PHYSICAL REVIEW D 86, 123002 (2012) Evolution of the chameleon scalar field in the early universe
David F. Mota1,* and Camilla A. O. Schelpe2,† 1Institute of Theoretical Astrophysics, University of Oslo, 0315 Oslo, Norway 2Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Cambridge CB3 0WA, United Kingdom (Received 3 August 2011; revised manuscript received 18 July 2012; published 3 December 2012) In order to satisfy limits on the allowed variation of particle masses from big bang nucleosynthesis until today, the chameleon scalar field is required to reach its attractor solution early on in its cosmological evolution. Brax et al. [Phys. Rev. D 70, 123518 (2004)] have shown this to be possible for certain specific initial conditions on the chameleon field at the end of inflation. However the extreme fine-tuning necessary to achieve this poses a problem if the chameleon is to be viewed a natural candidate for dark energy. In this paper we revisit the behavior of the chameleon in the early universe, including the additional coupling to electromagnetism proposed by Brax et al. [Phys. Lett. B 699, 5 (2011)]. Solving the chameleon evolution equations in the presence of a primordial magnetic field, we find that the strict initial conditions on the chameleon field at the end of inflation can be relaxed, and we determine the associated lower bound on the strength of the primordial magnetic field.
DOI: 10.1103/PhysRevD.86.123002 PACS numbers: 98.62.En, 04.50.Kd, 98.80.Cq
for this direct chameleon-photon interaction have been I. INTRODUCTION investigated extensively [8–18]. We discuss the chameleon Gravity theories which extend general relativity by model in more detail in Sec. II. introducing a new degree of freedom have received The early universe behavior of the chameleon, consid- increased attention lately due to combined motivation ering only the interaction of the chameleon with matter coming from high-energy physics, cosmology and astro- species, was analyzed by Brax et al. in 2004 [19]. They physics; see Ref. [1] for a recent review. If they claim to assumed that the chameleon field was generated at some account for dark energy or dark matter, then they are only phase transition during inflation, and that at the end of valid if they pass the many weak-field limit tests. For many inflation the field value was left at some arbitrary position this is only possible if they have a chameleon mechanism within the effective potential. The chameleon then rolls to [1–5]. Such mechanisms have the effective mass of the the minimum of its effective potential and in some cases scalar degree of freedom being a function of the curvature oscillates about it. Brax et al. [19] assumed a 10% limit on (or energy density) of the local environment, so that in the variation of particle masses from big bang nucleosyn- effect the mass is large at Solar System and terrestrial thesis (BBN) until today which requires the maximum curvatures and densities, but small at cosmological curva- amplitude of the chameleon oscillations to be below a tures and densities. certain threshold at BBN. Their analysis, which neglected The original chameleon model was formulated by the electromagnetic interaction, found that the chameleon Khoury and Weltman [5,6] as a scalar field model for field initial conditions needed to be strongly fine-tuned if dynamical dark energy. It is a scalar-tensor theory in which the chameleon’s approach to the minimum were to satisfy the new degree of freedom is identified with the chameleon the constraints placed on the field at BBN. In this paper we field in the Einstein frame. The interaction of the chame- reanalyze the early universe behavior of the chameleon, leon to matter is through the conformal metric but this time including the direct coupling between the 2 ð Þ Am g . It was later shown in Ref. [7] that the coupling chameleon and electromagnetic fields. We find that a pri- of the chameleon to charged matter naturally generates a mordial magnetic field (PMF), produced from some direct coupling between the chameleon and electromag- symmetry-breaking phase during inflation, can drive the ð Þ netic field of the form AF F F . This extra term chameleon towards the minimum of its potential more should not be neglected in these models. The interaction quickly. Thus satisfying the constraints at BBN for a between the chameleon and electromagnetic fields leads to much wider range of initial conditions. conversion between chameleons and photons in the pres- The existence of a large-scale PMF in the early universe ence of a magnetic field, which has the potential to alter the is a matter of current debate. It has so far gone undetected intensity and polarization of radiation passing through by observations and only upper bounds at the OðnGÞ level magnetic regions. The astrophysical and laboratory tests have been placed on its magnitude. Theories explaining the origin of the PMF are speculative, but it has been suggested *[email protected] that a large-scale magnetic field could be produced during †[email protected] inflation if the conformal invariance of the electromagnetic
1550-7998=2012=86(12)=123002(14) 123002-1 Ó 2012 American Physical Society DAVID F. MOTA AND CAMILLA A. O. SCHELPE PHYSICAL REVIEW D 86, 123002 (2012) field is broken [20–26]. See Ref. [27] for a recent review. A cosmological evolution. This analysis includes the direct PMF is often invoked to explain the order G magnetic coupling between the chameleon and electromagnetic fields observed in nearly all galaxies and galaxy clusters. fields. We consider the effects of a background primordial The origin of these galactic magnetic fields is unclear, but a magnetic field and ignore the second order contribution natural argument is that they develop by some form of from the interaction of the chameleon with electromag- amplification from a pregalactic cosmological magnetic netic radiation. field. Two popular formation scenarios are either some The action describing the chameleon scalar field model exponential dynamo mechanism which amplifies a very is that of a generalized scalar-tensor theory: 30 small seed field of order 10 G as the galaxy evolves or Z pffiffiffiffiffiffiffi 4 1 2 1 the adiabatic collapse of a larger existing cosmological field S ¼ d x g M R g @ @ Vð Þ ð 10 9Þ 2 Pl 2 of order 10 10 G. See Refs. [28–32] for reviews on Z the subject. The generic model for the primordial magnetic 1 ð Þ A ð ÞF F þ d4xL ðc i ;A2ð Þg Þ; field is of a stochastic field parametrized by a power-law 4 F matter i ð Þ¼ nB power spectrum, P k ABk , up to a cutoff scale k 123002-2 EVOLUTION OF THE CHAMELEON SCALAR FIELD IN ... PHYSICAL REVIEW D 86, 123002 (2012) 1 defined the conserved energy density in the electromag- h ¼ € _ þ r2 3H ; ð Þðj j2 þj j2Þ a2 netic field by EM AF B E =2. Combining ð Þ these two equations leads to the conservation equation where H t a=a_ is the Hubble expansion rate. The for each species, stress-energy tensor in the Jordan frame is defined by ð þ Þ ¼ a 3 1 wi ; (6) ð Þ 2 L i i0 T i pffiffiffiffiffiffiffiffiffiffiffi matter ; ¼ gðiÞ gðiÞ where p wi defines the equation of state parameter wi. For radiation we have w ¼ 1=3, while for nonrelativistic in which particle masses are constant and independent matter w ¼ 0. In general the energy density in the back- of . This leads to ground electromagnetic fields is much less than the energy density of relativistic matter in the Universe, r, and so 0 1 2 0 3 0 ðiÞ h ¼ V ð Þþ F A ð Þ Amð ÞAmð Þgð Þ T : 4 F i 2 2ð Þ¼ þ 3 þ 4 3MPlH a m0a r0a ; (7) The stress-energy tensor for a perfect fluid with density which is the standard expression for the Hubble expansion and pressure p is in the presence of dynamical dark energy. T ¼ð þ pÞu u þ pg ; Equations (3) and (7) form the starting point for our analysis of the chameleon evolution in the early universe. where the 4-velocity of the cosmological fluid is The remainder of this section is dedicated to a brief ¼ð 0Þ ¼ ð 2 u 1; in this metric, and so T diag ; a p; summary of the properties of the chameleon model, and a 2p; a 2pÞ. The contraction of the stress-energy tensor the experimental bounds that have been placed on its is then T ¼ þ 3p. The energy densities in the physi- parameters. pffiffiffiffiffi2 Lmatter cal Einstein frame, defined by T g g , are re- The self-interaction potential of the scalar field, Vð Þ, 4 ðiÞ lated to those in the Jordan frame by T ¼ AmT . Thus determines whether a general scalar-tensor theory is chameleon-like or not. For a chameleon we require the 1 h ¼ 0ð Þþ 2 0 ð Þ 1ð Þ 0 ð Þ mass of small perturbations about the minimum to depend V F AF T Am Am : (3) 4 on the surrounding matter density. This imposes that Vð Þ In general is in the weak-field limit and so we can is of runaway form; in other words it is monotonically 1 decreasing and all of V,(V =V) and (V =V ) tend to approximate Am ð Þ 1. This allows us to define an ; ; ; effective potential that the chameleon moves in, zero as tends to infinity, and to infinity as tends to zero. A typical choice can be described by 1 ð Þ ð Þþ 2 ð Þ ð Þ Veff V F AF T Am : (4) n 4 Vð Þ¼ 4 exp ; (8) 0 n These equations determine the behavior of the chameleon in the early universe in the presence of a primordial mag- where n Oð1Þ. This is also the desired form for quintes- 2 2 2 netic field. In general F ¼ 2ðjBj jEj Þ and T ¼ sence models of dark energy [38]. For the chameleon to ¼ m, where E and B are the background electric and be a suitable dark energy candidate, we require 0 3 magnetic fields respectively and m is the density of the ð2:4 0:3Þ 10 eV, and we assume Oð 0Þ for surrounding matter. The mass of small perturbations, m , reasons of naturalness. The resulting chameleon effective in the chameleon field is given by potential is plotted for two different values of m in Fig. 1. The shape of the effective potential depends on both the m2 Veff ð ; F 2; Þ; (5) ; min m background electromagnetic (EM) fields and the density of the surrounding matter. In this plot we neglect the EM where F and m are the background values of the fields, contribution and assume A ð Þ’expð =MÞ. From this and min is the minimum of the effective potential. m The Friedmann equations governing the expansion of we can see that the curvature of the effective potential at the Universe are also modified by the presence of chame- the minimum increases as the density of the surrounding leon dark energy: matter, m, increases. Thus, in the low density environ- ments of space the chameleon is very light and can drive a_ 2 2 ¼ þ þ the cosmic acceleration, while in high density environ- 3MPl EM ; a ments, such as in the laboratory, the chameleon is much a€ a_ 2 heavier and evades detection. 6M2 þ ¼ð 3p Þþð 3pÞ; Pl a a Nonlinear self-interactions in the chameleon model also help the scalar field satisfy fifth-force constraints through a where we have defined the energy density and pressure in property termed the thin-shell effect [6,39]. In the thin- the scalar field, neglecting spatial variations in ,as shell effect the bulk of the variation in the field, from its _ 2 _ 2 =2 þ Vð Þ and p =2 Vð Þ respectively, and equilibrium value inside a test body to its equilibrium value 123002-3 DAVID F. MOTA AND CAMILLA A. O. SCHELPE PHYSICAL REVIEW D 86, 123002 (2012) limits on the production of starlight polarization in the 9 galactic magnetic field is MF * 1:1 10 GeV [17], while measurements of the Sunyaev–Zel’dovich effect in galaxy clusters place a lower bound on MF in the range 9 MF * ð0:25–1:14Þ 10 GeV, depending on the model ) φ ( assumed for the cluster magnetic field [14,15]. eff V ρ exp(φ/M) m III. THE CHAMELEON EVOLUTION In this section we proceed to solve the chameleon evo- lution equations, developed in the previous section, for a V(φ) range of initial field values at the end of inflation. This is a φ nontrivial task since the equations governing the chame- leon are highly nonlinear. While it is possible to approxi- mate the behavior of the chameleon near the minimum of its potential by a harmonic oscillator, the field at the end of inflation is likely to lie far away from this point and so the assumption of linearity breaks down. A numerical approach to solving the evolution equations exactly is ) also extremely difficult due to the very different scales φ ( eff governing the two sides of the effective potential. In this V analysis we find a semianalytical solution to the chameleon ρ exp(φ/M) evolution, using the shape of the potential to our advantage m in making a few simplifying assumptions. There are a number of steps necessary to build up a V(φ) framework for solving the chameleon behavior: in φ Sec. III A we determine that an attractor solution exists for the chameleon in which the field rolls along the slowly FIG. 1 (color online). The chameleon effective potential, Veff, varying minimum of its effective potential. In Sec. III B we is the sum of the scalar potential, Vð Þ, and a density-dependent solve for the initial trajectory of the field towards the term. In high density environments (upper plot) the mass of the minimum and discuss the necessary assumptions that are chameleon is much greater than in low density environments applied to its effective potential. In Sec. III C we iterate this (lower plot). solution to solve for the subsequent oscillations of the chameleon. An extra contribution to the chameleon evolu- in the surrounding medium, occurs in a narrow band at the tion comes from particle species dropping out of thermal surface of the body. The acceleration felt by a test mass equilibrium, which we include in our calculations in from the force mediated by the chameleon particle is Sec. III D. 1 r proportional to M . Thus for a thin-shelled object, it is as if the chameleon field only interacts with the narrow A. Evolution along the attractor band of matter near the surface of the test body where the The effective potential describing the chameleon behav- variation in occurs, and the resulting fifth force mediated ior was defined in Eq. (4). The minimum of the potential by the chameleon is strongly suppressed. In general large, will evolve over time as the energy density in the matter dense objects have thin shells while smaller objects do not. and electromagnetic fields decreases with the expansion of The strength of the chameleon-matter coupling 1=M is the Universe. We begin our analysis by checking that the constrained by existing laboratory experiments [39–43]. minimum provides a viable attractor solution for the cha- The best direct bound comes from particle physics experi- meleon when the electromagnetic interaction is included in ments, M * 104 GeV [39,42,43]. the chameleon model. This requires the evolution of the The strength of the chameleon to photon interaction potential to be slow enough for the chameleon to respond 1=MF is more tightly constrained. Laser-based laboratory to these changes and follow the minimum. We follow a searches for these particles, such as PVLAS [44,45] and similar procedure to that outlined in Ref. [19]. GammeV [46], place an upper bound on the strength of the The shift in the minimum of the potential is as a result of 6 photon interaction, MF * 10 GeV. However, the effects the expansion of the Universe, and so the characteristic of chameleon-photon mixing will accumulate most during time scale for the change is approximately a Hubble time propagation through large-scale astrophysical magnetic H 1. The characteristic response time of the chameleon is fields, and so these scenarios offer the best testing ground the period of oscillations about the minimum which is 1 for the theory. The constraint coming from considering the given by m . For the chameleon to adjust quickly enough 123002-4 EVOLUTION OF THE CHAMELEON SCALAR FIELD IN ... PHYSICAL REVIEW D 86, 123002 (2012) to changes in the position of the minimum, the ratio m =H The function ð H Þ2ð þ Þnþ2 is minimized at late H0 PMF m needs to be greater than one. times. It is greater than 10 5 for all n Oð1Þ. Again taking 3 The minimum of the effective potential occurs when M MPl and 10 eV, we expect V0 ð Þ¼0, for which eff 2 m 35 * 31 nþ 0ð Þþ PMF þ m ¼ 10 1: V min 0; H2 MF M Thus, we can confirm that at all times the ratio m =H is where we have defined the energy density in the PMF, jBj2 much greater than unity, and so the minimum does provide PMF =2. We assume min M, MF such that 0 0 a viable attractor solution for the chameleon. Once the A 1=M and A 1=M . For the self-interaction m F F chameleon has settled to the minimum of its potential it potential given in Eq. (8) we have will always be able to track the minimum as the Universe n ðnþ1Þ ð Þ¼ PMF þ m expands. n min V min : (9) MF M Substituting into the expression for the chameleon mass (5) B. Approaching the minimum of the potential and rearranging, We now turn our attention to the approach of the cha- meleon to the attractor solution. Although the origin of the 1 nþ1 2 ¼ PMF þ m ð þ Þ þ m 1 n n chameleon field is speculative, it is reasonable to assume MF M min min that it is produced at some phase transition during inflation, þ PMF þ m and that at the end of inflation the field value is left at some 2 2 : MF M arbitrary position within the effective potential. We expect the field to roll to the minimum of the potential, and Under the assumption M MF and taking min M,we oscillate about the minimum before then settling to its approximate equilibrium value. The presence of a large-scale magnetic 2 2 ð þ Þ nþ1 field provides an additional source term driving the cha- m 3MPl PMF m ð1 þ nÞ þ n ; meleon to its minimum. The Universe becomes a good 2 M H min min conductor during the reheating phase at the end of infla- 2 2 2 2 where PMF PMF=3H MPl and m m=3H MPl are tion, after which any electric fields in the Universe will be the fractional energy densities in the magnetic field and in dissipated and the primordial magnetic field becomes nonrelativistic (matter) degrees of freedom respectively. In frozen-in, so that from then on jBj2 / a 4. the limit min , The equations governing the chameleon evolution were derived in Sec. II. Assuming the field is spatially homoge- m2 3M2 ð þ Þ nþ1 Pl PMF m n neous, Eq. (3) simplifies to H2 M min d2 d 2 0 0 0 3nM ð þ Þ 3H ¼ V ð Þþ A ð Þ T A ð Þ; * Pl PMF m : dt2 dt PMF F m M (10) þ The smallest value of the function, PMF m, which describes a weakly damped oscillator, where the occurs at early times ( 10 7 at a 10 23) and m friction term proportional to H is due to the expansion of for the limiting case of ¼ 0. The largest natural PMF the Universe. The evolution of HðtÞ was given in Eq. (7). value for the mass parameter of the chameleon coupling Prior to BBN the Universe is radiation dominated and so M is OðM Þ. Taking 10 3 eV and n Oð1Þ,we Pl we can approximate expect H2ðaÞ’H a 4; (11) m2 0 r0 * 1024: 2 2 H where r0 8 G r0=3H0 is the fractional energy density ¼ in radiation. In general the stress-energy tensor T Similarly for the limit when min , m, where m is the energy density in nonrelativistic 2 2 ð þ Þ matter. Soon after reheating, all particle species are rela- m 3MPl PMF m ð1 þ nÞ ; tivistic and so the driving term in the chameleon effective H2 M min potential has T 0. However, as the Universe cools the ð Þ 4 and we can approximate V min in Eq. (9) to deter- particles drop out of thermal equilibrium. This causes a mine min. Then, short-lived contribution to T which can significantly affect the chameleon as it rolls to the minimum. We follow 2 2 2 2 1 m 3ð1 þ nÞM 3M H nþ1 nþ2 Pl Pl ð þ Þnþ [19] and refer to these contributions as ‘‘kicks.’’ The first PMF m 1: H2 M nM 3 kick occurs at a redshift of approximately z 1014 as the 123002-5 DAVID F. MOTA AND CAMILLA A. O. SCHELPE PHYSICAL REVIEW D 86, 123002 (2012) top quark drops out of thermal equilibrium. To start with potential before coming to a halt. The self-interaction we neglect this contribution and solve for the initial behav- potential Vð Þ dominates the initial roll to the minimum. ior of the chameleon. We include the contribution from From Eq. (10), these kicks in Sec. III D. Neglecting the contribution from the kicks for now, the 2 d ’ 0ð Þ effective potential characterizing the chameleon motion in V ; dt2 Eq. (4) can be expressed as n and so the velocity as it shoots past the minimum is ’ V ð Þ¼ 4 exp þ exp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;t eff PMF ð Þ ð Þ ð Þ MF 2V i [since V i V min ]. We treat this as instan- ¼ taneous and occurring at a ai. Beyond the minimum the þ exp : m M evolution is determined by Eq. (12p).ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The initial conditions, ða Þ¼ ða Þ and ða Þ¼ 2Vð Þ=a Hða Þ, deter- i min i ;a i i i i Notice that M, MF and so the left-hand side of the mine the constants A and B. Substituting, we find that the potential with < min is many orders of magnitude field comes to a halt at steeper than that for > min. The exponential decay of the functions means that for < the potential is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi min ð Þ almost entirely dominated by Vð Þ, while for > ’ þ 2V i min amax ai ai; the dominant driving force is from the and Hða Þ m PMF i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi terms. This strong asymmetry in the chameleon effective a 2Vð Þ potential allows us to make a few simplifying assump- ¼ max þ i max ln ð Þ ; tions so that the chameleon evolution can be determined ai H ai analytically. ð Þ Whenever > min we assume V is negligible. where we have assumed max min. The chameleon Approximating A ð Þ, A ð Þ 1, we can solve for the will then start to fall back towards the minimum and evolve m F evolution along this side of the potential so that Eq. (10) identically to the case when i min, but with amax and becomes max as the initial conditions. Depending on the size of the background magnetic field and the initial starting value for 1 d d H a4H ’ PMF0 a 4 m0 a 3; the chameleon field, either the field falls to the minimum 2 a da da MF M and starts oscillating about it, or the friction term domi- nates and the field remains frozen at its initial value until where we have converted from cosmological time to the the first kick occurs as the top quark drops out of thermal scale factor using the relation HðaÞ a=a_ . The contribu- equilibrium. tion from (nonrelativistic) matter, m, will be negligible prior to BBN and can be omitted. Substituting Eq. (11) for the evolution of the Hubble expansion, we find C. Oscillations about the minimum d In general, the velocity of the chameleon when it first ¼ a 1 þ Aa 2; reaches the minimum will be very large and cause it to (12) da overshoot. It will then oscillate about the minimum with a 1 ðaÞ¼ loga Aa þ B; large amplitude which gradually decays due to the damp- ing term from the Hubble expansion. We apply the where approach outlined in the previous section to iterate over 2 multiple oscillations and determine the evolution of the PMF0= r0H0MF: (13) amplitude of the chameleon oscillations. The constants A and B are determined by the initial Let us consider a single oscillation which starts from rest ¼ conditions. at 1 at some time a1 with 1 min. The field is Conversely, when < min the PMF term is negligible governed by Eq. (12) as it rolls to the minimum. compared to Vð Þ. The friction term proportional to H in Substituting the initial conditions to determine A and B Eq. (10) is also negligible since the steepness of the po- gives tential dominates. The roll up or down on this side of the potential will be very rapid and we treat it as instantaneous. d a1 Thus as the chameleon approaches from larger field values, ’ a 1 1 ; da a Vð Þ acts as a perfect elastic collision. We assume the chameleon starts from rest at some initial ’ a þ a1 1 log 1 : 23 a a value i at the end of inflation, ai 10 . For starting 1 & values with i min, the chameleon falls very quickly to the minimum and overshoots to the other side of the At the minimum, 123002-6 EVOLUTION OF THE CHAMELEON SCALAR FIELD IN ... PHYSICAL REVIEW D 86, 123002 (2012) d a D. Kicks ’ 1 1 amin 1 ; da amin In addition to the oscillations described in Sec. III C, the min a a (14) chameleon behavior is modified by an extra contribution to ’ log min þ 1 1 ; min 1 a a T as particle species drop out of thermal equilibrium. 1 min This manifests itself as a short-lived boost to the driving which form the initial conditions for rolling back up. term on the right-hand side of Eq. (10). The contribution as Equation (14) can be inverted to determine amin under each particle, labeled by k, goes nonrelativistic is the assumption . Following the approximation min 1 ð Þ 45 g m T k ¼ H2ðaÞM2 k k ; (18) discussed in the previous section, we assume the field 4 Pl ð Þ undergoes an instantaneous perfect elastic collision at g? T T the minimum and rebounds with the same velocity. where Substituting into Eq. (12), the chameleon as it rolls back pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z to larger field values is governed by 1 u2 x2 ðxÞ x2 du ; eu 1 d x ’ 2ð Þ a 2amin a a1 ; and the sign is for fermions and bosons respectively da a a a a [19,47,48]. The function is plotted in Fig. 2. It is ap- ’ log þ 2 1 þ min 3 1 : Oð Þ 1 a a a a proximately 1 for x 1 and negligible otherwise. The 1 min ¼ 1 temperature of the Universe redshifts as T T0a . The maximum contribution to T occurs when the temperature It comes to a halt at the retracement point 2 at some of the Universe has cooled sufficiently for it to match the time a2: mass of the particle, mk T. The mass of the different ’ a2 2amin a1; (15) species, along with the ratio of the number of degrees of freedom to the effective number of relativistic degrees of freedom, gk=g?, is listed in Table I. ’ a2 þ a1 An exact solution to the chameleon evolution with the 1 2 log 2 1 : (16) a1 amin kicks is not possible. Instead we consider the two limiting cases: when the duration of the kick is very much less than ð Þ This evolution from one retracement point 1;a1 to the oscillation period of the chameleon, we can approxi- another ð ;a Þ can be iterated to determine the chame- 2 2 mate the contribution to T as a delta function similar to leon behavior. the procedure outlined in Ref. [19]; on the other hand An analytic approximation for the evolution of the when the chameleon is oscillating rapidly about the mini- retracement point max exists in the limit of fast oscilla- mum, we can make an adiabatic approximation and assume þ tions when amin=a1 1 , 1. In this limit, Eq. (14) the function is constant over one oscillation. What is a becomes reasonable estimate for the duration of the kick? Referring to Fig. 2, the width of the function runs from approxi- 1 1 2 3 þ 1 ’ 1 þ þ Oð Þ; mately a=ak ¼ 0:1–10, beyond which it has dropped to 2 less than 10% of its maximum value. We therefore assume pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi which implies ’ 2 = . Similarly Eqs. (15) and (16) 1 1.4 can be approximated in the limit of small , such that fermions bosons 1.2 ’ a a2 a1 2 a1; 2 1 ’ 3 max 2 1 : 3 0.8 (x) τ In the limit of fast oscillations we can approximate 0.6 d 2 2 max max 1 ) log ’ loga þ const: 0.4 da a 3 a max 3 (17) 0.2 0 This power-law behavior is independent of the strength of 0246810 x the magnetic field. In this regime, the magnetic field R pffiffiffiffiffiffiffiffiffiffi strength will dictate how rapidly the chameleon oscillates ð Þ¼ 2 1 u2 x2 FIG. 2 (color online). The function x x x du eu 1 , but not how long it takes to settle to the minimum. where the is for fermions and bosons respectively. 123002-7 DAVID F. MOTA AND CAMILLA A. O. SCHELPE PHYSICAL REVIEW D 86, 123002 (2012) TABLE I. List of particle species that provide a kick to the where A and B are constants that are determined from the chameleon evolution as they drop out of thermal equilibrium. chameleon evolution prior to the kick. The change in the ¼ chameleon from its free evolution without the kick is Particle Mass, mk gk=g? (at T mk) Type a t 173 GeV 12=106:75 Fermion ¼ M 1 k ; (19) Z 91 GeV 3=95:25 Boson k Pl a W 80 GeV 6=92:25 Boson which tends to M in the limit a a . The above b 4 GeV 12=86:25 Fermion k Pl k 1.7 GeV 4=75:75 Fermion result assumes the particle drops out of thermal equilib- c 1.27 GeV 12=72:25 Fermion rium instantaneously relative to the background evolution, 0.14 GeV 3=17:25 Boson which is only valid when the duration of the kick is much 0.105 GeV 4=14:25 Fermion smaller than the oscillation period. e 0.5 MeV 4=10:75 Fermion F. Case 2: akick aosc For the situation when the duration of the kick is sig- the transition between the two regimes occurs when nificantly greater than the oscillation period of the chame- the period of oscillation satisfies a2=a1 100, where leon, we use an adiabatic approximation treating ða=a Þ ¼ k a2 a1 aosc. as constant over one oscillation. We follow the same analysis as in Secs. III B and III C but with the extra source E. Case 1: akick aosc term from T : For the situation when the duration of the kick is much 1 d d X a H a4H ’ PMF0 a 4 H2M 1 ; less than the chameleon period of oscillation, we take the 2 Pl k a da da MF k ak contribution to T to be approximated by a delta function. The chameleon spends most of its time during oscillations where a1 is the scale factor at the start of the oscillation and on the right-hand side of the potential ( > min), and so we are assuming does not change significantly from a1 to we make the reasonable assumption that the chameleon a2 (the scale factor at the end of the oscillation). will be rolling along this side of the potential when the kick Substituting in Eq. (11) for the Hubble expansion, and occurs. As before, Eq. (10) is approximated by solving, we find 1 d d 1 d H a4H ’ PMF0 a 4 þ T ; ’ a 1 þ Aa 2; 2 eff a da da MF M da (20) ðaÞ’ loga Aa 1 þ B; and HðaÞ is given by Eq. (11). Approximating ðxÞ as a eff ¼ delta function centered on a ak for each kick, Eq. (18) which is identical to Eq. (12) but with replaced by becomes X a þ M 1 : ðkÞ 45 g eff Pl k 2ð Þ 2 k ð Þ ak T 4 H a MPl ak a ak ; k g?ðakÞ In this regime the effect of the kicks is similar to a tempo- where ak T0=mk, and 4:9 4:6 is the total rary increase in the magnetic field strength, and generally area under the ðxÞ curve. Note that the delta function boosts the oscillations into the ‘‘fast oscillations’’ regime ð Þ¼ ð Þ satisfies x=x0 1 x0 x x0 . Then, defining k described by Eq. (17). ð Þ 4 45MPl gk=g? = M, d d a2 ’ M a ða a Þ; IV. RESULTS a da k Pl k k d The equations derived in Secs. III B to III D provide a 2 where as before PMF0= r0H0MF. Integrating, framework for determining the evolution of the chameleon we find field given a specific starting value at the end of inflation. 8 Figure 3 contains a few examples of the resulting predic- < 1 2 d a þ Aa ;a 123002-8 EVOLUTION OF THE CHAMELEON SCALAR FIELD IN ... PHYSICAL REVIEW D 86, 123002 (2012) 5 10 0 5 0 −5 −5 −10 /M) /M) −10 φ φ ( ( −15 10 10 −15 log −20 log −20 −25 −25 −30 −30 −35 −35 −24 −22 −20 −18 −16 −14 −12 −10 −8 −24 −22 −20 −18 −16 −14 −12 −10 −8 log (a) log (a) 10 10 20 10 15 5 10 0 5 −5 /M) /M) 0 −10 φ φ ( ( 10 10 −15 −5 log log −20 −10 −25 −15 −30 −20 −35 −25 −24 −22 −20 −18 −16 −14 −12 −10 −8 −6 −24 −22 −20 −18 −16 −14 −12 −10 −8 −6 log (a) log (a) 10 10 5 0 −5 −10 /M) φ ( −15 10 log −20 −25 −30 −35 −24 −22 −20 −18 −16 −14 −12 −10 −8 −6 log (a) 10 FIG. 3 (color online). Chameleon evolution from the end of inflation to BBN, for different B, M and i. The dashed blue line is the evolution of min. Green circles mark the actual location of the chameleon field including oscillations between its maximum amplitude and the minimum, while the solid green line is the evolution of max once it enters the fast oscillations regime. The red crosses mark the location of the kicks as different particle species drop out of thermal equilibrium. The behavior of the chameleon is strongly dependent on on the strength of the primordial magnetic field were its starting value and the strength of the chameleon inter- discussed in the introduction: observations of the action with the primordial magnetic field. The strength of CMB limit the magnitude of B0 to be no greater than the magnetic interaction depends on both the size of the approximately 5 nG. The strength of the electromagnetic magnetic field B0 and the coupling strength between the interaction in the chameleon model is constrained by chameleon and electromagnetic field 1=MF. The bounds observations of starlight polarization in our galaxy to 123002-9 DAVID F. MOTA AND CAMILLA A. O. SCHELPE PHYSICAL REVIEW D 86, 123002 (2012) * 9 MF 1:1 10 GeV [17] (see Sec. II). The chameleon period is given by Eqs. (14) and (15). In the limit a2 a1 evolution after inflation depends on the parameter , we can approximate this by defined in Eq. (13), and as such is degenerate between B0 a2 and M . We define ’ 2 exp 1 þ 1 : F a 1 B 2 M 1 ¼ 23 ¼ 0 F ; Substituting for a1 ai 10 and a2 akick B 9 15 5nG 1:1 10 GeV 3:77 10 leads to the condition 1 18 , which implies the transition occurs for to characterize this dependence. Given the constraints on ð Þ & crit 5 B0 and MF, it is only realistic to consider the range B 1. i 3:8 B 10 MPl: (21) We assume the following values for the cosmological When the chameleon field value at the end of inflation is ’ 2 ’ 2 ’ 5 parameters: h 0:7, m0h 0:13 and r0h 2:5 10 ðcritÞ 1 1 less than i the chameleon will immediately begin to [49], where H0 100h km sec Mpc , and we take ’ roll towards the minimum of its potential and start oscillat- n 2 in the model for the self-interaction potential. ðcritÞ The plots in Fig. 3 illustrate the two distinct regimes that ing. For field values greater than i the chameleon is the chameleon evolution falls into, depending on the values effectively frozen at this position until particle species we choose for the different parameters. For larger magnetic begin dropping out of thermal equilibrium. In the next fields and/or when the chameleon lies closer to its mini- section we discuss the implication of these results on the mum at the end of inflation, we find that the chameleon ability of the chameleon to satisfy constraints on the varia- starts to roll towards the minimum of its potential and tion of particle masses after BBN. begins oscillating. See for example Figs. 3(a) and 3(b). Often the behavior satisfies the fast oscillations regime so V. BBN CONSTRAINTS that the amplitude of the chameleon oscillations decays according to the power law defined in Eq. (17). In this The analysis of Brax et al. [19], which neglected the scenario the effect of the kicks is to boost the chameleon chameleon interaction with electromagnetism, imposed a into the fast oscillations regime, or to have no effect if the constraint of 10% on the allowed variation of particle field is already oscillating rapidly. masses after BBN. This places a limit on the maximum For smaller magnetic field values, or when the chame- amplitude of the chameleon oscillations at the onset of leon is released far away from its minimum, we find that BBN. The bound on the allowed variation of particle the field remains stuck at its initial value until species start masses coming from big bang nucleosynthesis is sensitive dropping out of thermal equilibrium and push the chame- to a number of parameters which are not well understood, leon towards the minimum. This process, however, is and so the constraint of 10% does not reflect a hard highly unstable. If the kick is sufficient to overshoot the experimental bound. We discuss the BBN constraints in minimum then the chameleon effectively bounces off the more detail below, but it is beyond the scope of this paper steep left-hand side of the potential and can shoot back to to perform a full analysis. Instead, we investigate the even larger field values. The chameleon to matter coupling effects on the chameleon model of imposing a fractional strength described by M determines the energy introduced constraint, BBN, on the allowed variation of particle ¼ to the system by the kicks. We saw in Eq. (19) that in the masses, and explicitly consider the case BBN 0:1 for a ¼ / direct comparison to the results of Brax et al. [19]. limit a ak, kMPl , where 1=M. Thus the stronger the matter coupling, the greater will be the jump in The BBN constraints are relevant because the amplitude the evolution as a result of the kicks. For example, of the chameleon oscillations can potentially result in a large variation in the masses of fundamental particles. The Figs. 3(d) and 3(e) have the same B and i but two choices of M. The smaller M value sends the field to matter particles follow geodesics of the conformal metric, 2 ð Þ much larger values as a result of the kicks. It is this second Am g . In the conformal (Jordan) frame, in which the regime that exists in the case of no primordial magnetic chameleon is nonminimally coupled to gravity, the particle field or when the electromagnetic interaction of the cha- masses mJ are independent of . However in the physical meleon is ignored. For this reason, the initial conditions Einstein frame they acquire a dependence: mð Þ¼ ð Þ considered by Brax et al. [19] had to be finely tuned so that Am mJ [50]. Variation in , therefore, leads to a varia- the chameleon ended up near the minimum of its potential tion in the observed mass of particles: at the onset of BBN. ð Þ 0 ð Þ m Am 1 The ratio of to dictates whether the chameleon has ¼ j j¼ j j: i B mð Þ A ð Þ M started to oscillate before the first kick when the top quark m drops out of thermal equilibrium. We approximate the The earliest time at which cosmological observations can transition between these two regimes by the requirement constrain the variation in particle masses is at big bang that the field has oscillated exactly once between its release nucleosynthesis. The primordial element abundances are at the end of inflation and the first kick. The oscillation sensitive to variation in the parameters of the Standard 123002-10 EVOLUTION OF THE CHAMELEON SCALAR FIELD IN ... PHYSICAL REVIEW D 86, 123002 (2012) Model of particle physics. Dent et al. [51,52] investigated B 2 M 1 0 F * 2:4 10 6 i : (23) the response of the element abundances to individual var- 9 5nG 1:1 10 GeV MPl iations in the fundamental parameters as well as consider- ing unified models which combined the variation of more Even when the primordial magnetic field is large enough to than one parameter. Varying only one of the parameters at a cause the chameleon to oscillate prior to the onset of the time, they were able to place 2 bounds on the allowed kicks, there is a limit on the speed with which the ampli- variation for the different fundamental couplings from tude of the oscillations can decay, which further constrains BBN until today. The allowed range for the mass of the the parameters. We consider the case when the chameleon þ electron was found to be 17% lnme 9%. is in the fast oscillations regime from the very start of its Although this result neglects the complex interactions evolution. In this situation the chameleon converges most from multiple fundamental parameters varying, it allowed rapidly on the minimum following the power-law behavior Brax et al. [19] to estimate a constraint on the variation in described by Eq. (17). Although it requires a large ratio of particle masses arising from the chameleon model to be = i to place us in this regime, the convergence is inde- less than approximately 10%. In our analysis we generalize pendent of . If the initial conditions for the field satisfy this constraint to some fraction BBN, which imposes i= 1, then the subsequent evolution is determined by max ¼ 2 a ðBBNÞ & M; (22) log log ; BBN i 3 ai ðBBNÞ 10 ðtodayÞ and so 10 . To satisfy the BBN constraints, since 108 M. The strength of the mat- max i min BBN this imposes ter coupling described by M has a significant impact on whether the constraints at BBN are satisfied. There are & 10 i 10 BBNM: (24) subtler issues of whether the chameleon itself influences the primordial element abundances; see Ref. [53] for a Figure 4 shows the exclusion bounds determined em- general review of primordial nucleosynthesis and the ef- pirically from requiring the chameleon to fall below the ¼ fects of nonstandard physics. However, as first shown by BBN 0:1 threshold at BBN. These are presented for Brax et al. [19], the deviations of the chameleon model two different choices of the matter coupling strength: ¼ 9 ¼ from standard LCDM are insignificant at the scales rele- M 1:1 10 GeV and M MPl, motivated by the con- vant for the expansion rate and baryon/photon ratio. strained value of MF assuming M MF and a natural In the previous section we were able to categorize the gravitational scale respectively. The bounds agree closely chameleon behavior into two distinct groups depending on with the limiting cases given in Eqs. (23) and (24). The whether the parameter values were such that the chame- transition between when the chameleon is oscillating and leon was oscillating about the minimum of its potential prior to the top quark dropping out of thermal equilibrium, BBN exclusion bounds for M=1.1×109 GeV and M=M Pl or whether the field was frozen at its initial value. It is this 6 second scenario that occurs when there is no interaction 4 between the chameleon and electromagnetic field, and corresponds to the situation analyzed by Brax et al. [19]. 2 When the particle species drop out of thermal equilibrium ) 0 Pl they introduce a a short-lived boost of energy into the /M i φ system and kick the chameleon field towards its minimum. ( −2 The initial field value of the chameleon needs to be 10 log −4 strongly fine-tuned for the cumulative effect of the kicks to result in the field being positioned close to the minimum −6 of its potential, rather than overshooting and rolling back −8 to even larger field values, as shown by the plots in −10 Figs. 3(c)–3(e). We find that in general, whenever the −14 −12 −10 −8 −6 −4 −2 0 log (ξ ) chameleon starts out frozen at its initial value, it is impos- 10 B sible to satisfy the constraints on the chameleon field at BBN for all < Oð1Þ without extreme fine-tuning. FIG. 4 (color online). The parameter space of i and B that is BBN excluded by BBN constraints when ¼ 0:1. The dashed Only if the allowed variation of particles is a few 100% BBN blue line marks the excluded region for M ¼ 1:1 109 GeV, will the chameleon fall below the appropriate threshold at and the dotted red for M ¼ MPl. The allowed parameter space BBN. The parameter space boundary that separates the two lies to the bottom right of the exclusion lines. The solid green regimes was derived in Eq. (21). For a given initial field line marks the transition from the chameleon oscillating prior to value i, this implies the following constraint on B0 and the kicks, and the one in which the field is frozen prior to the Oð Þ MF for all BBN < 1 : kicks. It is independent of M. 123002-11 DAVID F. MOTA AND CAMILLA A. O. SCHELPE PHYSICAL REVIEW D 86, 123002 (2012) when it is frozen at its initial value marks the BBN exclu- the most conservative bounds on the magnetic field sion bound for small i and, as we can see from Fig. 4, strength), we find the following allowed band for the there is a saturation point at i 1MPl for the case of M ¼ chameleon-photon coupling strength, 1:1 109 GeV, which is dependent on M but independent of B, which excludes larger values of i. There is a small 1 B 2 M B 2 0 & F & 4 0 range of intermediate B values for which the field oscil- 600 9 1:6 10 : lates more than once before the first kick, but is not actually 1nG 10 GeV 1nG in the fast oscillations regime from the start of its evolution [Fig. 3(b) for example], which forms a smooth transition For primordial magnetic fields smaller than 10 10 G, there between the two limiting cases in the exclusion plot. are no coupling strengths that can satisfy these combined The bounds in Eqs. (23) and (24) lead to a constraint on bounds. We note that the allowed degree of chameleon- B0, MF and M if we impose a range of i that we consider photon mixing in the early universe places an upper bound likely for the chameleon field at the end of inflation. They on the PMF strength and chameleon-electromagnetic in- only apply if the constraints at BBN limit the variation of teraction, while for the chameleon to be well behaved in particle masses after BBN to be less than Oð1Þ, but they are the early universe and satisfy constraints at BBN requires a not sensitive to the exact value of 10% imposed by Brax much stronger magnetic interaction to drive the chameleon et al. [19]. We consider a few different physical energy to its minimum in time. There is only a narrow band of scales which could act as natural initial conditions for the parameter values that satisfy these constraints and in all * chameleon field at the end of inflation. The resulting cases they require B0 0:1nG. For different values of the bounds are magnetic field spectral index the bounds become even stronger with B * 2nGfor n ¼ 1:0. 0 B These bounds hold for any 123002-12 EVOLUTION OF THE CHAMELEON SCALAR FIELD IN ... PHYSICAL REVIEW D 86, 123002 (2012) In this article, we have demonstrated that the presence of natural value for the chameleon at the end of inflation is Oð Þ a primordial magnetic field can help drive the chameleon up to and including i MPl . Combined with the field to the minimum of its potential. Without a primordial tentative <100% constraint at BBN, this places a degen- magnetic field (or at least with a very weak one), the erate constraint on the PMF strength and chameleon- chameleon is frozen at its initial value at the end of photon coupling, inflation and requires the energy from particle species B 2 M 1 dropping out of thermal equilibrium to drive it towards 0 F * 2:4 10 6: 9 the minimum. However, this process is very unstable since 5nG 1:1 10 GeV the energy introduced by particle species going nonrela- The degeneracy of the constraint can be broken by con- tivistic is akin to a delta function and easily causes the field sidering the effects of chameleon-photon mixing in the to overshoot the minimum and oscillate more violently. By primordial magnetic field [16]. This imposes a narrow contrast, a strong PMF causes the field to oscillate about band of allowed parameters which can only be satisfied * the minimum of its potential with a decaying amplitude as for B0 0:1nG. These magnetic field values are close to the Universe expands, and can satisfy constraints at BBN the level that could be detected or ruled out by the next for a range of initial conditions. generation of CMB experiments. We also find that the The paper by Brax et al. [19] that originally examined chameleon to matter coupling strength is constrained by the cosmological evolution of the chameleon in the case of the limit at BBN to have zero EM coupling imposed an upper limit on the chame- * 8 leon field at BBN arising from a 10% constraint on the M 10 GeV; variation of particle masses. This constraint was suggested which is many orders of magnitude greater than the by Dent et al. [51,52] based on an analysis of the primor- existing direct constraints on M. dial element abundances at BBN. However, the degener- Our analysis has been with specific reference to the acies between the different factors influencing the element chameleon model [5,6]. However many theories of modi- abundances are not well understood and so this can only be fied gravity are required to have some form of chameleon taken as an estimate. mechanism if they are to satisfy the many weak-field limit In this work we have developed a semianalytical solu- tests and explain dark energy or dark matter [1]. It would tion to the early universe evolution of the chameleon as it be interesting to investigate the role that can be played by falls towards the minimum of its potential, allowing for primordial magnetic fields in these alternative theories, in the coupling of the chameleon to any electromagnetic the light of our results. fields. This provides a valuable framework for anyone wishing to analyze the evolution of the chameleon in the light of new experimental bounds that may be placed ACKNOWLEDGMENTS upon the theory. 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