From the Horndeski Lagrangian to observations
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Horndeski Lagrangian: too big to fail? Luca Amendola University of Heidelberg in collaboration with Martin Kunz, Mariele Motta, Ippocratis Saltas, Ignacy Sawicki Benasque 2012 Observations are converging… …to an unexpected universe Benasque 2012 Classifying the unknown, 1 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. Cosmological constant Dark energy w=const Dark energy w=w(z) quintessence scalar-tensor models coupled quintessence mass varying neutrinos k-essence Chaplygin gas Cardassian quartessence quiessence phantoms f(R) Gauss-Bonnet anisotropic dark energy brane dark energy backreaction void models degravitation TeVeS oops....did I forget your model? Benasque 2012 Classifying the unknown, 2 a) b) c) d) Lambda and w(z) models (i.e. change only the expansion) modified matter (i.e. change the way matter clusters) modified gravity (i.e. change the way gravity works) non-linear effects (i.e. change the underlying symmetries) Benasque 2012 Prolegomena zu einer künftigen jedenjeden künftigen DarkMetaphysik Energy physik ©Kant Observational requirements: Physical requirements: A) Isotropy Scalar field B) Large abundance C) Slow evolution D) Weak clustering Benasque 2012 Vπ ≈ π»2 0 Vπ′ βͺπ Vπ′′ βͺπ Theorem 1: A quintessential scalar field The most general 4D scalar field theory with second order equation of motion ο© οΉ 4 dx ο g L + L matter οΊ οͺο₯ i ο² ο« i ο» οΌ First found by Horndeski in 1975 οΌ rediscovered by Deffayet et al. in 2011 οΌ no ghosts, no classical instabilities οΌ it modifies gravity! οΌ it includes f(R), Brans-Dicke, k-essence, Galileons, etc etc etc Benasque 2012 Simplest MG: f(R) The simplest Horndeski model which still produces a modified gravity: f(R) ο² dx 4 g ο f ο¨R ο©+ Lmatter ο οΌ equivalent to a Horndeski Lagrangian without kinetic terms οΌ easy to produce acceleration (first inflationary model) οΌ high-energy corrections to gravity likely to introduce higherorder terms οΌ particular case of scalar-tensor and extra-dimensional theory Benasque 2012 Theorem 2: the Yukawa correction Every Horndeski model induces at linear level, on sub-Hubble scales, a Newton-Yukawa potential GM οr /ο¬ ο (r ) ο½ ο (1 ο« ο‘ e ) r where α and λ depend on space and time Every consistent modification of gravity based on a scalar field generates this gravitational potential Benasque 2012 The next ten years of DE research Combine observations of background, linear and non-linear perturbations to reconstruct as much as possible the Horndeski model … or to rule it out! Benasque 2012 The great Horndeski Hunt Let us assume we have only 1) pressureless matter 2) the Horndeski field and Benasque 2012 Background: SNIa, BAO, … Then we can measure H(z) and 1 dz D( z ) ο½ sinh( H 0 οο k 0 ο² ) H ( z) H 0 οο k 0 and therefore Ωπ0 Then we can measure everything up to Ωπ0 Benasque 2012 Two free functions The most general linear, scalar metric ds 2 ο½ a 2 [(1 ο« 2ο)dt 2 ο (1 ο« 2ο)(dx 2 ο« dy 2 ο« dz 2 )] At linear order we can write: ο§ Poisson’s equation π» 2 Ψ = 4ππΊ π π, π ππ πΏ π ο§ anisotropic stress ο ο«ο ο ο¨ (k0, aο½) ο½ ο οο Benasque 2012 Modified Gravity at the linear level ο§ standard gravity ο§ scalar-tensor models ο§ f(R) ο§ DGP ο§ coupled Gauss-Bonnet Y (k , a) ο½ 1 ο¨ (k , a) ο½ 1 G* 2( F ο« F '2 ) Y (a) ο½ FGcav ,0 2 F ο« 3F '2 F '2 ο¨ (a) ο½ 1 ο« F ο« F '2 k2 1 ο« 4m 2 * G a R, Y (a) ο½ k2 FGcav ,0 1 ο« 3m 2 a R Boisseau et al. 2000 Acquaviva et al. 2004 Schimd et al. 2004 L.A., Kunz &Sapone 2007 k2 a2 R ο¨ (a) ο½ 1 ο« k2 1 ο« 2m 2 a R m 1 ; ο’ ο½ 1 ο« 2 Hrc wDE 3ο’ 2 ο¨ (a) ο½ 1 ο« 3ο’ ο 1 Bean et al. 2006 Hu et al. 2006 Tsujikawa 2007 Y (a) ο½ 1 ο Y (a) ο½ ... ο¨ (a) ο½ ... Benasque 2012 Lue et al. 2004; Koyama et al. 2006 see L. A., C. Charmousis, S. Davis 2006 Modified Gravity at the linear level Every Horndeski model is characterized in the linear regime and for scales ππ π β« 1 by the two functions a1 ο½ 2 B6 B8ο2 a2 ο½ B8 B6ο1 de Felice, Tsujikawa 2011 Benasque 2012 Linear observables Matter conservation equation If we could observe directly the growth rate… 3 − π(π, π§)Ωπ 2 we could test the HL… Benasque 2012 πΏ′ π= πΏ Reconstruction of the metric ds2 ο½ a2[(1 ο« 2ο)dt 2 ο (1 ο« 2ο)(dx2 ο« dy 2 ο« dz 2 )] massive particles respond to Ψ massless particles respond to Φ-Ψ Benasque 2012 Linear observables πΏπππ π, π§ = πΊ(π, π§) π(π, π§) π8 πΏπ‘,0 (π) Benasque 2012 Three linear observables Amplitude Redshift distortion clustering lensing Σ = π(1 + π) Lensing Benasque 2012 Two model-independent ratios Amplitude/Redshift distortion Lensing/Redshift distortion Benasque 2012 f P1 ο½ b οm0ο P2 ο½ f Observing the HL Lensing/Redshift distortion οm0ο P2 ο½ f If we can obtain an equation for P2 then we can test the HL Benasque 2012 Two model-independent ratios We combine now growth rate and lensing Benasque 2012 A consistency equation οm0ο P2 ( k , z ) ο½ f Differentiating P2 and combining with the growth rate and lensing equations we obtain a consistency relation for the HL valid for every k that depends on 8 functions of z πΆ π; ππ π§ = Benasque 2012 A consistency equation ππ = If we estimate P2(k,z) for many k’s we have an overconstrained system of equations πΆ(π1 ; ππ (π§)) = 0 πΆ(π2 ; ππ (π§)) = 0 πΆ(π3 ; ππ (π§)) = 0 πΆ(π4 ; ππ (π§)) = 0 …….. If there are no solutions, the HL is disproven! Benasque 2012 Reconstructing the HL ? We can estimate ππ = And therefore partially reconstruct the HL but the reconstruction is not unique: an infinite number of HL will give the same background and linear dynamics Benasque 2012 Conclusions • The (al)most general dark energy model is the Horndeski Lagrangian • It contains a specific prescription for how gravity is modified, the Yukawa term • Linear cosmological observations constrain a particular combination of the HL functions • Quantities like Ωπ0 , π, ππ8 (π§) are unobservable with linear observations • In principle, observations in a range of scales and redshifts can rule out the HL The HL is not too big to fail! (but it is too big to be reconstructed) Benasque 2012
