Cosmological constraints on post-Newtonian parameters in effectively massless scalar-tensor theories of gravity

TL;DR

This paper investigates how scalar-tensor theories of gravity affect cosmological observations, constraining deviations from general relativity using CMB and BAO data, and finds no significant evidence for deviation.

Contribution

It provides new cosmological constraints on post-Newtonian parameters in massless scalar-tensor gravity models beyond Jordan-Brans-Dicke theory.

Findings

No significant deviation from Einstein's general relativity.

Constraints on scalar-tensor coupling parameter b4 < 0.064 at 95% CL.

Post-Newtonian parameters b3_{PN} and \u03b2_{PN} are tightly constrained near their GR values.

Abstract

We study the cosmological constraints on the variation of the Newton's constant and on post-Newtonian parameters for simple models of scalar-tensor theory of gravity beyond the extended Jordan-Brans-Dicke theory. We restrict ourselves to an effectively massless scalar field with a potential $V \propto F^2$, where $F(\sigma)=N_{pl}^2+\xi\sigma^2$ is the coupling to the Ricci scalar considered. We derive the theoretical predictions for cosmic microwave background (CMB) anisotropies and matter power spectra by requiring that the effective gravitational strength at present is compatible with the one measured in a Cavendish-like experiment and by assuming adiabatic initial condition for scalar fluctuations. When comparing these models with $Planck$ 2015 and a compilation of baryonic acoustic oscilation (BAO) data, all these models accomodate a marginalized value for $H_0$ higher than in $\Lambda$CDM. We find no evidence for a statistically significant deviation from Einstein's general relativity. We find $\xi < 0.064$ ($|\xi| < 0.011$) at 95 % CL for $\xi > 0$ (for $\xi < 0$, $\xi \ne -1/6$). In terms of post-Newtonian parameters, we find $0.995 < \gamma_{\rm PN} < 1$ and $0.99987 < \beta_{\rm PN} < 1$ ($0.997 < \gamma_{\rm PN} < 1$ and $1 < \beta_{\rm PN} < 1.000011$) for $\xi >0$ (for $\xi < 0$). For the particular case of the conformal coupling, i.e. $\xi=-1/6$, we find constraints on the post-Newtonian parameters of similar precision to those within the Solar System.

Tables

Table 1. Table 1: Constraints on main and derived parameters for P ​ l ​ a ​ n ​ c ​ k 𝑃 𝑙 𝑎 𝑛 𝑐 𝑘 Planck TT + lowP + lensing + BAO (at 68% CL if not otherwise stated). In the first column we report the results obtained for the branch with ξ > 0 𝜉 0 \xi>0 and in the second the branch for ξ < 0 𝜉 0 \xi<0 . In the first column we report the results obtained for the Λ Λ \Lambda CDM model with the same dataset Ade  et al. ( 2016b ) and in the second column IG case, i.e. N p ​ l = 0 subscript 𝑁 𝑝 𝑙 0 N_{pl}=0 , for comparison Ballardini  et al. ( 2016 ) .

	$Planck$ TT + lowP	$Planck$ TT + lowP	$Planck$ TT + lowP	$Planck$ TT + lowP
	+ lensing + BAO	+ lensing + BAO	+ lensing + BAO	+ lensing + BAO
	$\Lambda$CDM	IG	($\xi >0$)	()
${\omega }_b$	$0.02225\pm 0.00020$	${0.02224}_{-0.00021}^{+0.00020}$	$0.02226\pm 0.00019$	$0.02226\pm 0.00021$
${\omega }_c$	$0.1186\pm 0.0012$	$0.1191\pm -0.0014$	$0.1190\pm 0.0015$	$0.1189\pm 0.0015$
$H_0$ [km s-1 Mpc-1]	$67.78\pm 0.57$	${69.4}_{-0.9}^{+0.7}$	${69.2}_{-1.1}^{+0.8}$	${69.2}_{-1.0}^{+0.7}$
${\tau }_{\mathrm{re}}$	$0.066\pm 0.012$	${0.063}_{-0.014}^{+0.012}$	$0.068\pm 0.014$	$0.069\pm 0.013$
$\ln \left({10}^{10}{A}_s\right)$	$3.062\pm 0.024$	${3.059}_{-0.026}^{+0.022}$	${3.069}_{-0.027}^{+0.023}$	$3.071\pm 0.024$
$n_s$	$0.9675\pm 0.0045$	${0.9669}_{-0.0047}^{+0.0042}$	$0.9674\pm 0.0046$	$0.9728\pm 0.0043$
$\xi$	$\dots$	(95% CL)	(95% CL)	$>-0.011$ (95% CL)
$N_{pl}\left[{\text{M}}_{pl}\right]$	$\dots$	$0$	$>0.81$ (95% CL)	(95% CL)
${\gamma }_{\mathrm{PN}}$	$1$	$>0.9970$ (95% CL)	$>0.995$ (95% CL)	$>0.997$ (95% CL)
${\beta }_{\mathrm{PN}}$	$1$	$1$	$>0.99987$ (95% CL)	(95% CL)
$\delta G_N/G_N$	$\dots$	$-{0.009}_{-0.009}^{+0.003}$	$>-0.027$ (95% CL)	$>-0.027$ (95% CL)
${10}^{13}{\dot{G}}_N\left(z=0\right)/G_N$ [yr-1]	$\dots$	$-{0.37}_{-12}^{+34}$	$>-1.4$ (95% CL)	$>-0.97$ (95% CL)

Table 2. Table 2: Constraints on main and derived parameters for P ​ l ​ a ​ n ​ c ​ k 𝑃 𝑙 𝑎 𝑛 𝑐 𝑘 Planck TT + lowP + lensing + BAO in the case of the CC model (at 68% CL if not otherwise stated).

	$Planck$ TT + lowP	$Planck$ TT + lowP
	+ lensing + BAO	+ lensing + BAO
		+ HST
${\omega }_b$	$0.02223\pm 0.00021$	$0.02228\pm 0.00021$
${\omega }_c$	${0.1188}_{-0.0015}^{+0.0014}$	$0.1187\pm 0.0015$
$H_0$ [km s-1 Mpc-1]	${69.19}_{-0.93}^{+0.77}$	$70.20\pm 0.83$
${\tau }_{\mathrm{re}}$	${0.068}_{-0.014}^{+0.012}$	${0.070}_{-0.015}^{+0.013}$
$\ln \left({10}^{10}{A}_s\right)$	$3.070\pm 0.024$	$3.074\pm 0.024$
$n_s$	$0.9699\pm 0.0045$	$0.9728\pm 0.0043$
$N_{pl}\left[{\text{M}}_{pl}\right]$	(95% CL)	${1.000028}_{-0.000014}^{+0.000012}$
${\gamma }_{\mathrm{PN}}$	$>0.99996$ (95% CL)	$1.00003\pm 0.00001$
${\beta }_{\mathrm{PN}}$	(95% CL)	$0.999998\pm 0.000001$

Equations

$$S=$$

$$+G_4(\sigma ,\chi )R-2G_{4,\chi }(\sigma ,\chi )\left(\square {\sigma }^2-{\sigma }^{;\mu \nu }{\sigma }_{;\mu \nu }\right)$$

$$+G_5(\sigma ,\chi )G_{\mu \nu }{\sigma }^{;\mu \nu }+\frac{1}{3}{G}_{5,\sigma }(\sigma ,\chi )(\square {\sigma }^3$$

$$\displaystyle\left.-3\sigma_{;\mu\nu}\sigma^{;\mu\nu}\square\sigma+2\sigma_{;\mu\nu}\sigma^{;\nu\rho}{\sigma^{;\mu}}_{;\rho}\right)+{\cal L}_{m}\Bigr{]}\,,$$

$$2G_4\equiv F(\sigma )=N_{\mathrm{pl}}^2+\xi {\sigma }^2,$$

$$\begin{array}{rl}{G}_{\mu \nu }& =\frac{1}{F(\sigma )}[T_{\mu \nu }+{\partial }_{\mu }\sigma {\partial }_{\nu }\sigma -\frac{1}{2}{g}_{\mu \nu }{\partial }^{\rho }\sigma {\partial }_{\rho }\sigma \\ & \frac{}{}-g_{\mu \nu }V(\sigma )+({\nabla }_{\mu }{\nabla }_{\nu }-g_{\mu \nu }\square )F(\sigma )].\end{array}$$

$$R=\frac{1}{F}\left[-T+{\partial }_{\mu }\sigma {\partial }^{\mu }\sigma +4V+3\square F\right],$$

$$-\square \sigma -\frac{1}{2}{F}_{,\sigma }R+V_{,\sigma }=0,$$

$$\begin{array}{rl}& -\square \sigma \left(1+\frac{3}{2}\frac{F_{,\sigma }^2}{F}\right)+V_{,\sigma }-2\frac{V{F}_{,\sigma }}{F}\\ & +\frac{F_{,\sigma }}{2F}\left[T-{\partial }_{\mu }\sigma {\partial }^{\mu }\sigma \left(1+3F_{,\sigma \sigma }\right)\right]=0.\end{array}$$

$$\mathrm{d}{s}^2=-\mathrm{d}{t}^2+a(t{)}^2\mathrm{d}{x}_i\mathrm{d}{x}^i.$$

$$3H^{2}F$$

$$-2\dot{H}F$$

$$\begin{split}\ddot{\sigma}=&-3H\dot{\sigma}+\frac{\xi\sigma}{F+6\xi^{2}\sigma^{2}}\Bigl{[}\rho_{m}+4V-\frac{FV_{,\sigma}}{\xi\sigma}\\ &-(1+6\xi)\dot{\sigma}^{2}\Bigr{]}.\end{split}$$

$${\rho }_{\sigma }$$

$$=\frac{{\dot{\sigma }}^2}{2}+V(\sigma )-6H\xi \sigma \dot{\sigma },$$

$$\begin{array}{rl}{p}_{\sigma }=& \frac{{\dot{\sigma }}^2}{2}\left[\frac{F(1+4\xi )+2{\xi }^2{\sigma }^2}{F+6{\xi }^2{\sigma }^2}\right]-2H\xi \sigma \dot{\sigma }\\ & +\frac{2{\xi }^2{\sigma }^2}{F+6{\xi }^2{\sigma }^2}\left({\rho }_m+4V-\frac{F{V}_{,\sigma }}{\xi \sigma }\right)-V,\end{array}$$

$${\rho }_{\mathrm{DE}}$$

$$p_{\mathrm{DE}}$$

$$G_{\mathrm{eff}}=\frac{1}{8\pi F}\left(\frac{2F+4F_{,\sigma }^2}{2F+3F_{,\sigma }^2}\right).$$

$${\tilde{\sigma }}_0^2=\frac{1}{\xi }\frac{1+8\xi }{1+6\xi },$$

$${\tilde{\sigma }}_0^2=\frac{18{\tilde{N}}_{pl}^2({\tilde{N}}_{pl}^2-1)}{1+3{\tilde{N}}_{pl}^2};$$

$$\begin{array}{rl}{\tilde{\sigma }}_0^2=& \frac{1-2{\tilde{N}}_{pl}^2+2\xi (4-3{\tilde{N}}_{pl}^2)}{2\xi (1+6\xi )}\\ & \pm \frac{\sqrt{1-4\xi (5{\tilde{N}}_{pl}^2-4)+4{\xi }^2(3{\tilde{N}}_{pl}^2-4{)}^2}}{2\xi (1+6\xi )}.\end{array}$$

$${\tilde{N}}_{pl}$$

$$\mathrm{d}{s}^2=-(1+2\Phi -2{\beta }_{\mathrm{PN}}{\Phi }^2)\mathrm{d}{t}^2+(1-2{\gamma }_{\mathrm{PN}}\Phi )\mathrm{d}{x}_i\mathrm{d}{x}^i.$$

$${\gamma }_{\mathrm{PN}}$$

$${\beta }_{\mathrm{PN}}$$

$$\mathrm{d}{s}^2=a(\tau {)}^2\left[-\mathrm{d}{\tau }^2+({\delta }_{ij}+h_{ij})\mathrm{d}{x}^i\mathrm{d}{x}^j\right],$$

$$\displaystyle h_{ij}^{S}=\int\mathrm{d}^{3}k\,e^{i\vec{k}\cdot\vec{x}}\Biggl{[}$$

$$\displaystyle+\left(\hat{k}_{i}\hat{k}_{j}-\frac{1}{3}\delta_{ij}\right)\eta(\vec{k},\tau)\Biggr{]},$$

$$\delta \sigma =\int {\mathrm{d}}^{3}k{e}^{i\vec{k}\cdot \vec{x}}\delta \sigma (\vec{k},\tau ).$$

$$\frac{k^2}{a^2}\eta -\frac{1}{2}H\dot{h}=-\frac{1}{2F}\left[\delta \rho +\dot{\sigma }\delta \dot{\sigma }+V_{,\sigma }\delta \sigma -\frac{F_{,\sigma }}{F}\left(\rho +\frac{{\dot{\sigma }}^2}{2}+V-3H\dot{F}\right)\delta \sigma \right],$$

$$\frac{k^2}{a^2}\dot{\eta }=\frac{1}{2F}\left[\sum _i({\rho }_i+p_i){\theta }_i+k^2\left(\dot{\sigma }\delta \sigma +\delta \dot{F}-H\delta F\right)\right],$$

$$\ddot{h}+3H\dot{h}-2\frac{k^2}{a^2}\eta =-\frac{3}{F}[p+\dot{\sigma }\delta \dot{\sigma }-V_{,\sigma }\delta \sigma -\frac{F_{,\sigma }}{F}\left(p+\frac{{\dot{\sigma }}^2}{2}-V+\ddot{F}+2H\dot{F}\right)\delta \sigma$$

$$+\frac{2}{3}\frac{k^2}{a^2}\delta F+\delta \ddot{F}+2H\delta \dot{F}+\frac{1}{3}\dot{h}\dot{F}],$$

$$\ddot{h}+6\ddot{\eta }+3H(\dot{h}+6\dot{\eta })-2\frac{k^2}{a^2}\eta =-\frac{3}{F}\left[\sum _i({\rho }_i+p_i){\sigma }_i+\frac{2}{3}\frac{k^2}{a^2}\delta F+\frac{\dot{F}}{3}(\dot{h}+6\dot{\eta })\right],$$

$$\begin{split}\delta\ddot{\sigma}=&-\delta\dot{\sigma}\left[3H+\frac{2(1+6\xi)\xi\sigma\dot{\sigma}}{F+6\xi^{2}\sigma^{2}}\right]-\delta\sigma\biggl{\{}\frac{k^{2}}{a^{2}}+\frac{FV_{,\sigma,\sigma}}{F+6\xi^{2}\sigma^{2}}-\frac{2\xi\sigma V_{,\sigma}}{F+6\xi^{2}\sigma^{2}}\biggl{[}1+\frac{F(1+6\xi)}{F+6\xi^{2}\sigma^{2}}\biggr{]}\\ &+\frac{\xi}{F+6\xi^{2}\sigma^{2}}\biggl{[}1-\frac{2(1+6\xi)\xi\sigma^{2}}{F+6\xi^{2}\sigma^{2}}\biggr{]}\Big{[}(1+6\xi)\dot{\sigma}^{2}-4V+(3p-\rho)\Big{]}\bigg{\}}-\frac{\left(3\delta p-\delta\rho\right)\xi\sigma}{F+6\xi^{2}\sigma^{2}}-\frac{1}{2}\dot{h}\dot{\sigma}.\end{split}$$

Full text

I Introduction

The astrophysical and cosmic tests for the change of the fundamental physical constants are improving thanks to the increasing precision of observations Uzan (2011); Ade et al. (2014). In most of the cases these tests cannot compete with the precision which can be achieved in laboratories, but can probe lengths and/or timescales otherwise unaccessible on ground. There are however exceptions: for instance, current cosmological data can constrain the time variation of the Newtonian constant at the same level of experiments within the Solar System such as the Lunar Laser ranging Umiltà et al. (2015); Ballardini et al. (2016).

As far as cosmological tests are concerned, one of workhorse model to test deviations from general relativity (GR) is the extended Jordan-Brans-Dicke (eJBD) Jordan (1949); Brans and Dicke (1961) theory, which has been extensively studied Chen and Kamionkowski (1999); Nagata et al. (2004); Acquaviva et al. (2005); Avilez and Skordis (2014); Li et al. (2013); Ooba et al. (2016); Umiltà et al. (2015); Ballardini et al. (2016). eJBD is perhaps the simplest extension of GR within the more general Horndeski theory Horndeski (1974):

[TABLE]

where $\chi=-g^{\mu\nu}\partial_{\mu}\sigma\partial_{\nu}\sigma$, ”;” denotes the covariant derivative, $R$ is the Ricci scalar, $G_{\mu\nu}=R_{\mu\nu}-g_{\mu\nu}R/2$, and ${\cal L}_{m}$ is the density Lagrangian for the rest of matter. The eJBD theory corresponds to $G_{3}=G_{5}=0$, $G_{2}=\omega_{\rm BD}\chi/\sigma-V(\sigma)$, $G_{4}=\sigma$ (in the equivalent induced gravity (IG) formulation with a standard kinetic term the two last conditions become $G_{2}=\chi/2-V(\sigma)$, $G_{4}=\xi\sigma^{2}/2$ with $\xi=1/(4\omega_{\rm BD})$).

Cosmology puts severe test on eJBD theories. The constraints from $Planck$ 2015 and a compilation of baryon acoustic oscillations (BAO) data lead to a 95% CL upper bound $\xi<0.00075$, weakly dependent on the index for a power-law potential Ballardini et al. (2016) (see Umiltà et al. (2015) for the $Planck$ 2013 constraints obtained with the same methodology). In terms of the first post-Newtonian parameter $\gamma_{\rm PN}=(1+\omega_{\rm BD})/(2+\omega_{\rm BD})=(1+4\xi)/(1+8\xi)$, the above 95% CL constraint read as $|\gamma_{\rm PN}-1|<0.003$ Ballardini et al. (2016). With the same data, a 95% CL bound is obtained on the relative time variation of the effective Newton’s constant $10^{13}|\dot{G}_{\textup{eff}}/G_{\textup{eff}}|\lesssim 6\times 10^{-3}\ H_{0}$ at 95% CL with an index for a power-law potential in the range $[0,8]$. The combination of future measurement of CMB anisotropies in temperature, polarization and lensing with Euclid-like (galaxy clustering and weak lensing) data can lead to constraints on $\gamma_{\rm PN}$ at a slightly larger level than the current Solar System constraints Ballardini et al. (2019) (see also Alonso et al. (2017) for forecasts for different experiments with different assumptions).

However, theoretical priors can play an important role in the derivation of the cosmological constraints and need to be taken into account in the comparison with other astrophysical or laboratory tests. Indeed, for eJBD theories only the first post-Newtonian parameter $\gamma_{\rm PN}$ is nonzero and fully encodes the deviations from GR, being the second post-Newtonian parameter $\beta_{\rm PN}\propto\mathrm{d}\gamma_{\rm PN}/\mathrm{d}\sigma$. In this paper we wish to go beyond the working assumption of $\beta_{\rm PN}=0$ implicit within in eJBD theories. For this purpose we therefore consider nonminimally coupled (NMC) scalar fields with $2G_{4}=N_{\rm pl}^{2}+\xi\sigma^{2}$ as a minimal generalization of the eJBD theories. NMC with this type of coupling are also known as extended quintessence models in the context of dark energy Uzan (1999); Perrotta et al. (1999); Bartolo and Pietroni (2000); Amendola (1999); Chiba (1999). As for eJBD, NMC are also within the class of Horndeski theories consistent with the constraints on the velocity of propagation of gravitational waves Baker et al. (2017); Creminelli and Vernizzi (2017); Ezquiaga and Zumalacárregui (2017) which followed the observation of GW170817 and its electromagnetic counterpart Abbott et al. (2017) (see also Lombriser and Taylor (2016); Lombriser and Lima (2017)).

The outline of this paper is as follows. In Section II we discuss the background dynamics and the post-Newtonian parameters $\gamma_{\rm PN}$ and $\beta_{\rm PN}$ for this class of scalar-tensor theories. We study the evolution of linear fluctuations in Section III. We show the dependence on $\xi$ of the CMB anisotropies power spectra in temperature and polarization in Section IV. We present the $Planck$ and BAO constraints on these models in Section V. We conclude in Section VI. The initial conditions for background and cosmological fluctuations are collected in Appendix A.

II Dark Energy as an effectively massless scalar field non-minimally coupled to gravity

We study the restriction of the Horndeski action (I) to a standard kinetic term and $G_{3}=G_{5}=0$. We also assume:

[TABLE]

where $\xi$ is the coupling to the Ricci scalar which is commonly used in extended quintessence Uzan (1999); Perrotta et al. (1999); Bartolo and Pietroni (2000); Amendola (1999); Chiba (1999). For simplicity we denote by a tilde the quantities normalized to $M_{pl}\equiv 1/\sqrt{8\pi G}$, where $G=6.67\times 10^{-8}$ cm3 g*-1* s*-2* is the gravitational constant measured in a Cavendish-like experiment. We also introduce the notation $\tilde{N}_{pl}\equiv 1\mp\Delta\tilde{N}_{pl}$ for $\xi\gtrless 0$.

The field equations are obtained by varying the action with respect to the metric:

[TABLE]

The Einstein trace equation results:

[TABLE]

where $T$ is the trace of the energy-momentum tensor. The Klein-Gordon (KG) equation can be obtained varying the action with respect to the scalar field:

[TABLE]

and substituting the Einstein trace equation one obtains:

[TABLE]

In this paper, we do not consider a quintessence-like inverse power-law potential (see for instance Uzan (1999); Perrotta et al. (1999); Bartolo and Pietroni (2000); Chiba (1999)), but we restrict ourselves to a potential of the type $V\propto F^{2}$ in which the scalar field is effectively massless. This case generalizes the broken scale invariant case Wetterich (1988a); Cooper and Venturi (1981); Finelli et al. (2008) to NMC and is a particular case of the class of models with $V\propto F^{M}$ admitting scaling solutions Amendola (1999). Note that though for the form of $F(\sigma)$ used in the paper and for large values of $\sigma$, this potential looks similar to that in the Higgs inflationary model Bezrukov and Shaposhnikov (2008), in fact it is crucially different, since it is exactly flat in the Einstein frame***Although our work is based in the original Jordan frame, it is also useful to think about this class of theories in the dual Einstein frame where $\hat{g}_{\mu\nu}\propto Fg_{\mu\nu}\,,\hat{V}={V}/{F^{2}}$. in the absence of other matter and cannot support a metastable inflationary stage in the early Universe. Contrary, this model may be used for description of dark energy in the present Universe.

II.1 Background cosmology

We consider cosmic time and a flat FLRW metric, for which the unperturbed cosmological spacetime metric is given by:

[TABLE]

The Friedmann and the KG equations are then given by:

[TABLE]

[TABLE]

In Fig. 1 the evolution of the scalar field $\sigma$ is shown for different values of $\xi$ for both positive and negative values of the coupling. The natural initial conditions for the background displayed in Appendix A neglect the decaying mode which would be rapidly dissipated, but would have destroyed the Universe isotropy at sufficiently early times otherwise (see for instance Finelli et al. (2008)). With this natural assumption the scalar field is nearly at rest deep in the radiation era whereas it grows (decreases) for positive (negative) couplings during the matter era and it reaches a constant value at recent times. During the matter dominated era in the regime $\xi\sigma^{2}\ll N_{\mathrm{p}l}^{2}$ (which is the only one allowed by observations, see Section V), the evolution of the scalar field can be approximated as $\sigma\sim\sigma_{i}\left[1+2\xi(\ln a+8)\right]$, with $\sigma_{i}$ being the initial value of the scalar field in the radiation era. In the bottom panel, we show the evolution of the scalar field for the conformal coupling (CC) case with $\xi=-1/6$ for different values of $N_{\mathrm{p}l}$. In this case the field is always sub-Planckian for $\Delta\tilde{N}_{\mathrm{p}l}\lesssim 0.0005$.

The above equations lead to the straightforward associations:

[TABLE]

[TABLE]

where in the equation for $p_{\sigma}$ we have explicitly substituted the KG equation. We can recover an expression for the dark energy (DE) density parameter dividing $\rho_{\sigma}$ for the quantity $3H^{2}F$ which represents the critical density.

Alternatively, it is also convenient to define new density parameters in a framework which mimics Einstein gravity at present and satisfy the conservation law $\dot{\rho}_{\rm DE}+3H(\rho_{\rm DE}+p_{\rm DE})=0$ Boisseau et al. (2000); Gannouji et al. (2006):

[TABLE]

The effective parameter of state for DE can be defined as $w_{\rm DE}\equiv p_{\rm DE}/\rho_{\rm DE}$.

In Fig. 2 the evolution of this effective parameter of state is shown for different values of the parameters $N_{pl}$ and $\xi$. In all the cases the parameter of state $w_{\rm DE}$ mimics $1/3$ ($-1$) in the relativistic era (at late times): this behaviour can be easily understood from Eq. (15) when $\rho_{r}$ ($V(\sigma)$) dominates over the energy densities of other components. The behavious of $w_{\rm DE}$ at the onset of the matter dominated era is instead model dependent: for $\xi\neq-1/6$, we see that $w_{\rm DE}>0$ from the upper two panels in Fig. 2, whereas for $\xi=-1/6$ we obtain $w_{\rm DE}\sim 1/3$ when $\sigma_{0}\ll\sigma$. The absence of an intermediate phase of a matter dominated era for $\xi=-1/6$ is also clear in the initial conditions for the scale factor reported in Appendix A. It can be seen from Fig. 2 that there is no phantom behaviour of the effective DE component at small redshifts in contrast to more general scalar-tensor DE models studied in Gannouji et al. (2006). Indeed a phantom behaviour with $w_{\rm DE}<-1$ Gannouji et al. (2006) is barely visible in the transient regime from the tracking value to $w_{\rm DE}\approx-1$ because of the small coupling $\xi$ considered in Fig. 2 and cannot occur in the stable accelerating regime for these models with $V(\sigma)\propto F^{2}(\sigma)$.

In Figs. 3-4-5, we show the evolution of the density parameters $\Omega_{i}$, corresponding to an Einstein gravity system with a Newton’s constant given by the current value of the scalar field today $G_{N}=1/(8\pi F_{0})$ Boisseau et al. (2000) (also used in Finelli et al. (2008); Umiltà et al. (2015)) for $\xi>0$, $\xi<0$ and $\xi=-1/6$, respectively.

II.2 Boundary conditions for the the scalar field

As boundary conditions we impose that the effective Newton’s constant at present is compatible with the Cavendish-like experiments. The effective gravitational constant for NMC is given by Boisseau et al. (2000):

[TABLE]

In Fig. 6 is shown the evolution of the relative effective gravitational constant (17). We can see that the effective gravitational constant decreases in time for all the choices of both $N_{pl}$ and $\xi$.

We can distinguish three different cases beyond GR:

• •

$\tilde{N}_{pl}\rightarrow 0$ which is the IG case. This leads to:

[TABLE]

which is the same result as obtained in Umiltà et al. (2015);

• •

$\xi\rightarrow-1/6$ which is the CC. In this particular case the polynomial equation (17) in $\sigma_{0}$ in quadratic and we have:

[TABLE]

• •

a general NMC case for $\xi\neq-1/6$:

[TABLE]

By requiring $\tilde{\sigma}^{2}\geq 0$ and $F\geq 0$, we obtain conditions on the two parameters $\tilde{N}_{pl}$ and $\xi$ for the physical solution:

[TABLE]

II.3 Comparison with general relativity

The deviations from GR for a theory of gravitation are described by the so called post-Newtonian parameters. For NMC only the parameters $\gamma_{\rm PN}$ and $\beta_{\rm PN}$ differ from GR predictions, for which they both equal unity. In terms of these parameters the line element can be expressed as:

[TABLE]

These parameters are given within NMC by the following equations Boisseau et al. (2000):

[TABLE]

We have $\gamma_{\rm PN}\leq 1$ and $\beta_{\rm PN}\leq 1$ for $\xi>0$, whereas $\gamma_{\rm PN}\leq 1$ and $\beta_{\rm PN}\geq 1$ for $\xi<0$.

In Figs. 7, 8 and 9, we show the evolution of these parameters for different values of $N_{pl}$ and $\xi$. It is interesting to note how in the CC case $\gamma_{\rm PN}$ and $\beta_{\rm PN}$ approach the GR value more rapidly than for $\xi\neq-1/6$.

III Linear perturbations

We study linear fluctuations around the FRW metric in the synchronous gauge:

[TABLE]

where $\tau$ is the conformal time and $h_{ij}$ include both the scalar ($h_{ij}^{S}$) and the tensor ($h_{ij}^{T}$) part. We follow the conventions of Ref. Ma and Bertschinger (1995) for scalar metric perturbations $h_{ij}$ and scalar field perturbation $\delta\sigma$:

[TABLE]

[TABLE]

In Fig. 10, we show the evolution of the scalar field perturbation $\delta\sigma$ at $k=0.05$ Mpc*-1* for different values of $N_{pl}$ and $\xi$.

The modified Einstein equations in Eq. (3) at first order for scalar perturbations are:

[TABLE]

where all perturbations are considered in the Fourier configuration. The quantities $\theta_{i}$ and $\sigma_{i}$ represent the velocity potential and the anisotropic stress, respectively. It can be seen from the last of these equations that the coupling function acts also as a source for the anisotropic stress.

The perturbed Klein-Gordon equation is:

[TABLE]

As for the homogeneous KG Eq. (12), the choice $V\propto F^{2}$ also leads to an effectively massless scalar field fluctuation. Both initial conditions for the background and for the linear perturbations at the next-to-leading order in $\tau$ are shown in the Appendix A. We consider adiabatic initial condition for the scalar cosmological fluctuations Rossi (2016); Umiltà et al. (2015).

Analogously, the transverse and traceless part of the metric fluctuation $h_{ij}^{T}$ is expanded as:

[TABLE]

where $h_{+}\,,h_{-}$ and $e^{+}\,,e^{\times}$ are the amplitude and normalized tensors of the two independent states to the direction of propagation of gravitational waves in Fourier space. The evolution equation for the amplitude is:

[TABLE]

where $s$ denote the two polarization state of the two independent modes ($s=+,\times$) and the right hand side denotes the contribution of the traceless and transverse part of the neutrino anisotropic stress. The importance of the extra-damping term in the evolution equation for gravitational waves has been previously stressed Riazuelo and Uzan (2000); Amendola et al. (2014). The example of the impact of this term with respect to GR is depicted in Fig. 11. Note that the parameters chosen are compatible with the previous figures in this paper and we are therefore in a regime in which $\dot{F}/F\ll 3H$.

IV CMB anisotropies and matter power spectra

The footprints of these scalar-tensor theories into the CMB anisotropies angular power spectra can be understood as a generalization of the effects in eJBD or equivalently IG theories. The redshift of matter-radiation equality is modified in scalar-tensor theories by the motion of the scalar field driven by pressureless matter and this results in a shift of the CMB acoustic peaks for values $\xi\neq 0$, as for the IG case Liddle et al. (1998); Chen and Kamionkowski (1999). In addition, a departure from $\tilde{N}_{pl}=1$ induces a further change both in the amplitude of the peaks and their positions. We note that decreasing the value of $\tilde{N}_{pl}$ is possible to suppress the deviations with respect to the $\Lambda$CDM model allowing for larger values of the coupling $\xi$ compared to the IG case.

We show the relative differences with respect to the $\Lambda$CDM model for the lensed CMB angular power spectra anisotropies in temperature and E-mode polarization, and the CMB lensing angular power spectra for different values of $N_{pl}$ for $\xi>0$ in Fig. 12, $\xi<0$ in Fig. 13, and the CC case $\xi=-1/6$ in Fig. 14. We show also the absolute difference of the TE cross-correlation weighted by the square root of the product of the two auto-correlators.

In Fig. 15 we show the relative differences for the matter power spectrum at $z=0$ with respect to the $\Lambda$CDM model for different values of the parameters. In all the cases the $P(k)$ is enhanced at small scales, i.e. $k\gtrsim 0.01$ h Mpc*-1*, compared to the $\Lambda$CDM model.

We end this section by discussing the B-mode polarization power spectra resulting from the evolution of tensor fluctuations in Eq. 35. Fig. 16 shows the comparison of the tensor and lensing contributions to B-mode polarization in $\Lambda$CDM GR and the scalar-tensor cases of IG ($N_{\mathrm{p}l}=0$), CC ($\xi=-1/6$), positive and negative $\xi$ for a value of a tensor-to-scalar ratio $r=0.05$, compatible with the most recent constraints Akrami et al. (2018a); Ade et al. (2018). It is important to note that for the values of the couplings chosen in Fig. 16 the main differences in the tensor contribution to B-mode polarization with respect to $\Lambda$CDM GR case is due to the different evolution in the Hubble parameter and in the transfer functions in the definition of CMB anisotropies.

V Constraints from cosmological observations

We perform a Monte Carlo Markov Chain analysis by using the publicly available code MontePython †††https://github.com/brinckmann/montepython_public Audren et al. (2013); Brinckmann and Lesgourgues (2018) connected to our modified version of the code CLASS ‡‡‡https://github.com/lesgourg/class_public Blas et al. (2011), i.e. CLASSig Umiltà et al. (2015).

We use $Planck$ 2015 and BAO likelihoods. We combine the $Planck$ high-$\ell$ ($\ell>29$) temperature data with the joint temperature-polarization low-$\ell$ ($2\leq\ell\leq 29$) likelihood in pixel space at a resolution of 3.7 deg, i.e. HEALPIX Nside=16 Aghanim et al. (2016). The $Planck$ CMB lensing likelihood in the conservative multipoles range, i.e. $40\leq\ell\leq 400$ Ade et al. (2016a) from the publicly available $Planck$ 2015 release is also combined. We use BAO data to complement CMB anisotropies at low redshift: we include measurements of $D_{V}/r_{s}$ at $z_{\mathrm{eff}}=0.106$ from 6dFGRS Beutler et al. (2011), at $z_{\mathrm{eff}}=0.15$ from SDSS-MGS Anderson et al. (2014), and from SDSS-DR11 CMASS and LOWZ at $z_{\mathrm{eff}}=0.57$ and $z_{\mathrm{eff}}=0.32$, respectively Ross et al. (2015).

We sample with linear priors the six standard cosmological parameters, i.e. $\omega_{\rm b}\equiv\Omega_{\rm b}h^{2}$, $\omega_{\rm c}\equiv\Omega_{\rm c}h^{2}$, $H_{0}$, $\tau_{\mathrm{re}}$, $\ln\left(10^{10}A_{\rm s}\right)$, and $n_{\rm s}$, plus the two extra parameters for a non-minimally coupled scalar field, i.e. $\Delta\tilde{N}_{\mathrm{p}l}$ and $\xi$. In the analysis we assume massless neutrinos and marginalize over $Planck$ high-$\ell$ likelihood foreground and calibration nuisance parameters Aghanim et al. (2016) which are allowed to vary.

As in Ballardini et al. (2016), we take into account the change of the cosmological abundances of the light elements during Big Bang Nucleosynthesis (BBN) induced by a different gravitational constant during the radiation era with respect the theoretical prediction obtained from the public code PArthENoPE Pisanti et al. (2008). We take into account the modified BBN consistency condition due to the different value of the effective gravitational constant during BBN, by considering this effect as modelled by dark radiation, since the latter effect is already tabulated as $Y_{\textup{P}}^{\textup{BBN}}(\omega_{b},\,N_{\textup{eff}})$ Hamann et al. (2008) in the public version of the CLASS code. As in Ballardini et al. (2016), the posterior probabilities for the primary cosmological parameters are hardly affected by the modified BBN consistency condition, and we report a small shift for the primordial Helium abundance to higher values.

V.1 Results

The results from our MCMC exploration are summarized in Table 1. We find for the positive branch of the coupling at 95% CL:

[TABLE]

We show in Fig. 17 a zoom of the 2D parameter space ($H_{0}$, $\xi$) comparing the result of NMC to IG, i.e. $N_{pl}=0$. The constraint on $\xi$ is degradated by almost two order of magnitude ($\xi<0.0075$ at 95% CL for IG Ballardini et al. (2016)) due to the strong degeneracy between $N_{pl}$ and $\xi$, see Fig. 18.

The constraints for the negative branch are (see Figs. 19-20):

[TABLE]

at the 95% CL for $Planck$ TT + lowP + lensing + BAO.

We quote also the derived constraints on the change of the effective Newton’s constant (17) evaluated between the radiation era and the present time, and also its derivative at present time at 95% CL:

[TABLE]

for $\xi>0$, and:

[TABLE]

for $\xi<0$.

For the CC case, i.e. fixing $\xi=-1/6$, results are listed in Tab. 2. This model is severely constrained by data leading to tight upper bound on $N_{pl}$ at 95% CL:

[TABLE]

where $\tilde{N}_{pl}$ can take only values larger than one in this case.

All these models provide a fit to $Planck$ 2015 and BAO data very similar to $\Lambda$CDM: we report $\Delta\chi^{2}\sim-2.6$ for all the models considered in this paper. Due to the limited improvement in $\Delta\chi^{2}$, none of these models is preferred at a statistically significant level with respect to $\Lambda$CDM.

V.2 The Hubble parameter

We find constraints compatible with the $\Lambda$CDM values for the standard cosmological parameters. However, the shifts in $H_{0}$ deserve a particular mention: as already remarked in Umiltà et al. (2015); Ballardini et al. (2016) for the IG case, the mean values for $H_{0}$ are larger for all the models studied here. Fig. 21 shows how the 2D marginalized contours for ($H_{0}$, $N_{pl}$) have a degeneracy. We find:

[TABLE]

This value is larger, but compatible at 2$\sigma$ level with the $\Lambda$CDM value ($H_{0}=67.78\pm 0.57\ [\text{km/s/Mpc}]$). However, it is still lower than the local measurement of the Hubble constant Riess et al. (2018) ($H_{0}=73.52\pm 1.62\ [\text{km/s/Mpc}]$) obtained by including the new MW parallaxes from HST and Gaia to the rest of the data from Riess et al. (2016). Therefore the tension between the model dependent estimate of the Hubble parameter from $Planck$ 2015 plus BAO data and the local measurement from Riess et al. (2018) decreases to 2.3$\sigma$ from the 3.3$\sigma$ of the $\Lambda$CDM model. For comparison, by varying the number degree of relativistic species $N_{\rm eff}$ in Einstein gravity, a lower value for the Hubble parameter, i.e. $H_{0}=68.00\pm 1.5\ [\text{km/s/Mpc}]$ (with $N_{\rm eff}=3.08^{+0.22}_{-0.24}$) for $Planck$ TT + lowP + lensing + BAO at 68% CL, is obtained compared to the CC case reported in Eq. (45). When the local measurement of the Hubble constant Riess et al. (2018) is included in the fit we obtain:

[TABLE]

Since the marginalized value for $H_{0}$ in either eJBD and NMC models is larger than in common extensions of the $\Lambda$CDM model Ade et al. (2016b), such as $\Lambda$CDM + $N_{\rm eff}$, it is useful to understand how the evolution of the Hubble parameter differ at early and late times. The differences at early time can be easily understood: since the effective Newton’s constant can only decrease, if we consider the same $H_{0}$, this will correspond to an higher $H(z)$ or to a larger $N_{\rm eff}$ in the radiation era compared to the $\Lambda$CDM. A second effect around recombination is the motion of the scalar field driven by pressureless matter. At lower redshifts, the differences with respect to $\Lambda$CDM are originated by the onset of the acceleration stage by $\sigma$. The upper panel of Fig. 22 shows relative differences of $H(z)$ with respect to the $Planck$ TT + lowP + lensing + BAO $\Lambda$CDM best-fit: best-fit (for IG) or NMC models within the 1$\sigma$ contours are compared with $\Lambda$CDM + $N_{\rm eff}$ or $w$CDM. This plot shows how in these scalar-tensor models both early and late time dynamics can contribute to a larger value for $H_{0}$ than in $\Lambda$CDM + $N_{\rm eff}$, for example.

However, because of this contribution from late time dynamics, the change in $H_{0}$ cannot be interpreted only as a proportional decrement in the comoving sound horizon at the baryon drag epoch $r_{s}$, which is the quantity used to calibrate the BAO standard ruler and is 147.6 Mpc for $\Lambda$CDM with the data considered. The bottom panel of Fig. 22 shows $D_{V}(z)/r_{s}\equiv\frac{[cz(1+z)^{2}D_{A}(z)^{2}H(z)^{-1}]^{1/3}}{r_{s}}$, with $D_{A}$ as the angular diameter distance, normalized to its $\Lambda$CDM value, and the value of $r_{s}$. It is easy to see that both $r_{s}$ and $H_{0}$ are lower for $\Lambda$CDM + $N_{\rm eff}$ than for the scalar-tensor models studied here and the eJBD model. These scalar-tensor models therefore differ from those which aim in reducing the tension between CMB anisotropies and the local measurements of $H_{0}$ through a decrement of $r_{s}$ Cuesta et al. (2015); Bernal et al. (2016); Aylor et al. (2019), such as those in which ultralight axion fields move slowly around recombination and then dilute away Poulin et al. (2018a, b); Agrawal et al. (2019). In the scalar-tensor models considered here the scalar field moves naturally around recombination since is forced by pressureless matter and dominates at late time acting as DE.

V.3 Constraints on the post-Newtonian parameters

Finally, we quote the derived constraints on the post-Newtonian parameters. In this class of models $\gamma_{\rm PN},\,\beta_{\rm PN}\neq 1$ according to Eqs. (24)-(25) at 95% CL:

[TABLE]

[TABLE]

See Fig. 23 for the 2D marginalized constraints in the $(\gamma_{\rm PN},\beta_{\rm PN})$ plane. See Fig. 24 for the 2D marginalized constraints in the $(H_{0},\gamma_{\rm PN})$ plane for $\xi>0$ compared to the IG case studied in Ballardini et al. (2016).

The tight constraint on $N_{pl}$ for the CC case correspond at 95% CL to:

[TABLE]

for $Planck$ TT + lowP + lensing + BAO, where the latter is tighter than the constraint from the perihelion shift $\beta_{\rm PN}-1=(4.1\pm 7.8)\times 10^{-5}$ Will (2014) and the former is twice the uncertainty of the Shapiro time delay constraint $\gamma_{\rm PN}-1=(2.1\pm 2.3)\times 10^{-5}$ Bertotti et al. (2003).

VI Conclusions

We have expanded on our previous study of the observational predictions within the eJBD theory or, equivalently, IG Umiltà et al. (2015); Ballardini et al. (2016), to the case of a scalar field nonminimally coupled to the Einstein gravity as in Eq. (I) with $G_{3}=G_{5}=0$ and $2G_{4}=F(\sigma)=N_{pl}^{2}+\xi\sigma^{2}$. We have studied this class of model under the assumption that the effective gravitational constant in these scalar-tensor theories is compatible with the one measured in a Cavendish-like experiment. Whereas in the eJBD theory only the first post-Newtonian parameter $\gamma_{\rm PN}$ ($=1-\frac{F_{,\sigma}^{2}}{F+2F_{,\sigma}^{2}}$) is not vanishing, in this simple extension both the first and second post-Newtonian parameter $\beta_{\rm PN}$ ($=1+\frac{F\,F_{,\sigma}}{8F+12F_{,\sigma}^{2}}\frac{d\gamma_{\rm PN}}{d\sigma}$) are non-zero. The second post-Newtonian parameters encodes the sign of the coupling to gravity, i.e. $\beta_{\mathrm{PN}}>0$ ($<0$) for $\xi>0$ ($\xi<0$).

For the sake of semplicity, we have restricted ourselves to the class of potential $V(\sigma)\propto F^{2}(\sigma)$, which makes the field effectively massless Amendola (1999) and allows for a direct comparison with the IG model for $N_{pl}=0$ Cooper and Venturi (1981); Wetterich (1988b); Finelli et al. (2008); Umiltà et al. (2015); Ballardini et al. (2016). For this choice of potential $V(\sigma)\propto F^{2}(\sigma)$, the scalar field is effectively massless. By assuming natural initial conditions in which the decaying mode is negligible, the scalar field starts at rest deep in the radiation era and is pushed by pressureless matter to the final stage in which it drives the Universe in a nearly de Sitter stage at late times with $\sigma={\rm const}$. In general the effective parameter of state $w_{\rm DE}$ for $\sigma$ defined in Boisseau et al. (2000) tracks the one of the dominant matter component before reaching $-1$ once the Universe enters in the accelerated stage as for the IG case. We find that the conformal case $\xi=-1/6$ is an exception to this general trend: for such a value the effective parameter of state $w_{\rm DE}$ interpolates between $1/3$ and $-1$ without an intermediate pressureless stage. Irrespective of the sign of the coupling $\xi$, $G_{N}(a)=1/(8\pi F)$ decrease with time for this class of potential.

As in our previous works in IG, we have considered adiabatic initial conditions for fluctuations Umiltà et al. (2015); Ballardini et al. (2016); Paoletti et al. (2018) which are derived in this work for a non-minimally coupled scalar field. By extending the modification of CLASSig Umiltà et al. (2015) to a generic coupling $F(\sigma)$, we have derived the CMB temperature and polarization anisotropies and the matter power spectrum. Since the effective Newton’s constant decrease in time after the relativistic era, we observe a shift of the acoustic peaks to higher multipoles and an excess in the matter power spectrum at $k\gtrsim 0.01$ Mpc*-1* proportional to the deviation from GR.

We have used $Planck$ 2015 and BAO data to constrain this class of models. As for IG, we obtain a marginalized value for $H_{0}$ higher than in $\Lambda$CDM for all these models, potentially alleviating the tension with the local measurement of the Hubble parameter obtained by calibrating with the Cepheids Riess et al. (2018). The goodness of fit to $Planck$ 2015 plus BAO data provided by the models studied in this paper is quantitative similar to $\Lambda$CDM: since they have one (for the conformal coupled case $\xi=-1/6$) or two (for $\xi$ allowed to vary) extra parameters, these models are not preferred with respect to $\Lambda$CDM. We have derived 95% CL upper bounds $\xi<0.064$ ($|\xi|<0.11$) and $0.81<N_{pl}<1$ ($1<N_{pl}<1.39$) for $\xi>0$ ($\xi<0$). It is interesting to note that the bound on $\gamma_{\rm PN}$ and $\delta G_{\rm eff}/G$ have just a small degradation with respect to eJBD with the same data set ($0.997<\gamma_{\rm PN}<1$ Ballardini et al. (2016)). Overall, some cosmological constraints do not seem strongly dependent on the assumption $\beta_{\rm PN}=0$ and have a large margin of improvement with future observations Ballardini et al. (2019). Although model dependent, cosmological observations seem more promising than other independent ways to test scalar-tensor theories in the strong gravity regime as the search for the presence of scalar polarization states of gravitational waves Du (2019), which is also strongly constrained by LIGO/Virgo Abbott et al. (2018).

The conformal value $\xi=-1/6$ is an interesting and particular case which stands out within the general class of non-minimally coupled scalar fields. In addition to what already remarked about its effective parameter of state, we find that $Planck$ 2015 + BAO data constrain quite tightly the conformal case with $V(\sigma)\propto F^{2}(\sigma)$: as 95% CL intervals, we find $1<10^{5}\Delta\tilde{N}_{\rm Pl}<3.8$, or equivalently $0.99996<\gamma_{\rm PN}<1$, $1<\beta_{\rm PN}<1.000003$, in terms of the post-Newtonian parameters. These tight cosmological constraints for the conformal case are comparable to those of obtained within the Solar System bounds Bertotti et al. (2003).

As from Figs. 12-13-14-16 CMB polarization anisotropies have a greater sensitivity to the variation of the gravitational strength in these models. It will be therefore interesting to see the impact of the latest and more robust measurement of CMB polarization anisotropies from Planck Akrami et al. (2018b, c); Aghanim et al. (2018) and from BICEP2/Keck Array Akrami et al. (2018b) as well as of the more recent BAO data on the constraints of these models.

Acknowledgements.

We would like to thank Lloyd Knox and Vivian Poulin for discussions. MBa was supported by the South African Radio Astronomy Observatory, which is a facility of the National Research Foundation, an agency of the Department of Science and Technology and he was also supported by the Claude Leon Foundation. MBa, MBr, FF and DP acknowledge financial contribution from the agreement ASI/INAF n. 2018-23-HH.0 ”Attività scientifica per la missione EUCLID – Fase D”. FF and DP acknowledge also financial support by ASI Grant 2016-24-H.0. AAS was partly supported by the program KP19-270 ”Questions of the origin and evolution of the Universe” of the Presidium of the Russian Academy of Sciences. This research used computational resources of the National Energy Research Scientific Computing Center (NERSC) and of INAF OAS Bologna.

Appendix A Initial Conditions

Here we report the initial conditions adopted in this paper for a non-minimally coupled scalar field, which generalize the case of adiabatic initial conditions for IG presented in Paoletti et al. (2018). These quantities reduces to IG and general relativity cases for $N_{pl}=0$ and $(N_{pl}=M_{pl},\xi=0)$, respectively.

For the background cosmology we have as initial conditions:

[TABLE]

[TABLE]

[TABLE]

where $\omega=\frac{\rho_{m0}}{\sqrt{3\rho_{r0}}}\frac{\sqrt{F_{i}}}{F_{i}+6\xi^{2}\sigma_{i}^{2}}$.

For cosmological fluctuations in the synchronous gauge we have as adiabatic initial conditions:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $R_{\nu}=\frac{\rho_{\nu 0}}{\rho_{r}0}$ and $R_{b}=\frac{\rho_{b0}}{\rho_{m0}}$.