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PHYSICAL REVIEW LETTERS
Cosmological Constraints on Brans-Dicke Theory A. Avilez* and C. Skordis† School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom (Received 18 March 2013; revised manuscript received 13 February 2014; published 2 July 2014) We report strong cosmological constraints on the Brans-Dicke (BD) theory of gravity using cosmic microwave background data from Planck. We consider two types of models. First, the initial condition of the scalar field is fixed to give the same effective gravitational strength Geff today as the one measured on Earth, GN . In this case, the BD parameter ω is constrained to ω > 692 at the 99% confidence level, an order of magnitude improvement over previous constraints. In the second type, the initial condition for the scalar is a free parameter leading to a somewhat stronger constraint of ω > 890, while Geff is constrained to 0.981 < Geff =GN < 1.285 at the same confidence level. We argue that these constraints have greater validity than for the BD theory and are valid for any Horndeski theory, the most general second-order scalar-tensor theory, which approximates the BD theory on cosmological scales. In this sense, our constraints place strong limits on possible modifications of gravity that might explain cosmic acceleration. DOI: 10.1103/PhysRevLett.113.011101
PACS numbers: 04.50.Kd, 98.70.Vc, 98.80.−k
The Brans-Dicke theory of gravity (BDT) [1,2] is one of the simplest extensions of general relativity (GR) depending on one additional parameter, ω. In addition to the metric, the gravitational field is further mediated by a scalar field ϕ whose inverse plays the role of a spacetime-varying gravitational strength. The importance of the BDT lies beyond its level of simplicity, in that it is the limit of more sophisticated but also more realistic and physically motivated theories. Its immediate generalizations, the so-called scalar-tensor theories [3,4], have had strong theoretical support from a variety of perspectives. For example, they manifest in the low-energy effective action for the dilaton-graviton sector in supergravity [5]. More generally, in compactifications of theories with extra dimensions, for instance, Kaluza-Klein (KK) type theories [6] or the Dvali-Gabadadze-Porrati theory [7], the extra-dimensional spacetime metric is decomposed in KK modes acting as effective scalars on our four-dimensional spacetime in the same way that occurs in scalar-tensor theories. The BDT is a close cousin of the so-called Galileon theories [8], recently proposed to explain cosmic acceleration while evading Solar System constraints. In the absence of matter fields, the scalar-tensor action also arises as a special sector of the Plebanski action when the trace component of the simplicity constraints is relaxed [9,10]. Finally, as we discuss further below, the BDT arises as a particular limit of the Horndeski theory [11,12], the most general scalar-tensor theory having second-order field equations in four dimensions. Solar System data put very strong constraints on the BD parameter ω. The measurement of the parameterized postNewtonian parameter γ (see [13,14]) from the Cassini mission gives ω > 40 000 at the 2σ level [14,15]. On cosmological scales, however, the story is somewhat different. Nagata, Chiba, and Sugiyama [16] report that 0031-9007=14=113(1)=011101(5)
ω > f50; 1000g at 4σ and 2σ, respectively, by using Wilkinson Microwave Anisotropy Probe (WMAP) firstyear data (WMAP-1). However, as argued in Ref. [17], their 2σ result is not reliable as the reported χ 2 has a sharp step form, and, rather, one should take the 4σ result as a more conservative estimate. Better constraints come from Acquaviva et al. [17], who report ω > f80; 120g at the 99% and 95% level, respectively, by using a combination of Cosmic Microwave Background (CMB) data from WMAP1 and a set of small-scale experiments as well as large-scale structure (LSS) data. Wu and Chen [18] report ω > 97.8 at the 95% level by using a combination of CMB data from 5 years of WMAP, other smaller-scale CMB experiments, and LSS measurements from the Sloan Digital Sky Survey (SDSS) release 4 [19]. Their constraint is weaker than Ref. [17] even though newer data are used. As argued in Ref. [18], this is due to the use of flat priors on lnð1 þ ð1=ωÞÞ rather than − lnð1=4ωÞ of Ref. [17]. Finally, Ref. [20] improves to ω > 181.65 (95% level) by using Planck data [21] with the same priors as Ref. [18]. Given that Solar System data provide a far superior bound on ω, why constrain the BDT with cosmological uBD planck + wmap(pol) rBD planck + wmap(pol) uBD wmap rBD wmap
wmap+hst planck+wmap(pol) planck+lens planck+wmap(pol)+hl
0.4
0.6
0.8
1 ξ
1.2
1.4
1.6
1
1.5
2
2.5 3 Log ω
3.5
4
4.5
FIG. 1. Left: The 1D marginalized posterior for ξ ¼ Geff =G. Right: The 1D marginalized posterior of ln ω for different BDT models.
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data? There are two reasons why this is important. First, cosmological constraints on the BDT are important as they concern very different spatial and temporal scales. Second, as we discuss further below, the BDT can be considered as an approximation to a subset of Horndeski-type theories, and, thus, cosmological constraints on the BDT can be interpreted in a more general setting. More specifically, while on cosmological scales the BDT emerges as an approximation to the Horndeski theory, the derivative selfinteractions of the Horndeski scalar become larger as one moves to smaller scales (higher curvature than cosmological environments), and this leads to the screening of the scalar resulting at the same time to the recovery GR. The model.—The BDT is described by the action 1 S¼ 16 πG
Z
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ω pffiffiffiffiffiffi 2 d x −g ϕR − 2Λ − ð∇ϕÞ þ Sm ; ð1Þ ϕ 4
where g is the metric determinant, R is the scalar curvature, Λ is the cosmological constant, G is the bare gravitational constant, and Sm is the matter action. Since Sm is independent of ϕ, the weak equivalence principle is satisfied. We have chosen ϕ to be dimensionless by convention. The relevant equations to be solved may be found in Ref. [2], and here we quote only the Friedman equation, which is _¯ 2 2 1 ϕ_¯ 8πG 1 ϕ 3 Hþ ¯ ¼ ¯ ρ þ ð2ω þ 3Þ ¯ ; 2ϕ 4 ϕ ϕ
ð2Þ
where H is the Hubble rate and ρ is the total matter density including Λ, and the background scalar equation ̈ ¯ þ 3Hϕ_¯ ¼ 8πG ðρ − 3PÞ; ϕ 2ω þ 3
ð3Þ
where P is the matter pressure including Λ. We consider only ω > −ð3=2Þ, since otherwise the scalar is a ghost. As it happens, the field stays constant during the radiation era, because (3) is sourced by ρ − 3P ¼ 0 [since P ¼ ð1=3Þρ for photons], resulting in ϕ¯ behaving like a massless scalar. As the Universe enters the matter era, however, ϕ¯ grows but only logarithmically with the scale factor a. Thus, the scalar field today, ϕ¯ 0 , is expected to be within a few percent of its initial value in the deep radiation era. If the scalar is approximately constant, then the Friedman equation becomes 3H2 ≈ 8πGeff ρ, where the effective cosmological gravitational strength is given by ξ ¼ Geff =G ¼ 1=ϕ. For bound systems in the quasistatic regime, e.g., our Solar System, the effective Newton’s constant is GN ¼ Gð2ω þ 3Þ=ð2ω þ 4Þ [1,2]; thus, observers in a bound system which formed today will measure the same cosmological and local gravitational strength if
2ω þ 4 ϕ¯ 0 ¼ : 2ω þ 3
ð4Þ
We call such models restricted (RBD), since to achieve the above condition the initial value of the scalar field, ϕ¯ i , must be appropriately fixed. Models for which ϕ¯ i is a free parameter will be called unrestricted (UBD). Analysis and methodology.—We numerically solved the BDT background (2) and (3) and the linearized equations in the Jordan frame where the matter equations are unchanged from GR, which ensures that the effective gravitational strength is correctly implemented in the code. To test the numerical results, we implemented the synchronous gauge equations for scalar modes in a modified version of the CAMB package [22] and compared with our own Boltzmann code (derived from CMBFAST [23] and DASH [24]) in which both the synchronous gauge and the conformal Newtonian gauge were used. We generated a chain of steps in parameter space by employing the Markov-chain Monte Carlo method [25,26] implemented in cosmology via COSMOMC [27] (see also [28]). Our chains were long enough to pass the convergence diagnostics and also give very accurate 1D and 2D marginalized posteriors. The main data sets we used are from the Planck satellite [21], WMAP-7/9 [29], the South Pole Telescope (SPT) [30], and the Atacama Cosmology Telescope (ACT) [31]. We also use data from big-bang nucleosynthesis (BBN) light element abundances [32]. The chains were generated for two types of models: the RBD and the UBD models. The RBD models have seven parameters which are the dimensionless baryon and dark matter densities ωb and ωc , respectively, the ratio of the angular diameter distance to the sound horizon at recombination θ, the reionization redshift zre , the amplitude and spectral index of the primordial power spectrum As and ns , respectively, and the BDT parameter ω. The Hubble constant H0 and the (dimensionless) cosmological constant density ωΛ are derived parameters. The UBD models have one additional parameter, which is the initial condition ϕ¯ i . When generating likelihoods for the Planck data, 11 astronomical parameters to model foregrounds and three instrumental calibration and beam parameters (3) were used as described in Ref. [21]. When ACT and SPT were also included with Planck, 17 more calibration parameters were used [21]. In order to sample efficiently the large number of “fast” parameters, we used the speed-ordered Cholesky parameter rotation and the dragging scheme described by Neal and Lewis [33,34] implemented in the latest version of COSMOMC. We now turn to the issue of priors. For the non-BDT parameters we assume the same priors as for ΛCDM, since the two types of cosmological evolution are very similar. A prior on H 0 (HST) from the measurement of the angular diameter distance at redshift z ¼ 0.04 [35] is also imposed for some chains. For the BDT parameters we impose flat
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TABLE I. Summary of our results for the restricted (R) and the unrestricted (U) BD models for a variety of data sets. Limits for ω and ξ_ at the 95% and 99% level are shown as well as 1σ level constraints for ξ. ξ
ωBD U
R
95% 99% 95% 99% WMAP7
99
55
90
51
WMAP7 þ HST
126
62
177 120
WMAP7 þ SPT þ HST
269
148
157 114
PLANCKTEMP þ WMAP9pol
1834
890 1808 692
PLANCKTEMP þ WMAP9pol þ HL
1923
843 1326 213
PLANCKTEMP þ PLANCKLENS
2441 1033 1901 420
PLANCKTEMP þ PLANCKLENS þ HL 1939
829 1408 330
priors on ϕ¯ i and on − lnðωÞ. This prior on ω is more convenient for sampling the chains [17]. However, the choice of prior is not important, as our constraints strongly improved compared to past experiments when Planck data were used in the analysis. Results and discussion.—We first discuss the restricted models, which contain only ω as an additional parameter to ΛCDM. Varying ω changes the background expansion history H which results in a shift of the peak locations and peak heights. It makes more sense, however, to consider the changes in the spectrum at fixed θ, as it was used as a parameter. In that case, the dominant effect is on the largescale temperature spectrum due to the integrated SachsWolfe effect, while small scales are affected due to perturbations in ϕ (indirectly through the effect of ϕ on the potential wells) and are less important. Using WMAP7 alone, we find ω > f90; 51g at the 95% and 99% level, respectively, while for WMAP7+HST we improve to ω > f126; 62g. This is a significant improvement over Ref. [17], driven mainly by the inclusion of polarization data which break the degeneracy between zre and ns and allow the measurements of the damping tail to improve the determination of the other parameters and limit the freedom of ω to vary. For WMAP7 þ SPT þ HST data, change this to ω > f157; 114g at the same confidence levels. The use of Planck temperature (PLANCKTEMP) data greatly improves the measurement of the damping tail and together with WMAP 9-year polarization (WMAP9pol) further improves the constraint to ω > f1808; 692g. This is in line with the forecasting of Ref. [36]. Our results and more data combinations are summarized in Table I and some representative posterior distributions are shown in Fig. 1. Constraints on UBD models, which contain ξ as a further parameter, have not been presented before. As discussed in Ref. [37], the main effect of increasing ξ is to increase the width of the visibility function which in turn increases photon diffusion and damps the CMB temperature
95%
99%
ξ from CMB þ BBN ξ_ × 10−13 years−1 Upper 1σ 95% 99%
þ0.98 0.98þ0.67 −0.55 0.98−0.63
1.113 0.156
2.45
3.37
1.07þ0.22 −0.24 1.10þ0.13 −0.14 1.12þ0.11 −0.11 1.11þ0.13 −0.11 1.12þ0.10 −0.14 1.07þ0.11 −0.10
0.991 0.036
2.85
3.59
0.996 0.029
1.92
2.81
1.006 0.018
0.93
1.78
1.009 0.014
0.74
1.75
0.998 0.029
0.63
1.51
0.999 0.024
0.36
0.83
þ0.40 1.07−0.43 þ0.17 1.10−0.19 þ0.16 1.12−0.14 þ0.17 1.11−0.14 þ0.16 1.12−0.17 þ0.14 1.07−0.13
anisotropies on small scales. Thus the main constraints on ξ from CMB temperature come from fitting the damping envelope with measurements of the CMB at small scales from ACT, SPT, and Planck. Increasing ξ has a slightly different effect on polarization. The same damping effect occurs on small scales, but on large scales we get an enhancement, as a thicker last scattering surface increases the amplitude of the local quadrupole, producing a larger polarization signal [37]. Using WMAP7 alone, we find ω > f99; 55g and þ0.98 ξ ¼ f0.98þ0.67 −0.55 ; 0.98−0.63 g at the 95% and 99% level, respectively, which changes to ω > f269; 148g and ξ ¼ þ0.17 f1.10þ0.13 −0.14 ; 1.10−0.19 g at the same confidence levels with WMAP7 þ SPT þ HST data. Using PLANCKTEMP þ WMAP9pol improves the constraint to ω > f1834; 890g þ0.16 and ξ ¼ f1.12þ0.11 −0.11 ; 1.12−0.14 g. These results and more data combinations are summarized in Table I. In both types of models, constraints on ω using PLANCKTEMP þ WMAP9pol improve by a factor of 6 over WMAP7 þ SPT þ HST. It is interesting to notice, however, that in the restricted models ω is less constrained than in the unrestricted class which has one more parameter. This is further pronounced when other data combinations with PLANCKTEMP are used (see below). The reason is because the extra parameter ξ helps to fit the data better; the best-fit sample for the unrestricted model has a slightly better χ 2 than the restricted model. From Table I, we observe that including ACT and SPT (HL) data does not improve the constraints on ω, but rather, in the case of the restricted models, the constraints become weaker. The marginalized distribution of ln ω for WMAP7 þ SPT þ HST exhibits a peak around ω ∼ 400. The difference in likelihood between the GR limit and this peak is very small, which renders this “detection” insignificant; however, its presence makes it difficult to improve the lower bound on ω. When we use PLANCKTEMP þ HL data, the peak is washed out; however, a small effect
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still remains. It seems that there is a small discrepancy between the PLANCKTEMP and HL data. PLANCKTEMP together with lensing potential reconstruction (PLANCKLENS) gives the tightest constraint on ω. However, PLANCKLENS displays small discrepancies from PLANCKTEMP [21], and once again we opt for not using it when we report our final result. Interestingly, when HL data are added, both PLANCKTEMPþPLANCKLENS and PLANCKTEMP þ WMAP9pol give very similar constraints. Cosmological constraints on G from the CMB can be found in Ref. [38], where 0.74 ≤ ξ ≤ 1.66 is found from WMAP-1 alone at the 95% of confidence level while including BBN data the tighter bounds 0.95 ≤ ξ ≤ 1.01 are obtained at 1σ. Using the same methodology as in Ref. [38], we find 0.998 ≤ ξ ≤ 1.024 at 1σ with PLANCKTEMP þ WMAP9pol þ BBN using more recent measurements of Ref. [32]. Our results improve on Ref. [38] and further put them in the context of a realistic theory. In a more recent work by the authors of Ref. [20], constraints for G are obtained in the context of the restricted BDT; however, in their analysis they left out a possible rescaling of GN . We also report a strong upper bound on the time variation of G from the CMB alone around ∼10−13 =year, as in Table I. Implications for modified gravity.—Regardless of the simplicity of this model, our constraints are more generally valid, as the BDT can be considered an approximation to the Horndeski theory, the most general second-order scalar-tensor theory [11,12], above some very large length l . The gravitational action is R 4 scale pffiffiffiffiffiffi P 3 1 S½g; ψ ¼ 16 πG d x −g I¼0 LðIÞ , where ψ is by convention dimensionless and LðIÞ are given by Lð0Þ ¼ K ð0Þ ;
the complete set of Horndeski theories is defined by four free functions of ψ and X, by restricting the set as above we get eight free constant parameters rather than functions. Thus, our results hold for any Horndeski theory which can be approximated in the above form on cosmological scales. Choosing l ∼ 1=16H0 (Hubble scale at recombination) and using the BDT solution as ϕ_¯ ¼ H=ω and ϕ¯ ∼ 1, we find conservative estimates for the coefficients as ϵi ≪ f10−2 ; 105 ; 10; 105 ; 1; 103 ; 103 ; 105 g. Theories within these limits can be well approximated by the BDT. Conclusions.—We found strong constraints on the BDT parameter ω > 890 and effective gravitational strength 0.981 ≤ ξ ≤ 1.285 at the 99% confidence level, significantly improving on previous work. Improvement on these bounds is expected through the next data release of Planck and with the inclusion of LSS and redshift-space distortions data which is left to future work. We thank Tessa Baker, Jo Dunkley, Pedro Ferreira, Adam Moss, Yong-Seon Song, and Tony Padilla for useful discussions. A. A. acknowledges support from CONACYT. C. S. acknowledges support from the Royal Society.
*
[1] [2] [3] [4] [5]
[6]
ð5Þ
Lð2Þ ¼ K ð2Þ R þ K X ½ð□ψÞ2 − ðDψÞ2 ;
ð2Þ
ð6Þ
[7]
Lð3Þ ¼ −6K ð3Þ Gμν ∇μ ∇ν ψ
ð7Þ
[8]
ð3Þ
ð8Þ
where X ¼ − 12 gμν ∇μ ψ∇ν ψ, Dψ ¼ ∇ ⊗ ∇ψ, and K ðIÞ are functions of X and ψ. The general functions K ðIÞ may be expanded as an analytical series and whose lowest-order terms are K ð0Þ ≈ −2Λ þ 8 ωX þ ε1 ψ 2 =l 2 þ ε2 l 2 X2 , K ð1Þ ≈ ϵ3 ψ 2 þ ϵ4 l 2 X, K ð2Þ ≈ ψ 2 þ ϵ5 ψ 4 þ ϵ6 l 2 X, and K ð3Þ =l 2 ≈ ϵ7 ψ 2 þ ϵ8 l 2 X. We have ignored the constant terms in K ð1Þ and K ð3Þ , as they lead to total derivatives. The constant term in K ð2Þ cannot be ignored, in general, but would lead to GR coupled to a massless scalar as ϵi → 0 and is irrelevant to our work. As the coefficients of the expansion ϵi → 0 and further performing the field redefinition ϕ ¼ ψ 2 , we recover the BDT. We see that, although
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†
Lð1Þ ¼ K ð1Þ □ψ;
þK X ½ð□ψÞ3 − 3ðDψÞ2 □ψ þ 2ðDψÞ3 ;
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[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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[20] Y.-C. Li, F. Q. Wu, and X. Chen, Phys. Rev. D 88, 084053 (2013). [21] P. A. R. Ade et al. (Planck Collaboration), arXiv:1303.5075 [Mon. Not. R. Astron. Soc. (to be published)]. [22] A. M. Lewis, http://camb.info. [23] U. Seljak and M. Zaldarriaga, Astrophys. J. 469, 437 (1996). [24] M. Kaplinghat, L. Knox, and C. Skordis, Astrophys. J. 578, 665 (2002). [25] R. M. Neal, Technical Report No. cRG-TR-93-1, Department of Computer Science, University of Toronto, 1993 (available online). [26] W. R. Gilks, S. Richardson, and D. J. Speigelhalter, Markov Chain Monte Carlo in Practice (Chapman and Hall, London, 1995). [27] A. Lewis and S. Bridle, Phys. Rev. D 66, 103511 (2002). [28] J. Dunkley, M. Bucher, P. G. Ferreira, K. Moodley, and C. Skordis, Mon. Not. R. Astron. Soc. 356, 925 (2005).
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[29] E. Komatsu et al. (WMAP Collaboration), Astrophys. J. Suppl. Ser. 192, 18 (2011). [30] K. Schaffer et al., Astrophys. J. 743, 90 (2011). [31] J. Dunkley et al., J. Cosmol. Astropart. Phys. 07 (2013) 025. [32] F. Iocco, G. Mangano, G. Miele, O. Pisanti, and P. Serpico, Phys. Rep. 472, 1 (2009). [33] R. M. Neal, technical report, 2005. [34] A. Lewis, Phys. Rev. D 87, 103529 (2013). [35] A. G. Riess et al., Astrophys. J. 699, 539 (2009). [36] X. Chen and M. Kamionkowski, Phys. Rev. D 60, 104036 (1999). [37] O. Zahn and M. Zaldarriaga, Phys. Rev. D 67, 063002 (2003). [38] K. I. Umezu, K. Ichiki, and M. Yahiro, Phys. Rev. D 72, 044010 (2005).
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Cosmological Constraints on Brans-Dicke Theory A. Avilez* and C. Skordis† School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom (Received 18 March 2013; revised manuscript received 13 February 2014; published 2 July 2014) We report strong cosmological constraints on the Brans-Dicke (BD) theory of gravity using cosmic microwave background data from Planck. We consider two types of models. First, the initial condition of the scalar field is fixed to give the same effective gravitational strength Geff today as the one measured on Earth, GN . In this case, the BD parameter ω is constrained to ω > 692 at the 99% confidence level, an order of magnitude improvement over previous constraints. In the second type, the initial condition for the scalar is a free parameter leading to a somewhat stronger constraint of ω > 890, while Geff is constrained to 0.981 < Geff =GN < 1.285 at the same confidence level. We argue that these constraints have greater validity than for the BD theory and are valid for any Horndeski theory, the most general second-order scalar-tensor theory, which approximates the BD theory on cosmological scales. In this sense, our constraints place strong limits on possible modifications of gravity that might explain cosmic acceleration. DOI: 10.1103/PhysRevLett.113.011101
PACS numbers: 04.50.Kd, 98.70.Vc, 98.80.−k
The Brans-Dicke theory of gravity (BDT) [1,2] is one of the simplest extensions of general relativity (GR) depending on one additional parameter, ω. In addition to the metric, the gravitational field is further mediated by a scalar field ϕ whose inverse plays the role of a spacetime-varying gravitational strength. The importance of the BDT lies beyond its level of simplicity, in that it is the limit of more sophisticated but also more realistic and physically motivated theories. Its immediate generalizations, the so-called scalar-tensor theories [3,4], have had strong theoretical support from a variety of perspectives. For example, they manifest in the low-energy effective action for the dilaton-graviton sector in supergravity [5]. More generally, in compactifications of theories with extra dimensions, for instance, Kaluza-Klein (KK) type theories [6] or the Dvali-Gabadadze-Porrati theory [7], the extra-dimensional spacetime metric is decomposed in KK modes acting as effective scalars on our four-dimensional spacetime in the same way that occurs in scalar-tensor theories. The BDT is a close cousin of the so-called Galileon theories [8], recently proposed to explain cosmic acceleration while evading Solar System constraints. In the absence of matter fields, the scalar-tensor action also arises as a special sector of the Plebanski action when the trace component of the simplicity constraints is relaxed [9,10]. Finally, as we discuss further below, the BDT arises as a particular limit of the Horndeski theory [11,12], the most general scalar-tensor theory having second-order field equations in four dimensions. Solar System data put very strong constraints on the BD parameter ω. The measurement of the parameterized postNewtonian parameter γ (see [13,14]) from the Cassini mission gives ω > 40 000 at the 2σ level [14,15]. On cosmological scales, however, the story is somewhat different. Nagata, Chiba, and Sugiyama [16] report that 0031-9007=14=113(1)=011101(5)
ω > f50; 1000g at 4σ and 2σ, respectively, by using Wilkinson Microwave Anisotropy Probe (WMAP) firstyear data (WMAP-1). However, as argued in Ref. [17], their 2σ result is not reliable as the reported χ 2 has a sharp step form, and, rather, one should take the 4σ result as a more conservative estimate. Better constraints come from Acquaviva et al. [17], who report ω > f80; 120g at the 99% and 95% level, respectively, by using a combination of Cosmic Microwave Background (CMB) data from WMAP1 and a set of small-scale experiments as well as large-scale structure (LSS) data. Wu and Chen [18] report ω > 97.8 at the 95% level by using a combination of CMB data from 5 years of WMAP, other smaller-scale CMB experiments, and LSS measurements from the Sloan Digital Sky Survey (SDSS) release 4 [19]. Their constraint is weaker than Ref. [17] even though newer data are used. As argued in Ref. [18], this is due to the use of flat priors on lnð1 þ ð1=ωÞÞ rather than − lnð1=4ωÞ of Ref. [17]. Finally, Ref. [20] improves to ω > 181.65 (95% level) by using Planck data [21] with the same priors as Ref. [18]. Given that Solar System data provide a far superior bound on ω, why constrain the BDT with cosmological uBD planck + wmap(pol) rBD planck + wmap(pol) uBD wmap rBD wmap
wmap+hst planck+wmap(pol) planck+lens planck+wmap(pol)+hl
0.4
0.6
0.8
1 ξ
1.2
1.4
1.6
1
1.5
2
2.5 3 Log ω
3.5
4
4.5
FIG. 1. Left: The 1D marginalized posterior for ξ ¼ Geff =G. Right: The 1D marginalized posterior of ln ω for different BDT models.
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data? There are two reasons why this is important. First, cosmological constraints on the BDT are important as they concern very different spatial and temporal scales. Second, as we discuss further below, the BDT can be considered as an approximation to a subset of Horndeski-type theories, and, thus, cosmological constraints on the BDT can be interpreted in a more general setting. More specifically, while on cosmological scales the BDT emerges as an approximation to the Horndeski theory, the derivative selfinteractions of the Horndeski scalar become larger as one moves to smaller scales (higher curvature than cosmological environments), and this leads to the screening of the scalar resulting at the same time to the recovery GR. The model.—The BDT is described by the action 1 S¼ 16 πG
Z
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ω pffiffiffiffiffiffi 2 d x −g ϕR − 2Λ − ð∇ϕÞ þ Sm ; ð1Þ ϕ 4
where g is the metric determinant, R is the scalar curvature, Λ is the cosmological constant, G is the bare gravitational constant, and Sm is the matter action. Since Sm is independent of ϕ, the weak equivalence principle is satisfied. We have chosen ϕ to be dimensionless by convention. The relevant equations to be solved may be found in Ref. [2], and here we quote only the Friedman equation, which is _¯ 2 2 1 ϕ_¯ 8πG 1 ϕ 3 Hþ ¯ ¼ ¯ ρ þ ð2ω þ 3Þ ¯ ; 2ϕ 4 ϕ ϕ
ð2Þ
where H is the Hubble rate and ρ is the total matter density including Λ, and the background scalar equation ̈ ¯ þ 3Hϕ_¯ ¼ 8πG ðρ − 3PÞ; ϕ 2ω þ 3
ð3Þ
where P is the matter pressure including Λ. We consider only ω > −ð3=2Þ, since otherwise the scalar is a ghost. As it happens, the field stays constant during the radiation era, because (3) is sourced by ρ − 3P ¼ 0 [since P ¼ ð1=3Þρ for photons], resulting in ϕ¯ behaving like a massless scalar. As the Universe enters the matter era, however, ϕ¯ grows but only logarithmically with the scale factor a. Thus, the scalar field today, ϕ¯ 0 , is expected to be within a few percent of its initial value in the deep radiation era. If the scalar is approximately constant, then the Friedman equation becomes 3H2 ≈ 8πGeff ρ, where the effective cosmological gravitational strength is given by ξ ¼ Geff =G ¼ 1=ϕ. For bound systems in the quasistatic regime, e.g., our Solar System, the effective Newton’s constant is GN ¼ Gð2ω þ 3Þ=ð2ω þ 4Þ [1,2]; thus, observers in a bound system which formed today will measure the same cosmological and local gravitational strength if
2ω þ 4 ϕ¯ 0 ¼ : 2ω þ 3
ð4Þ
We call such models restricted (RBD), since to achieve the above condition the initial value of the scalar field, ϕ¯ i , must be appropriately fixed. Models for which ϕ¯ i is a free parameter will be called unrestricted (UBD). Analysis and methodology.—We numerically solved the BDT background (2) and (3) and the linearized equations in the Jordan frame where the matter equations are unchanged from GR, which ensures that the effective gravitational strength is correctly implemented in the code. To test the numerical results, we implemented the synchronous gauge equations for scalar modes in a modified version of the CAMB package [22] and compared with our own Boltzmann code (derived from CMBFAST [23] and DASH [24]) in which both the synchronous gauge and the conformal Newtonian gauge were used. We generated a chain of steps in parameter space by employing the Markov-chain Monte Carlo method [25,26] implemented in cosmology via COSMOMC [27] (see also [28]). Our chains were long enough to pass the convergence diagnostics and also give very accurate 1D and 2D marginalized posteriors. The main data sets we used are from the Planck satellite [21], WMAP-7/9 [29], the South Pole Telescope (SPT) [30], and the Atacama Cosmology Telescope (ACT) [31]. We also use data from big-bang nucleosynthesis (BBN) light element abundances [32]. The chains were generated for two types of models: the RBD and the UBD models. The RBD models have seven parameters which are the dimensionless baryon and dark matter densities ωb and ωc , respectively, the ratio of the angular diameter distance to the sound horizon at recombination θ, the reionization redshift zre , the amplitude and spectral index of the primordial power spectrum As and ns , respectively, and the BDT parameter ω. The Hubble constant H0 and the (dimensionless) cosmological constant density ωΛ are derived parameters. The UBD models have one additional parameter, which is the initial condition ϕ¯ i . When generating likelihoods for the Planck data, 11 astronomical parameters to model foregrounds and three instrumental calibration and beam parameters (3) were used as described in Ref. [21]. When ACT and SPT were also included with Planck, 17 more calibration parameters were used [21]. In order to sample efficiently the large number of “fast” parameters, we used the speed-ordered Cholesky parameter rotation and the dragging scheme described by Neal and Lewis [33,34] implemented in the latest version of COSMOMC. We now turn to the issue of priors. For the non-BDT parameters we assume the same priors as for ΛCDM, since the two types of cosmological evolution are very similar. A prior on H 0 (HST) from the measurement of the angular diameter distance at redshift z ¼ 0.04 [35] is also imposed for some chains. For the BDT parameters we impose flat
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TABLE I. Summary of our results for the restricted (R) and the unrestricted (U) BD models for a variety of data sets. Limits for ω and ξ_ at the 95% and 99% level are shown as well as 1σ level constraints for ξ. ξ
ωBD U
R
95% 99% 95% 99% WMAP7
99
55
90
51
WMAP7 þ HST
126
62
177 120
WMAP7 þ SPT þ HST
269
148
157 114
PLANCKTEMP þ WMAP9pol
1834
890 1808 692
PLANCKTEMP þ WMAP9pol þ HL
1923
843 1326 213
PLANCKTEMP þ PLANCKLENS
2441 1033 1901 420
PLANCKTEMP þ PLANCKLENS þ HL 1939
829 1408 330
priors on ϕ¯ i and on − lnðωÞ. This prior on ω is more convenient for sampling the chains [17]. However, the choice of prior is not important, as our constraints strongly improved compared to past experiments when Planck data were used in the analysis. Results and discussion.—We first discuss the restricted models, which contain only ω as an additional parameter to ΛCDM. Varying ω changes the background expansion history H which results in a shift of the peak locations and peak heights. It makes more sense, however, to consider the changes in the spectrum at fixed θ, as it was used as a parameter. In that case, the dominant effect is on the largescale temperature spectrum due to the integrated SachsWolfe effect, while small scales are affected due to perturbations in ϕ (indirectly through the effect of ϕ on the potential wells) and are less important. Using WMAP7 alone, we find ω > f90; 51g at the 95% and 99% level, respectively, while for WMAP7+HST we improve to ω > f126; 62g. This is a significant improvement over Ref. [17], driven mainly by the inclusion of polarization data which break the degeneracy between zre and ns and allow the measurements of the damping tail to improve the determination of the other parameters and limit the freedom of ω to vary. For WMAP7 þ SPT þ HST data, change this to ω > f157; 114g at the same confidence levels. The use of Planck temperature (PLANCKTEMP) data greatly improves the measurement of the damping tail and together with WMAP 9-year polarization (WMAP9pol) further improves the constraint to ω > f1808; 692g. This is in line with the forecasting of Ref. [36]. Our results and more data combinations are summarized in Table I and some representative posterior distributions are shown in Fig. 1. Constraints on UBD models, which contain ξ as a further parameter, have not been presented before. As discussed in Ref. [37], the main effect of increasing ξ is to increase the width of the visibility function which in turn increases photon diffusion and damps the CMB temperature
95%
99%
ξ from CMB þ BBN ξ_ × 10−13 years−1 Upper 1σ 95% 99%
þ0.98 0.98þ0.67 −0.55 0.98−0.63
1.113 0.156
2.45
3.37
1.07þ0.22 −0.24 1.10þ0.13 −0.14 1.12þ0.11 −0.11 1.11þ0.13 −0.11 1.12þ0.10 −0.14 1.07þ0.11 −0.10
0.991 0.036
2.85
3.59
0.996 0.029
1.92
2.81
1.006 0.018
0.93
1.78
1.009 0.014
0.74
1.75
0.998 0.029
0.63
1.51
0.999 0.024
0.36
0.83
þ0.40 1.07−0.43 þ0.17 1.10−0.19 þ0.16 1.12−0.14 þ0.17 1.11−0.14 þ0.16 1.12−0.17 þ0.14 1.07−0.13
anisotropies on small scales. Thus the main constraints on ξ from CMB temperature come from fitting the damping envelope with measurements of the CMB at small scales from ACT, SPT, and Planck. Increasing ξ has a slightly different effect on polarization. The same damping effect occurs on small scales, but on large scales we get an enhancement, as a thicker last scattering surface increases the amplitude of the local quadrupole, producing a larger polarization signal [37]. Using WMAP7 alone, we find ω > f99; 55g and þ0.98 ξ ¼ f0.98þ0.67 −0.55 ; 0.98−0.63 g at the 95% and 99% level, respectively, which changes to ω > f269; 148g and ξ ¼ þ0.17 f1.10þ0.13 −0.14 ; 1.10−0.19 g at the same confidence levels with WMAP7 þ SPT þ HST data. Using PLANCKTEMP þ WMAP9pol improves the constraint to ω > f1834; 890g þ0.16 and ξ ¼ f1.12þ0.11 −0.11 ; 1.12−0.14 g. These results and more data combinations are summarized in Table I. In both types of models, constraints on ω using PLANCKTEMP þ WMAP9pol improve by a factor of 6 over WMAP7 þ SPT þ HST. It is interesting to notice, however, that in the restricted models ω is less constrained than in the unrestricted class which has one more parameter. This is further pronounced when other data combinations with PLANCKTEMP are used (see below). The reason is because the extra parameter ξ helps to fit the data better; the best-fit sample for the unrestricted model has a slightly better χ 2 than the restricted model. From Table I, we observe that including ACT and SPT (HL) data does not improve the constraints on ω, but rather, in the case of the restricted models, the constraints become weaker. The marginalized distribution of ln ω for WMAP7 þ SPT þ HST exhibits a peak around ω ∼ 400. The difference in likelihood between the GR limit and this peak is very small, which renders this “detection” insignificant; however, its presence makes it difficult to improve the lower bound on ω. When we use PLANCKTEMP þ HL data, the peak is washed out; however, a small effect
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still remains. It seems that there is a small discrepancy between the PLANCKTEMP and HL data. PLANCKTEMP together with lensing potential reconstruction (PLANCKLENS) gives the tightest constraint on ω. However, PLANCKLENS displays small discrepancies from PLANCKTEMP [21], and once again we opt for not using it when we report our final result. Interestingly, when HL data are added, both PLANCKTEMPþPLANCKLENS and PLANCKTEMP þ WMAP9pol give very similar constraints. Cosmological constraints on G from the CMB can be found in Ref. [38], where 0.74 ≤ ξ ≤ 1.66 is found from WMAP-1 alone at the 95% of confidence level while including BBN data the tighter bounds 0.95 ≤ ξ ≤ 1.01 are obtained at 1σ. Using the same methodology as in Ref. [38], we find 0.998 ≤ ξ ≤ 1.024 at 1σ with PLANCKTEMP þ WMAP9pol þ BBN using more recent measurements of Ref. [32]. Our results improve on Ref. [38] and further put them in the context of a realistic theory. In a more recent work by the authors of Ref. [20], constraints for G are obtained in the context of the restricted BDT; however, in their analysis they left out a possible rescaling of GN . We also report a strong upper bound on the time variation of G from the CMB alone around ∼10−13 =year, as in Table I. Implications for modified gravity.—Regardless of the simplicity of this model, our constraints are more generally valid, as the BDT can be considered an approximation to the Horndeski theory, the most general second-order scalar-tensor theory [11,12], above some very large length l . The gravitational action is R 4 scale pffiffiffiffiffiffi P 3 1 S½g; ψ ¼ 16 πG d x −g I¼0 LðIÞ , where ψ is by convention dimensionless and LðIÞ are given by Lð0Þ ¼ K ð0Þ ;
the complete set of Horndeski theories is defined by four free functions of ψ and X, by restricting the set as above we get eight free constant parameters rather than functions. Thus, our results hold for any Horndeski theory which can be approximated in the above form on cosmological scales. Choosing l ∼ 1=16H0 (Hubble scale at recombination) and using the BDT solution as ϕ_¯ ¼ H=ω and ϕ¯ ∼ 1, we find conservative estimates for the coefficients as ϵi ≪ f10−2 ; 105 ; 10; 105 ; 1; 103 ; 103 ; 105 g. Theories within these limits can be well approximated by the BDT. Conclusions.—We found strong constraints on the BDT parameter ω > 890 and effective gravitational strength 0.981 ≤ ξ ≤ 1.285 at the 99% confidence level, significantly improving on previous work. Improvement on these bounds is expected through the next data release of Planck and with the inclusion of LSS and redshift-space distortions data which is left to future work. We thank Tessa Baker, Jo Dunkley, Pedro Ferreira, Adam Moss, Yong-Seon Song, and Tony Padilla for useful discussions. A. A. acknowledges support from CONACYT. C. S. acknowledges support from the Royal Society.
*
[1] [2] [3] [4] [5]
[6]
ð5Þ
Lð2Þ ¼ K ð2Þ R þ K X ½ð□ψÞ2 − ðDψÞ2 ;
ð2Þ
ð6Þ
[7]
Lð3Þ ¼ −6K ð3Þ Gμν ∇μ ∇ν ψ
ð7Þ
[8]
ð3Þ
ð8Þ
where X ¼ − 12 gμν ∇μ ψ∇ν ψ, Dψ ¼ ∇ ⊗ ∇ψ, and K ðIÞ are functions of X and ψ. The general functions K ðIÞ may be expanded as an analytical series and whose lowest-order terms are K ð0Þ ≈ −2Λ þ 8 ωX þ ε1 ψ 2 =l 2 þ ε2 l 2 X2 , K ð1Þ ≈ ϵ3 ψ 2 þ ϵ4 l 2 X, K ð2Þ ≈ ψ 2 þ ϵ5 ψ 4 þ ϵ6 l 2 X, and K ð3Þ =l 2 ≈ ϵ7 ψ 2 þ ϵ8 l 2 X. We have ignored the constant terms in K ð1Þ and K ð3Þ , as they lead to total derivatives. The constant term in K ð2Þ cannot be ignored, in general, but would lead to GR coupled to a massless scalar as ϵi → 0 and is irrelevant to our work. As the coefficients of the expansion ϵi → 0 and further performing the field redefinition ϕ ¼ ψ 2 , we recover the BDT. We see that, although
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†
Lð1Þ ¼ K ð1Þ □ψ;
þK X ½ð□ψÞ3 − 3ðDψÞ2 □ψ þ 2ðDψÞ3 ;
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