Scalar-tensor gravity in an accelerating universe

TL;DR

This paper investigates how a accelerating universe constrains scalar-tensor gravity by linking late-time observations to the underlying Lagrangian. It develops a reconstruction framework that uses $D_L(z)$ and $H(z)$, together with perturbation data, to infer the functions $F(\Phi)$ and $U(\Phi)$ (or $A(\varphi)$ and $V(\varphi)$) in the Jordan and Einstein frames, respectively. A key finding is that a potential-free scalar-tensor theory is generically inconsistent for GR-like expansion histories beyond a small redshift, due to positivity requirements on the graviton and scalar energies; including curvature can somewhat ameliorate this, but overall the observed $H(z)$ up to $z\sim2$ places strong constraints on such theories. The work also analyzes massless-scalar scenarios with a cosmological constant, showing that to fit the data the scalar must contribute negligibly or the universe must be marginally closed, and demonstrates that, in general, cosmological observations can be more constraining than solar-system tests for these theories, enabling a path to reconstruct the full Lagrangian from cosmological data.

Abstract

We consider scalar-tensor theories of gravity in an accelerating universe. The equations for the background evolution and the perturbations are given in full generality for any parametrization of the Lagrangian, and we stress that apparent singularities are sometimes artifacts of a pathological choice of variables. Adopting a phenomenological viewpoint, i.e., from the observations back to the theory, we show that the knowledge of the luminosity distance as a function of redshift up to z ~ (1-2), which is expected in the near future, severely constrains the viable subclasses of scalar-tensor theories. This is due to the requirement of positive energy for both the graviton and the scalar partner. Assuming a particular form for the Hubble diagram, consistent with present experimental data, we reconstruct the microscopic Lagrangian for various scalar-tensor models, and find that the most natural ones are obtained if the universe is (marginally) closed.

Figures (6)

• Figure 1: Reconstructed $F(z)$ [ i.e., Brans-Dicke scalar $\Phi_{\rm BD}(z)$] and Einstein-frame scalar $\varphi$ as functions of the Jordan-frame ( i.e., observed) redshift $z$, for the maximum value of $|\alpha_0|$ allowed by solar-system experiments, and for a vanishing potential. The helicity-0 degree of freedom $\varphi$ diverges at $z_{\rm max} \approx 0.68$.
• Figure 2: Maximum redshift $z$ consistent with the positivity of energy of both the graviton and the scalar field, as a function of the parameter $|\alpha_0|$. This figure corresponds to the case of a vanishing scalar-field potential, and we fit the exact $H(z)$ predicted by general relativity plus a cosmological constant (GR $+\Lambda$).
• Figure 3: Two versions of the reconstructed coupling function $\ln A(\varphi)$ for $|\alpha_0| = 1$, the dashed one looking bi-valued, but the (single-valued) solid one giving the same predicted $H(z)$. This figure still corresponds to the case of a vanishing scalar-field potential, and we fit the exact $H(z)$ predicted by GR $+\Lambda$.
• Figure 4: Random deformations of the $H(z)$ predicted by GR $+\Lambda$ (with $\Omega_{\Lambda,0} = 0.7$), and corresponding maximum value of the redshift $z$ consistent with the positivity of energy. The dashed lines indicate the region in which random points have been chosen at regular intervals of $z$. The thin solid lines correspond to two polynomial fits of such random points. Note that they can differ from the GR $+\Lambda$ curve even more than the dashed lines. The dotted line labeled simply "GR" corresponds to a vanishing cosmological constant $\Lambda$. Such a bias of the GR $+\Lambda$ curve changes $z_{\rm max}$ much more that the random noise we considered.
• Figure 5: Maximum redshift $z$ consistent with the positivity of energy, as a function of the value of a constant potential $V$ (case of a massless helicity-0 degree of freedom $\varphi$).