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Detecting Solar Chameleons Through Radiation Pressure

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Detecting Solar Chameleons Through Radiation Pressure Physics Letters B 739 (2014) 167–173 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Detecting solar chameleons through radiation pressure a,b, c,d e d,f g,h S. Baum ∗, G. Cantatore , D.H.H. Hoffmann , M. Karuza , Y.K. Semertzidis , A. Upadhye i, K. Zioutas b,j a Uppsala Universitet, Box 516, SE 75120, Uppsala, Sweden b European Organization for Nuclear Research (CERN), Gèneve, Switzerland c Università di Trieste, Via Valerio 2, 34127 Trieste, Italy d INFN Trieste, Padriciano 99, 34149 Trieste, Italy e Institut für Kernphysik, TU-Darmstadt, Schlossgartenstr. 9, D-64289 Darmstadt, Germany f Phys. Dept. and CMNST, University of Rijeka, R. Matejcic 2, Rijeka, Croatia g Center for Axion and Precision Physics Research (IBS), Daejeon 305-701, Republic of Korea h Department of Physics, KAIST, Daejeon 305-701, Republic of Korea i Physics Department, University of Wisconsin–Madison, 1150 University Avenue, Madison, WI 53706, USA j University of Patras, GR 26504 Patras, Greece a r t i c l e i n f o a b s t r a c t Article history: Light scalar fields can drive the accelerated expansion of the universe. Hence, they are obvious dark Received 18 September 2014 energy candidates. To make such models compatible with tests of General Relativity in the solar system Received in revised form 19 October 2014 and “fifth force” searches on Earth, one needs to screen them. One possibility is the so-called “chameleon” Accepted 24 October 2014 mechanism, which renders an effective mass depending on the local matter density. If chameleon Available online 28 October 2014 particles exist, they can be produced in the sun and detected on Earth exploiting the equivalent of a Editor: A. Ringwald radiation pressure. Since their effective mass scales with the local matter density, chameleons can be reflected by a dense medium if their effective mass becomes greater than their total energy. Thus, under appropriate conditions, a flux of solar chameleons may be sensed by detecting the total instantaneous momentum transferred to a suitable opto-mechanical force/pressure sensor. We calculate the solar chameleon spectrum and the reach in the chameleon parameter space of an experiment using the preliminary results from a force/pressure sensor, currently under development at INFN Trieste, to be mounted in the focal plane of one of the X-Ray telescopes of the CAST experiment at CERN. We show, that such an experiment signifies a pioneering effort probing uncharted chameleon parameter space. 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3. 1. Introduction frame reads: R The standard model of cosmology, the ΛCDM model, describes 4 1 2 J S d x √ g (∂φ) V (φ) Sm g , (1) the history of our universe based on Einstein’s theory of general = − 16π G − 2 − + relativity (GR), cold dark matter (CDM), and a cosmological con- stant (Λ). Whilst being in good agreement with astronomical and where φ is the gravitationally coupled scalar field with potential astrophysical observations, it provides no explanation for the value V (φ), g gµν the determinant of the metric, R the Ricci scalar, =| | of its parameters. Although accounting for the accelerated expan- G the gravitational constant, and Sm the matter action depend- J sion of the universe by a cosmological constant is the simplest ing on the Jordan frame metric g , which is related to gµν by the model, its value must be fine-tuned. Such problems of the ΛCDM J 2 conformal transformation gµν A (φ)gµν . In general, A(φ) is al- model motivate modified gravity models (MOG). One such model is lowed to be different for the va=rious matter fields, thus giving rise scalar-tensor gravity, where GR is modified by introducing a grav- to violations of the weak equivalence principle. itationally coupled scalar field φ.A general action in the Einstein However, to be in agreement with GR tests in the solar system, the effects of such scalar fields must be screened on corresponding Corresponding author. length scales. The chameleon-model [1,2] realizes such a mecha- * E-mail addresses: [email protected] (S. Baum), [email protected] nism by rendering an effective mass of the scalar depending on (K. Zioutas). the local matter density. http://dx.doi.org/10.1016/j.physletb.2014.10.055 0370-2693/ 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3. 168 S. Baum et al. / Physics Letters B 739 (2014) 167–173 The equation of motion for φ derived from the action is 2 n 1there is no chameleon effect. For n between 1 and − 1≤/2t≤−he scaling is very rapid, leading to very strongly scr−eened 2 ∂ φ V ,φ(φ) A,φρ, (2) c≈hameleons which are best probed cosmologically [6]. Thus, for = + where ρ is the matter density, related to the Einstein frame den- this work we are primarily interested in n 1/2 and n < 2. − sity ρ E and the Jordan frame density ρ J by ρ A 1ρ E A 3ρ J . Chameleons would be produced in regions of high photon den- − − The dynamics of φ are governed by the matter= density dependent= sity and strong magnetic field, e.g. inside the sun. One can search effective potential V (φ) V (φ) A(φ)ρ. for solar chameleons on Earth by detecting their radiation pres- eff For suitable V (φ) and A≡(φ), the+ field will have a local minimum sure [7]: In a dense medium chameleons get large effective mass. If their total energy is smaller than their effective mass in a medium at some value φmin φmin(ρ). The effective mass is calculated = they try to penetrate, they will get reflected, resulting in the equiv- from the curvature of the effective potential at φmin: alent of radiation pressure. In the following, we calculate the sen- 2 m V eff,φφ V ,φφ(φmin) A,φφ(φmin)ρ, (3) sitivity of such an experiment in three parts: φ,eff = = + which itself is explicitly dependent on the local mass density ρ. chameleon production in the sun It has been shown that chameleons cannot explain the observed • the chameleons’ journey to the detector accelerated expansion of the universe as true MOG effect [3], how- • detecting chameleons on Earth. ever, they can act as dark energy. In the following, we work with • an inverse power law potential 2. Chameleon production in the sun 4 n Λ + V (φ) λ , (4) Armed with our chameleon model, we can study chameleon n = φ production in the sun. We build our calculations on the previous where Λ is a mass scale. For n 4, the λ-parameter can be ab- works of Brax et al. [7–9]. Photons mix with chameleons in regions = − sorbed into Λ. If one adds a constant term, the chameleon fields of strong magnetic field, cf. the case for Axions. In the sun, strong acts like a cosmological constant on large scales, hence becomes a magnetic fields are found in the so-called tachocline, the thin tran- dark energy model. Such a potential can also be understood as the sition zone between the radiative core of the sun exhibiting almost first (non-trivial) order approximation of an exponential potential: solid rotation and the convective envelope rotating differentially. n Here, the large shear causes strong magnetic fields [10]. The con- n n Λ V (φ) Λ4eΛ /φ Λ4 1 . (5) version probability for photons of energy ω in a magnetic field B = ≈ + φn traveling by a length L is given by [12]: Since the constant term has no effect on the dynamics of the field, 2 2 we neglect it in the following and absorb λ into Λ for n 4. The Pchameleon(ω) sin (2θ)sin , (10) =βm−φ = cos 2θ chameleon-matter coupling is parametrized by A(φ) e MPl . Our 2Bωβ = where (m2 ω2 )L/4ω and tan(2θ) γ is the effective potential now reads: eff pl m2 2 M = − = ( eff ωPl) pl 2 − 4 n β βγ mixing angle. 4 n m is the plasma frequency with n Λ m φ φ ωpl πα e/ e e + M M = V eff(φ) e Pl ρm e Pl ργ , (6) = φn + + the electron number density, me the electron mass, and α the fine structure constant. If chameleons are predominantly produced where ρ is the local matter density. We assume a univer- m in the sun’s tachocline, this simplifies greatly. Assuming ρtacho sal chameleon-matter coupling β β for simplicity. ρ 3 6 = m,i m γ 0.2g/cm , Btacho 30 T, and Ttacho 2 10 K [10], we find 1 ≡ = µν 20 =2 βγ = × 4 Fµν F is the Lagrangian density of the electromagnetic field. tan(2θ) 10 eV . Chameleons can only travel in the − m2 ω2 Whether the chameleon-photon coupling exists at tree level ∼ × eff Pl − or arises through fermion loops is a matter of ongoing de- sun for an effective momentum bate [4,5]. Here, we add the coupling by hand through a term 2 2 2 2 k ω meff ωpl 0. (11) βm/MPlφ 1 µν e 4 Fµν F in the action and consider βγ to be indepen- = − − ≥ 10 dent of βm. From this effective potential we find As we willsee below,stellar evolution constraints βγ 10 for large parts of the parameter space.
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