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Multiscalar-Metric Gravity: Cosmological Constant Screening and Emergence of Massive-Graviton Dark Components of the Universe

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Abstract

In the multiscalar-metric frameworks, the issues of the vacuum energy/cosmological constant (CC) screening due to Weyl-scale enhancement of the Diff gauge symmetry, along with emergence of the massive dark gravity components through the gravitational Higgs mechanism are considered. A generic dark gravity model is developed, with two extreme versions of the model of particular interest based on general relativity (GR) and its classically equivalent Weyl transverse alternative, being compared and argued to be, generally, inequivalent. The so constructed spontaneously broken Weyl Transverse Relativity (WTR) is proposed as a viable beyond-GR effective field theory of gravity, with screening of the Lagrangian CC, superseded by the induced one, and emergence of the massive tensor and scalar gravitons as dark gravity components. A basic concept with the spontaneously broken Diff gauge symmetry/relativity—in particular, WTR vs. GR—as a principle source of the emergent dark gravity components of the Universe is put forward.

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Notes

  1. For an original treatment of the vacuum energy/CC problem, see [1].

  2. For the observational status of CC see, e.g., [2]. For a modern view of the vacuum energy/CC problem, see, e.g., [3, 4].

  3. For the importance of the EFT approach to the CC problem, cf., e.g., [5].

  4. For modifications of gravity see, e.g., [6–8], and in particular, [9, 10] for massive gravity.

  5. For the relevance of TDiff gauge symmetry as a substitute of the full Diff one for a consistent description of the massless tensor graviton, see [12].

  6. The notation in Introduction is in accord with what follows.

  7. For a development of WTR, known also as Weyl Transverse Gravity, cf., e.g., [15, 16].

  8. This fact alone could, conceivably, justify WTR (and its siblings) to be further explored at least on par with GR (and its siblings).

  9. This, in turn, ensures one more important ingredient for solving the CC problem—a dynamical non-gravitating space-time measure \(\sqrt{-{\kappa}}\). For the relevance of such a measure built from the so-called “measure fields” within modified GR, still with the conventional metric \(g_{\mu\nu}\), cf., e.g., [33, 34].

  10. Though adhering primarily to the space-time dimension \(n=4\), not to be formally confined to this value, the term “scalar quartet,” valid at \(n=4\), is replaced in what follows with a more general one—the “multiscalar”—and, accordingly, “quartet-metric” with “multiscalar-metric,” etc.

  11. Here we start directly from the EFT level. For a justification of EFT of the multiscalar-metric gravity as an affine-Goldstone nonlinear model, see [37].

  12. To this end, we assume the (patchwise) Lorentz-preservation in the field space of \(X^{a}\), up to the physically equivalent affine redefinitions of the fields \(X^{a}\) and the metric \(\eta_{ab}\).

  13. The dependence \(\bar{w}(\sigma)\) presents the simplest case. More generally, one could envisage within EFT the dependence of the effective metric scale \(\bar{w}\) on the matter fields, the environment, etc, with the ultraviolet and infrared behavior of the effective metric being a priori quite different.

  14. By default, the space-time indices are now lowered (raised) through the effective metric \(\bar{g}_{\mu\nu}\) (\(\bar{g}^{\mu\nu}\)) if not stated explicitly otherwise.

  15. A priori, the Weyl scale symmetry is assumed to be specifically a metric one, with the conventional matter fields, like \(X^{a}\), to be inert under Weyl rescalings, unless stated otherwise.

  16. For definiteness, we restrict the consideration here and in what follows to the space-time dimension \(n=4\).

  17. Note, though, that the exclusively derivative couplings of \(\sigma\) still admit the residual global Weyl scale symmetry \(g_{\mu\nu}\to e^{\zeta_{0}}g_{\mu\nu}\), with constant shifts \(\sigma\to\sigma+2\zeta_{0}\).

  18. In the spirit of EFT, \(\bar{L}_{G}\) is assumed to be of the order of the Plank mass squared, \(M_{\textrm{Pl}}^{2}\), while \(\bar{L}_{M}\) and \(\bar{\Delta}L\) to be relatively suppressed at the scale of \(M_{\textrm{Pl}}\).

  19. In principle, any modification of \(\bar{L}_{g}\), like \(f(\bar{R})\), etc., is conceivable too.

  20. At that, is explicitly missing, being expressed, in principle, through \(\sigma\) accounted in \(\bar{L}_{M}\).

  21. A priori, one may admit another mode of spontaneous breaking, with the potential depending, instead of , on its inverse , subject to the analyticity requirement near and possessing by the flat background at the symmetric point .

  22. At that, the scalar-graviton FE as such is absent, with the composite \(\sigma\) being, generally, off-mass-shell, \(\delta\bar{L}_{M}/\delta\sigma\neq 0\).

  23. Note that the value \(\bar{\theta}_{s\mu\nu}\neq 0\) additionally closes or opens a space-time region, i.e., acts in this region as an effective dark matter (DM) or DE, depending on \(\delta\bar{L}_{M}/\delta\sigma\) being positive or negative, respectively.

  24. To be more concise, the scalar graviton, being a measure of the (Weyl) scale transformations, may be called the systolon, with the tensor graviton being conventionally the graviton.

  25. For the reduced case , with the (ultralight) physical \(\sigma\) serving as a scalar DE superseding the Lagrangian CC, see [38]. For the similar case with \(\sigma\) as a scalar DM, cf. [39].

  26. For viability of the trace-free tensor-gravity FE in cosmology, cf., e.g., [40, 41].

  27. In addition to the scalar graviton \(\sigma\), the extended matter Lagrangian \(\hat{L}_{M}=\hat{L}_{M}(\hat{g}_{\mu\nu},\sigma,\phi_{I})\) could, in principle, account for the (more conventional) DM components. In the absence of matter, the purely scalar-graviton Lagrangian \(\hat{L}_{M}=\hat{L}_{s}(\hat{g}_{\mu\nu},\sigma)\) could be sought for, e.g., in a most general form for the scalar field \(\sigma\) satisfying the second-order FE to avoid the explicit Ostrogradsky instabilities [42, 43].

  28. Otherwise, the multiscalar-modified UR could be obtained as a reduction of the multiscalar-modified WTR by imposing, ab initio, the constraint \(g={\kappa}\), consistent with \(\hat{g}={\kappa}\). This would imply, in particular, the explicit absence of the scalar graviton \(\sigma\), which is to be added ad hock as an additional scalar particle.

  29. Note that imposing ab initio the UR restriction \(g={\kappa}\), eliminating, in fact, one independent component would result in the LA in \(h=\hat{h}=2\partial\chi\) (still preserving \(\hat{f}=0\)), and \(\hat{h}_{\alpha\beta}=h_{\alpha\beta}\), as well as \({s}=0\), with lost Weyl scale symmetry.

  30. In particular, a similar potential at a proper \(\beta_{4}\) is used for the consistent description of the massive tensor graviton within spontaneously broken GR [28].

  31. In fact, due to \(\hat{f}=0\) in the LA, the second term in (46) in the quadratic approximation is irrelevant, with the potential being (quasi-)stable irrespective of \(\hat{\beta}_{4}\), admitting, in particular, \(\hat{\beta}_{4}=0\) and a purely quadratic .

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ACKNOWLEDGMENTS

The author is sincerely grateful to S. S. Gershtein for encouraging discussions.

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  1. Theory Division, Logunov Institute for High Energy Physics of NRC “Kurchatov Institute”, 142281, Protvino, Moscow region, Russia

    Yury F. Pirogov

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Pirogov, Y.F. Multiscalar-Metric Gravity: Cosmological Constant Screening and Emergence of Massive-Graviton Dark Components of the Universe. Gravit. Cosmol. 28, 263–274 (2022). https://doi.org/10.1134/S0202289322030070

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