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1 3,4 2,1 1 1 2 L. Bernus , O. Minazzoli , A. Fienga , M. Gastineau , J. Laskar , P. Deram IMCCE, Observatoire de Paris, PSL University, CNRS, Sorbonne Universit´ e, 77 avenue Denfert-Rochereau, 75014 Paris, France G´ eoazur, Observatoire de la Cˆ ote d’Azur, Universit´ e Cˆ ote d’Azur, IRD, 250 Rue Albert Einstein, 06560 Valbonne, France Centre Scientifique de Monaco, 8 Quai Antoine 1er, Monaco Artemis, Universit´ e Cˆ ote d’Azur, CNRS, Observatoire de la Cˆ ote d’Azur, BP4229, 06304, Nice Cedex 4, France We use the planetary ephemeris INPOP17b to constrain the mass of the graviton in the Newtonian limit. We also give an interpretation of this result for a specific case of fifth force framework. We find that the residuals for the Cassini spacecraft significantly (90% C.L.) degrade for Compton wavelengths of the graviton smaller than 1.83×10 km, corresponding to a graviton mass bigger than −23 2 6.76 × 10 eV/c . This limit is comparable in magnitude to the one obtained by the LIGO-Virgo collaboration in the radiative regime. We also use this specific example to illustrate why constraints on alternative theories of gravity obtained from postfit residuals are generically overestimated. INTRODUCTION phenomenology in the Newtonian regime can reduce to the one of general relativity to any given level of accuracy. Another generic feature of many massive gravity theo- From a particle physics point of view, general relativity ries is that, if the graviton is massive, its dispersion rela- can be thought as a theory of a massless spin-2 particle 2 2 2 2 4 tion may be modified according to E = p c + m c — hereafter named graviton. From this perspective, it [2], such that the speed of a gravitational waves de- is legitimate to investigate whether or not the graviton 2 2 2 2 2 pends on its energy (or frequency) v /c = c p /E ' could actually possess a mass — even if minute. Such an 2 2 2 2 1 − h c /(λ E ). Therefore, the waveform of gravita- eventuality has been scrutinized from a theoretical point tional waves would be modified during their propagation, of view since the late thirties, with the pioneer work of while at the same time, sources of gravitational waves Fierz and Pauli [1]. There is a wide set of massive gravity have been seen up to more than 1420 Mpc (at the 90% theories, which lead to various phenomenologies [2]. One C.L.) [4]. As a consequence, waveform match filtering of the generic prediction from several models — although can be used to constrain the graviton mass from gravita- not all of them — is that the usual 1/r falloff of the tional waves detections [5, 6]. Newtonian potential acquires a Yukawa suppression [2]. Combining bounds from several events in the cata- In the present manuscript, we aim to test this particular log GWTC-1 [4] leads to λ ≥ 2.6 × 10 km (resp. phenomenology, regardless of the specificity of the the- −23 2 2 3 m ≤ 5.0 × 10 eV/c [4, 7] ) at the 90% C.L . It oretical model that produced it. For more information is important to keep in mind that this limit is obtained on the status of current theoretical models, we refer the in the radiative regime, while we focus here on the New- reader to [2, 3]. As a consequence, we assume that the tonian regime. Although, one could expect λ to have line element in a space-time curved by a spherical massive the same value in both regimes for most massive gravity object at rest, at leading order in the Newtonian regime, theories, it may not be true for all massive gravity the- reads ories. Therefore, both constraints should be considered independently from an agnostic point of view. See, e.g., 2GM 2 −R/λ 2 2 [2] for a review on the graviton mass constraints. ds = −1 + e c dT c R 2GM −R/λ 2 + 1 + e dL , (1) IMPORTANCE OF A GLOBAL FIT ANALYSIS c R 2 2 2 2 2 2 2 with dL ≡ dX + dY + dZ , R ≡ X + Y + Z and Twenty years ago [5] and more recently [8], Will ar- λ the Compton wavelength of the graviton — although g gued that Solar System observations could be used to we will see that our constraints can be applied on a wider improve — or at least be comparable with — the con- range of massive and non-massive gravity metrics. Ob- straints on λ obtained from the LIGO-Virgo Collabo- viously, as long as λ is big enough, the gravitational With the definition m = h/(cλ ). g g 1 3 In particular, usually not for models prone to the Vainshtein Assuming that the graviton mass affects the propagation only, mechanism [2]. and not the binaries dynamics. arXiv:1901.04307v2 [gr-qc] 1 Aug 2019 2 λ a Mercury a Mars a Saturn a Venus a EMB GM benefits of an improved modeling of the Earth-Moon λ 1 0.50 0.49 0.04 0.39 0.05 0.66 system, as well as an update of the observational sample a Mercure · · · 1 0.21 0.001 0.97 0.82 0.96 used for the fit [14] — especially including the latest a Mars · · · · · · 1 0.03 0.29 0.53 0.06 Mars orbiter data. For this work we use an extension of a Saturn · · · · · · · · · 1 0.003 0.02 0.01 INPOP17a, called INPOP17b, fitted over an extended a Venus · · · · · · · · · · · · 1 0.86 0.94 sample of Messenger data up to the end of the mission, a EMB · · · · · · · · · · · · · · · 1 0.73 provided by [16]. GM · · · · · · · · · · · · · · · · · · 1 In the present communication, our goal is to use the TABLE I. Examples of correlations between various IN- latest planetary ephemeris INPOP17b in order to con- POP17b parameters and the Compton wavelenght λ . a, g strain a hypothetical graviton mass directly at the level EMB and M state for semi-major axes, the Earth-Moon of the numerical integration of the equations of motion barycenter and the mass of the Sun respectively. and the resulting adjusting procedure. By doing so, the various correlations between the parameters are intrin- sically taken into account, such that we can deliver a ration — assuming that the parameters λ appearing in g conservative constraint on the graviton mass from Solar both the radiative and Newtonian limits are the same. System obervations — details about the global adjusting A graviton mass would indeed lead to a modification of procedure are given in Supplemental Material. the perihelion advance of Solar System bodies. Hence, based on current constraints on the perihelion advance MODELISATION FOR SOLAR SYSTEM of Mars — or on the post-Newtonian parameters γ and PHENOMENOLOGY β — derived from Mars Reconnaissance Orbiter (MRO) data, Will estimates that the graviton’s Compton wave- Following Will [8], we develop perturbatively the po- length should be bigger than (1.2− 2.2)× 10 km (resp. −24 2 tential in terms of r/λ , such that the line element (1) m < (5.6− 10)× 10 eV/c ), depending on the spe- g g now reads cific analysis. However, as an input for his analysis, Will uses results based on the statistics of residuals of the So- lar System ephemerides that are performed without in- 2GM 1 r 2 2 2 cluding the effect of a massive graviton. But various pa- ds = −1 + 1 + c dt (2) 2 2 c r 2 λ rameters of the ephemeris (eg. masses, semi-major axes, g Compton parameter, etc.) are all more or less correlated 2GM 1 r 2 −3 −2 + 1 + 1 + dl +O(c λ ), to λ (see Table I). Therefore, any signal introduced by 2 2 c r 2 λ λ < +∞ — for instance, a modification of a perihelion advance — can in part be re-absorbed during the fit of albeit with a change of coordinate system other parameters that are correlated with the mass of the graviton. (See Supplemental Materials). This necessar- t x q q T = , X = (3) ily leads to a decrease of the constraining power of the GM GM 1 + 1− 2 2 c λ c λ g g ephemeris on the graviton mass with respect to postfit estimates. As a corollary, all analyses based solely on The change of coordinate system is meant to get rid of postfit residuals tend to overestimate the constraints on the non-observable constant terms that appear in the line alternative theories of gravity due to the lack of informa- −1 element Eq. (1) after expanding in terms of λ . Con- tion on the correlations between the various parameters. sidering a N-body system, the resulting additional accel- Eventually, one cannot produce conservative estimates of eration to incorporate in INPOP’s code is any parameter without going through the whole proce- dure of integrating the equations of motion and fitting i i 1 GM x − x i P −3 δa = +O(λ ), (4) the parameters with respect to actual observations — 2 λ r which is the very raison d’ˆetre of the ephemeris INPOP. INPOP (Int´ egrateur Num´ erique Plan´ etaire de i where M and x are respectively the mass and the po- l’Observatoire de Paris) [9] is a planetary ephemeris sition of the gravitational source P . In what follows, we that is built by integrating numerically the equations of make the standard assumption that the underlying the- motion of the Solar System following the formulation of ory is such that light propagates along null geodesics [3]. [10], and by adjusting to Solar System observations such as lunar laser ranging or space missions observations. In addition to adjusting the astronomical intrinsic param- eters, it can be used to adjust parameters that encode 4 We assume that the underlying theory of gravity is covariant, deviations from general relativity [11–14], such as λ . such that this change of coordinates has no impact on the deriva- tion of the actual observables. The latest released version of INPOP, INPOP17a [15], 3 From the null condition ds = 0 and Eq. (2), the re- To quantify the statistical meaning of this degradation, sulting additional Shapiro delay at the perturbative level we perform a Pearson [19] χ test between both residuals reads in order to look at the probability that they were both built from the same distribution. To compute the χ , we X build an optimal histogram with the Cassini residuals of 1 GM ~ ~ ~ δT = N · R R − R R (5) ER ER PR PR PE PE INPOP17b using the method described in [20], assum- 3 2 2 c λ ing the gaussianity of the distribution of the residuals. !# ~ ~ We determine the optimal bins in which are counted the R + R · N PR PR ER 2 −3 −3 + b ln +O(c λ ), P g residuals to build the histogram. Then, using the same ~ ~ R + R · N PE PE ER bins, we build an histogram for the Cassini residuals ob- ~ ~ ~ tained by the solution to be tested with a given value of where R = ~x − ~x , R = |R |, N = XY Y X XY XY XY λ . Note that the first bin left-borned is −∞ and the last 2 2 ~ ~ ~ R /R and b = R − (R · N ) . One can XY XY P PE ER PE bin right-borned is +∞. Let (C ) be the bins in which i i notice that the correction to the Shapiro delay scales as I G are counted the values of the residuals and N , N be i i (L /λ ) with respect to the usual delay, where L is c g c the number of residuals of INPOP17b and the solution a characteristic distance of a given geometrical config- to be tested, respectively, counted in bin number i. One uration. Given the old acknowledged constraint from can then compute Solar System observations on the graviton mass (λ > G I 2 2.8 × 10 km [5, 8, 17]), one deduces that the correc- X (N − N ) 2 i i χ (λ ) = (7) tion from the Yukawa potential on the Shapiro delay is i=1 negligible for past, current and forthcoming radio-science observations in the Solar System . For Cassini data, it occurs that the optimal binning gives On the other hand, the fifth force formalism predicts 2 2 10 bins. As a result, this χ follows a χ law with 10 de- an additional Yukawa term to the Newtonian potential grees of freedom. If the computed χ is then greater than [18] its quantile for a given confidence probability p, we can say that the distribution of the residuals obtained for λ Gm −r/λ V = (1 + αe ) (6) is different from the residuals obtained by the reference solution with a probability p. This test can be done for If we assume that λ r and α > 0, we can also expand both a positive detection of a physical effect and a rejec- the Yukawa term, such that our result on λ can be trans- tion of the existence of a physical effect. If the computed posed to λ/ α — although, one first has to rescale the χ (λ ) becomes then greater than its critical value for a gravitational constant to G = G(1+α), and then to make probability p, one has to check if residuals are smaller or the same coordinates change as in Eq. (3), but substitut- bigger than those obtained by the reference solution. In ing λ by λ/α. Note that a fifth force is also one of the the first case (smaller – or better – residuals), it means generic features of several massive gravity theories [2]. that the added effect increases significantly the quality of the residuals and is probably (with a probability p) a true physical effect. On the contrary, in the second EVALUATION OF THE SIGNIFICANCE OF THE case (bigger – or degraded – residuals), it means that the RESIDUALS DETERIORATION added effect is probably physically false. In our work, the critical increasing of χ (λ ) corresponds to a degradation To give a confidence interval for λ , we proceed as fol- of the residuals (see Supplemental Material for a detailed lows. For each value of λ , we perform a global fit of analysis). The massive graviton can then be rejected for all other parameters to observations using the same data high enough values of the mass (or low enough values of that for the reference solution INPOP17b — therefore, λ ). for the same number of observations. After the global fit procedure, we compute the residuals at the same dates that for the reference solutions and look how they are RESULTS degraded or improved with respect to λ . The result is that Cassini residuals are the first to degrade signifi- In Fig. 1 we plot the χ as a function of λ . In this plot, cantly while λ decreases (see Supplemental Material for g we give two values of quantiles associated to two proba- details). bilities of significance, p = 90% and p = 99, 9999999%, which correspond to critical values of χ equal to 15.99 and 62.94 respectively for a 10 degrees of freedom χ dis- However, note that the scaling of the correction to the Shapiro −1 tribution. 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We also provide a zoom of the main figure in laboration, “Tests of General Relativity with the Bi- order to show that the χ is not monotonic for small dif- nary Black Hole Signals from the LIGO-Virgo Cata- ferences of λ . However, if a given limit is crossed several log GWTC-1,” arXiv e-prints , arXiv:1903.04467 (2019), times, our algorithm automatically takes the most con- arXiv:1903.04467 [gr-qc]. servative value in the discrete set of λ , as can be seen in [8] Clifford M Will, “Solar system versus gravitational-wave Fig. 1. bounds on the graviton mass,” Classical and Quantum ——————————————————————— Gravity 35, 17LT01 (2018). [9] A. Fienga, H. Manche, J. Laskar, and M. Gastineau, “INPOP06: a new numerical planetary ephemeris,” As- tronomy and Astrophysics 477, 315–327 (2008). CONCLUSION [10] T. D. Moyer, Deep Space Communications and Naviga- tion Series , Vol. 2 (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2003). In the present manuscript, we deliver the first conser- [11] A. 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As previously explained, in to improve planetary ephemeris and to test general rel- terms of a fifth force, the constraint on λ can be trans- ativity,” Astronomy and Astrophysics 561, A115 (2014), lated into a constraint on λ/ α, simply by substituting √ arXiv:1306.5569 [astro-ph.EP]. λ by λ/ α, if α > 0. [13] A. Fienga, J. Laskar, P. Exertier, H. Manche, and The fact that our 90% C.L. bound is comparable in M. Gastineau, “Numerical estimation of the sensitivity of magnitude to the one obtained by the LIGO-Virgo col- INPOP planetary ephemerides to general relativity pa- rameters,” Celestial Mechanics and Dynamical Astron- laboration in the radiative regime [7, 21] is a pure coin- omy 123, 325–349 (2015). cidence: the two bounds rely on totally different types [14] V. Viswanathan, A. Fienga, O. Minazzoli, L. Bernus, of observation — gravitational waves versus radioscience 5 J. Laskar, and M. Gastineau, “The new lunar ephemeris ris, S. Sakai, R. Sch¨ odel, and G. Witzel, “Testing Gen- INPOP17a and its application to fundamental physics,” eral Relativity with Stellar Orbits around the Supermas- MNRAS 476, 1877–1888 (2018), arXiv:1710.09167 [gr- sive Black Hole in Our Galactic Center,” Physical Review qc]. Letters 118, 211101 (2017), arXiv:1705.07902. [15] V. Viswanathan, A. Fienga, M. Gastineau, and [19] Karl Pearson, “On the criterion that a given system J. Laskar, “INPOP17a planetary ephemerides,” Notes of deviations from the probable in the case of a corre- Scientifiques et Techniques de l’Institut de Mecanique lated system of variables is such that it can be reason- Celeste 108 (2017), last Accessed: 2018-11-13. ably supposed to have arisen from random sampling,” [16] A. K. Verma and J.-L. Margot, “Mercury’s gravity, tides, in Breakthroughs in Statistics: Methodology and Distri- and spin from MESSENGER radio science data,” Jour- bution , edited by Samuel Kotz and Norman L. Johnson nal of Geophysical Research (Planets) 121, 1627–1640 (Springer New York, New York, NY, 1992) pp. 11–28. (2016), arXiv:1608.01360 [astro-ph.EP]. [20] DAVID W. SCOTT, “On optimal and data-based his- [17] C. Talmadge, J.-P. Berthias, R. W. Hellings, and E. M. tograms,” Biometrika 66, 605–610 (1979). Standish, “Model-independent constraints on possible [21] B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, modifications of Newtonian gravity,” Physical Review K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Ad- Letters 61, 1159–1162 (1988). hikari, V. B. Adya, and et al., “GW170104: Observa- [18] A. Hees, T. Do, A. M. Ghez, G. D. Martinez, S. Naoz, tion of a 50-Solar-Mass Binary Black Hole Coalescence E. E. Becklin, A. Boehle, S. Chappell, D. Chu, A. De- at Redshift 0.2,” Physical Review Letters 118, 221101 hghanfar, K. Kosmo, J. R. Lu, K. Matthews, M. R. Mor- (2017), arXiv:1706.01812 [gr-qc]. Supplemental materials: Constraining the mass of the graviton with the planetary ephemeris INPOP 1 3,4 2,1 1 1 2 L. Bernus , O. Minazzoli , A. Fienga , M. Gastineau , J. Laskar , P. Deram IMCCE, Observatoire de Paris, PSL University, CNRS, Sorbonne Universit´ e, 77 avenue Denfert-Rochereau, 75014 Paris, France G´ eoazur, Observatoire de la Cˆ ote d’Azur, Universit´ e Cˆ ote d’Azur, IRD, 250 Rue Albert Einstein, 06560 Valbonne, France Centre Scientifique de Monaco, 8 Quai Antoine 1er, Monaco Artemis, Universit´ e Cˆ ote d’Azur, CNRS, Observatoire de la Cˆ ote d’Azur, BP4229, 06304, Nice Cedex 4, France NUMERICAL ANALYSIS To model and confront the massive graviton to Solar System observations, we add its contribution to the IN- POP17b Solar System model (including asteroid masses), and then fit the newly obtained ephemeris according to the procedure described in [1]. For a fixed value of λ , we fit the parameters of the whole model to the data. We have performed an iterative fit of the INPOP17b parameters of the ephemeris for each given values of λ . In order to compare the tested and the reference solu- tions, one needs to apply exactly the same procedure in each case. Accordingly, the same data and weights FIG. 1. Plot of the standard deviations with respect to λ are used. As usual, weights are representative of data after 12 iterations of the data fit. The values of the refer- uncertainty and distribution. The comparison between ence solution INPOP17b standard deviations (given in Table the residuals of a solution with a massive graviton to the I) have been removed. The colored areas correspond to zones where the standard deviations of each observable are con- residuals of the reference solution gives a measurement of sidered a priori as being marginally to not significant. The the sensibility of the ephemeris to the effects of a massive limits come from the estimation of the internal accuracy of graviton. For each value of λ we compute the standard the ephemeris obtained through the comparison with DE436, deviation of the residuals of range observations. More see Table I as well as text below. [A rigorous evaluation of specifically, we exhibit residuals obtained with observa- the significance of the deterioration is made in the main part tions for Cassini mission, Messenger mission, and Mars of the manuscript.] Odyssey and Mex mission. Other observations are less relevant due to less accurate data and/or high correla- tion with λ . This algorithm processes for 1024 different standard deviations of the last iteration (the 13th) in Fig. 13 13 fixed values of λ between 1×10 and 8 ×10 km. We 1. plot the different standard deviations with respect to λ An important point to have in mind is that Mars data for each iteration. We remove the values of the refer- constrain the global fit of parameters to observations — ence solution standard deviations listed in Table I. In thanks to the important weight in the fit given to the Table I, we indeed give the 1σ standard deviations of Mars Odyssey and MEX missions accurate data in the INPOP17b residuals together with the 1σ differences ob- reference solution . So, while in Will’s analysis [3], the tained between INPOP17b and the Jet Propulsion Lab- high quality of Martian data is expected to allow the oratory ephemeris DE436 geocentric distances on the best constraints on λ , this high quality actually helps to time interval of the data sample. The latter gives an idea better adjust the whole set of parameters — but they do of the internal accuracy of the reference ephemeris itself. not significantly constrain λ . Alternatively, given the At about the 10th iteration, the standard deviations for fact that Saturn’s semi-major axe is less correlated to the three sets of residuals stop evolving — meaning that λ (see Table I in the main part of the manuscript), it the adjustment has converged. We report the plot of the is not surprising to see in Fig. 1 that Saturn positions 1 3 i.e. for λ = +∞ As mentioned above, weights are representative of data uncer- Which is based on DE430 ([2]). tainty and distribution. arXiv:1901.04307v2 [gr-qc] 1 Aug 2019 2 Observations Time # (O-C) INPOP17b INPOP17b-DE436 Intervals 1σ 1σ [m] [m] Mercury (Messenger) 2011 : 2013.2 950 7.2 3.9 Mars (Ody, MEX) 2002 : 2016.4 52946 5.0 1.4 Saturn (Cassini) 2004 : 2014 175 32.1 11.7 TABLE I. Summary of data selected for monitoring INPOP sensitivity to λ . Columns 2 and 3 provide the time coverage of the sample and the number of observations per sample respectively. Column 4 gives the 1σ standard deviation of residuals obtained with INPOP17b. These standard deviations are taken as reference values in the text. The last column indicates the 1σ differences between INPOP17b and DE436 for geocentric distances and for interval of time covered by the two ephemeris adjustments. deduced from the Cassini observations are actually the most constraining on λ . It is found that the variation of the standard deviations of the selected residuals dominates compared to the vari- ations of the mean values of the residuals, for all values of λ . Therefore, in the following, we focus on standard deviations of data. As one can see in Fig. 1, after 12 iterations, only residuals deduced from Saturn positions obtained with Cassini show important deviations at a high value of λ . On the other hand, all the values of standard deviation of Mars data are below 1.5 m higher than the reference value. Around λ = 1.5 × 10 km, Messenger data go FIG. 2. Plot of the standard deviations variations with re- a little above 3 m higher than the reference value, but spect to λ without adjusting the Solar System parameters. decrease then as λ decreases, while Mars standard devi- As expected, λ seems to have a much more important im- ation does exactly the opposite — indicating a compen- pact compared to the case in Fig. 1, where one has adjusted sating mechanism between the two sets of standard de- the Solar System parameters after adding the λ parameter. [Note that the scale of the x-axis here is different with respect viations, whose controlling parameters are indeed highly to Fig. 1]. correlated. Of course, the compensating mechanism is actually across the whole set of residuals and depends on both the weights attributed to the different data and to the effect of λ < +∞ on the ephemeris, when the Solar the correlations between various parameters. In the end, g System parameters are (wrongly) considered to be known only residuals deduced from Cassini show a significant beforehand. The standard deviations for Saturn Cassini (and monotonic) increase as λ decreases, as one can see residuals are shown in Fig. 2. If one applies the same in Fig. 1. statistical analysis on those residuals — instead of on the residuals obtained after adjusting the Solar System pa- rameters as in Fig. 1 of the main part of the manuscript DISCUSSION — one would mistakenly get at an improvement of more than a factor 20 on the constraint of λ , as it can be seen The significant level of correlation between λ and the 14 in Fig. 3 (the postfit analysis gives λ ≤ 4.98× 10 km Solar System parameters indicates that any signal intro- 13 instead of 1.83× 10 km, for the 90% C.L. bound). duced by λ < +∞ can in part be re-absorbed during the fit of all the parameters. As a consequence, as explained previously, analyses based on postfit residuals tend to overestimate the constraint on λ . We can illustrate this further with the following example. [1] V. Viswanathan, A. Fienga, M. Gastineau, and J. Laskar, “INPOP17a planetary ephemerides,” Notes Scientifiques We look at the standard deviations of residuals with et Techniques de l’Institut de Mecanique Celeste 108 respect to λ without re-adjusting all the Solar System (2017), last Accessed: 2018-11-13. parameters — which effectively corresponds to do a sort [2] W. M. Folkner, J. G. Williams, D. H. Boggs, R. S. Park, of postfit analysis, in the sense that one considers Solar and P. Kuchynka, “The Planetary and Lunar Ephemerides System parameters to be given beforehand by previous DE430 and DE431,” Interplanetary Network Progress Re- analyses. This gives an indication of the amplitude of port 196, 1–81 (2014). 3 [3] Clifford M Will, “Solar system versus gravitational-wave bounds on the graviton mass,” Classical and Quantum Gravity 35, 17LT01 (2018). FIG. 3. Plot of χ (λ ), and the constraints one would have deduced for λ without re-adjusting the Solar System param- eters. It would lead to a spurious improvement of a factor 27.2 and 22.4 for the 90% and 99.9999999% C.L. bounds re- spectively, compared to the global fit analysis.
General Relativity and Quantum Cosmology – arXiv (Cornell University)
Published: Jan 9, 2019
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