Neutron Limit on the Strongly-Coupled Chameleon Field

1.1 MB·

by  K. Li

Neutron Limit on the Strongly-Coupled Chameleon Field
K. Li,1,2 M. Arif,3 D. G. Cory,4,5,6,7 R. Haun,8 B. Heacock,9,10 M. G. Huber,3,∗ J. Nsofini,11,6 D. A.
Pushin,6,11,† P. Saggu,4 D. Sarenac,6,11 C. B. Shahi,8 V. Skavysh,9 W. M. Snow,1,2,‡ and A. R. Young9,10,§
(The INDEX Collaboration)
1Department of Physics, Indiana University, Bloomington, Indiana 47408, USA
2Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47408, USA
3National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
4Department of Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
5Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
6 6Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
1 7Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada
0 8Department of Physics, Tulane University, New Orleans, LA 70118, USA
2 9Department of Physics, North Carolina State University, Raleigh, NC 27695, USA
n 10Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708, USA
a 11Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
J (Dated: January 27, 2016)

6
The physicalorigin of thedark energythat causes theaccelerated expansion rateof theuniverse
2
is one of the major open questions of cosmology. One set of theories postulates the existence of a

self-interactingscalarfieldfordarkenergycouplingtomatter. Inthechameleondarkenergytheory,
]
O thiscouplinginducesascreeningmechanismsuchthatthefieldamplitudeisnonzeroinemptyspace
butisgreatly suppressedinregionsofterrestrialmatterdensity. Howevermeasurementsperformed
C
under appropriate vacuum conditions can enable the chameleon field to appear in the apparatus,
h. where it can be subjected to laboratory experiments. Here we report the most stringent upper
p boundonthefreeneutron-chameleoncouplinginthestrongly-coupledlimitofthechameleontheory
- using neutron interferometric techniques. Our experiment sought the chameleon field through the
o relative phase shift it would inducealong one of the neutron paths inside a perfect crystal neutron
r interferometer. Theamplitudeofthechameleonfieldwasactivelymodulatedbyvaryingthemillibar
t
s pressures inside a dual-chamberaluminum cell. Wereport a 95% confidence level upperbound on
a theneutron-chameleoncouplingβ rangingfrom β<4.7×106 foraRatra-Peeblesindexofn=1in
[ thenonlinearscalarfieldpotentialtoβ<2.4×107 forn=6,oneorderofmagnitudemoresensitive

1 thanthemostrecentfreeneutronlimitforintermediaten. Similarexperimentscanexplorethefull
v parameter range for chameleon dark energy in the foreseeable future.
7
9
8 INTRODUCTION and dynamical screening mechanism to explain why it
6 hasnotbeenobservedinpreviousprecisiongravitational
0
measurements.
.
1 The discovery of the accelerated expansion of the uni-
0 Inthispaperwespecificallyaddressaparticularexam-
verse[1,2]incombinationwithothercosmologicalobser-
6 ple of such a screened scalar field called the chameleon
vations implies that a component of the universe called
1 field [8–11]. The chameleon field has a nonlinear poten-
darkenergyconstitutesabout70%ofthe energydensity
v: tial of the form V(φ)=Λ4+ Λ4+n [7] with n the Ratra-
of the universe. This original work has been confirmed φn
i Peebles index and Λ=2.4 meV based on the acceleration
X by more sensitive observations [3–6]. It is known that
rate of the universe expansion. Additionally, it has an
ar anecgoanttivriebpurteiosnsutroe itnheEvinasctueuinm’sefineelrdgyeqdueantsioitnys,aacntsdlsikinecae extra term that couples to matter field A(φ) = MβPLφ,
whereβisadimensionlesscouplingtomatterandM is
pressuregravitatesitcancausethe expansionoftheuni- PL
thereducedPlanckmass. Sotheeffectivepotentialtakes
verse to accelerate. There is no consensus on the na-
theformV (φ)=V(φ)+A(φ)ρ. Theappearanceofthe
ture and physical origin of dark energy, and most of the eff
local matter density ρ in the effective potential makes
proposed research in this area consists of astronomical
observations to more precisely characterize its effects on the effective mass m2eff ∼ n(n + 1)Λ−14++nn(nMβρPL)nn++12
theuniverse’sexpansionrateandothercosmologicaland of the chameleon field density-dependent, allowing the
astronomical observables. However there is an interest- chameleon to evade many of the existing experimental
ing subset of ideas for the origin of dark energy which tests of gravity. This also causes the chameleon field to
can be addressed in laboratory experiments. One set of be highly suppressed in the presence of even the mod-
such ideas postulate that dark energy is due to a scalar estly low matter density environment present in most
field φ which adopts a nonzero value in the vacuum of terrestrial lab experiments where the effective mass is
outer space. For this scalar field to evolve into the dark extremely heavy, thus further escaping detection. As a
energy seen today, one must postulate a self-interaction result, the chameleon field, along with a range of simi-
2
lar theories, has yet to be ruled out by experiment. A tron tests involving an apparatus used to test the weak
very extensive review of the chameleon field within the equivalence principle for free neutrons [27] and a Lloyd’s
broadercontextofmodifiedgravitytheorieshasappeared mirrortypeofneutroninterferometer[28]havealsobeen
recently [13]. proposed. A recent review of calculations performed to
This paper presents the result of a recent experiment search for dark energy of various types using neutrons,
using a perfect crystal neutron interferometer to place laboratoryexperiments to searchfor Casimir forces,and
a limit on the chameleon field’s coupling to matter β gravitational inverse square law violations has recently
anddoessoinaparticularlydirect,transparentmanner. appeared [29].
We actively modulate the amplitude of the chameleon Calculations which showed that atoms can feel an un-
scalar field in a gas cell in one arm of our interferome- screened chameleon field [20] have encouraged experi-
ter and exploit the unique ability of neutrons to coher- ments to search for chameleons using atom interferom-
ently penetrate the cell wall and access the phase shift etry. The first result of an atom interferometry experi-
from the neutron-chameleon coupling. This research ac- ment conducted to search for chameleons has appeared
tivity brings together several physics subfields (gravita- very recently [21]. This experiment looked for a phase
tionalphysics,atomicphysics,condensedmatterphysics) shift in a cesium atom interferometer operated in ultra-
andintheneutroncaseemployscentralizeduserfacilities highvacuumnearasphericalmasswhichcanbeasource
constructedmainlyformaterialssciencestudies,thereby ofachameleonfield. Alreadytheconstraintsonthe cou-
involving an uncommon diversity of scientific research pling β from this atom interferometry experiment are
techniques and environments in the quest to experimen- quite strong. The prospects for further improvement in
tally address what is perhaps the most exciting issue in the atom interferometry experiments are very encourag-
cosmology. ing [22, 23].
In the regime of small matter coupling β the best lab- As the chameleon fields must couple directly to mass
oratory constraints on chameleons over a wide range of to be relevant for the observed universe expansion ac-
n come from laboratory tests of the inverse square law celeration,other possible chameleonprobes such as pho-
ofgravitywith sensitivity atthe 100microndark energy tons possess a more model-dependent coupling to these
scale[14]. Experimentaltestsofthegravitationalinverse dark energy scalar fields. Strong experimental con-
squarelawoperatingoverotherdistanceregimes,suchas straints on chameleon-photon couplings already exist.
those designed originally to measure the Casimir inter- Examples of photon-based searches for chameleons in-
action, can also constrain β. The sensitivity of a force clude the CHASE (the GammeV CHameleon Afterglow
sensor specially designed to search for chameleons has SEarch)experiment[16],theAxionDarkMattereXperi-
recently been analyzed [15]. ment(ADMX) [17], asearchfor chameleonparticlescre-
In the regime of large β techniques which employ two atedviaphoton-chameleonoscillationswithinamagnetic
largemassesstarttobecomeinsensitivebecauseboththe field[18], and the CAST CernAxionSolarTelescope ex-
sourceandsinkofthechameleonfieldemanateonlyfrom periment [19].
athinregionofthesurfaceoftheobjectsduetothenon- Mostof the chameleonexperiments performed to date
linearchameleonself-shielding,anditisthereforeprefer- using neutrons and atoms have sought the chameleon
able in this regime to employ test particles whose pres- field by passing the probe close to a dense mass in-
encedoesnotsuppressthe chameleonfield. Neutronand side a high vacuum environment and searching for the
atom interferometry can be used to searchfor the possi- phase shift from the chameleon-matter coupling β in a
bleexistenceofchameleonandrelatedscalarfieldsinthis chameleon field gradient. The chameleon field profile
strongly-coupledchameleonregimesincetheseprobesdo is obtained by solving the appropriate nonlinear Klein-
not locally suppress the chameleon field. Gordonequationforφusingtheboundaryconditionsset
Slow neutrons can be used to perform sensitive by the experimental apparatus. A special feature of the
searches for dark energy scalar fields [24]. The fact that experiment reported in this work is that we periodically
neutron experiments are a sensitive method to probe introduce a nonzero matter density into the experimen-
chameleon fields was pointed out by [25] in the context tal chamber to actively suppresses the chameleon field
of an analysis of measurements on the quantum states in the measurement. To perform this type of search we
of bouncing ultra-cold neutrons. The disturbance of the exploit the fact that, unlike atoms, neutrons are able to
chameleon field near the surface of the flat neutron mir- pass through matter at densities (gas pressures of a few
ror employed in these measurements modifies the neu- mbarsuffice)forwhichthechameleonfieldisgreatlysup-
tron bound state energies and wave functions as well pressed. We were able to realize an experiment in which
as the relative phase of coherent superpositions of the thechameleonfieldintheapparatusseenbytheneutrons
neutron gravitational bound states. Recent experiments is repeatedly “turned off” by the addition of a small gas
conductedinthissystem[26]havebeenusedtoconstrain pressureintheapparatusand“turnedon”byevacuation
chameleon fields, and elaborations of this method are in of the chamber. Moreover, using a neutron interferome-
principle capable ofmuchgreatersensitivity. Other neu- ter provides another advantage in that we can keep the
3
pressure difference between the two arms of interferome- in the quantum amplitude of a spin-1/2 particle rotated
ter constant. This provides a direct measurementof any by 2π, (3) the most precise determinations of neutron-
background that is independent of gas pressure, such as nucleusscatteringamplitudes,(4)sensitivetestsofquan-
neutron scattering off the cell walls etc. tum entanglement predictions such as the Bell inequal-
ities and the Greenberger-Horne-Zeilinger inequalities,
and (5) subtle effects in neutron optics, most recently
NEUTRON INTERFEROMETRY the successful manipulation of the orbital angular mo-
mentum quantum number of a neutron beam [30].
The experiment was performed at the National Insti- Our experiment searches for the neutron phase shift
tute of Standards and Technology’s (NIST) Center for between the two coherent paths of the interferome-
NeutronResearch(NCNR)locatedinGaithersburg,MD. ter arising from the coupling of the neutron to the
AttheNCNRfreeneutronsaregeneratedusinga20MW chameleon field. The neutron phase shift Φ due to
cham
researchreactorwhichfeedsovertwodozenindividualin- the chameleon scalar field is
strumentsthatareprimarilytailoredformaterialscience
β m2φ(x)
applications. Here we used monochromatic 11.1 meV Φ =− dx (3)
neutrons and interferometric techniques similar to that cham Z MPL ¯h2k
of a Mach-Zehnder interferometer for light optics [36].
wheremisthemassoftheneutron,φ(x)isthechameleon
Theperfectcrystalneutroninterferometerusedinthis
field, k is the neutron wave vector, and the integration
experiment consistsof three crystalblades on a common
is performed over the neutron’s trajectory. We use the
crystalbase;aschematicofwhichisshowninFig.1. The
signconventionthat defines the neutronphase suchthat
monolithic silicon base below the blades ensures proper
positive potentials give negative phases. By measuring
arcsecondalignmentbetweenthelatticeplanesofeachof
Φ , we can then limit β for a given Ratra-Peebles
cham
the threeblades. Thefirstbladeservesto spatiallysepa-
index n.
ratetheneutron’swavefunctionψe−iΦ intotwocoherent
The calculation of the chameleon phase shift requires
paths (A and B). In order for the two paths interfere, a
integratingover the neutron trajectoryof the chameleon
centralcrystalbladeactsasalossymirroranddirectsthe
field profile φ(x) inside the cell as a function of the gas
paths back together onto the third blade. Neutrons exit
density. Previous calculations have shown [32] that φ(x)
the interferometer along either one of two paths labeled
is a rapidly varying function of the density of the gas.
traditionallyas‘O’and‘H’andaredetectedusinghighly
When the gas density is low the chameleon field φ(x)
efficient 3He-filled proportional counters. It should be
develops a nonzero amplitude for distances sufficiently
noted that there is only one neutronat a time inside the
far from the walls of the vacuum chamber. As the gas
interferometerandthusitisaelegantexampleofmacro-
density is raised into a critical regime, which depends
scopicselfinterference. Differencesinphase∆Φbetween
on β and the geometry of cell, the chameleon field φ(x)
the paths A andB modulates the intensities recordedby
is suppressed and tends toward zero. We can therefore
the detectors as
activelymodulatethe chameleonfieldintheexperiment.
I =A +Bcos[ξ(δ)+∆Φ] (1) The first experiment to use perfect crystalneutronin-
O O
I =A −Bcos[ξ(δ)+∆Φ] (2) terferometrytosearchforchameleonscalarfieldswasre-
H H
cently performed at Institut Laue-Langevin (ILL) [35].
In order to determine ∆Φ and the other fit parameters In this experiment neutrons also passed through a vac-
(AO,H and B) one could vary the cosine term in a con- uum chambermounted in the perfect crystalinterferom-
trollable way (ξ(δ)). This is done by the adding what is eter. The effect of the chameleon field was sought both
calleda ‘phaseflag’inside the interferometer. The phase by varying the relative separation of the neutron paths
flag used here is a 1.5 mm thick × 50 mm wide piece fromthevacuumcellwallsbytranslatingthecellrelative
of optically flat quartz and is illustrated in Fig. 1b. By totheincidentbeamandalsobyvaryingthepressurein-
rotating the phase flag an angle δ a phase shift of ξ(δ) sidethecell. Sincetheformerislimitedbytheuniformity
is caused due to the effective path length difference be- of a machined cell, in our experiment we employed the
tweenpathsAandB.Rotatingδ by±2.5degreescreates pressure variation method.
an interferogram like the one shown in Fig 2.
The perfect crystal neutron interferometery technique
employed in this work has been used to conduct a num- EXPERIMENTAL DESIGN AND SETUP
ber of textbook experiments in gravitation, neutron op-
tics,andquantumentanglement[31]. Theseexperiments A two-chamber vacuum cell is placed in the perfect
include, but are not limited to, (1) the first demonstra- crystal neutron interferometer (see Fig. 1b) with inter-
tionthatthegravitationalfieldaffectsneutronwavefunc- nal pressures controlled at different values. The vacuum
tions asexpectedinnon-relativisticquantummechanics, cell as well as the two neutron beams are both symmet-
(2) clear demonstrations of the fascinating minus sign ric about the center walls of vacuum cell, so those two
4
The helium gas pressures in the cell in either configu-
ration are low enough that the equation of state of the
helium gas is well-described by the ideal gas law. The
gas density and resulting neutron phase shift from the
neutron optical potential of the helium gas is then pro-
portional to the gas pressure. At these low gas densities
the neutron phase shift from the helium gas is a few or-
dersofmagnitudesmallerthantheultimatesensitivityof
FIG. 1: Left is a 3D schematic of the neutron our experiment to phase shifts from the chameleon field,
interferometer seen in profile with the two coherent so in practice Φgas can be neglected. Furthermore, even
beam paths; right shows the top view of the ifthephaseshiftfromthegaswaslarger,ouractivecon-
two-chamber gas cell for the experiment, which fits trol of the pressure difference between the two sides of
around the central blade of the interferometer crystal. the cell in the two configurationswould cause this phase
shift difference (Φ −Φ )−(Φ −Φ )
gas,1A gas,1B gas,2A gas,2B
0.45 to cancel to high accuracy. The phase shift difference
from the two cell walls in the two different configura-
0.40
tions, (Φ − Φ ) − (Φ − Φ ) is also
cell,1A cell,1B cell,2A cell,2B
0.35
) negligible: the only physically plausible mechanism that
H
+I 0.30 mightcauseadifference,namelysomeabsolute-pressure-
O
(I 0.25 dependent change of the phase shift from the neutron
/O optical potential of the aluminum cell walls, is known to
I 0.20
benegligiblefrompreviousmeasurementsatmuchhigher
0.15 Conf. 1 gaspressuresattheNCNRoftheneutron-heliumoptical
Conf. 2
0.10 potential. Undertheseconditions,∆Φ isacleanand
cham
-2 -1 0 1 2 direct measurement of the chameleon phase shift. Ta-
d [deg] ble I shows the two sets of pressure configurations that
are used in the experiment. The “setpoint” in the table
FIG. 2: A typical pair of O-beam interferograms, refers to a low enough pressure at which the chameleon
normalized to the total sum of counts, corresponding to fieldmayproduceanextraphaseshift. Thechoiceofgas
the two run configurations in the experiment. pressures used in this experiment are also low enough
Uncertainties are purely statistical. that, for the chameleon coupling strengths to which we
aresensitive,thechameleonfieldseestheheliumgasused
in the cell as a homogeneous medium based on previous
neutron beams should feel exactly the same chameleon analysis of this issue [32].
field if the pressure inside is same. However the neutron In the experiment we choose to control the absolute
interferometer works in such way that it detects the dif- pressure (the low pressure side in Fig. 1) and the differ-
ference between the two neutron beams so we need to entialpressureacrosstwochamberstoeliminate possible
controlthe pressureofthe twochambersatdifferentlev- systematiceffectsasdiscussedabove. Theabsolutepres-
els. First (Conf. 1), the pressure in the right chamber sures in Conf. 2 will deviate slightly from the expected
is kept low so that the chameleon field can develop a 1.33 mbar and 2.67 mbar shown in Table I because the
nonzerovalue. Meanwhile,the leftchamberisfilledwith absolute pressure gauge used in this experiment has a
gas at a higher pressure so that the chameleon field is maximummeasuringrangeof0-0.133mbar. Soinstead
highly suppressed. Then (Conf. 2), the gas pressure in of measuring the pressures in Conf. 2 directly, the dif-
each chamber is raised by the same amount so that the ferential pressure gauge, which can measure up to 1.33
chameleonfieldgetssuppressedinbothchambers. Sothe mbar,is usedto achievethe desiredpressuresinConf. 2.
four phase shift neutron picks up during the experiment We measured the associated pressure uncertainty to be
are, smaller than 0.01 mbar. This gives a negligible system-
atic uncertainty since the chameleon field amplitude is
Φ =Φ +Φ +Φ (4)
1,A cham,1A cell,1A gas,1A closetozeroatsuchhighpressureanditisonlylogarith-
Φ1,B =Φcham,1B+Φcell,1B+Φgas,1B (5) mically sensitive to the pressure in this regime.
Φ =Φ +Φ +Φ (6) Fig. 3 shows the schematic of the gas handling system
2,A cham,2A cell,2A gas,2A
Φ =Φ +Φ +Φ (7) (GHS)usedinthisexperiment. Thevacuumcellismade
2,B cham,2B cell,2B gas,2B
of aluminum alloy 7075. Two CF flanges are machined
Ideally, the phase shift due to chameleon in Φ , Φ , on the two ends of the cell to accommodate aluminum
1,B 2,A
Φ shouldbeclosetozerosowedefinethephaseshiftof gasket seals with negligible outgassing. A wall of thick-
2,B
chameleontobe∆Φ =(Φ −Φ )−(Φ −Φ ). ness3mmseparatesthecellintotwochamberswhichcan
cham 1,A 1,B 2,A 2,B
5
be filled by gas at different pressures. The GHS employs
metal seals (CF 1-1/3”) and ultra-high vacuum (UHV) RunCycle Configurations Pressure(High) Pressure(Low)
compatible components which are helium leak tight as Conf. 1 0.67 mbar Setpoint
Run1
verified by measurements using a helium leak detector.
Conf. 2 0.79 mbar 0.133 mbar
All the vacuumtubes andthe mechanicalbellows havea
Conf. 1 1.33 mbar Setpoint
high-conductance path to the cell (tubing diameters are Run2
Conf. 2 2.67 mbar 1.33 mbar
greater than 2 cm) so the pressure gradient inside the
gas handling system is minimized. Two absolute pres-
TABLE I: Pressure configurations used for two different
sure capacitance transducers and one differential capaci-
measurement runs.
tance transducer areplacedclose to the vacuum cell and
used to monitor the absolute pressure and differential
pressure across the vacuum cell. Two types of feedback
loops controlthe pressures inside the cells. One controls CALCULATION OF THE CHAMELEON FIELD
PHASE SHIFT IN THE CELL
the differential pressure across the chamber using a mo-
torized edge-welded stainless UHV bellows according to
readings from the differential pressure gauge. The other Tocalculatethechameleonscalarfieldinsideavacuum
feedbackcontrolemploysanabsolutepressuregaugeand cell, one could solve the Klein-Gorden equation,
a vacuum compatible sensitive mass flow controller to
control the pressure in the low pressure chamber. All
∂V −nΛ4 βρ
valves are controlled by air-actuated switches to reduce ∆φ= eff = + (8)
∂φ φn+1 M
possible electromagnetic or vibrational noise in the in- PI
terferometer environment. Both absolute pressure and
Unfortunately there is no known analytical solution
differential pressure are controlled with fractional fluc-
to this non-linear second order partial differential equa-
tuations below 1%. Fig. 5 shows the pressure stability
tion so we had to use a finite difference method to ob-
data.
tain a numerical solution of the 3D field profile inside
HP LP HP: High Pressure chamber the experiential cell chamber. We built a 3D mesh with
PG
LP: Low Pressure chamber roughly 500×500×500 nodes in total and solved the
PG: Pressure Gauge non-linearPoissonequationiterativelyusingtheformula
DFPG DFPG: Differential Pressure Gauge below (Eqn. 9, where i,j,k denotes the node index in
PV PV PV: Pneumatic valve each dimension and l denotes the iteration number). To
PG TP: TurbomolecularPump accelerate the convergence of the calculation we further
MB PV MB: Motor and Bellow exploitedtheGauss-SeidelmethodshowninEqn.10. To
(cid:0) get a precise solution, the number of grid points must
PV MFC MFC: Mass Flow Controller
be fairly large because the chameleonfield growsrapidly
He
TP close to the walls. The gradientof the chameleon field is
close to infinity at such places, which inevitably causes
FIG. 3: Schematic diagram of the gas handling system
instabilityintheiterationmethod. Toaddressthisprob-
which maintains a constant pressure in one chamber
lem, we implemented an uneven grid with more grid
and a controlled pressure difference between the two
points where the field changes dramatically and fewer
chambers.
grid points where the field does not change much. The
explicit formula is lengthy but is similar to Eqn. 10.
φ(l) +φ(l) −2φ(l+1) φ(l) +φ(l) −2φ(l+1) φ(l) +φ(l) −2φ(l+1) −nΛ4 βρ
i+1,j,k i−1,j,k i,j,k i,j+1,k i,j−1,k i,j,k i,j,k+1 i,j,k−1 i,j,k
+ + = + (9)
∆h2x ∆h2y ∆h2z φi(,lj),nk+1 MPI
( nΛ4 − βρ )+ φi(+l)1,j,k+φi(−l+11,)j,k + φi(l,j)+1,k+φi(l,j+−11),k + φi(l,j),k+1+φi(l,j+,k1)−1
φ(l+1) = φ(il,j)n,k+1 MPI ∆h2x ∆h2y ∆h2z (10)
i,j,k 2 + 2 + 2
∆h2 ∆h2 ∆h2
x y z
Having solved for the chameleon field inside vacuum cell, the extra phase shift picked up by neutrons due to
6
chameleonfieldis computedaccordingtoEqn.3by inte- torsreduceslightlytheamplitudeofthelineintegralused
gratingthechameleonfieldovertheneutronpathlength. tocomputethechameleonphaseΦ (showninFig.4)
cham
Fig. 4 shows the calculated phase shift caused by the andthereforereducesthecalculatedupperlimitofβ pro-
chameleon field with different Ratra-Peebles model pa- portionally. These scaling corrections are also negligible
rameters n and β. This result agrees well a previous compared to the uncertainty in our upper limits for β.
calculation in the literature [32].
systematic correction uncertainty
-2
10 Helium nuclear scattering 0.002 rad/bar 2.0E-6 rad
4 Pressure gauge accuracy 0.3% FS 1.2E-4 rad
d] -32 Vacuum cell misalignment 1◦ rotation 0.0005Φcham
ra 10 1 mm translation 0.02Φcham
| [m 4 Neutron beam divergence 1.5◦ 0.006Φcham
a 2
ch 10-4
TABLE II: Estimates of systematic uncertainties.
|F 4 n=1, b =10✆ n=1, b =10☎
2 n=6, b =10✆ n=6, b =10☎
10-5 Before fitting the measured phase shift to the calcu-
-6 -5 -4 -3 -2 -1 lated phase shift due to chameleon field, the sequence of
10 10 10 10 10 10
Pressure [ ] phaseshiftdifferences isfilteredusinga time-seriesanal-
(cid:0)✁✂✄ ysis algorithm designed to remove slow zero-point drifts
FIG. 4: The calculated phase shift caused by chameleon in data sequences like ours which oscillate between two
field versus gas pressure for various values of the states with equal measurement times [37]. To do this
Ratra-Peebles index n and β. one takes a weighted average from neighboring points
from the original sequence, y = p c u , where u
i k=0 k i+k i
is the original data sequence andPy is the combined se-
i
quence. A covariance matrix must be used to properly
estimate the uncertainties after the correlations induced
DATA ACQUISITION AND ANALYSIS
by the weighting algorithm. The weights c satisfy the
k
equation,
Environmental factors are known to cause the phase
of a neutron interferometer to drift. To isolate the
chameleon phase from environmental phase drifts, we 1 1 1 1 ··· 1 c 0
0
switched between Conf. 1 and Conf. 2 after taking each 0 1 2 3 ··· p  c  0
1
interferogram. The contrast (or B/A from Eqn. 1) of
tfehreonmeeutterronwainstearrfoeurenndce37p%attaenrndwisitchotnOhseiscteenlltinwtihtheionttheerr- 0... 1... 4... 9... ·.·.·. p...2  c...2 =0... (11)
    
Al cells used inside the interferometer (the empty inter- 0 1 2p−1 3p−1 ··· pp−1c  0
  p−1  
ferometer contrast is 85%). Since the chameleon field 1 0 1 0 ··· 0  c  1
  p   
is a function of pressure as shown in Fig. 4, we also
varied the pressure setpoint in Table I to look for any which is designed so that a zero point drift in the signal
pressure dependence of the phase shift. Two sequences withapolynomialtime dependence uptoorderp willbe
ofmeasurements weretakenduring two adjacentreactor cancelled by combining each p+1 items in the sequence
cycles. Theonlydifferencebetweenthetworunsisinthe while a true signal correlated with the difference in the
pressure configurations used. In the first run, nine pres- two configurations is kept unchanged. For comparison
suresetpointswerescannedintherangefrom3.33×10−4 we present both the filtered mean and unfiltered mean
mbar to 2.67×10−3 mbar. In the second run, we chose in Fig. 6. The good agreementbetween the mean values
three pressure setpoints that span over a wider range, andstatisticaluncertaintiesofthe filteredandunfiltered
3.33 × 10−4 mbar, 3.33 × 10−3 mbar and 2.00 × 10−2 phase shift data shows that any possible effects of inter-
mbar. The measuredphase shift alongwith the pressure ferometer phase drift are negligible in our measurement.
and differential pressure stability is shown in Fig.5. We use the filtered data to extract our limit.
Systematic uncertainties in our measurement are neg- Toestablishanupperlimitofβ atthe95%confidence
ligible compared to the statistical uncertainty. Table II level,thesquareoftheweightedresidualsissummedover
lists the major systematic uncertainties and corrections. all measurements (χ2(β))
The first two items could lead to a nonzero phase shift
even in the absence of a chameleon field but both are
[ζ(β) −∆Φ ]2
much smaller than the statistical uncertainty (typically χ2(β)= i i (12)
0.0025 rad) in the chameleon phase. The last three fac- Xi σi2
7
a)
mbar] 00..666666661150 Z0χ2(βlim)p12(χ2′)dχ2′ =0.95 (13)
[
P 0.666605 where p (χ2) is the χ2 distribution with 12 degrees of
12
-2 freedom,correspondingtothe12pressuresetpoints. The
10
] calculated limit is shown in Table III and the excluded
r
ba 10-3 areaisshowninFig. 7asafunctionoftheRatra-Peebles
m
[ index n.
P 0.2
a)
] 0.1 0.03
d
a
r 0.02
[
m 0.0 ]
ha ad 0.01
c r
[
-0.1 m 0.00
a
h
c
-0.01
-0.2 =1.0X107 Unfiltered mean
0 2 4 6 8 10 -0.02 ✢ 6 Filtered mean
=5.0X10
Days ✢ 6
b) -0.03 =2.0X10
✢
2 3 4 5 6 7 8 9 2 3 4 5
] 1.3333 0.001
ar Setpoint Pressure [mbar]
b
m 1.3332
[ b)
P 0.008 Unfiltered mean
1.3331 Filtered mean
-2 7
10 =1.0X10
P [mbar] 100-.32 [rad]ham 00..000004 ✢✢✢==52..00XX110066
c
d] 0.1 -0.004
a
r
[
m 0.0
a -0.008
h
c 2 4 6 8 2 4 6 8 2
-0.1 0.0001 0.001 0.01
Setpoint Pressure [mbar]
-0.2
FIG. 6: a) and b) shows measured phase in two runs
0 10 20 30
compared to calculated phase with different values of β
Days
(n=1).
FIG. 5: a) and b) shows the pressure difference between
the two cells (∆P), the Conf. 1 pressure in the low
pressure chamber (P), and the measured chameleon n 1 2 3 4 5 6
phase shift in the first and second runs (∆Φcham). βlimit×106 4.7 8.2 12.7 17.9 20.4 23.8
TABLE III: The calculated upper limit on β with 95%
confidence level.
where ζ(β) is the expected chameleon phase shift and
i
∆Φ is the measured phase shift with uncertainty σ for
i i
the ith pressure setpoint. To estimate β, χ2(β) is then
minimizedwithrespecttoβ foragivenRatra-Peeblesin- CONCLUSION
dex n. However,the typical computation of a fit param-
eter confidence interval is not valid in this case, because We have conducted a search for chameleon dark en-
for our measurements this function reaches its minimum ergy fields using neutron interferometry. We realized an
at β = 0 for all values of n due to the constraint β > 0. experiment in which the chameleon field is periodically
Tofindthe95%confidenceinterval,χ2(β)wassolvedfor varied in magnitude with no change in the experimen-
the value of χ2 that gives: tal geometry. Our upper bound of β < 4.7×106 for a
8
tal refutation of a plausible dark energy theory. Scalar
field candidates for dark energy which employ other
screening methods, such as symmetrons, might also be
constrained by this and other experiments with further
analysis.
ACKNOWLEDGEMENTS
We acknowledge the support of the National Institute
of Standards and Technology, US Department of Com-
FIG. 7: The excluded regions in (β, n) parameter space merce, in providing the neutron facilities used in this
work. This work was supported by NSF grants NSF
compared to other experiments. From bottom to top:
torsion pendulum experiment [12]; atom interferometer PHY-1205342, PHY-1068712, PHY-1307426 and DOE
experiment [21]; this work; ILL neutron interferometer awardDE-FG02-97ER41042. Financialsupportprovided
by the NSERC CREATE and DISCOVERY programs,
experiment [35]. Other experimental constraints in the
regime of large β are less stringent. CERC, and the NIST Quantum Information Program
are gratefully acknowledged. K. Li and W. M. Snow
acknowledge the support of the Indiana University Cen-
Ratra-Peeblesindexofn=1toβ <2.4×107forn=6is terforSpacetimeSymmetriesandtheIndianaUniversity
Faculty Research Support Program.
the most sensitive direct constraint on the free neutron-
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5forn=1toafactorof30atn=4,anditcutsintopart ∗ Corresponding author: [email protected]
of the projected sensitivity of a proposedexperiment us- † Corresponding author: [email protected]
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