Theoretical Viability and Comparison with Observations

A cosmological model is not viable merely because it reproduces an expansion history or admits a stable perturbation at one epoch. Viability is a chain: a defined action and matter frame, a regular background branch, a well-posed perturbation problem, a cutoff above every predicted scale, a controlled nonlinear completion, and initial data compatible with the solution being evolved. Only after that chain is documented is there a prediction packet suitable for empirical analysis.

Helpful background. Einstein Equations supplies the background and perturbative constraints; Gravitational Actions and Boundary Terms supplies the action-level definition of the theory; Formulations of General Relativity supplies well-posed initial-data criteria; Gravity Beyond General Relativity supplies degrees of freedom, branches, and EFT power counting; Causality and Global Structure supplies hyperbolicity and causal-cone requirements; and Gravitational Radiation and Waveforms supplies the source/propagation separation for tensor predictions.

A hierarchy of theoretical filters

Let $\theta$ denote both parameters and time-dependent functions. For a specified redshift and wavenumber domain $\mathcal D$, the usable region is schematically

$$\mathcal V(\mathcal D)= \mathcal B\cap\mathcal K\cap\mathcal G\cap \mathcal H\cap\mathcal C\cap\mathcal N,$$

where $\mathcal B$ denotes regular backgrounds and constraints, $\mathcal K$ positive kinetic matrices, $\mathcal G$ acceptable gradient matrices, $\mathcal H$ a hyperbolic initial-value problem, $\mathcal C$ validity below the cutoff, and $\mathcal N$ a controlled nonlinear or screened regime where one is used. The intersection depends on $\mathcal D$ and on the selected branch.

These filters answer different questions. A positive quadratic kinetic matrix removes linear ghosts on that background but does not prove nonlinear hyperbolicity. Positive short-wavelength sound-speed squares remove rapid gradient growth below the cutoff but do not exclude a slow tachyon. A regular FLRW branch does not guarantee a regular screened solution around a halo. An EFT coefficient that is harmless at cosmological frequency need not have the same value near its cutoff.

Conditions also require physical normalization. The effective Planck mass must not pass through zero; constraints must be solved without dividing by a vanishing coefficient; matter must remain coupled to the metric used to define clocks, redshift, and rulers. A field redefinition can simplify the gravity action, but it moves the matter coupling and cannot erase a physical prediction.

What a testable prediction must specify

The theory calculation should deliver quantities with enough metadata that an empirical analysis never has to guess their domain:

Prediction object	Required theoretical information	Why it is needed
Background $H(a)$ and distances	action, matter frame, branch, initial constraint, parameter range	fixes geometry without disguising modified terms as a unique material fluid
Linear scalar transfer functions	gauge-invariant variables, species equations, initial conditions, stable $a$–$k$ domain, cutoff	distinguishes a dynamical response from an algebraic quasistatic fit
Tensor transfer	$M_*^2$, speed, mass, mixing, polarization basis, frequency range	separates source generation from cosmological propagation
Nonlinear or screened response	environment, boundary data, matching prescription, branch, convergence test	prevents a cosmological linear function from being applied inside a screened object
Failure mask	instability surfaces, singular denominators, strong-coupling scales, excluded branches	stops an analysis from interpolating through regions where no prediction exists

For tensor signals, this packet accepts source amplitudes and source-frame validity information from Binary Dynamics and Gravitational Radiation Beyond GR and returns a background-dependent propagation transfer; it does not recompute compact-binary generation. Likelihoods, survey systematics, population models, and numerical bounds are not part of this packet. They are developed in Tests of Gravity, which consumes the theory prediction together with an explicit measurement model.

Results and open questions

The literature status below is fixed at 2026-08-19. It concerns the construction and limitations of cosmological theory predictions, not the numerical outcome of any data fit.

Question	Evidence status at the cutoff	Scope and limitation
Can a broad single-scalar theory class be compressed into common linear functions?	Yes for Horndeski perturbations: $M_*^2$ and the four $\alpha$ functions organize the quadratic dynamics Bellini and Sawicki 2014, §§ 3.1 and 3.3, Eqs. (3.12)–(3.14), and Appendix A, Eqs. (A.6)–(A.10)	The compression is background- and frame-dependent; it excludes additional scalars, vectors, nonlocal prescriptions, and generic nonlinear dynamics
Can parameter sampling be built to satisfy linear scalar stability by construction?	Yes within a specified luminal-Horndeski basis using positive kinetic and sound-speed variables Lombriser et al. 2019, §§ 2.2–2.3 and 4.1	“Inherently stable” refers to the encoded linear conditions; it is not a proof of a UV completion, nonlinear well-posedness, or screening
Are phenomenological and theory-informed time functions interchangeable priors?	No. A current comparison finds materially different theory-space weighting when $\mu$–$\Sigma$ and EFT-inspired functions or time dependences are used Shah, Koyama, and Noller 2026, §§ II–III and V–IX	This establishes prior sensitivity, not a uniquely correct parameterization; its numerical likelihood results are outside this page’s scope
Does a low-energy tensor speed determine propagation at all frequencies?	No. Higher-derivative operators near the cutoff can change the dispersion and restore high-frequency luminality in explicit partial completions de Rham and Melville 2018, pp. 2–4, Eqs. (6)–(9) and (13)–(18), PDF	The example demonstrates scale dependence but does not provide a universal UV completion or license arbitrary interpolation
Is modified friction a well-defined cosmological tensor prediction?	Yes in the massless, unmixed, adiabatic regime; it gives a calculable GW/EM luminosity-distance ratio Belgacem et al. 2018, § II, Eqs. (9)–(13), pp. 1–2	Source generation, screening, mass, mixing, lensing, and detector/population likelihoods require separate treatment
Does a stable linear cosmology determine its nonlinear observable window?	No general closure is known. Screening and quasistatic reductions remain mechanism-, environment-, and scale-dependent Koyama 2016, §§ 4–6	A simulation or matching prescription validated for one action and branch cannot be transferred to another by reusing the same $\mu$ function

Negative results in this matrix are scientifically useful. There is no one-to-one map from an arbitrary pair $\{\mu,\Sigma\}$ to a healthy covariant action; linear no-ghost and no-gradient conditions do not establish nonlinear health; and a low-energy propagation coefficient cannot be assumed constant across twenty or more decades in frequency. These failures delimit what a model-independent analysis can honestly claim.

The status of a universal viability prescription is therefore unresolved. No universal verdict covers every alternative theory: the sufficient conditions depend on its fields, derivative order, degeneracy structure, matter coupling, background branch, and cutoff. Individual model classes can be decided more sharply than this frontier-wide statement.

Competing strategies and an adversarial check

One strategy samples flexible phenomenological functions and postpones theory restrictions, aiming to cover unexpected departures from GR. Its strength is breadth; its weakness is that the sampling measure can heavily weight functions that have no stable action, valid cutoff, or nonlinear completion. A second strategy samples only coefficients or covariant models that pass chosen theoretical priors. Its strength is interpretability; its weakness is that the result is conditional on a narrower theory class and may exclude consistent theories not represented by the basis.

The defensible practice is to label both. A phenomenological result should state its function basis and regularity assumptions. A theory-filtered result should state the action class, stability tests, and cutoff. Agreement between the two is informative; disagreement diagnoses prior or coverage dependence rather than automatically favoring either interpretation.

An adversarial test illustrates the point. Imagine a sampled model for which the scalar kinetic coefficient and sound speed are positive throughout the survey redshift interval, but $M_*^2$ crosses zero just before that interval and the desired maximum physical wavenumber exceeds the EFT cutoff at one intermediate epoch. A linear solver initialized after the crossing could return smooth transfer functions. The model is still inadmissible for the advertised prediction: its initial history is branch-singular, and part of the requested scale range lies outside its theory. Smooth numerical output is not evidence against those failures.

The same reasoning applies to screening. Recovering GR in a static, spherical, high-density solution does not prove recovery in a time-dependent nonspherical environment, nor that cosmological initial data evolve onto that branch. Predictions used in data analysis must identify where the theory calculation fails rather than silently filling missing regions with GR.

Open questions. The conclusions could change if a broader covariant parameterization supplies a demonstrably complete stable map beyond the present single-scalar bases; if a nonlinear well-posedness theorem or validated simulation suite closes the cosmology-to-screened-regime matching problem; if a UV calculation fixes the cross-frequency tensor transfer; if the 2026 theoretical-prior analysis is corrected, superseded, or withdrawn; or if a new result changes which perturbative stability conditions are necessary or sufficient.

Exercises

A model has $M_*^2\gt0$, a positive scalar kinetic coefficient, and $c_s^2\gt0$ for every sampled time and wavenumber. At $a=a_*$, however, its physical cutoff satisfies $\Lambda_{\mathrm{EFT}}(a_*)\lt k_{\max}/a_*$. May its transfer function be handed off over the full requested range? Give the minimal repair.

Solution

No. The positivity conditions show that the retained quadratic equations are linearly stable where they apply, but they do not extend those equations above their cutoff. Modes with $k/a_*$ greater than $\Lambda_{\mathrm{EFT}}(a_*)$ probe omitted operators with uncontrolled size.

The minimal defensible repair is to restrict the prediction domain to

$$\frac{k}{a}\lt\Lambda_{\mathrm{EFT}}(a)$$

at every epoch used in the transfer, and to pass that time-dependent mask with the output. Extending to $k_{\max}$ instead requires a higher-cutoff completion or an enlarged EFT with its new operators matched and revalidated. Merely damping or extrapolating the transfer function does not repair the theoretical loss of control.

References

• Belgacem, Enis, Yves Dirian, Stefano Foffa, and Michele Maggiore. “The Gravitational-Wave Luminosity Distance in Modified Gravity Theories.” Physical Review D 97, 104066 (2018). DOI. Open PDF
• Bellini, Emilio, and Ignacy Sawicki. “Maximal Freedom at Minimum Cost: Linear Large-Scale Structure in General Modifications of Gravity.” Journal of Cosmology and Astroparticle Physics 2014 (07), 050 (2014). DOI. Open PDF
• de Rham, Claudia, and Scott Melville. “Gravitational Rainbows: LIGO and Dark Energy at Its Cutoff.” Physical Review Letters 121, 221101 (2018). DOI. Open PDF
• Koyama, Kazuya. “Cosmological Tests of Modified Gravity.” Reports on Progress in Physics 79, 046902 (2016). DOI. Open PDF
• Lombriser, Lucas, Charles Dalang, Joe Kennedy, and Andy Taylor. “Inherently Stable Effective Field Theory for Dark Energy and Modified Gravity.” Journal of Cosmology and Astroparticle Physics 2019 (01), 041 (2019). DOI. Open PDF
• Shah, Neel, Kazuya Koyama, and Johannes Noller. “Dark Energy Constraints in Light of Theoretical Priors.” Journal of Cosmology and Astroparticle Physics 2026 (01), 054 (2026). DOI. Open PDF