Conformal Killing Gravity Cosmology

• Conformal Killing Gravity Cosmology is a framework that reformulates third-order gravity into a second-order Einstein-like system by integrating a divergence-free conformal Killing tensor.
• The modified Friedmann equations introduce an extra geometric dark sector term proportional to a² or (1+z)⁻², allowing interpolation between decelerating matter eras and late-time acceleration.
• Observational fits show matter density close to ΛCDM, while the split between ΩΛ and the geometric dark term leads to distinct future expansion scenarios.

Conformal Killing Gravity cosmology denotes the cosmological sector of Harada’s third-order gravitational theory, later reformulated as Einstein gravity supplemented by a divergence-free conformal Killing tensor. In this framework, the cosmological constant can appear as an integration constant, the field equations can be reduced from third to second order on Robertson–Walker and generalized Robertson–Walker backgrounds, and the additional conformal-Killing contribution behaves as a geometric dark sector rather than as an independently postulated dark-energy fluid. The resulting cosmologies admit modified Friedmann equations with an extra term proportional to $a^{2}$ or, equivalently in the usual redshift parametrization used in several works, $(1+z)^{-2}$; they can interpolate between decelerating matter eras and late-time acceleration, but they also raise selection and consistency problems that remain active subjects of discussion (Mantica et al., 2023, Mantica et al., 2024, Hervik et al., 2024, Altas et al., 10 Feb 2025).

1. Foundational equations and Einstein-type reformulation

Harada’s starting point is a third-order tensor equation

$H_{jkl}=8\pi G\,T_{jkl},$

with

$$H_{jkl} = \nabla_j R_{kl} +\nabla_k R_{lj} +\nabla_l R_{jk} -\frac13\left(g_{kl}\nabla_jR+g_{lj}\nabla_kR+g_{jk}\nabla_lR\right).$$

Because $H_{jkl}$ contains derivatives of the Ricci tensor, the theory is third order in the metric. The central result of Mantica and Molinari is that, in the cosmological setting considered there, this equation can be integrated once and rewritten as

$R_{jk}-\frac12 g_{jk}R = T_{jk}+K_{jk},$

where $K_{jk}$ is an arbitrary symmetric tensor required to be both divergence-free and conformal Killing. In that notation,

$K_{jk}=R_{jk}-\frac12 g_{jk}R-T_{jk}.$

The Bianchi identity then gives

$\nabla^j K_{jk}=0,$

and therefore

$\nabla^j T_{jk}=0,$

so energy-momentum conservation remains a consequence of the field equations in the integrated formulation (Mantica et al., 2023).

This reduction is the decisive conceptual simplification in conformal Killing gravity cosmology. It turns Harada’s equations from third order to second order and places the theory within a standard Einstein-type cosmological workflow, with $(1+z)^{-2}$0 acting as an additional geometric source. In parallel presentations of the theory, Harada’s field equations are also written in trace-modified form as

$(1+z)^{-2}$1

with $(1+z)^{-2}$2 and $(1+z)^{-2}$3. In this language, any GR solution of $(1+z)^{-2}$4 and any Einstein-$(1+z)^{-2}$5 solution of $(1+z)^{-2}$6 also solves conformal Killing gravity, which explains why the theory inherits the standard homogeneous and isotropic sector while extending it by additional integration constants and source ambiguities (Barnes, 2024).

2. Conformal Killing tensors, perfect-fluid form, and Robertson–Walker geometry

A symmetric tensor $(1+z)^{-2}$7 is conformal Killing when it satisfies

$(1+z)^{-2}$8

where $(1+z)^{-2}$9 is the associated conformal vector. If $H_{jkl}=8\pi G\,T_{jkl},$0, one recovers an ordinary Killing tensor; if $H_{jkl}=8\pi G\,T_{jkl},$1, one has a gradient conformal Killing tensor. In the divergence-free case, the contraction formula used in the literature gives

$H_{jkl}=8\pi G\,T_{jkl},$2

with $H_{jkl}=8\pi G\,T_{jkl},$3 (Mantica et al., 2023).

For cosmology, the relevant tensors have perfect-fluid form,

$H_{jkl}=8\pi G\,T_{jkl},$4

or, in the explicit cosmological notation used in the integration theorem,

$H_{jkl}=8\pi G\,T_{jkl},$5

In generalized Robertson–Walker spacetime, such a tensor exists precisely when the spacetime has the GRW structure and the velocity field is torse-forming,

$H_{jkl}=8\pi G\,T_{jkl},$6

with $H_{jkl}=8\pi G\,T_{jkl},$7 in comoving coordinates. This geometrizes the Hubble function directly at the level of the conformal Killing tensor (Mantica et al., 2023).

A more specialized construction introduces Sinyukov-like tensors. In the perfect-fluid ansatz

$H_{jkl}=8\pi G\,T_{jkl},$8

$H_{jkl}=8\pi G\,T_{jkl},$9 is conformal Killing iff the velocity field is shear-free. If it is also divergence-free, then

$$H_{jkl} = \nabla_j R_{kl} +\nabla_k R_{lj} +\nabla_l R_{jk} -\frac13\left(g_{kl}\nabla_jR+g_{lj}\nabla_kR+g_{jk}\nabla_lR\right).$$0

and

$$H_{jkl} = \nabla_j R_{kl} +\nabla_k R_{lj} +\nabla_l R_{jk} -\frac13\left(g_{kl}\nabla_jR+g_{lj}\nabla_kR+g_{jk}\nabla_lR\right).$$1

If, in addition, the flow is acceleration-free, the tensor becomes a special divergence-free CKT of Sinyukov-like type, and the spacetime is generalized RW; with conformal flatness it is RW. In RW spacetime the tensor is fixed by the scale factor $$H_{jkl} = \nabla_j R_{kl} +\nabla_k R_{lj} +\nabla_l R_{jk} -\frac13\left(g_{kl}\nabla_jR+g_{lj}\nabla_kR+g_{jk}\nabla_lR\right).$$2 and two constants $$H_{jkl} = \nabla_j R_{kl} +\nabla_k R_{lj} +\nabla_l R_{jk} -\frac13\left(g_{kl}\nabla_jR+g_{lj}\nabla_kR+g_{jk}\nabla_lR\right).$$3 and $$H_{jkl} = \nabla_j R_{kl} +\nabla_k R_{lj} +\nabla_l R_{jk} -\frac13\left(g_{kl}\nabla_jR+g_{lj}\nabla_kR+g_{jk}\nabla_lR\right).$$4: $$H_{jkl} = \nabla_j R_{kl} +\nabla_k R_{lj} +\nabla_l R_{jk} -\frac13\left(g_{kl}\nabla_jR+g_{lj}\nabla_kR+g_{jk}\nabla_lR\right).$$5 For $$H_{jkl} = \nabla_j R_{kl} +\nabla_k R_{lj} +\nabla_l R_{jk} -\frac13\left(g_{kl}\nabla_jR+g_{lj}\nabla_kR+g_{jk}\nabla_lR\right).$$6,

$$H_{jkl} = \nabla_j R_{kl} +\nabla_k R_{lj} +\nabla_l R_{jk} -\frac13\left(g_{kl}\nabla_jR+g_{lj}\nabla_kR+g_{jk}\nabla_lR\right).$$7

This is the tensorial origin of the cosmological dark term in the Robertson–Walker sector (Mantica et al., 2024).

3. Modified Friedmann equations and the geometric dark sector

With RW geometry and ordinary matter

$$H_{jkl} = \nabla_j R_{kl} +\nabla_k R_{lj} +\nabla_l R_{jk} -\frac13\left(g_{kl}\nabla_jR+g_{lj}\nabla_kR+g_{jk}\nabla_lR\right).$$8

the conformal-Killing contribution converts the Einstein equation into modified Friedmann relations. In the notation with $$H_{jkl} = \nabla_j R_{kl} +\nabla_k R_{lj} +\nabla_l R_{jk} -\frac13\left(g_{kl}\nabla_jR+g_{lj}\nabla_kR+g_{jk}\nabla_lR\right).$$9 restored, Mantica and Molinari derive

$H_{jkl}$0

and

$H_{jkl}$1

Here $H_{jkl}$2 is the scalar curvature of the spatial slices, $H_{jkl}$3, $H_{jkl}$4, and $H_{jkl}$5 is the integration constant generated by the conformal-Killing integration. Eliminating the conformal-tensor parameter gives

$H_{jkl}$6

while the continuity equation retains the standard form

$H_{jkl}$7

for $H_{jkl}$8 (Mantica et al., 2023).

In the equivalent Sinyukov-like description, the same cosmology is written as

$H_{jkl}$9

$R_{jk}-\frac12 g_{jk}R = T_{jk}+K_{jk},$0

For dust plus radiation,

$R_{jk}-\frac12 g_{jk}R = T_{jk}+K_{jk},$1

with

$R_{jk}-\frac12 g_{jk}R = T_{jk}+K_{jk},$2

the generalized Friedmann equation becomes

$R_{jk}-\frac12 g_{jk}R = T_{jk}+K_{jk},$3

with

$R_{jk}-\frac12 g_{jk}R = T_{jk}+K_{jk},$4

In redshift space,

$R_{jk}-\frac12 g_{jk}R = T_{jk}+K_{jk},$5

The associated effective dark density and pressure are

$R_{jk}-\frac12 g_{jk}R = T_{jk}+K_{jk},$6

Thus the conformal Killing sector is represented as a geometric dark fluid whose density grows as $R_{jk}-\frac12 g_{jk}R = T_{jk}+K_{jk},$7 (Mantica et al., 2024).

A closely related exact solution writes the cosmological evolution in Friedmann form as

$R_{jk}-\frac12 g_{jk}R = T_{jk}+K_{jk},$8

with

$R_{jk}-\frac12 g_{jk}R = T_{jk}+K_{jk},$9

Since $K_{jk}$0, the continuity equation fixes

$K_{jk}$1

This is the origin of the phantom-like effective equation of state often associated with the conformal Killing dark term (Harada, 2023).

4. Cosmological solution space and late-time evolution

The flat, dust-dominated FLRW sector is the canonical example. Setting

$K_{jk}$2

one obtains

$K_{jk}$3

equivalently

$K_{jk}$4

with $K_{jk}$5, and

$K_{jk}$6

This is precisely the form that yields decelerating early evolution and late-time acceleration when the $K_{jk}$7 term becomes dominant; in the notation used there, $K_{jk}$8 corresponds to accelerated expansion at late times. This is the standard conformal Killing gravity realization of “accelerating expansion without dark energy” in a matter-dominated universe (Mantica et al., 2023).

A second exact sector is the constant-curvature case. The scale factor is written as

$K_{jk}$9

with

$K_{jk}=R_{jk}-\frac12 g_{jk}R-T_{jk}.$0

and the effective equation of state tends to

$K_{jk}=R_{jk}-\frac12 g_{jk}R-T_{jk}.$1

and

$K_{jk}=R_{jk}-\frac12 g_{jk}R-T_{jk}.$2

or, in the Harada extension discussed there, to a phantom-like regime with $K_{jk}=R_{jk}-\frac12 g_{jk}R-T_{jk}.$3 in the far future when radiation is included. This provides a radiation-to-dark-energy interpolation generated by geometry (Mantica et al., 2023).

Vacuum or sourceless FLRW solutions further enlarge the cosmological phase space. With

$K_{jk}=R_{jk}-\frac12 g_{jk}R-T_{jk}.$4

the new integration constant $K_{jk}=R_{jk}-\frac12 g_{jk}R-T_{jk}.$5 measures the deviation from GR; $K_{jk}=R_{jk}-\frac12 g_{jk}R-T_{jk}.$6 reproduces the standard Friedmann equation. In vacuum,

$K_{jk}=R_{jk}-\frac12 g_{jk}R-T_{jk}.$7

The classification by $K_{jk}=R_{jk}-\frac12 g_{jk}R-T_{jk}.$8, $K_{jk}=R_{jk}-\frac12 g_{jk}R-T_{jk}.$9, and $\nabla^j K_{jk}=0,$0 yields singularity-free eternal cosmologies, closed oscillating universes, and universes evolving symmetrically from a big bang to a big crunch within a finite lapse of time. For example, with $\nabla^j K_{jk}=0,$1, $\nabla^j K_{jk}=0,$2, and $\nabla^j K_{jk}=0,$3, the scale factor oscillates periodically between two finite radii; with $\nabla^j K_{jk}=0,$4 and $\nabla^j K_{jk}=0,$5, the universe evolves symmetrically from a big bang to a big crunch in finite time (Clément et al., 2024).

Late-time fate depends sharply on the sign of the dark parameter in data-driven RW analyses. If $\nabla^j K_{jk}=0,$6, then as $\nabla^j K_{jk}=0,$7,

$\nabla^j K_{jk}=0,$8

and the model predicts a big-rip singularity; one estimate quoted for the future time interval is

$\nabla^j K_{jk}=0,$9

If $\nabla^j T_{jk}=0,$0, then $\nabla^j T_{jk}=0,$1 reaches zero at finite scale factor, and in the CC+BAO fit this occurs around

$\nabla^j T_{jk}=0,$2

This suggests that conformal Killing cosmology naturally supports both phantom-ending and finite-turnaround futures, depending on the sign of the geometric dark term (Mantica et al., 2024).

5. Observational fits, growth of structure, and early-universe consistency

The Robertson–Walker version with

$\nabla^j T_{jk}=0,$3

was tested on cosmic chronometers and BAO. In the flat, late-time approximation $\nabla^j T_{jk}=0,$4, the CC-only fit gave approximately

$\nabla^j T_{jk}=0,$5

while the CC+BAO fit gave approximately

$\nabla^j T_{jk}=0,$6

The matter density remains close to the $\nabla^j T_{jk}=0,$7CDM value, whereas $\nabla^j T_{jk}=0,$8 and $\nabla^j T_{jk}=0,$9 have large uncertainties and can change sign; their sum remains broadly close to the $(1+z)^{-2}$00CDM dark-energy fraction. In the same framework, the linear matter contrast in the matter-dominated era obeys

$(1+z)^{-2}$01

and the numerical conclusion is that the conformal Killing dark sector gives no significant deviation from $(1+z)^{-2}$02CDM and GR results during matter domination (Mantica et al., 2024).

A later Bayesian analysis used COBAYA with DESI DR2 or SDSS DR16 BAO data and Pantheon+ or Union3 supernova compilations in a spatially flat FRW background. In that presentation, the late-time Hubble law was written as

$(1+z)^{-2}$03

with $(1+z)^{-2}$04. The reported constraints were

$(1+z)^{-2}$05

for DESI+Union3,

$(1+z)^{-2}$06

for DESI+Pantheon+,

$(1+z)^{-2}$07

for DR16+Union3, and

$(1+z)^{-2}$08

for DR16+Pantheon+. The same study reported $(1+z)^{-2}$09 values near $(1+z)^{-2}$10, present dark-energy parameters $(1+z)^{-2}$11 in the quintessence regime, and first-acoustic-peak estimates $(1+z)^{-2}$12 extraordinarily near the Planck best value $(1+z)^{-2}$13 (Capozziello et al., 4 Aug 2025).

Taken together, these studies indicate that conformal Killing gravity can reproduce a background expansion history close to $(1+z)^{-2}$14CDM, with the main empirical novelty concentrated in the split between $(1+z)^{-2}$15 and the geometric dark parameter $(1+z)^{-2}$16, rather than in large deviations of $(1+z)^{-2}$17 or in large departures of linear growth. A plausible implication is that the model’s observational distinctiveness lies primarily in late-time background evolution and future asymptotics rather than in matter-era structure formation (Mantica et al., 2024, Capozziello et al., 4 Aug 2025).

6. Symmetry-based extensions and non-conservative generalizations

A symmetry-oriented extension studies scalar-field flat FLRW cosmology through the Eisenhart lift. The lifted minisuperspace metric is

$(1+z)^{-2}$18

with null Hamiltonian constraint

$(1+z)^{-2}$19

A non-trivial conformal Killing vector of this lifted geometry supplies an extra conserved quantity and complete integrability. The determinant condition for the prolonged conformal Killing equations reduces to a nonlinear second-order ODE for

$(1+z)^{-2}$20

and its regular branch yields the most general local potential in the $(1+z)^{-2}$21-independent sector,

$(1+z)^{-2}$22

The singular branch solves the determinant equation but is incompatible with the full conformal Killing equations. This establishes a precise integrable class of conformal-Killing-invariant flat FLRW scalar cosmologies (Chiba et al., 24 Apr 2026).

A different extension abandons separate matter conservation. In non-conservative conformal Killing gravity,

$(1+z)^{-2}$23

and the Einstein-like form becomes

$(1+z)^{-2}$24

Here

$(1+z)^{-2}$25

is the effective dark-sector tensor. In FRW, CMB considerations are used to impose $(1+z)^{-2}$26, so the dark sector couples only with the trace of the stress-energy tensor. Dust then satisfies

$(1+z)^{-2}$27

with solution

$(1+z)^{-2}$28

and the Hubble function becomes

$(1+z)^{-2}$29

The corresponding effective dark fluid has $(1+z)^{-2}$30 at early times and $(1+z)^{-2}$31 at late times, so the interacting variant preserves the phantom-like asymptotics of the conservative model while changing matter dilution (Capozziello et al., 6 Jun 2026).

7. Open problems, selection issues, and formal criticisms

The cosmological promise of conformal Killing gravity is counterbalanced by substantial open problems. In vacuum, assuming constant scalar curvature reduces the field equations to

$(1+z)^{-2}$32

so the Ricci tensor becomes a Killing tensor. This admits Einstein metrics

$(1+z)^{-2}$33

symmetric spaces, Ricci-symmetric spaces, and many non-Einstein Kundt solutions. In the Kundt sector, the metric function satisfies a linear third-order PDE, and examples include Siklos and Defrise geometries, $(1+z)^{-2}$34, $(1+z)^{-2}$35, $(1+z)^{-2}$36, and higher-dimensional $(1+z)^{-2}$37. The resulting “Pandora’s box” problem is that the theory contains a myriad of vacuum solutions, but does not by itself provide a criterion selecting the observed isotropic and homogeneous FLRW branch with positive effective cosmological term (Hervik et al., 2024).

A second issue is consistency. One detailed criticism is that a gravitational theory based on a rank-3 field equation does not arise from a diffeomorphism-invariant action with the metric as dynamical field, so the usual route from Bianchi identities and Noether charges to conserved quantities is unavailable. In this analysis, Schwarzschild remains a solution, but the usual mass parameter $(1+z)^{-2}$38 has no clear interpretation as a conserved charge of the theory. Attempts to build conserved currents yield quantities that vanish identically on Einstein manifolds, including Schwarzschild and Kerr-type solutions. This undermines standard notions of energy, angular momentum, and gravitational-wave flux, and therefore raises doubts about the physical completeness of the cosmological framework (Altas et al., 10 Feb 2025).

A related structural caveat is matter-source non-uniqueness. In the pp-wave sector, the general vacuum solution is

$(1+z)^{-2}$39

so the GR wave profile is supplemented only by a non-propagating quadratic term. Yet the same metric can correspond to different matter sources, because if two trace-modified tensors differ by a Killing tensor, they generate the same $(1+z)^{-2}$40. Even without additional symmetries, one may add

$(1+z)^{-2}$41

without changing the three-tensor source, and with Killing vectors the ambiguity increases further. This suggests that the “dark sector” of conformal Killing cosmology is geometrically natural but not uniquely tied to a single matter interpretation (Barnes, 2024).

These objections do not eliminate the cosmological constructions summarized above, but they fix the present status of the subject. Conformal Killing gravity cosmology is a mathematically explicit program in which accelerated expansion can arise from a divergence-free conformal Killing tensor, $(1+z)^{-2}$42 can emerge as an integration constant, and RW dynamics can be treated within a second-order Einstein-like system. At the same time, vacuum multiplicity, source ambiguity, and the absence of a fully satisfactory conserved-charge structure remain central obstacles to a definitive physical interpretation (Mantica et al., 2023, Hervik et al., 2024, Altas et al., 10 Feb 2025).