Chasing chameleons L. Kraiselburd 1 , 4 , S. Landau 2 , D. Sudarsky 3 , M. Salgado 3 and H. Vucetich 1 . 1 Facultad de Ciencias Astron´ omicas y Geof´ ısicas, UNLP , Argentina. 2 Instituto de F´ ısica, CONICET-UBA, Argentina. 3 Instituto de Ciencias Nucleares, UNAM, Mexico. 4 CONICET, Argentina. GGI-Workshop. Firenze, 2016. April,2016 – p. 1
Motivations • The greatest surprise of modern cosmology was the observation that the Universe is accelerating in its expansion. • While the data are consistent with the expansion being driven by a Λ , dark energy is more generally modeled by a scalar field rolling down an almost flat potential. ◦ It is expected that such field to be essentially massless on solar system scales. • If this field exists, why it has not been detected in local tests of the EP and 5 th force searches? April,2016 – p. 2
Motivations • Khoury & Weltman 2004, proposed novel solution to this problem, the “chameleon effect” whereby the coupling of a light scalar field to matter is effectively suppressed via a background dependent induced effective mass for these fields: ◦ in places where ρ matter is high, the particle interaction is weak; ◦ in places where ρ matter is low, the particle interaction is strong; The Universe could be being pushed by the Chameleon´s force. April,2016 – p. 3
Motivations • According to Mota & Shaw 2007 update; ◦ The most simple models break the Weak Equivalence Principle (WEP). ◦ This violation does not happen in the no-linear regimen; the chameleon fields and/or their interactions with matter are independent of the composition of bodies in free fall because these effects are only relevant in a small region on the surface of bodies. THIN SHELL April,2016 – p. 4
Motivations • The WEP is incorporated ab initio by pure metric-based theories while it is violated by construction by models such as “chameleons” even when referring to point test particles . • This violation might not be observable in experiments due to the “screening phenomenon” BUT can be exacerbated when considering test bodies . • We shall analyze the two body problem (both extended) embedded in a light medium. Preliminary results show detectable violations of the WEP . However, when considering the test body encased in a shell of dense material (like the chamber in the experiment) this violations are strongly supressed. April,2016 – p. 5
Motivations • With similar arguments to those proposed by Hui et al., we want to show: ◦ difference in acceleration depends on the properties of the test bodies even when the coupling β i is universal; ◦ when the thin shell effect becomes relevant, the physical objects must be considered as extendend bodies, and an effective violation of the WEP appears. April,2016 – p. 6
Chameleon models • In this scenario, the action is given by: d 4 x √− g � M pl � � � 2 R − ( ∂ Φ) 2 − V (Φ) � � d 4 xL m Ψ ( i ) m , g ( i ) S = − µν L m is the lagrangian of the matter fields and g ( i ) µν = exp [ 2 β i Φ M pl ] g µν . Φ − n ; being M Λ ∼ 10 − 3 eV the dark The potential V (Φ) ∝ M n +4 Λ energy scale; n y β i constant dimensionless parameters of the theory. The key of the model: The no-linears effects are only relevant in a very small zone near the surface of the body called thin shell ; Φ ∞ − Φ C = ∆ R << 1 6 βM pl Φ N R April,2016 – p. 7
Chameleon models V eff = V (Φ) + A (Φ) V (Φ) = λM n +4 Φ − n , A (Φ) = − T m e β Φ /M pl Λ April,2016 – p. 8
Chameleon models • The equation of motion is: ✷ Φ = ∂V eff ∂ Φ , V eff (Φ) ≃ V eff (Φ min ) + 1 2 ∂ ΦΦ V eff (Φ min )[Φ − Φ min ] 2 . • Defining the “effective mass”: m 2 eff = ∂ ΦΦ V eff (Φ min ) , 1 r ∂r [ r 2 ∂r Φ] = m 2 eff [Φ − Φ min ] . • The thin-shell condition becomes: m eff R >> 1 April,2016 – p. 9
Chameleon models • The force mediated by the chameleon is: F Φ = − β M tp � ∇ Φ . (1) M pl • The force due to a compact body of radius R and mass M c is generated by the gradient of the chameleon field outside the body which interpolates between the minimum inside and outside the body. • Inside the solution is nearly constant up to the boundary of the object and jumps over a thin shell ∆ R R . • Outside the field is given by, 3∆ R β M c Φ ≈ Φ ∞ − (2) M pl R r April,2016 – p. 10
The situation to analyze is given by, �� �� � � � ��������� �� �� �� � � � � � � � � ���������� � April,2016 – p. 11
Our proposal • We take the complete solution of ✷ Φ = m 2 eff [Φ − Φ min ] in 3 regions: ◦ Inside the massive body (MB) Φ 1 , and the test body (TB) Φ 2 ; and outside both bodies Φ 3 • We analyze the case when the 2 bodies contribute to the external field. • The boundary conditions are : r → 0 ∂ r Φ 1 , 2 = 0 lim so as r → 0 Φ 1 , 2 = Φ C 1 , 2 ; lim r →∞ ∂ r Φ 3 = 0 lim so as r →∞ Φ 3 = Φ ∞ ; lim ∂ Φ j ∂r = ∂ Φ 3 Φ j = Φ 3 | R j ; ∂r | R j , j = 1 , 2 April,2016 – p. 12
Our proposal The most general solution is; lm C 1 Φ 1 = � lm i l ( µ 1 r ) Y lm ( θ, φ ) + Φ C 1 r ≤ R 1 lm C 3 . 1 Φ 3 = � lm k l (ˆ µr ) Y lm ( θ, φ ) + outside both Φ = C 3 . 2 µr ′ ) Y lm ( θ ′ , φ ′ ) + Φ ∞ lm k l (ˆ bodies r ′ ≤ R 2 lm C 2 lm i l ( µ 2 r ′ ) Y lm ( θ ′ , φ ′ ) + Φ C 2 Φ 2 = � µ = m 3 eff . We calculate the C j µ 1 , 2 = m 1 , 2 eff and ˆ lm thanks to the next transformations with | r | ≤ | D | y | r ′ | ≤ | D | ; and we truncate the series with N = e ˆ µ | D | ; 2 vw ( � vw α ∗ lm µr ′ ) Y vw ( θ ′ , φ ′ ) µr ) Y lm ( θ, φ ) = � k l (ˆ D ) i v (ˆ vw ( � µr ′ ) Y lm ( θ ′ , φ ′ ) = � vw α lm k l (ˆ D ) i v (ˆ µr ) Y vw ( θ, φ ) , April,2016 – p. 13
Our proposal � � r � � � r � 10 000 10 000 8000 8000 6000 6000 4000 4000 2000 2000 r r 500 1000 1500 1 2 3 4 5 6 For the same length of interval, the “thin shell effect” is more notorious in the large body (hill) that in the test body (small sphere of aluminum). For this case, n = β = 1 and the bodies are immersed in the Earth’s atmosphere. April,2016 – p. 14
Our proposal In order to calculate the force chameleon, we calculate the energy of the whole system which depends on Φ , � T Φ 00 + T m = U Φ 00 dV, V 2 ∇ 2 Φ + V eff (Φ) + ρ + β Φ T m � − Φ � � = dV, M pl V V eff (Φ) + (3 + n ) β Φ T m � − (2 + n ) � � = + ρ dV 2 2 M pl V and derive it respect to the position between the bodies F Φ = − ∂U � . ∂ � D Taking the limit R TB → 0 , we recover the predictions for the “test particle” model. April,2016 – p. 15
Results • We get the acceleration due to chameleon force � a C and so we can evaluate the expression η ∼ | � a T A − � a T B | | � a T A + � a T B | ( � a T = � a C + � g ) to compare with Eöt-Wash torsion-balance experiments (WEP) (Be-Al-Hill). • We use two different environments; the Earth´s atmosphere and the chamber´s vacumm. • The test bodies no longer have thin shell for β ≤ 10 − 1 . 5 , while in the cases of the massive body β can be much more smaller. April,2016 – p. 16
Results -6 -3 n=1 n=2 n=3 -4 -7 n=4 Eot-Wash bound -5 -8 -6 -9 -7 Log10( η ) Log10( η ) -10 -8 -11 -9 -12 -10 n=1 -13 n=2 -11 n=3 n=4 Eot-Wash bound -14 -12 -5 -4 -3 -2 -1 0 -6 -4 -2 0 2 4 Log10( β ) Log10( β ) In the left figure ρ out is the Earth’s atmosphere, and in the right one is the chamber’s vacuum. April,2016 – p. 17
Results Brax made us notice that in these particular models, the effect of the layer of the vacuum chamber should be taken into account, Upadhye (2012). 0 0 n=1 n=1 n=2 n=2 n=3 n=3 n=4 n=4 -5 -5 Eot-Wash Bound Eot-Wash Bound -10 -10 -15 -15 Log10( η ) Log10( η ) -20 -20 -25 -25 -30 -30 -35 -35 -40 -40 -5 -4 -3 -2 -1 0 1 2 3 -5 -4 -3 -2 -1 0 1 2 3 Log10( β ) Log10( β ) In the left figure ρ out is the Earth’s atmosphere, and in the right one is the chamber’s vacuum. The force suppression factor ∼ sech(2 m layer eff d ) , being d the diameter of the layer. April,2016 – p. 18
Results The LLR experiment test the WEP without the shielding between the test bodies and (Earth-Moon) and the source (Sun). -8 -8 n=1 n=1 n=2 n=2 -10 n=3 n=3 -10 n=4 n=4 LLR bound LLR bound -12 -12 -14 -14 -16 -16 Log10( η ) Log10( η ) -18 -18 -20 -20 -22 -22 -24 -24 -26 -26 -28 -30 -28 -3 -2 -1 0 1 2 -3 -2 -1 0 1 2 Log10( β ) Log10( β ) In the left figure the Earth and the Sun are surrounded by their atmospheres, and in the right not. In both cases ρ out is density of the interstellar medium and the three bodies have thin shell. April,2016 – p. 19
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