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Comment on “Measurements of Newton's gravitational constant and the length of day” by Anderson J. D. et al.

EPL (Europhysics Letters)

August 2015
111(3)

DOI:10.1209/0295-5075/111/30002

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CC BY 3.0

Authors:

Matthew Pitkin

University of Glasgow

There have been recent claims (Anderson et al., EPL, 110, 10002) of a 5.9 year periodicity in measurements of Newton's gravitational constant, G, which show a very strong correlation with observed periodic variations in the length of the day. I have used Bayesian model comparison to test this claim compared to other hypotheses that could explain the variation in the G measurements. I have used the data from the initial claim, and from an updated set of compiled G measurements that more accurately reflect the experimental dates, and find that a model containing an additional unknown Gaussian noise component is hugely favoured, by factors of

\gtrsim e^{30}

, over two models allowing for a sinusoidal component.

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August 2015

EPL, 111 (2015) 30002 www.epljournal.org

doi: 10.1209/0295-5075/111/30002

Comment

Comment on “Measurements of Newton’s gravitational constant

and the length of day” by Anderson J. D. et al.

M. Pitkin

SUPA, School of Physics & Astronomy, University of Glasgow - Glasgow, G12 8QQ, UK

received 29 May 2015; accepted in ﬁnal form 30 July 2015

published online 18 August 2015

PAC S 04.80.-y – Experimental studies of gravity

PAC S 06.30.Gv – Velocity, acceleration, and rotation

PAC S 96.60.Q- – Solar activity

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Published by the EPLA under the terms of the Creative Commons Attribution 3.0 License (CC BY).

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Introduction. – In [1] the authors claim to observe a

periodic signal in measurements of Newton’s gravitational

constant, G. Speciﬁcally they ﬁnd a 5.9-year-period signal

that is strongly correlated with variations in the observed

length of day [2]. They do not suggest that Gactually

varies on these time scales, but rather that there could be

some systematic eﬀect on the measurement process that is

correlated with the mechanism that leads to the variation

in the length of day. Here I present a reanalysis of the

data used in [1] using Bayesian model selection to test

the hypothesis that the data contains a periodic signal

compared to other potential models. In light of updated

information on the times of the various Gmeasurements

given in [3] I also reanalyse this new dataset with the same

method. In both datasets I have found that a model for the

variations in Gthat only contains an additional Gaussian

noise term is hugely favoured, by factors of e30,over

models containing a sinusoid term1.

Analysis method. – Bayesian model selection pro-

vides a natural way to test multiple hypotheses by forming

the odds ratio of evidences for the diﬀerent hypothe-

ses. The odds ratio for two hypotheses Hiand Hjis

given by

Oij =(p(d|Hi,I)/p(d|Hi,I)) ×(p(Hi|I))/(p(Hj|I)) ,(1)

where p(d|Hi,I) is the evidence for hypothesis Higiven

some data d,p(Hi|I) is the prior probability for Hi,andI

1The code, data tables, ﬁgures and prior ranges for this analysis

can be found at https://github.com/mattpitkin/periodicG.

is information concerning any other assumptions. When

comparing hypotheses I assume that they are equally

probable a priori, so the prior ratio is unity. Therefore,

I just calculate the ratio of evidences for each hypothesis.

If a given hypothesis is deﬁned by a set of parameters, θi,

with their own priors, p(θi|Hi,I), then to calculate the

evidence the parameters must be marginalised over, e.g.

p(d|Hi,I)=θi

p(d|θi,H

i,I)p(θi|Hi,I)dθi,(2)

where p(d|θi,H

i,I) is the likelihood function of the data

given a set of model parameters θi.

The general model that I use for my hypotheses is

m(μG,A,P,φ

0,T

k)=Asin (φ0+2π(Tk−t0)/P )+μG,

(3)

where μGis an oﬀset value, Ais the sinusoid amplitude,

φ0is an initial phase at an epoch t0,Pis the sinusoid

period, and Tkis the time.

In this analysis I have compared four diﬀerent hypothe-

ses, Hi, to explain the measurements of G:H1) the data

is consistent with Gaussian errors, given by the experi-

mental error bars σe,k, about an unknown μG;H2)as

for H1, but also including an unknown common Gaussian

noise term σsys;H3)asforH1, but also including a si-

nusoid with unknown A,φ0and P; and, H4)asforH3,

but also including an unknown σsys. Theseeachcorre-

spond to a diﬀerent set of parameters required in θand

also the number of parameters required in the integral

of eq. (2).

30002-p1

Comment

Table 1: Log odds ratios for the four hypotheses (irepresents

rows and jrepresents columns) when using the data used in [1],

with those when using the data from [3] in parentheses.

ln Oij H2H3H4

H1−133 (−140) −102.2(−66) −103 (−110)

H230 (74) 30 (30)

H3−0.3(−45)

For an initial examination of the claim in [1] I have used

their ﬁg. 1 to read-oﬀ the experimental times and then

used table XVII of [4] for values of G(see footnote 2).

In [1] the experiment times are given no associated

error. However, many of the times used correspond to the

received date of the respective paper rather than the date

of the actual experiment. In analysing this data I speciﬁed

uncertainties on the experiment times of σt=0.25 years

(with the exception of the JILA-10 and LENS-14 measure-

ments for which I use uncertainties of one week) before the

given time. I have taken this time uncertainty into account

by marginalising over it for each data point.

Results. – The odds ratios comparing hypotheses

when using the Gdataset of [1] are summarised in table 1.

It is clear that hypotheses including extra parameters over

that for H1are hugely favoured by factors of e100.The

two hypotheses, H3and H4, containing a sinusoidal signal

are both approximately equally probable. However, H2,

just containing the additional unknown noise term σsys

and the unknown oﬀset μG, is hugely favoured by factors

∼e30 over H3and H4. This shows that the simple model

for which variations are just due to an unknown Gaus-

sian noise term is far more likely to be the cause of the

variations than an additional sinusoidal variation. This

is due to Bayesian model selection naturally applying a

penalty for including additional parameters that do not

signiﬁcantly increase the evidence.

I have also looked at the posterior probability distribu-

tions for the period and for H3I see a clear lone spike in

probability around the claimed period of 5.9 years. A sim-

ilar spike shows up for H4, but is much less pronounced.

I have assessed the signiﬁcance of this period probability

peak for H3by rerunning that analysis 20 times, but each

time randomly shuﬄing the Gvalues to remove any real

periodicity in the data. Out of these 20 runs there is one

time when the hypothesis using the shuﬄed data is more

favoured than when using the unshuﬄed data and another

couple that are within a factor of two. The posteriors for

these cases also show very similar spikes in the period to

that from the unshuﬄed data.

Since the acceptance of [1] Schlamminger et al. [3] ex-

amined the claim, in particular noting that the experi-

mental times in the original work are not accurate. They

examined the literature to compile a more complete list

2For the BIPM-13 measurements I used the combined servo and

Cavendish value from [5] and for the LENS-14 measurements I used

the values from [6].

of experiments with information on the actual dates on

which the experiments were performed. I have reanalysed

this new dataset for each of the four hypotheses. When

marginalising over the time error I have now set the error

to be symmetric around the mean experiment times. For

all other parameters I have used the same prior ranges as

in the initial analysis. The odds ratios for each of these

cases are also given in table 1 from which it can be seen

that H2is still favoured over all other hypotheses by a huge

amount. However, H3is now hugely disfavoured over H4,

i.e. just including a sinusoid, but adding no additional

noise term does far worse at ﬁtting the data than also

including the noise term.

Conclusions. – I have reanalysed the data consisting

of measurements of Gfrom [1] and [3] to asses the claim

of a periodic component with a period of 5.9 years.

Using Bayesian model selection, and four diﬀerent hy-

potheses to describe the variations in the data, and in-

cluding uncertainties on the experimental times, I have

found that the best model is one in which there is an

additional unknown Gaussian noise term on top of the ob-

served experimental errors. This is favoured over a model

also containing a sinusoidal term by factors of e30. I also

ﬁnd that periodic signals can easily be found in random

permutations of the data suggesting that the observed

periodicity seen in [1] is just a random artifact of the

data.

Following the publication of [1] the authors have taken

into account the work of [3] (see [7]). They ﬁt an ad-

ditional sinusoid to the updated data and note that the

signiﬁcance of the correlation with the length of day de-

creases. I expect that calculating the evidence for a model

including two sinusoids would not cause such a model to be

favoured over the simpler model containing just the extra

Gaussian noise term, as the increase in parameter space

will be penalised if the ﬁt does not signiﬁcantly improve.

I note that if there were good a priori reasons to expect

a periodic component with a speciﬁc period in the data

(i.e. if there were a good reason why the mechanism lead-

ing to changes in the length of day could couple into mea-

surements of G), then the evidence for models containing

such a periodic signal might dramatically increase. How-

ever, without such prior knowledge using such a constraint

would strongly bias us.

∗∗∗

I would like to thank Prof. J. Faller for useful discus-

sions, Prof. C. Speake for putting me in contact with

Dr. S. Schlamminger, and Dr. S. Schlamminger

for providing me with a data ﬁle of their compiled G

measurements. I am funded by the STFC under grant

ST/L000946/1.

Additional remark (16th July 2015): In responding to

my Comment the authors of the original article put for-

ward two related pieces of evidence to suggest that models

30002-p2

Comment

for the time variations of Gmeasurements containing a

constant oﬀset and one or two periodic components are

favoured over a model with no periodic component. They

show that once the best ﬁts for the models containing one

or two periodic components are removed from the data the

residuals have smaller variance, and have a distribution

that is closer to a Gaussian distribution, than for the

model containing no periodic component. These ﬁndings

are not at all surprising. If one adds more parameters to

the model then one can almost always ﬁnd a better ﬁt with

smaller residuals —in cases where the additional model

complexity adds no extra information this is commonly

known as over-ﬁtting and is related to the concept of Oc-

cam’s razor. A Bayesian analysis, such as I performed,

naturally allows one to penalise extra complexity when

it is unnecessary through the incorporation of an Occam

factor. This Occam factor penalty comes about through

use of the prior volume of the parameter space for each

model. Each parameter has a range over which it is a pri-

ori thought to exist, and in my analysis the prior volume

is just the product of these ranges. So, a model with more

parameters will often have a larger prior volume. This nat-

urally incorporates a penalty for over-ﬁtting in that larger

prior volumes will down-weight the evidence for a model,

so if the likelihood does not compensate enough for this

down-weighting then the evidence will be reduced.

In my analysis this penalty far outweighs the slightly

better ﬁt that can be achieved with a more complex model

containing a sinusoid. It should also be noted that my

most favoured model does not just contain a constant oﬀ-

set, but also contains an additional unknown noise term.

This may suggest that the quoted errors on the Gmea-

surements are underestimates of the true noise.

A further addition that I was able to include in my

analysis, but which the authors of the original article do

not appear to have addressed is that there are also error

bars on the experimental times. This may also weaken

their ﬁt.

Further Gmeasurement data may change my conclu-

sions, but with the current data I stand by my result that

the data is best explained by just a constant oﬀset and an

additional unknown noise term, and no periodic compo-

nent is required.

REFERENCES

[1] Anderson J. D., Schubert G., Trimble V. and

Feldman M. R.,EPL,110 (2015) 10002.

[2] Holme R. and de Viron O.,Nature,499 (2013) 202.

[3] Schlamminger S., Gundlach J. H. and Newman

R. D.,Phys. Rev. D,91 (2015) 121101.

[4] Mohr P. J., Taylor B. N. and Newell D. B.,Rev.

Mod. Phys.,84 (2012) 1527.

[5] Quinn T., Speake C., Parks H. and Davis R.,Phys.

Rev. Lett.,113 (2014) 039901.

[6] Rosi G., Sorrentino F., Cacciapuoti L., Prevedelli

M. and Tino G. M.,Nature,510 (2014) 518.

[7] Anderson J. D., Schubert G., Trimble V. and

Feldman M. R., arXiv:1504.06604v2 (2015).

30002-p3

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The universal Newtonian gravitational constant is being redetermined at the National Bureau of Standards with use of the method of Boyes in which the period of a torsion pendulum is altered by the presence of two 10.5-kg tungsten balls. The difference in the squares of the frequencies with and without the balls is proportional to G. The resulting value of G is (6.6726\ifmmode\pm\else\textpm\fi{}0.0005)\ifmmode\times\else\texttimes\fi{}{10}^{-{}11}

{\mathrm{m}}^{3}

\ifmmode\cdot\else\textperiodcentered\fi{} {\mathrm{sec}}^{-{}2}\ifmmode\cdot\else\textperiodcentered\fi{} {\mathrm{kg}}^{-{}1}\ifmmode\cdot\else\textperiodcentered\fi{}

Measurement of Newton's Constant Using a Torsion Balance with Angular Acceleration Feedback

Article

Nov 2000

We measured Newton's gravitational constant G using a new torsion balance method. Our technique greatly reduces several sources of uncertainty compared to previous measurements: (1) It is insensitive to anelastic torsion fiber properties; (2) a flat plate pendulum minimizes the sensitivity due to the pendulum density distribution; (3) continuous attractor rotation reduces background noise. We obtain G = (6.674215+/-0.000092) x 10(-11) m3 kg(-1) s(-2); the Earth's mass is, therefore, M = (5.972245+/-0.000082) x 10(24) kg and the Sun's mass is M = (1.988435+/-0.000027) x 10(30) kg.

A New Determination of G Using Two Methods

Article

Oct 2001

We present the results of a measurement of G made with a torsion-strip balance used in two substantially independent ways. The two results agree to within their respective uncertainties; the correlation coefficient of the two methods is -0.18. The result is G = 6.675 59(27)x10(-11) m(3) kg(-1) s(-2) with a standard uncertainty of 4.1 parts in 10(5). Our result is 2 parts in 10(4) higher than the recent result of Gundlach and Merkowitz.

Comment on “Measurements of Newton's gravitational constant and the length of day” by Anderson J. D. et al.

Abstract

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