No-Go Theorem for Cosmology

• No-go theorems in cosmology are rigorous results derived under strict physical and mathematical assumptions that delineate which cosmic acceleration and modified gravity scenarios are feasible.
• They demonstrate that extra-dimensional and screening models, such as those involving chameleon mechanisms, require ad hoc constraints or SEC violations to achieve sustained cosmic acceleration.
• These theorems also highlight inherent conflicts in constructing non-singular, bouncing, or self-tuning cosmologies, often resulting in unavoidable instabilities or the need for unconventional modifications.

A cosmological no-go theorem is a rigorous result, derived under well-specified physical and mathematical assumptions, that forbids the realization of a particular cosmological property or scenario in broad classes of theories. In modern cosmology, no-go theorems have become central to the appraisal of modified gravity models, extra-dimensional scenarios, effective field theories, energy condition constraints, and the search for consistent singularity-free or accelerating cosmologies. These results delineate the boundaries of what is achievable within specified frameworks, directly regulating the space of valid cosmological models and informing both theoretical searches and the interpretation of observational data.

1. No-Go Theorems in Extra-Dimensional Cosmology

A prominent application of no-go theorems is in the paper of extra-dimensional models attempting to generate or sustain four-dimensional cosmic acceleration—including inflation and dark energy—using smooth, compactified extra dimensions. Classic theorems, such as those due to Steinhardt & Wesley, claim that in Kaluza–Klein (KK) compactifications that satisfy the higher-dimensional null energy condition (NEC), any accelerated expansion in four dimensions is necessarily transient and cannot be sustained for sufficiently many e-folds. These results employ a block-diagonal metric ansatz,

$$ds^2 = e^{2\Omega(t,y)} g^E_{\mu\nu}(t) dx^\mu dx^\nu + g^{(d)}_{mn}(t,y) dy^m dy^n$$

with the external 4d metric $g^E_{\mu\nu}$ describing a homogeneous and isotropic universe and the internal metric $g^{(d)}_{mn}$ capturing the compactified directions.

However, it was demonstrated (Koster et al., 2011) that such no-go theorems critically depend on an additional, ad hoc metric restriction—specifically, the imposition of a constraint

$\partial_t \Omega + \partial_t \sigma = 0$

relating the time derivative of the warp factor and the volume modulus of the extra dimensions (see Eq. (2.23) in (Koster et al., 2011)). Without this supplementary constraint—which is neither enforced by coordinate invariance nor by the requirement of a finite, ghost-free 4d Planck mass—the machinery for obtaining robust no-go results collapses. The derivation relies on an averaging procedure over the internal space and particular sign properties that fail absent this ad hoc assumption. Notably, explicit counterexamples exist with time-dependent, non-separable metrics violating this restriction, demonstrating a finite 4d Planck mass and no ghost instability. Therefore, the power of the higher-dimensional Einstein equations to forbid cosmic acceleration is, in practice, much weaker than previously asserted unless the theory structure directly imposes this extra constraint.

These findings have two primary implications: (i) Kaluza–Klein models without the restrictive metric assumption potentially permit sustained cosmic acceleration even with NEC-preserving stress-energy; (ii) brane world setups, where boundary terms or the Einstein-frame identification may fail, can avoid these obstructions entirely.

2. Chameleon and Screening Mechanism No-Go Theorems

Screening mechanisms—most notably, the chameleon, symmetron, and dilaton models—use light scalar fields whose properties depend on local matter density, hiding their effects at high densities to evade local gravity constraints. A comprehensive no-go theorem (Wang et al., 2012) establishes that in such models:

1. The Compton wavelength (inverse effective mass) of the scalar at present cosmic density is bounded by $\sim 1$~Mpc, meaning any scalar fifth force can only affect non-linear, small scales. Their modifications cannot influence the linear regime of cosmic large-scale structure.
2. The conformal factor $A(\phi)$ that relates Einstein and Jordan frames is essentially constant over the last Hubble time, $\Delta A/A \ll 1$. This prohibits self-acceleration from these modified gravity degrees of freedom.

Consequently, chameleon-like mechanisms cannot generate cosmological acceleration as a genuine modification of gravity. Any cosmic acceleration must instead arise from a true dark energy component (such as vacuum energy or quintessence), not from fifth forces generated by screened scalars in these frameworks.

3. No-Go Theorems for Non-Singular and Geodesically Complete Cosmologies

Singularity theorems constrain the conditions under which a cosmology can be both non-singular (all curvature invariants remain finite) and geodesically complete (all null and timelike geodesics can be extended to arbitrary affine parameter). A recent result (Burwig et al., 15 Oct 2025) establishes that for non-static, spatially flat (k = 0) or open (k = –1) Friedmann–Robertson–Walker (FRW) universes, the requirements of nonsingularity, geodesic completeness, and compliance with the averaged null energy condition (ANEC) cannot be met simultaneously. The essential identity is

$$I_{\text{ANEC}} = \lim_{T_-\rightarrow -\infty, T_+\rightarrow +\infty} \int_{T_-}^{T_+} \frac{\rho(t)+p(t)}{a(t)} dt$$

For flat or open universes, provided curvature is bounded and null geodesics are complete, the integral $I_{\text{ANEC}}$ is necessarily negative due to the dominant contribution of the $-2 \int H^2/a\, dt$ term. As ANEC requires $I_{\text{ANEC}}\geq 0$, the only way out is either a singularity or ANEC violation.

Conversely, closed universes (k = +1) possess an additional positive-definite curvature term in the ANEC integral. In such cases, exemplified by global de Sitter space, nonsingular, geodesically complete, and ANEC-consistent solutions exist. This classification sharpens and supersedes the scope of the classical Hawking–Penrose singularity theorems by linking global topology (spatial curvature) to robustness against singularities and energy condition violations.

Moreover, positive spatial curvature can mimic the observational phenomenology of phantom dark energy ($w < -1$) in cosmological reconstructions that assume flatness. For small curvature parameter $|\Omega_{k,0}| \sim 0.004$, the effective equation-of-state parameter $w(z)$ inferred from flat-model fits can be biased at the sub-percent level, even if the actual model is $\Lambda$CDM with slight positive curvature.

4. No-Go Theorems in Modified Gravity and Self-Tuning Mechanisms

In brane world and other higher-dimensional settings, self-tuning solutions that attempt to absorb or cancel a bare cosmological constant using integration constants in the bulk equations have been studied. Traditional no-go results demonstrate that, for canonical (or suitably continuous) scalar or three-form bulk Lagrangians, either:

• Bulk singularities arise,
• The effective four-dimensional Planck mass diverges, or,
• The effective 4d theory contains ghost modes (wrong-sign kinetic terms) or tachyonic Kaluza-Klein (KK) excitations.

Even when employing unorthodox bulk Lagrangians—e.g., Lagrangians nonlinear or noncanonical in the bulk field strength $H^2$—recent work (Forste et al., 2013) shows that at the perturbative level, the resulting four-dimensional theory always includes either a continuum of ghosts or unstable KK modes. The requirements for successful self-tuning are thus in direct contradiction with the stability conditions for the mode spectrum, making perturbatively stable self-tuning in such scenarios unachievable without significant modification.

5. No-Go Theorems in Bouncing and Non-Singular Cosmologies

For non-singular bounce cosmologies, especially those constructed within Horndeski (generalized Galileon) theories, robust no-go theorems show that entirely ghost- and gradient-stable bounce solutions with flat spatial sections are impossible if the graviton speed remains subluminal (or even simply less than a critical value) throughout evolution, assuming the quadratic action coefficients remain well-behaved. The contradiction arises from integrated inequalities for background-dependent quantities (such as the function $\xi = a G_T \Theta$ (Kobayashi, 2016)). Either the instability appears as a gradient instability at some epoch or as a vanishing tensor kinetic term.

Extensions to generalized multi-Galileon frameworks show that, even in the presence of multiple scalars and the inclusion of covariantized new higher-order interaction terms, gradient instabilities persist under the requirement of non-singularity and geodesic completeness (Akama et al., 2017). In Galileon models with torsion, this no-go theorem holds unless a brief superluminal phase for the graviton is permitted, during which the speed of gravitational perturbations exceeds the luminal value by a significant factor (Mironov et al., 2023). With such a phase, the integrated constraints can be avoided, opening a loophole for constructing fully stable bouncing solutions at the expense of strict subluminality.

In the context of matter-bounce scenarios where cosmic perturbations and non-Gaussianities are computed in single-field setups, no-go theorems demonstrate a direct trade-off between the required suppression of the tensor-to-scalar ratio (to conform to observational bounds) and excessive enhancement of primordial non-Gaussianities ($f_{\rm NL}$). For canonical single-field (and even general $k$-essence) bounces, an irreducible tension appears: if the bounce boosts curvature perturbations enough to lower $r$ to allowed levels, $f_{\rm NL}$ is inevitably driven above experimental constraints (Quintin et al., 2015, Li et al., 2016).

6. Energy Condition-Based No-Go Theorems in Compactified and String Cosmologies

Many results on the possibility of cosmic acceleration in effective 4d cosmologies derived from higher-dimensional theories leverage the Raychaudhuri equation and associated energy conditions. The Gibbons-Maldacena-Núñez (GMN) no-go theorem (Faruk, 13 Feb 2024) is a geometric constraint showing that in any warped compactification with static extra dimensions, the existence of an accelerating four-dimensional external FRW universe requires violation of the strong energy condition (SEC) in the higher-dimensional theory. Explicitly, integrating the relevant Raychaudhuri equation components and using the compactification ansatz,

$$ds^2 = \Omega^2(y)\left[g_{\mu\nu}(x)dx^\mu dx^\nu + h_{mn}(y)dy^m dy^n\right],$$

one finds the constraint

$$3(\dot{H} + H^2)\left(\frac{G_D}{G_d}\right) = -\int d^n y \sqrt{h} \Omega^{D-2} R_{00}^{(D)}$$

and thus, to realize $\dot{H} + H^2 > 0$, the integrated $R_{00}^{(D)}$ must be negative, which is precisely a violation of the SEC. This result is independent of matter content and underscores that many string compactification attempts at de Sitter or accelerating cosmologies are obstructed without SEC-violating ingredients (e.g., specific fluxes, brane sources, or time-dependent internal spaces).

7. Outlook and Future Directions

Contemporary research continues to explore ways to evade or refine no-go theorems by relaxing underlying assumptions, incorporating quantum effects, utilizing brane sources or singularities, invoking nonlocal modifications to gravity, or constructing multifield or higher-derivative theories. For each no-go result, the domain of validity is precisely circumscribed by assumptions about the metric ansatz, energy conditions, locality, field content, and boundary conditions. Recent work has focused on formalizing nonlocal gravitational modifications that can bypass classical constraints, as explored in infinite derivative gravity frameworks and associated investigations into the cosmological constant problem (Capozziello et al., 11 Feb 2025).

Ongoing theoretical and observational developments will further test the sharpness of these results and probe the exceptional cases where new fundamental physics may lie. No-go theorems, by illuminating the limits of established frameworks, continue to provide essential guidance in the search for viable cosmological theories consistent with both high-energy physics and precision astrophysical data.