observers never reach curvature singularities in finite proper time.
• The big bang singularity in standard cosmology is mapped to a regular, conformally flat spacetime.
• Observational deviations from GR in weakly curved regimes are exponentially suppressed but may become relevant in high-curvature environments such as black-hole ringdown or early-universe epochs—testable via Planck-scale dispersion and ringdown signatures.
• Stability of the cosmological solutions is ensured by positivity of sound speed, absence of ghosts, and convergence to de Sitter attractor behavior; NEC and DEC are satisfied throughout, with SEC violation at late times precisely as needed for cosmic acceleration (Singh et al., 13 Oct 2025, Yadav et al., 18 Sep 2025).

A plausible implication is that Narlikar gravity models, especially those incorporating conformal invariance and nonlocality, offer a framework in which all classical singularities are generically resolved without resorting to exotic-matter sources or spacetime discreteness (Modesto et al., 2016).

7. Extensions, Open Problems, and Connections

The conformally finite quantum gravity models extend the original Narlikar paradigm by embedding it in a perturbatively unitary, anomaly-free setting, utilizing nonlocal operators to render the theory UV-complete and singularity-free (Modesto et al., 2016). This upgrade incorporates a finite number of “killer” terms in the action to set all beta functions to zero, ensuring the absence of conformal anomaly in all relevant spacetime dimensions.

Empirical application to cosmology via parameter fits demonstrates the flexibility and viability of the framework. Ongoing work investigates further observational consequences, constraints on creation field coupling constants, bulk-viscous extensions, and high-curvature strong-field predictions that can distinguish these models from $s=\pm1$9CDM.

The Narlikar gravity model thus provides both a technically robust and physically distinct alternative to standard approaches, merging classical singularity resolution, quantum finiteness, and observational viability within a unified scalar-tensor–conformal framework (Modesto et al., 2016, Singh et al., 13 Oct 2025, Yadav et al., 18 Sep 2025).