Cosmological Frame Ambiguity

• Cosmological-frame ambiguity is the non-uniqueness in selecting coordinate frames where physical observables remain invariant under proper conformal and gauge transformations.
• It arises from residual large diffeomorphisms, coordinate and gauge choices, and observer-dependent measurements that can alter quantities like the Hubble constant and redshift.
• Researchers address this ambiguity by consistently applying transformation rules and fixing gauge freedoms to ensure robust, frame-invariant predictions in both background and perturbative analyses.

Cosmological-frame ambiguity refers to the non-uniqueness inherent in the choice of mathematical frame, coordinates, or gauge for describing the geometry, physical fields, and observables of the universe—both at the level of the background and of perturbations. In modern cosmology, this ambiguity manifests in several distinct but interconnected ways: conformal frame freedom in scalar-tensor theories, residual large diffeomorphism (“large gauge transformation” or LGT) freedom at the perturbative level, reference-frame dependence in dynamical measurements, and observer-related kinematic ambiguities in the interpretation of key phenomena such as redshift, horizon structure, and the Hubble constant. While these ambiguities are largely unphysical—observable predictions are frame-invariant if calculations fully account for the proper transformations—they require a careful and consistent formal approach to ensure that theoretical and empirical analyses remain robust.

1. Conformal Frames and Field Redefinitions in Modified Gravity

In scalar-tensor theories, and more generally in modified gravity, the physical content of the model can be equivalently encoded in different “conformal frames,” most notably the Jordan frame (where the scalar field couples non-minimally to the Ricci scalar and matter) and the Einstein frame (where a local conformal rescaling and canonical field redefinition cast the action into manifestly Einstein–Hilbert gravity plus minimally coupled scalar fields) (Chiba et al., 2013, Hyun et al., 2017, Rondeau et al., 2017). The actions in the two frames are related via

$$\tilde g_{\mu\nu} = \Omega^2(\phi)\,g_{\mu\nu}\,,\qquad \phi\mapsto \phi(\tilde\phi)\,,$$

with associated transformations of all dynamical and matter fields. Although the underlying physics is not altered by such reparameterizations—classically the field equations are equivalent—naive computation of cosmological observables can yield superficially distinct results if one improperly applies or neglects the consequences of the conformal and field redefinitions. In particular, background quantities (scale factor, Hubble parameter, energy densities), perturbations (curvature, tensor, entropy modes), and the interpretation of measured quantities such as the redshift or luminosity distance all transform non-trivially under these mappings.

A key point is that observable, dimensionless quantities (e.g., the redshift $z$, CMB anisotropies, number counts) remain invariant under the full dictionary of conformal and field transformations provided that units and the change of clocks and rods due to the conformal rescaling are consistently tracked (Chiba et al., 2013, Francfort et al., 2019, Rondeau et al., 2017). Dimensionful quantities (e.g., $H$, $d_L$, $G_{\text{eff}}$) acquire frame-dependent factors, with physical relevance only in the context of frame-fixed units. The practical resolution is to consistently work in a single conformal frame when extracting or interpreting observational parameters, and to map between frames using the exact transformation rules when theoretical or computational convenience demands (Hyun et al., 2017, Chiba et al., 2020).

2. Frame Independence versus Ambiguity in Observable Cosmology

Despite the mathematical equivalence of frames, the notion of “cosmological-frame ambiguity” becomes physically significant where observable properties are directly or indirectly affected by the choice of frame. Several domains illustrate this:

• Gauge-invariant curvature and tensor perturbations: For adiabatic superhorizon modes and vanishing relative entropy perturbations, curvature perturbations $\zeta$ and their spectra are frame-invariant (Chiba et al., 2013, Prokopec et al., 2013). However, with non-adiabatic or entropy sources, the mapping becomes nonlinear and frame dependence enters at the level of $n$-point functions unless formulated in terms of covariant, frame-independent variables.
• Number counts and power spectra: Observable number-count fluctuations $\Delta(n, z)$—as opposed to the underlying unobservable matter power spectrum $P(k)$—are exactly invariant under conformal and disformal frame transformations. The frame dependence of $P(k)$ is suppressed on sub-horizon scales but can be relevant at the largest wavelengths (Francfort et al., 2019, Chiba et al., 2020).
• Dynamical frame principle and cosmic acceleration: The “cosmological-frame principle” asserts that any physical cosmological effect, such as late-time acceleration, must be present and stable as an attractor in all conformally related frames. Only such solutions are physically meaningful, resolving apparent ambiguity as a gauge artifact (Cotsakis et al., 2023).

The overall consensus in the context of conformal and disformal mappings is that observable relations are robust against frame shifts, provided the transformation properties of all relevant quantities (including units) are consistently applied (Chiba et al., 2013, Francfort et al., 2019, Chiba et al., 2020).

3. Large Gauge Transformations, Soft Modes, and Perturbative Frame Freedom

A distinct aspect of cosmological-frame ambiguity arises from residual large diffeomorphisms—coordinate transformations that are not connected to the identity or do not decay at infinity. In cosmological perturbation theory, these so-called “large gauge transformations” (LGTs) generate low-multipole (monopole/dipole) adiabatic (“soft”) modes that remain unconstrained by the local field equations. Such transformations carry nontrivial quasi-local charges (e.g., shifted mean-curvature energy or angular momentum of a cosmological patch), generating physical ambiguity in the bulk values of correlation functions and background quantities (Achour et al., 7 Jan 2026, Mitsou et al., 2022).

In practice, the Newtonian gauge admits a hierarchy of residual LGTs with unconstrained time dependence for the $\ell=0,1$ (monopole and dipole) sector: $$\xi^0(\eta, x)\equiv m(\eta) + g_i(\eta)x^i\,,\quad \xi^i(\eta, x)\equiv c^i(\eta) + \Omega^i{}_j(\eta)x^j\,,$$ with $m(\eta)$, $g_i(\eta)$, $c^i(\eta)$, and $\Omega^i{}_j(\eta)$ arbitrary (Achour et al., 7 Jan 2026). Unless boundary data or explicit gauge-fixing conditions are imposed (e.g., by demanding a vanishing net dipole or setting patch charges to FLRW reference values), any correlator or observable evaluated in a finite patch remains ambiguous at $\ell\leq1$.

The conceptual resolution is to either fix the large-gauge freedom via a physical or geometric prescription (e.g., matching bulk and boundary charges, using observational rest frames, or imposing charge-neutral initial data), or recognize that only quantities invariant under this extended symmetry group (genuine diffeomorphism invariants) have unambiguous meaning (Mitsou et al., 2022).

4. Observer Motion, Preferred Frames, and the Measured Hubble Constant

Cosmological-frame ambiguity also appears in the interpretation of reference-frame-dependent observables. In contrast to special relativity, cosmology features a distinguished rest frame defined by the vanishing of the CMB dipole. Observers (e.g., the Solar System barycenter) moving with respect to this frame experience anisotropies and measurable shifts in local expansion parameters: $$H_{\mathrm{obs}} = H_0\,\left[1 + \frac{\zeta^2}{6} + \cdots\right],\qquad \zeta = v/c,$$ where $H_0$ is the Hubble constant in the CMB frame and $\zeta$ is the peculiar velocity relative to the cosmic rest frame (Chang et al., 2019). This induces a $\sim 1\%$ systematic shift in $H_0$ determinations—a non-negligible error budget for precision cosmology.

Furthermore, the shape and volume of cosmological horizons as seen in a moving frame acquire a Lorentzian ellipsoidal deformation with eccentricity $e\sim v/c$, physically altering the causal structure and the calculation of horizon-bounded quantities (Dokuchaev et al., 2010).

In this context, the reference frame must be carefully specified in both theoretical and data analysis pipelines, or else “cosmological-frame ambiguity” leads to systematic errors or misidentification of true cosmological signals (Burde, 2016).

5. Ambiguity from Coordinate and Gauge Choices: Synchronous vs. Other Coordinates

Operationally, the choice of coordinates (e.g., synchronous, non-synchronous, conformal Newtonian) can introduce artifacts that mimic genuine physical effects. For instance, non-synchronous (e.g., proper distance) frames introduce inertial accelerations that can be misattributed to cosmological constants or dark energy, biasing the extracted value of $\Lambda$ by up to 20% if uncorrected (Grib et al., 2020). The same invariant redshift may be split into “cosmological” and “gravitational” components depending only on the frame, with no experiment able to distinguish the division (Toporensky et al., 2017). This further demonstrates that only the invariant relations—those insensitive to slicing or temporal gauge—carry true physical content.

6. Quantum Corrections, Anomalies, and Breaking of Classical Equivalence

While classical frame covariance holds in standard scalar–tensor models, quantum anomalies may break the equivalence. In specific cases, such as scale-invariant Jordan frame models, one-loop quantum corrections generate anomalous terms that cannot be absorbed by field redefinitions and persist after renormalization. The Jordan frame develops an anomalous $\ln\phi$ term, resulting in a non-vanishing contribution to the S-matrix not present in the Einstein frame (Herrero-Valea, 2016). As a consequence, classical frame ambiguity inflates into a genuine physical inequivalence at the quantum level, necessitating separate UV completions for the distinct frames.

7. Physical and Philosophical Implications

From a theoretical perspective, cosmological-frame ambiguity underscores that many aspects of cosmological modeling—expansion versus varying units, existence or removal of a cosmological constant problem (Lombriser, 2023), presence of apparent superluminality, or the “reality” of cosmic acceleration—are a matter of representation governed by the underlying symmetries and transformation rules of the theory. No observable experiment can distinguish a preference for the Jordan or Einstein frame, or for Minkowski rather than FLRW geometry, provided all transformations are consistently handled and all physical clocks and rulers are correctly rescaled (Lombriser, 2023, Mitsou et al., 2022). Nonetheless, practical data analysis, model building, and searches for new physics (e.g., in dark energy, dark matter, or inflation models) demand careful self-consistency within the chosen frame and explicit attention to the mathematical ambiguities discussed above.

In summary, “cosmological-frame ambiguity” is a multi-faceted issue arising from mathematical redundancies—conformal transformations, large diffeomorphisms, coordinate/gauge choices, and observer-dependent formulations—in the representation of cosmological phenomena. Although properly constructed observables are invariant under these ambiguities, explicit frame- or gauge-fixing is essential for unambiguous calculation. Recent advances refine these principles at both the perturbative and quantum level, and ongoing work addresses the practical ramifications for precision cosmology and the foundational interpretation of gravitational theories (Achour et al., 7 Jan 2026, Mitsou et al., 2022, Chiba et al., 2013, Hyun et al., 2017, Lombriser, 2023).