Dilatonic Dark Matter –Dilaton As GIMP Versus Axion and Wimps–

Dilatonic Dark Matter –Dilaton as GIMP versus Axion and WIMPs–

Yongmin Cho

Administration Building 310-4, Konkuk University and School of Physics and Astronomy College of Natural Science, Seoul National University

June 22, 2016

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 1 / 30 Motivation

Why Dilaton?

1 Appears in all modern higher-dimensional uniﬁed theories to represent the extra space

2 Modiﬁes Einstein’s gravity and generates the ﬁfth force as the fundamental scalar graviton

3 Explains the dark matter as the GIMP (gravitationally interacting massive particle)

4 Can explain the hierarchy problem and origin of mass

5 Can play the role of the quintessence

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 2 / 30 Outline

1. Introduction: Brans-Dicke Dilaton

2. Kaluza-Klein Dilaton

3. Constraints on Dilaton Mass

4. Experimental Veriﬁcation

5. Discussion

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 3 / 30 Introduction: Brans-Dicke Dilaton

A) Role of Extra Space

All higher dimensional uniﬁed theories (Kaluza-Klein theory, supergravity, and superstring) assume the existence of the extra space.

A most important consequence of the extra space is the appearance of the dilaton (and other internal gravitons) in 4-dimensional physics. So by detecting the dilaton we can conﬁrm the existence of the extra space.

The Jordan-Brans-Dicke dilaton is the oldest and prototype dilaton which comes from the 5-dimensional Kaluza-Klein uniﬁcation.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 4 / 30 The physics of higher dimensional theories (in particular the dilaton physics) crucially depends on how we perceive the extra space.

In the popular view where the extra space is assumed real but very small, the dimensional reduction is made by the mode expansion. In this case we have the massless zero mode and massive excited modes.

In the dimensional reduction by isometry, however, the isometry makes the extra space invisible. In this case we have only the dilaton (and internal gravitons), and there are no excited modes.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 5 / 30 B) Brans-Dicke Dilaton: Jordan frame versus Pauli frame

In Jordan frame the Brans-Dicke Lagrangian can be written as √ h ω i L = − −g˜ φ R˜ + (∂ φ)2 BD φ µ √ h 1 i − −g˜ (F )2 + ψiγ¯ µD ψ − mψψ¯ , 4 µν µ

where g˜µν is the Jordan metric, φ and ω are the Brans-Dicke scalar ﬁeld and its coupling constant.

The Lagrangian assures the weak equivalence principle in Jordan frame: The Jordan metric g˜µν couples minimally to the matter fields, and the scalar field φ does not couple to the matter fields.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 6 / 30 Make the conformal transformation and introduce the Pauli metric gµν and the Brans-Dicke dilaton σ by 1 φ = φ0 exp(ασ), gµν = exp(ασ)g ˜µν, α = √ . 2ω + 3 1 Identify φ = with the Newton’s constant G and ﬁnd 0 16πG √ −g h 1 i √ h1 L = − R + (∂ σ)2 − −g (F )2 BD 16πG 2 µ 4 µν −3ασ i + exp ψiγµD ψ − exp − 2ασ mψψ¯ . 2 µ

Pauli Frame

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 7 / 30 Notice that

1. It is the Pauli metric gµν, not the Jordan metric g˜µν, which describes the massless spin-two Einstein’s graviton.

2. The dilaton describes a massless scalar graviton. Moreover, it has direct coupling to the matter fields in the Pauli frame, with different strengths to different matters.

So the Brans-Dicke dilaton generates a composition-dependent long range ﬁfth force which violates the equivalence principle.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 8 / 30 C) Dirac’s Conjecture

In Brans-Dicke theory φ0 = hφi should be interpreted as the mean value of φ in the present universe. In this case G = 1/16πφ0 becomes time-dependent.

Dirac proposed to treat the Newton’s constant a time-dependent parameter to resolve the hierarchy problem. So the Brans-Dicke theory naturally realizes the Dirac’s conjecture.

Unfortunately the massless scalar graviton is clearly ruled out by experiment. This is a critical defect of the zero-mode approximation.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 9 / 30 Kaluza-Klein Dilaton

A) Dimensional Reduction by Isometry

All higher-dimensional uniﬁed theories (Kaluza-Klein theory, supergravity, and superstring) have the dilaton.

In the mode expansion the mass of the excited modes is ﬁxed by the size of extra space. But the zero-mode is ruled out by experiment, and we have no evidence of excited modes.

In the dimensional reduction by isometry, however, the curvature of the extra space generates the dilaton mass and makes it an excellent candidate of dark matter. Moreover, there are no excited modes.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 10 / 30 With the dimensional reduction by isometry the unified Lagrangian in 2 2 the Pauli frame simplifies to (with g0κ = 16πG) √ −g h 1 1 L = − R + (∂ σ)2 − (D ρab)(Dµρ ) CF 16πG 2 µ 4 µ ab Rˆ (ρ ) rn + 2 r n i + G ab exp − σ + Λ exp − σ κ2 n 0 n + 2 √ −g rn + 2 + exp σ F a F µν, 4 n µν a where κ is the scale of the extra space, n is the dimension of the extra space, ρab is the normalized internal metric (det ρab = 1), and σ is the dilaton field.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 11 / 30 B) Dilaton Mass

a With hgµνi = ηµν, hσi = 0, hρabi = δab, hAµ i = 0, the dilaton potential becomes

1 h rn + 2 V (σ) = exp(− σ (16πG)2 n n + 2 r n 2 i hRˆ i − exp(− σ) + G , n n + 2 n κ2

where hRˆGi is the (dimensionless) vacuum curvature of the extra space, 1 1 hRˆ i = − f df bδac − f mf nδacδbdδ . G 2 ab cd 4 ab cd mn

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 12 / 30 σ Identifying the dilaton to be σˆ = √ the dilaton mass µ is given 16πG by

d2V (0) hRˆ i g2 4G µ2 = 16πG = − G α m2, (α = 0 = ) dσ2 2n KK p KK 4π κ2

19 where mp ' 1.2 × 10 GeV is the Planck mass.

This demonstrates that, when hRˆGi= 6 0, a large extra space (small αKK ) makes the dilaton mass small.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 13 / 30 As importantly the scale of the extra space κ is ﬁxed by the dilaton mass, s 2hRˆ i 1 1 κ = − G ' . n µ µ

For the S3 compactiﬁcation of the 3-dimensional internal space we have hRˆGi = −3/2, so that κ = 1/µ.

Furthermore, the same mechanism also reduces the mass of the q internal gravitons ρab to µ˜ = hRˆGi/κ ' µ. So a large extra space reduces all mass scales of the theory, and thus naturally explains the huge gap in the mass hierarchy.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 14 / 30 C) Origin of Mass: Mass from Curvature 1 The Einstein’s equation R − R g = −8πG T has always had µν 2 µν µν one way interpretation, that the energy-momentum tensor creates the curvature.

The dilaton suggests us to interpret it the other way, that the curvature creates the energy-momentum and thus mass. This provides a geometric mass generation mechanism.

The hierarchy problem is the problem of the mass hierarchy. The dilaton resolves the hierarchy problem by explaining the origin of mass.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 15 / 30 Constraints on Dilaton Mass

There are two important constraints on the dilaton mass.

1. The ﬁfth force constraint: The dilaton mass determines the range of the ﬁfth force.

2. The cosmological constraint: The dilaton mass determines the dilatonic dark matter density of the universe.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 16 / 30 A) Fifth Force Constraint

As the scalar graviton the dilaton generate the ﬁfth force which α modiﬁes the Newton’s gravity. With V = − 5 exp(−µr) the total 5 r force in the Newtonian limit is expressed by α α F = F + F ' g + 5 1 + µr e−µr g 5 r2 r2 α = g 1 + β(1 + µr) e−µr, r2

where αg, α5 are the ﬁne structure constants of the gravitation and ﬁfth force, and β is the ratio between them.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 17 / 30 In Kaluza-Klein uniﬁcation we have β = n/(n + 2), but in general one may assume β ' 1 treating the dilaton as a GIMP.

A recent torsion-balance fifth force experiment puts the upper bound of the range of the fifth force to be around 56 µm with 95% confidence level. This puts the lower bound of dilaton mass to be around 10−2 eV.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 18 / 30 B) Cosmological Constraint

The dilaton in thermal equilibrium near the big bang decouples from other sources near the Planck time, and may easily survive to the present universe to become the dark matter.

In the linear approximation the dilaton interaction is given by √ g L ' − 16πG σˆ 1 F F µν + g mψψ¯ , int 4 µν 2

where g1 and g2 are the coupling constants, m is the fermion mass, σˆ is the dilaton ﬁeld. From this we have the following decay rates

2 3 2 2 3/2 g1 µ 2g2 m µh 2m2i Γσ→γγ = 2 , Γσ→ψψ¯ = 2 1 − . 16 mp mp µ

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 19 / 30 With this the present number density of the relic dilaton becomes −t n(t ) ' 7.5 exp 0 cm−3, t = 1.36 × 1010 yr, 0 τ(µ) 0

where t0 is the age of the universe and τ(µ) is the dilaton life-time.

Equating the dilaton mass density ρ(µ) with the dark matter density we have

−3 ρ(µ) = µ × n(t0) ' 1.2 keV cm .

Notice that the dark matter density is much bigger than the energy density of the electric ﬁeld of 1 V/m, which is 27.3 eV cm−3.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 20 / 30 With g1 ' g2 ' 1 we ﬁnd two solutions of the dilaton mass,

35 17 1. µ1 ' 160 eV, τ1 ' 3.8 × 10 sec ' 8.1 × 10 t0, 16 −2 2. µ2 ' 276 MeV, τ2 ' 3.3 × 10 sec ' 6.9 × 10 t0.

When µ < 160 eV or µ > 276 MeV, the dilaton undercloses the universe. But when 160 eV < µ < 276 MeV, it overcloses the universe. This rules out the dilaton with 160 eV < µ < 276 MeV.

But the dilaton with µ ' 276 MeV (with t0 ' 14.5 τ2) should have decayed to photons and light fermions which overclose the universe. So only the dilaton with µ ' 160 eV is acceptable as the dark matter.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 21 / 30 C) Size of Extra Space

The ﬁfth force experiments require the extra space can not be larger than 10−5 m, but the cosmological constraint excludes the extra space smaller than 10−9 m (20 times the Bohr radius).

From this we obtain the following ﬁgure which shows the allowed scale of the extra space. This excludes the large extra space of κ ' 10−5 nm advocated by the ADD.

But remember that our result is a rough approximation based on simple assumptions which need improvement.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 22 / 30 Figure : The scale of extra space κ versus β = α5/αg of the fifth force in 7-dimensional Kaluza-Klein unification with S3 compactification of extra space. The colored region marked by (–) is the excluded region.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 23 / 30 Experimental Veriﬁcation

The above discussion makes the experimental detection of the dilaton a most urgent issue in current physic.

The cosmological constraint on dilaton mass implies that it is futile to try to detect the dilaton by the ﬁfth force experiment, if the dilaton is indeed the dark matter.

We can think of two experiments, both based on two-photon decay of dilaton. 1. Resonant cavity experiment 2. Satellite experiment

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 24 / 30 A) Cavity Experiment

Consider a dilaton-photon conversion experiment and let V = LxLyLz be the volume of a rectangular resonant cavity. With B~ ext = B0 cos(qx)ˆz the photon detection power is given by

2 4g1 2 Pd = 2 ρdB0 LxV. πmp Compare this with the axion detection power,

2 gγα 2 Pa = 2 ρaB0 LxV, 4πfa

where gγ is the axion-photon coupling constant, α is the ﬁne structure constant and fa is the PQ symmetry breaking scale.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 25 / 30 Notice that the detection power is enhanced by 1. the strong magnetic background 2. the large halo density

12 Since fa(' 6 × 10 GeV) is much smaller than mp, we have

P g 2 f 2 d ' 1.9 × 106 1 a ' 4.7 × 10−7, Pa gγ mp

when g1 ' gγ. However, if g1 >> gγ, Pd can be as big as Pa.

The dilaton produces the TE mode (magnetic wave), but the axion produces the TM mode (electric wave). Moreover, for the dilaton the photon polarization is perpendicular to the external magnetic ﬁeld, but for the axion the polarization is parallel to the magnetic ﬁeld.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 26 / 30 B) Satellite Experiment

One could try to detect the two-photon decay of the relic dilaton directly from the sky. In spirit this is similar to COBE/WMAP experiment, except here we are looking for the x-ray and/or γ-ray, not the microwave. J-Webb telescope could do this.

The advantage of this experiment is that

1. We know exactly what kind of two-photon decay signal to look for. 2. We can estimate how much the detection rate we expect.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 27 / 30 Discussion

The dilaton exists in all higher dimensional uniﬁed theories and can not be ignored. The theoretical raison d’etre is too convincing!

The dilaton has a deep impact on cosmology. In particular, as the GIMP it becomes an excellent candidate of dark matter.

The cosmological constraint on dilaton mass excludes the extra space of the scale between 10−16m and 10−9m. Assuming that the dilaton is the dark matter, we can estimate the scale of the extra space to be of the order of 10−9 m.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 28 / 30 This makes the experimental detection of the dilaton very important. The cavity experiment or the satellite experiment appears most promising to detect the dilaton, and we can use the axion detector to detect the dilaton.

The dilaton can reveal the nature of the extra space. In the zero-mode approximation the massless zero-mode has always been a problem. If LHC does not produce the KK excited modes, the popular view that the extra space is real but small will be in serious trouble.

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 29 / 30 References

1 P.A.M. Dirac, Nature (London) 136, 323 (1937); P. Jordan, Ann. Phys. (Leibzig) 1, 218 (1947); C. Brans and R. Dicke, Phys. Rev. 124, 921 (1961). 2 Y.M. Cho, J. Math. Phys. 16, 2029 (1975); Y.M. Cho and P.G.O. Freund, Phys. Rev. D12, 1711 (1975). 3 Y.M. Cho, Phys. Rev. Lett. 68, 21 (1992). 4 Y.M. Cho and Y.Y. Keum, Class. Quant. Grav. 15, 907 (1998); Y.M. Cho and J.H. Kim, Phys. Rev. D79, 023504 (2009).

Y. M. Cho (Konkuk University) Dilatonic Dark Matter June 22, 2016 30 / 30