Connection between a Possible Fifth Force and the Direct Detection of Dark Matter

Jo Bovy and Glennys R. Farrar

Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, New York 10003, USA(Received 19 July 2008; revised manuscript received 12 August 2008; published 11 March 2009)

If there were a fifth force in the dark sector and dark matter (DM) particles interacted nongravitationally

with ordinary matter, quantum corrections generically would lead to a fifth force in the visible sector. We

show how the strong experimental limits on fifth forces in the visible sector produce bounds on the cross

section for DM detection and the strength of the fifth force in the dark sector. For a fifth force comparable

in strength to gravity, the spin-independent direct detection cross section must typically be & 10�55 cm2.

The anomalous acceleration of ordinary matter falling towards dark matter would also be constrained:

�OM�DM & 10�8.

DOI: 10.1103/PhysRevLett.102.101301 PACS numbers: 95.35.+d, 04.80.Cc, 14.80.�j

In recent years, the possibility has arisen that there mightbe a long-range ‘‘fifth’’ force acting in the dark sector. (Fora theoretical perspective see, e.g., [1,2] and referencestherein.) Such a force arises in many extensions of thestandard model. Most dark matter models predict that theforce between DM particles is short range, but the exis-tence of a very light boson, such as the dilaton that natu-rally occurs in supersymmetry and string theory, suggeststhat a long-range force could also be present. If such lightbosons exist, they are constrained to couple only veryweakly to ordinary matter by the Eot-Wash experimentsdiscussed below. In the following we show that a surprisingrelation exists between the bounds on the direct detectioncross section of DM and the strength of a possible fifthforce in the dark sector.

There are observational motivations to consider a long-range attractive nongravitational force between dark mat-ter (DM) concentrations. Such a force may resolve somediscomforts with conventional �CDM. If the range of thefifth force were large compared to the scale of structureformation, it would effectively increase the strength of thegravitational interaction for the dark matter by a factor �ð1þ �Þ, and thus would accelerate structure formation.This would be helpful because: (i) Recent cosmologicalsimulations using a reduced �8 value compatible with theWMAP year 5 cosmology [3], predict z ¼ 0 halo concen-trations distinctly lower than the Millennium Run using thelarger WMAP1 value of �8, and find ‘‘very significantdiscrepancies with x-ray observations of groups and clus-ters of galaxies.’’ Enhancing the effective gravitationalattraction with a fifth force would mimic the effect of alarger �8 at recombination and would alleviate this prob-lem. (ii) The number of superclusters observed appears tobe an order of magnitude larger than predicted by �CDMsimulations [4]; the accelerated structure formation due toa fifth force would reduce this discrepancy. (iii) As noted inRef. [2], a fifth force would tend to clear out the voids;Ref. [5] confirms this in a simulation. This may improveagreement with �CDM [6], although the existence of adiscrepancy has been challenged [7]. (iv) A variety of

observations, for instance the lack of evidence in theMilky Way for a major merger, is hard to reconcile withthe present day accretion predicted in�CDM. Acceleratedstructure formation reduces late-time accretion, simplybecause it leaves less to be accreted later [5]. (v) Thenumber of satellites in a galaxy such as the Milky Way ispredicted to be an order of magnitude larger than is ob-served. A fifth force would help by reducing the stellarcontent of dwarf galaxies and making them harder to find,because baryons—not feeling the fifth force—are rela-tively less bound to dark matter concentrations than inconventional theory. This would reduce the amount ofbound gas and lower the star formation rate in dwarfgalaxies, and increase the tidal loss of the stars that doform. The evidence above does not demand a fifth force,but certainly motivates exploring the implications of such aforce.Experimental constraints on the existence of

equivalence-principle-violating forces for dark matter arerelatively weak [8]. On a subgalactic scale, Kesden andKamionkowski pointed out that a difference in accelerationbetween DM and stars would change the distribution ofstars in the tidal tails of the Sagittarius dwarf galaxy andclaimed a limit about 10% the strength of gravity [9].However this claim is in limbo for the time being, becausethe distances of the stars in the Stream have been found tohave significant systematic error [10] and until those dis-tances are revised, it is difficult to draw any conclusionsusing the Sagittarius Stream. With the original distancesthere was a difficulty reconciling the observed line-of-sightvelocities of the stars in the leading stream, which implieda prolate Dark Matter halo, with the precession of thedebris orbital plane, which implied an oblate halo [11].This discrepancy might be resolved by an additional at-tractive force, or may disappear when the stellar distancesare corrected. On a galaxy-cluster scale, the apparent needfor a fifth force to account for the reported velocity [12] ofthe components of the merging ‘‘bullet’’ cluster, IE0657-56, [13] has been removed by a better determination of thevelocity [14].

PRL 102, 101301 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending

13 MARCH 2009

0031-9007=09=102(10)=101301(4) 101301-1 � 2009 The American Physical Society

While there is room to consider a fifth force in the darksector, constraints on new forces in the visible sector arevery strict. Precise experiments performed by the Eot-Wash Group used a torsion balance to study the differentialacceleration of a Be-Ti test-body pair in the fields of theEarth, the Sun, and our Galaxy [15]. They tested theuniversality of free fall, leading to a limit on a newYukawa-type force between ordinary matter particles onastronomical length scales

�5 � g2eff4�u2G

� 10�10 � �5;ul; (1)

where u is the atomic mass unit and geff is the Yukawacoupling of ordinary matter to a new light boson. Theupper limit �5;ul depends on the specific coupling of the

new force, and the quoted value corresponds to the leaststringent limit; more typical values are 2 orders of magni-tude smaller, from updating [16] with [15]. Such a stronglimit arises naturally if some symmetry forbids a tree-levelcoupling between ordinary matter and the light bosonwhich mediates the fifth force, and we assume here thatthis is the case. If the tree-level coupling does not vanish,then our arguments below become even stronger unlessthere is extreme fine tuning.

In this Letter we show that due to the stringency of thebound (1), the existence of a fifth force between darkmatter particles implies a strong constraints on the possibleDM direct detection cross section and vice versa. FromFigs. 1(a) and 2 it is clear how a fifth force can arise forordinary matter as a quantum effect involving a fifth forcebetween dark matter particles and the direct detectionvertex. The upper bound (1) on the effective Yukawainteraction between ordinary matter particles can thereforeimpose a limit on the product of the DM direct detectioncross section and the DM fifth force, which we determinebelow.

We assume the fifth force boson to be a light scalarparticle, because a pseudoscalar particle would give aspin-dependent, and thus nonmacroscopic, force, and avector or axial-vector carrier would give rise to a repulsiveforce. If dark matter is fermionic and there is a nongravita-

tional attractive fifth force between dark matter particles, itis most simply modeled as a long-range Yukawa force andwe adopt that description here. Reality may be more com-plex. For instance in chameleon models [17] the newinteraction is damped by sufficiently strong concentrationsof matter, so in principle it could have an impact onstructure formation but not be evident on subgalacticscales. The bounds derived here would need to be revisitedin such models.Consider a Yukawa interaction between a spin-1=2 dark

matter field � and a light scalar � (spin-0 dark matter isdiscussed below)

L int ¼ �gð����Þ ���: (2)

Here, g is a dimensionless constant and g�� is the mass ofa dark matter particle in the absence of a vacuum expec-tation value h�i of the scalar field, included for generality.The potential corresponding to this interaction is found

by comparing the amplitude of the process in Fig. 1(b) tothe Born approximation formula from nonrelativistic quan-tum mechanics: the scattering amplitude in the low-energyapproximation, normalized according to nonrelativisticconventions is given by

M ¼ g2

jqj2 þm2�

ðq ¼ p0 � pÞ; (3)

which is simply the Fourier transform of the potential,giving the familiar Yukawa potential (see, e.g., [18])

VðrÞ ¼ � g2

4�

1

re�m�r; (4)

in which m� is the mass of the scalar. At distances small

compared to the range of the force, m�1� in natural units,

the force is just proportional to the gravitational attraction,with proportionality constant �:

� � g2=4�

Gm2�

¼ 1

4�Gðh�i ���Þ2¼ m2

Pl

4�ðh�i ���Þ2; (5)

since the mass of the dark matter particle ism� ¼ gðh�i ���Þ. Notice how g drops out of Eq. (5), leaving the relativestrength of the fifth force compared to gravity and the massof the dark matter particles as two independent quantities.This relationship, derived here from the single particle

FIG. 1. (a) Direct detection four-fermion vertex couplingquarks and dark matter fermions �. (b) Fifth force Yukawainteraction for fermionic dark matter.

FIG. 2. An effective Yukawa interaction of ordinary matterparticles arises due to a quantum correction.

PRL 102, 101301 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending

13 MARCH 2009

101301-2

perspective, was found earlier from density perturbationconsiderations [2].

A few observations: (a) In order for the fifth force to beof the order of the gravitational force, i.e., to be astrophysi-cally relevant, h�i ��� should be of the order of thePlanck mass. This choice is quite natural, since there isno immediate relation between the scalar and any of thelower mass scales. Since h�i ��� appears in the denomi-nator, a smaller valuewould mean a much larger fifth force.(b) With h�i ��� of the order of the Planck scale, the darkmatter would be very heavy in most scenarios. To get amass in the electroweak region would require g�OðmW=mPlÞ, which presumably should be enforced bysome symmetry. For g closer to unity, we can infer thatdark matter must be a nonthermal relic. Nonthermal relicsare produced in sufficient abundance if their mass is oforder the inflaton mass (presumably around 1013 GeV)[19]. This would connect fifth force physics with inflationphysics, which would be interesting, but not within thescope of this work.

In direct detection experiments, evidence for dark matteris sought in the observation of ordinary matter particlesrecoiling due to their being scattered by dark matter parti-cles. The most general nonderivative four-fermion effec-tive operator is given by

M ¼ X

i

Aið �qOiqÞð ��Oi�Þ; (6)

where the sum is over scalar (S), pseudoscalar (P), vector(V), axial-vector (A), and tensor vertices (T), and the Ai areconstants with dimensions of energy�2. If the DM particleis Majorana, the vector and tensor terms are absent in (6);this is commonplace in supersymmetric theories. In thelimit of small momenta, the only contributing terms are thescalar, vector, axial-vector, and tensor interactions. Theselast two contribute to the spin-dependent cross section,which the reasoning below will fail to limit.

The spin-independent (SI) cross section for (fermionic)DM—quark scattering is

�SIq� ¼ ðAS þ AVÞ2

m2q

�; (7)

in the limit that the DM mass is much larger than mq (the

effective quark mass, ��QCD). The cross section for

nucleon-DM scattering is of the same order of magnitude.Figure 2 shows how an effective Yukawa-type fifth force

interaction of quarks arises as a consequence of a quantumcorrection. The correction is made up of two parts: first, thedirect detection vertex couples the two quarks to twovirtual dark matter particles; second, these dark matterparticles couple to the fifth force scalar. For scalar � thescalar and tensor contributions to the sum (6) give non-vanishing effective couplings

geff � AS;Tg�2; (8)

where� is the ultraviolet cut-off, since this diagram would

be quadratically divergent if the 4-fermion vertex werepointlike. In reality, we expect the short distance theoryto be renormalizable, giving the effective 4-fermi vertex aform-factor. This cuts off the divergence at some model-dependent scale leading to an expression like (8) with �the scale of new physics, possibly the TeV scale, the darkmatter scale, or the Planck scale.Naively, the vector part of the 4-fermi vertex, (6), would

not contribute to geff due to chirality. However, an insertionof the Higgs vacuum expectation value into the loop dia-gram breaks SUð2Þ �Uð1Þ, and gives rise to a nonvanish-ing effective vertex

geff � AVgmqm� ln�; (9)

since this diagram is now superficially logarithmicallydivergent. The quantum correction in Fig. 2 from theaxial-vector term in (6) vanishes. Therefore, AA is notconstrained and no relation between a fifth force for DMand a fifth force for ordinary matter ensues.Through (5) we reexpress the dark matter Yukawa cou-

pling as g2 ¼ 4��Gm2�, where �, the relative strength of

the fifth force with respect to gravity, is & Oð1Þ fromastrophysics [8]. Imposing the experimental limit (1) onthe effective vertex (8), leads to

A2S;T &

�5;ul

�

u2

m2��

4; (10)

or for the vector coupling

A2V &

�5;ul

�

u2

m4�m

2qln

2�: (11)

Since the bounds on Ai become ever more stringent forhigher DM mass scales, we use the smallest m�, i.e., weak

scale DM, to obtain the most conservative limits. If both AS

and AV are present in (7), the constraint (11) is dominantand from (7) and (11) we find an upper bound on theproduct of � and the SI cross section:

�

��SI

q�

10�43 cm2

�& 10�3

��5;ul

10�10

��100 GeV

m�

�4�

1

ln�

�2:

(12)

However, if only the scalar term in (6) is present in (7)—e.g., because DM is Majorana—then (10) leads to the morestringent bound:

�

��SI

q�

10�43 cm2

�& 10�12

��5;ul

10�10

��mq

�QCD

�2

��100 GeV

m�

�2�1 TeV

�

�4: (13)

The product of the spin-dependent (SD) cross sectionand � is similarly constrained in the case that only a tensorterm contributes to the matrix element (6). The SD crosssection is given by (7), but with ðAS þ AVÞ2 replaced by6A2

T . The bound on the product of the SD cross section and

PRL 102, 101301 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending

13 MARCH 2009

101301-3

� is then of the same form as Eq. (13), but with thecoefficient 10�11.

In the case of spin-0 dark matter, the effective inter-action Lagrangian corresponding to (2) is Lint ¼�~gm�ð����Þ�2

�, where �� is the spin-0 dark matter

field and the coupling ~g is chosen so that the amplitude ofthe process in Fig. 1(b) is again given by (3), and con-sequently the potential is the same as for fermionic DM,with g ! ~g. The spin-0 DM mass is m� ¼ 2~gðh�i ���Þ,so the relative strength of the fifth force with respect to

gravity at distances � m�1� is ~� ¼ m2

Pl

16�ðh�i���Þ2 , much like

for fermionic DM (5). The analog of (6) is M ¼~ASm� �qq�

2�, leading to the DM-quark cross section �SI

q� ¼~A2Sm

2q=4�. The diagram in Fig. 2 now is superficially

logarithmically divergent, which leads us to write geff �~AS~gm

2� ln�. As before, we get an upper bound on the four-

fermion vertex: ~A2S &

�5;ul

~�u2

m6�ln

2�, which for scalar DM

leads to

�

��SI

q�

10�43 cm2

�& 10�9

��5;ul

10�10

��mq

�QCD

�2

��100 GeV

m�

�6�

1

ln�

�2: (14)

Thus, we see that for either fermionic or scalar darkmatter, if there is an astrophysically relevant fifth force sothat � is of order 0.1 or greater, the bounds we haveobtained on the spin-independent direct detection crosssection are significantly lower than current experimentallimits. These restrict the SI scattering cross section of DMon ordinary matter to be less than 10�43 cm2 for m� ¼100 GeV and less than 10�30 cm2 for superheavy DM,m� ¼ 1013 GeV [20].

Before closing, we note that the effective vertex in Fig. 2discussed above also leads to a fifth force between ordinarymatter (OM) and dark matter proportional to ggeff . Thisimplies that the fractional anomalous acceleration of twomaterials falling towards DM is given by

�OM�DM & 10�8

��OM�OM ��ðq=�Þ � �

10�13 � 10�3 � 1

�1=2

; (15)

where �OM�OM is the fractional anomalous accelerationfor the two materials falling towards OM and �ðq=�Þ isthe difference of the ratio of the fifth-force charge to themass of the different materials, with the average q=� of thesun equal to one. This limit is more stringent than currentexperimental bounds, �OM�DM & 10�5 [15].

To summarize, we have shown here that if there is acosmologically important dark matter fifth force, then theexperimental limit on a fifth force for ordinary matteressentially excludes the direct detection of dark matter

particles in the near future, except through the spin-dependent cross section. Generically, for weak scale darkmatter and a gravitational-strength fifth force, �SI

q� �10�55 cm2; in the least constrained case the cross sectionmust be �10�46 cm2. For heavier dark matter, as in non-thermal relics scenarios, the bound is even lower.Conversely, if the spin-independent interaction of darkmatter is observed in laboratory experiments, our argumentshows that any fifth force in the dark sector must be tooweak to be astrophysically relevant. The tensor part of thespin-dependent cross section is likewise constrained. Wehave also given a model independent upper bound on anongravitational interaction between ordinary matter anddark matter, which is significantly better than current directexperimental limits.The research of G. R. F. has been supported in part by

NSF grants PHY-0245068, PHY-0701451 and NASANNX08AG70G.

[1] L. Amendola, Phys. Rev. D 62, 043511 (2000).[2] G. Farrar and P. Peebles, Astrophys. J. 604, 1 (2004).[3] A. R. Duffy et al., Mon. Not. R. Astron. Soc. 390, L64

(2008).[4] J. Einasto et al., Astron. Astrophys. 462, 397 (2007).[5] A. Nusser et al., Phys. Rev. D 71, 083505 (2005).[6] P. J. E. Peebles, Astrophys. J. 557, 495 (2001).[7] H. Mathis and S. White, Mon. Not. R. Astron. Soc. 337,

1193 (2002); D. Croton et al., ibid. 356, 1155 (2005); D.Croton and G. R. Farrar, ibid. 386, 2285 (2008).

[8] B. Gradwohl and J. Frieman, Astrophys. J. 398, 407(1992).

[9] M. Kesden and M. Kamionkowski, Phys. Rev. Lett. 97,131303 (2006).

[10] M.-Y. Chou et al., Astrophys. J. 670, 346 (2007).[11] D. R. Law et al., Astrophys. J. 619, 807 (2005).[12] M. Markevitch, in The X-ray Universe 2005, ESA SP-604,

edited by A. Wilson, (Garching, ESA, 2006), p. 723.[13] G. Farrar and R. Rosen, Phys. Rev. Lett. 98, 171302

(2007).[14] V. Springel and G. Farrar, Mon. Not. R. Astron. Soc. 380,

911 (2007).[15] S. Schlamminger et al., Phys. Rev. Lett. 100, 041101

(2008).[16] Y. Su et al., Phys. Rev. D 50, 3614 (1994).[17] J. Khoury and A. Weltman, Phys. Rev. D 69, 044026

(2004); B. Feldman and A. E. Nelson, J. High EnergyPhys. 08 (2006) 002.

[18] M. E. Peskin and D.V. Schroeder, An Introduction toQuantum Field Theory (Addison-Wesley, Reading, MA,1995).

[19] D. J. H. Chung et al., Phys. Rev. D 59, 023501 (1998).[20] J. Angle et al., Phys. Rev. Lett. 100, 021303 (2008);

I. F.M. Albuquerque and L. Baudis, ibid. 90, 221301(2003).

PRL 102, 101301 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending

13 MARCH 2009

101301-4