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Inflation in generalized unimodular gravity

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Pith's one-line read Generalized unimodular gravity can drive inflation with no inflaton field, its scalar graviton generating the red-tilted primordial spectrum that fits CMB observations.

desk verdict A solid formal derivation of GUMG inflation spectra, but the homogeneous-mode ghost is never convincingly decoupled; treat the predictions as conditional. read the letter →

arxiv 1908.05697 v1 pith:HKIPINIU submitted 2019-08-15 gr-qc astro-ph.COhep-th

A. O. Barvinsky , N. Kolganov This is my paper · ORCID
classification gr-qcastro-ph.COhep-th PACS 98.80.Cq04.20.Fy04.50.Kd
keywords generalizedunimodulargravityscalargravitoncosmologicalinflationprimordialpowerspectrumeffectiveperfectfluidMukhanov-Sasakiequationspeedofsounddarkenergy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that generalized unimodular gravity, a modification of Einstein gravity in which the lapse function is fixed to a function of the spatial metric determinant, can produce cosmological inflation from pure geometry. In the branch where a dark perfect fluid emerges, the theory's scalar graviton—gravity's own extra degree of freedom—becomes the driver of an almost exponentially expanding Universe and generates the nearly scale-invariant, red-tilted primordial spectrum that the cosmic microwave background requires. The authors derive the scalar and tensor power spectra, show they coincide in form with the standard hydrodynamic slow-roll result but with a nontrivial speed of sound, and sketch how the defining function of the model could be reconstructed from the observed tilt, amplitude, and tensor-to-scalar ratio. If correct, the paper removes the need for an inflaton field and connects inflation to the same dark sector originally proposed for dark energy.

What carries the argument

The machinery is the kinematical restriction $N = N(\gamma)$, identifying the lapse with a generic function of the determinant of the spatial metric, which converts the gravitational field into an effective perfect fluid with $p = w\varepsilon$ and $w = 2\, d\ln N/ d\ln\gamma$. The argument runs on two derived quantities: $\Omega = 1 + w + 2\, d\ln w/ d\ln\gamma$, which controls the coefficient of the kinetic term, and the scalar speed of sound $c_s^2 = w(1+w)/\Omega$. With these, the perturbation action reduces to a single canonically normalized field satisfying the Mukhanov-Sasaki equation, and matching the adiabatic vacuum across horizon crossing yields the power-spectrum amplitude (5.10). The same reduction reproduces the known tensor spectrum because tensor gravitons keep unit sound speed.

What would settle it

A concrete test would be to compute the leading cubic coupling between the homogeneous ghost mode $\psi_0$ and the inhomogeneous scalar modes $\vartheta_k$; if the ghost's negative energy leads to exponential growth of $\vartheta_k$ within a few e-folds, the computed power spectra are invalid. Observationally, the reconstruction relations (6.16)-(6.18) tie the tilt to a sound-speed parameter $\Delta\beta$, so a precise measurement of the scalar spectral index together with the tensor-to-scalar ratio—or a direct detection of a scale-dependent sound speed through non-Gaussianity—could rule the model in or out.

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Extended reading notes

Core claim

The central claim is that the second branch of GUMG—where the Hamiltonian constraint is replaced by the condition that the effective fluid be spatially homogeneous—supports an inflationary background driven by the global conformal mode, with perturbations described by a single scalar degree of freedom, the scalar graviton. The paper shows that this degree of freedom obeys a Mukhanov-Sasaki equation with sound speed $c_s^2 = w(1+w)/\Omega$, and that after horizon crossing its long-wavelength modes produce the scale-invariant spectrum (5.10), with spectral index and tensor-to-scalar ratio expressible through the functions $w$ and $\Omega$ evaluated at horizon crossing. A reconstruction of the lapse function near $\gamma\to 0$ with nonanalytic corrections (6.2) yields $n_s - 1 \simeq -3\Delta\beta/4$ and $r \simeq 10^{-3}$, in agreement with current bounds, indicating that the model can match the observed red tilt without an inflaton.

Load-bearing premise

The calculation assumes that the homogeneous-mode ghost—a scalar mode with a negative kinetic term that the paper itself identifies—never couples strongly to the inhomogeneous modes whose fluctuations produce the CMB spectrum, so that the vacuum initial conditions for those modes remain under control.

Editorial extensions

If this is right

  • Inflation in GUMG requires no extra matter field: the accelerating expansion is powered by the effective fluid of the dark gravitational sector, with initial conditions set by the constant $C$ in the Friedmann equation.
  • The scalar spectrum is red tilted with $n_s - 1$ given by (5.13); choosing $N(\gamma)$ as in (6.2) with $\beta = 3/2 - \Delta\beta$ reproduces $n_s \simeq 0.96$.
  • The tensor spectrum is the standard one with $c_s = 1$, and in the reconstructed model $r \simeq 10^{-3}$, below current observational upper bounds.
  • If $\Omega$ crosses zero during cosmic history, the scalar graviton enters strong coupling, which may trigger a quantum transition from the GUMG branch to the general-relativistic branch; this is a possible exit mechanism from the GUMG phase.
  • Adding ordinary matter to GUMG just adds its energy density to the Friedmann equation, so in principle the same construction can also describe radiation- and matter-dominated epochs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: because the scalar perturbations propagate with $c_s^2 = w(1+w)/\Omega < 1$, GUMG inflation should generate an observable equilateral-type non-Gaussianity in the scalar bispectrum; measuring this signal would directly probe the sound speed, a computation the paper does not perform.
  • Editorial inference: the reconstruction of $N(\gamma)$ is made only at the inflationary asymptote $\gamma \to 0$; a combined fit extending the ansatz (6.2) through reheating and into the late-time dark-energy epoch would determine whether one and the same function can sustain both early and late acceleration.
  • Editorial inference: the homogeneous ghost $\psi_0$ is the same mode that sets the background's integration constant; the paper's assumption that it decouples from perturbations could be probed by computing the next-order interaction between $\psi_0$ and the inhomogeneous modes $\vartheta_k$, which is not performed here.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proposes generalized unimodular gravity (GUMG) as a mechanism for cosmological inflation without an inflaton field. After reviewing the constrained dynamics and the Friedmann background, the authors develop quadratic cosmological perturbations, isolate the physical scalar and tensor modes, and derive a Mukhanov-Sasaki equation for the scalar graviton with a nontrivial sound speed c_s^2 = w(1+w)/Ω (Eqs. (4.26)–(4.31)). They then compute scalar and tensor power spectra (Eqs. (5.5) and (5.18)), express the observable curvature perturbation in terms of Bardeen invariants (Eq. (5.10)), and obtain spectral indices (Eqs. (5.13) and (5.19)). Section 6 proposes a class of functions N(γ) that yields a red-tilted spectrum with n_s ≈ 0.96, amplitude ~1e-10, and r ~ 1e-3 at the price of tuning β near 3/2. The paper concludes with speculations about transitions to the GR branch.

Significance. If the results hold, the paper provides a concrete alternative to inflaton-driven inflation in which the dark purely gravitational sector generates both the background expansion and the nearly scale-invariant perturbations. Its strengths are the explicit derivation of the perturbation action, the mapping to the standard Mukhanov–Sasaki formalism, the calculation of spectral indices from the GUMG action rather than by assumption, and the transparent treatment of stability conditions. The main technical risk is the status of the homogeneous ghost mode, which is acknowledged but not shown to decouple from the physical perturbations.

major comments (2)
  1. [§4, Eq. (4.11) and §5] Section 4 states that the spatially constant mode ψ0 has a negative kinetic term and is a ghost, while the stability conditions w/Ω > 0 and 1+w > 0 in Eq. (4.27) are imported from [6] for the inhomogeneous sector only. The paper never demonstrates that the homogeneous ghost decouples from the inhomogeneous modes beyond the quadratic action, nor does it estimate the nonlinear couplings that could allow the ghost to decay into ϑ quanta or backreact on the Friedmann background over the required ~60 e-folds. Since the full Hamiltonian is unbounded below, the adiabatic vacuum used in Section 5 for the Mukhanov–Sasaki variable is not automatically the vacuum of the interacting theory; until this is addressed, Eqs. (5.5), (5.10), and the reconstruction in Section 6 are not established physical predictions.
  2. [§6, Eqs. (6.16)–(6.18)] The reconstruction of N(γ) is a qualitative fit with several free parameters (A, B, B1, β, H0), and the comparison with CMB data relies on choosing β = 3/2 − Δβ with Δβ ≈ 0.05 to obtain n_s ≈ 0.96, while H0 and B are adjusted to match the amplitude and tensor-to-scalar ratio. The paper itself notes that the β = 3/2 limit requires exponentially small B to be consistent with N ≈ 60 e-folds, leading to a gigantic H0 and an inadmissibly large r. No uniqueness or error analysis is given for the tuned parameters, so the claim that GUMG 'can match' observations is an existence argument rather than a predictive reconstruction.
minor comments (4)
  1. [§6, text before Eq. (6.1)] The text says 'small γ = a3', but γ = det γij scales as a^6 in the Friedmann background (Eq. (3.4)); the relation w = (1/3) d ln N/d ln a in Eq. (3.11) confirms this. This typo is inconsistent with the w → −1 limit of N → 1/√γ and should be corrected.
  2. [§4, Eq. (4.1)] The notation H = aH is confusing because H and H differ by a factor of a and both appear in the same section; a distinct symbol, e.g. script H or calligraphic H, would improve readability.
  3. [§5, Eq. (5.20)] The symbol p is used both for the post-inflationary power-law index in a(τ) ∝ τ^p in Eq. (5.10) and for the radiation era value p = 1/2 in Eq. (5.20); the text should explicitly state that these are the same power-law index in different epochs.
  4. [Introduction, last paragraph] The phrase 'Two abstracts contain the formalism...' appears to be a typo; it should read 'Two appendices contain the formalism...'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the inflationary spectra are derived from the GUMG action and the Section 6 match to CMB data is explicitly a reconstruction, not a prediction disguised as derivation.

full rationale

The paper's central derivation starts from the GUMG action (2.6) and the kinematical lapse restriction (1.1), then obtains the quadratic perturbed action, reduces to the physical scalar sector, and arrives at a Mukhanov-Sasaki equation (4.31) and the power spectrum (5.5). This chain is internally self-contained: the sound speed (4.26), stability conditions (4.27), and canonical variables are stated as consequences of the action rather than as assumed inputs. The stability criteria are imported from the authors' prior work [6], but they are used to restrict the admissible domain, not to fix the values of the spectra; the same-sector derivation is reproduced in this paper. The close analogy with hydrodynamical inflation and k-inflation is acknowledged explicitly, and the final formula (5.11) is noted to coincide with the classical result, which is a consistency check rather than a circular reduction. Section 6 is openly a model-reconstruction exercise: parameters A, B, B1 are introduced to fit ns, delta^2_Phi, and r, and the text explicitly says the number of parameters is 'sufficient to fit basic estimates coming from observations'. This is standard model building, not a fitted input renamed as a prediction. The homogeneous zero-mode ghost noted in Section 4 is a physical concern about stability and vacuum choice, and the paper does not fully control its nonlinear couplings, but this is a correctness/robustness caveat, not a circular derivation step. There is no self-definitional input, no uniqueness theorem invoked to forbid alternatives, and no equation that is equivalent to its own output by construction. The paper is self-contained against the external benchmark formalism of inflaton- and k-inflation perturbation theory.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new entities; the scalar graviton was already part of GUMG. The central claim relies on stability conditions inherited from [6] and the reconstruction uses several free parameters to match observations. The ghost in the homogeneous sector is acknowledged but not resolved.

free parameters (6)
  • A = O(1), fitted to amplitude and tilt
    Prefactor in N(gamma) ansatz (6.2), appears in spectral amplitude (6.11) and (6.16).
  • B = O(1), fitted to amplitude and tilt
    Coefficient of power term in (6.2); controls sound speed and amplitude.
  • B1 = fitted in expansion (6.7)
    Subleading coefficient in N(gamma) expansion, affects tilt via (6.12).
  • beta = 3/2 - Delta beta, with Delta beta ~ 0.05
    Exponent in (6.2); chosen near 3/2 to yield ns ~ 0.96.
  • H0 = several orders of magnitude below Planck scale
    Initial Hubble scale from integration constant C, fitted to amplitude delta^2_Phi ~ 10^-10.
  • C = 3 H0^2
    Integration constant in Friedmann equation (3.8); sets energy scale of GUMG fluid.
assumptions (5)
  • domain assumption The GUMG action and its constraint structure, as derived in [5] and [6], are correct.
    The paper starts from the GUMG action (2.6) and constraint analysis (2.11)-(2.13) from prior work.
  • domain assumption Stability conditions w/Omega > 0 and 1 + w > 0 hold throughout inflation.
    Eq. (4.27) is imported from [6] and assumed to be satisfied for the chosen N(gamma).
  • standard math Perturbations start in the adiabatic vacuum.
    Standard choice in inflationary cosmology, used in Section 5.
  • domain assumption The Bardeen potentials Psi and Phi are the observable gravitational potentials and matter couples covariantly.
    Section 5 and Appendix A; the paper assumes typical matter coupling even though GUMG breaks diffeomorphisms.
  • standard math The decaying mode of long-wavelength perturbations can be discarded.
    Standard in cosmological perturbation theory, used throughout Section 5.

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Cite this review

Pith. "Pith review of Inflation in generalized unimodular gravity." pith.science (2026). https://pith.science/paper/HKIPINIU

@misc{pith2026190805697,
  author       = {Pith},
  title        = {Pith review of: Inflation in generalized unimodular gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HKIPINIU}},
  note         = {Machine review of arXiv:1908.05697}
}
read the original abstract

The recently suggested generalized unimodular gravity theory, which was originally put forward as a model of dark energy, can serve as a model of cosmological inflation driven by the effective perfect fluid -- the dark purely gravitational sector of the theory. Its excitations are scalar gravitons which can generate, in the domain free from ghost and gradient instabilities, the red tilted primordial power spectrum of CMB perturbations matching with observations. The reconstruction of the parametric dependence of the action of the theory in the early inflationary Universe is qualitatively sketched from the cosmological data. The alternative possibilities of generating the cosmological acceleration or quantum transition to the general relativistic phase of the theory are also briefly discussed.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Henneaux-Teitelboim Form of the Generalized Unimodular Gravity Action

    hep-th 2024-12 conditional novelty 6.0 of 10

    A generalized Henneaux-Teitelboim action for generalized unimodular gravity is constructed, introducing a spatially nonlocal operator and showing that the theory is not fully diffeomorphism invariant.

Reference graph

Works this paper leans on

27 extracted references · 12 canonical work pages · cited by 1 Pith paper

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    RECONSTRUCTION OF THE MODEL We will not try to reconstruct the GUMG functions N (γ) and w(γ) that would match with the observable cosmological data throughout the whole evolution of the Universe. But a remarkable similarity of the cosmolog- ical perturbation formalism in the GUMG model and GR inflationary cosmology suggests to consider a pos- sible choice ...

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    INTRODUCTION Cosmological acceleration phenomenon and fundamen- tal problems of quantum gravity produce a rich play- ground for modifications of Einstein general relativity (GR). Quite curiously, these modifications associated nowadays with the resolution of the dark energy problem were initiated by Einstein himself [1] in the form of what is presently know...

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    The action of this theory follows from the Einstein-Hilbert action SEH[gµν ] by the substitution of the kinematical restriction (1.1)

    GENERALIZED UNIMODULAR GRA VITY Dynamics of the generalized unimodular gravity [5] is described in much detail in [6]. The action of this theory follows from the Einstein-Hilbert action SEH[gµν ] by the substitution of the kinematical restriction (1.1). This ac- tion generates the equations of motion which effectively coincide with Einstein equations in th...

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    GUMG COSMOLOGY Cosmological applications of GUMG imply the neces- sity of working in various coordinate systems, especially in closed model when the homogeneity hypersurface is a three-dimensional sphere on which γ = det γij can- not be globally regular. On the other hand, the condi- tion (1.1) breaks both time and space diffeomorphisms, which seemingly pr...

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    GUMG COSMOLOGICAL PER TURBATION THEOR Y The theory of cosmological perturbations on the Fried- mann background (3.3) was built for GUMG model in [6]. Despite a big difference of GUMG model from the conventional GR, where inflationary expansion is usu- ally driven by the additional inflaton scalar field rather than by effective perfect fluid, the formalisms of c...

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    INFLATIONAR Y POWER SPECTRUM Suppose that the GUMG functions N (a) and w(a) are chosen so that they provide a sufficiently long quasi- exponential expansion, that is w(a) is slightly higher than −1. This can be considered as the inflation stage gener- ated by the global conformal mode – the scale factor a which, in contrast to GR, is a dynamical degree of fr...

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    (6.6) Thus, ns − 1 is negative for 1 / 2 < β < 3/ 2 and small when β tends to 3 / 2

    (6.2) As a result one has up to terms of higher order in powers of the ratio γ/γ 0 w ≃ −1 +A √ γ γ0 , (6.3) Ω ≃ −2β (2β − 1)B ( γ γ0 ) β , (6.4) c2 s ≃ A 2B 1 β (2β − 1) ( γ γ0 ) 1 2 −β , (6.5) ns − 1 ≃ 3 2β − 3 6β − 1. (6.6) Thus, ns − 1 is negative for 1 / 2 < β < 3/ 2 and small when β tends to 3 / 2. To demonstrate the difficulties in the reconstruction ...

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    CONCLUSIONS All this shows that inflation and more generally cosmo- logical acceleration stage of the Universe can, in principle, be driven by the scalar graviton of the vacuum GUMG theory without any extra matter constituents like infla- ton field. This enlarges the list of the models with a similar property, starting with UMG and including the theory of va...

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