Challenges in connecting modified gravity theory and observations

Eric V. Linder

Phys. Rev. D 95, 023518 – Published 27 January 2017

Abstract

Cosmic acceleration may be due to modifications of cosmic gravity and to test this we need robust connections between theory and observations. However, in a model independent approach like effective field theory or a broad class like Horndeski gravity, several free functions of time enter the theory. We show that simple parametrizations of these functions are unlikely to be successful; in particular the approximation $α_{i} (t) \propto Ω_{de} (t)$ drastically misestimates the observables. This holds even in simple modified gravity theories like $f (R)$ . Indeed, oversimplified approximations to the property functions $α_{i} (t)$ can even miss the signature of modified gravity. We also consider the question of consistency relations and the role of tensor (gravitational wave) perturbations.

Received 19 July 2016

DOI:https://doi.org/10.1103/PhysRevD.95.023518

Physics Subject Headings (PhySH)

Alternative gravity theories Dark energy

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Eric V. Linder

Berkeley Center for Cosmological Physics & Berkeley Lab, University of California, Berkeley, California 94720, USA

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Issue

Vol. 95, Iss. 2 — 15 January 2017

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Images

Figure 1
The property function $α_{M}$ (solid black) and its ratio relative to the effective dark energy density, $α_{M} / Ω_{de}$ (dashed blue), is plotted vs scale factor $a$ . It is well behaved and physical, with the expected early and late time limits. Note that the quantity $α_{M} / Ω_{de}$ is definitely not well approximated by a constant in the recent Universe (or the late time Universe). We also show the result (dotted red) when calculating $α_{M}$ using Eq. (5). It gives an excellent approximation at late times but not at early times; the text explains why any function of $Ω_{de}$ is likely to fail during the observational epoch.
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Figure 2
The fractional energy density in dark energy is plotted vs the log of the scale factor from $a = 1 0^{14}$ to the present, for the tracker assumption (thick, black curve) and the cosmological constant (thin, blue curve). Assuming that the behavior follows the tracker at early times (rather than only at late times as in the generic physics scenario) imposes severe fine-tuning, well beyond the fine-tuning of the cosmological constant problem.
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Figure 3
The time dependence of the property functions divided by the effective dark energy, $α_{i} (a) / Ω_{de} (a)$ , is exhibited for exact solutions corresponding to case 1 (the thicker curves, with larger variation) and case 2 (the thinner curves, with smaller variation). They are not constant, as the prop approximation of Eq. (4) assumes. Such an approximation is particularly inaccurate during the key observational epoch $z = 0 - 3$ ( $\log a \approx - 0.6 - 0$ , shaded).
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Figure 4
The time dependence of the gravitational slip $η$ is compared for the exact solution (solid curves) and the $α_{i} \propto Ω_{de} (t)$ approximation (dashed curves). The prop approximation gives completely different and inaccurate results for the physics. While the true behavior depends on the differing parameters of the theory (shown for the same two cases as Fig. 3), the prop approximation has nearly identical behavior since the models have nearly the same expansion history. The prop approximation also underestimates the deviation from general relativity, possibly missing detection of modified gravity, and has an incorrect de Sitter asymptote.
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Figure 5
The time dependence of the gravitational coupling $G_{eff}$ is compared for the exact solution (solid curves) and the $α_{i} \propto Ω_{de} (t)$ approximation (dashed curves). The left panel shows the global behavior of $G_{eff}^{Φ}$ while the right panel zooms in on the detail around the observational epoch and also shows $G_{matter}$ (dotted curves) and $G_{light}$ (dot-dashed curves). The linear proportionality approximation gives completely different and inaccurate results for the physics. While the true behavior depends on the parameters of the theory (shown for the same two cases as Fig. 3), the prop approximation has nearly identical behavior since the models have nearly the same expansion history. The prop approximation also underestimates the deviation from general relativity during the observable epoch, possibly missing detection of modified gravity, and has an incorrect, and divergent, de Sitter asymptote.
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