Fundamental Physics with the Euclid satellite. Luca Amendola University of Heidelberg

Transcription

1 Fundamental Physics with the Euclid satellite Luca Amendola University of Heidelberg 1

2 Outline Lecture 1 The Euclid satellite Lecture 2 Testing gravity with Euclid Suggested texts L.A. et al arxiv: Living Rev.Rel. 21 (2018) no.1, 2 L.A. and S. Tsujikawa, Dark Energy. Theory and Observations, Cambridge U. P. Proceedings of the Pedra Azul school 2017 Disclaimer: some material in these slides has been taken from other authors and is believed to be in the public domain. Whenever known, I hope I have always correctly credited the authors. I apologize if I have missed some.

3 L. Amendola, Vitoria,

4 Friedman equation!a a 2 = H 2 = 8π 3 ρ GR+Homogeneity+Isotropy: Friedmann equation Ω m,0 = 8πρ m0 3H, Ω = 8πρ rad,0, Ω 2 rad, H Λ,0 = 8πρ Λ, H 0 H 2 = H 2 0 Ω m,0 (1+ z) 3 + Ω rad,0 (1+ z) 4 + Ω Λ,0 1 = Ω m,0 + Ω rad,0 + Ω Λ,0, a = (1+ z) 1 H 2 = H 0 2 Ω i,0 (1+ z) 3(1+w i ) L. Amendola, Vitoria,

5 Cosmology Executive Summary Dark matter 27% Baryons 5% Massive neutrinos: 0.1% Photons: 0.01% Humans: % Spatial curvature: very close to 0 Something else: 70% L. Amendola, Vitoria,

6 Cosmic inventory Cosmic Inventory name density EOS w baryons CDM radiation /3 Massive neutrinos < Cosm. const curvature <0.01-1/3 Other??? L. Amendola, Vitoria,

7 Euclid Surveys Simultaneous (i) visible imaging (ii) NIR photometry (iii) NIR spectroscopy 15,000 square degrees 40 million redshifts, 2 billion images Median redshift z 1 PSF FWHM ~0.18 >1000 peoples, >10 countries Euclid in a nutshell Euclid satellite L. Amendola, Vitoria, arxiv Red Book

8 Euclid s goals Nature of dark energy (DE), test of gravity (MG) and of dark matter (DM) Distinguish DE, MG, DM effects on the geometry of the Universe: Weak Lensing (WL), Galaxy Clustering (GC), on structure formation: WL, Redshift-Space Distortion, Clusters of Galaxies controlling systematic residuals to a very high level of accuracy. (elaboration from a talk by Y. Mellier) L. Amendola, Vitoria,

9 Quantifying precision Parameterising our ignorance: DE equation of state: P/ρ = w and w(a) = w p + w a (a p -a) What precision should we aim to? From Euclid data alone, get FoM=1/(Dw a x Dw p ) > 400: if data consistent with Λ, and FoM > 400 then : Λ is favoured with odds of more than 100:1 = a decisive statistical evidence. (elaboration from a talk by Y. Mellier) L. Amendola, Vitoria,

10 WL and GC: optimal primary probes for Euclid WL and GC Weak Lensing (WL) 3-D cosmic shear measurements (tomography) over 0<z<3 probes distrib. of matter (D+L), expansion history, growth factor. photo-z sufficient Galaxy Clustering (GC) 3-D position measurements over 0.5<z<2 probes clustering history of galaxies induced by gravity spectroscopic redshifts needed. (elaboration from a talk by Y. Mellier) L. Amendola, Vitoria,

11 The Euclid Mission: baseline and options In ~5.5 years 15,000 deg 2 40 deg 2 Shapes + Photo-z of n = 1.5 x10 9 galaxies? z of n=5x10 7 galaxies Possibility to propose other surveys: SN and/or m-lens surveys, Milky Way? Ref: Euclid RB arxiv: L. Amendola, Vitoria,

12 Euclid: optimised for shape measurements M51 z=0.1 z=0.1 z=0.7 Euclid images of z~1 galaxies: same resolution as SDSS images at z~0.1 and at least 3 magnitudes deeper. Space imaging of Euclid will outperform any other surveys of weak lensing. (elaboration from a talk by Y. Mellier) 12

13 Third Euclid probe: Clusters of galaxies Clusters of galaxies: probe of peaks in density distribution number density of high mass, high redshift clusters very sensitive to primordial non-gaussianity and deviations from standard DE models Euclid data 60,000 clusters with a S/N>3 between 0.2<z<2 (obtained for free). more than 10 4 of these will be at z>1. ~ 5000 giant gravitational arcs very accurate masses for the whole sample of clusters (WL) dark matter density profiles on scales >100 kpc direct constraints on numerical simulations strong galaxy lensing giant arcs test of CDM : probe substructure and small scale density profile. (elaboration from a talk by Y. Mellier) L. Amendola, Vitoria,

14 Third Euclid probe: Clusters of galaxies Clusters of galaxies: probe of peaks in density distribution number density of high mass, high redshift clusters very sensitive to primordial non-gaussianity and deviations from standard DE models (elaboration from a talk by Y. Mellier) L. Amendola, Vitoria,

15 Predicted FoM of the Euclid mission Modified Gravity Dark Matter Initial Conditions Dark Energy Parameter g m n /ev f NL w p w a FoM Euclid primary (WL+GC) Euclid All Euclid+Planck Current (2009) ~10 Improvement Factor >10 >40 >400 Ref: Euclid RB arxiv: More detailled forecasts given in Amendola et al arxiv: , Living Rev.Rel. 21 (2018) no.1, 2 L. Amendola, Vitoria,

16 The two pillars of Euclid ds 2 = a 2 [(1+ 2Ψ)dt 2 (1+ 2Φ)(dx 2 + dy 2 + dz 2 )] Motion of a particle in a perturbed metric for v<<1, v depends only on Ψ for v 1, v depends on Ψ - Φ L. Amendola, Vitoria,

17 The two pillars of Euclid ds = a [(1 + 2 Ψ) dt (1 + 2 Φ )( dx + dy + dz )] Non-relativistic particles respond to Ψ δ ''+(1+ H' H )δ '= k2 Ψ δ = ρ ρ ρ Relativistic particles respond to Φ-Ψ α = perp ( Ψ Φ)dz clustering of galaxies lensing of galaxies L. Amendola, Vitoria,

18 WL and GC: optimal primary probes for Euclid The first pillar: Weak Lensing Light deflection L. Amendola, Vitoria,

19 WL and GC: optimal primary probes for Euclid Short intro to Weak Lensing Lensing, weak lensing, strong lensing L. Amendola, Vitoria,

20 WL and GC: optimal primary probes for Euclid Short intro to Weak Lensing Abell 2218 Credit: NASA, ESA, and Johan Richard (Caltech, USA) L. Amendola, Vitoria,

21 Weak Shear Lensing Weak Lensing Local Distortion NASA z

22 The Lens equation Propagation of light in a given metric General equations of motion

23 The Lens equation x 2 x 1 r Equations of motion on the plane in the linearly perturbed metric (see my book) lensing potential

24 The distortion matrix x i = rθ i Mapping of the position in the source plane to the observed position in the lens plane

25 The distortion matrix = κ γ 1 γ 2 γ 2 γ 1 = magn +shear

26 The distortion matrix Magnification Shear

27 An elliptical distortion

28 Measuring the shear intensity of light as a function of angles on the sky quadrupole ellipticity it turns out

29 The power spectrum of ellipticity Now, the ellipticity depends linearly on γ 1,2. These depend linearly on the metric potential Ψ,Φ. The metric potential depend linearly on the matter density contrast along the line of sight. Therefore, the power spectrum of the ellipticity depends linearly on the power spectrum of matter!

30 It turns out The power spectrum of ellipticity depends linearly on the power spectrum of (dark+luminous) matter! depends only on background quantities, i.e., on distances, H(z), and on the source distribution matter power spectrum

31 (elaboration from a talk by Y. Mellier) WL surveys

32 (elaboration from a talk by Y. Mellier) WL surveys

33 The second pillar: Galaxy clustering Modified-gravity sub-horizon linear matter equations θ = ik j v j ρ( x) ρ0 δ = ρ 0 continuity Euler Poisson This can be written as a single equation: δ ''+(1+ H' H )δ ' 3 2 Ω m δ = 0 prime is d/dlog(a) L. Amendola, Vitoria,

34 Galaxy clustering δ ''+(1+ H' H )δ ' 3 2 Ω m δ = 0 With this evolution equation, one can in principle predict the power spectrum of fluctuations today given the initial (inflationary) power spectrum P(k) k n k L. Amendola, Vitoria,

35 Galaxy clustering: BAO Baryon acoustic oscillations cause peaks in the power spectrum Bassett, Hlozek 2009

36 Galaxy clustering: BAO BAO peak in the galaxy correlation function Eisenstein+ 05

37 Galaxy clustering: Redshift Space Distortions r = v/h 0. Real space Redshift space 37

38 RSD on the correlation function Guzzo et al. 2008

39 From theory to observations ρ( x) ρ0 δ = ρ ds = a [(1 + 2Ψ) dt (1 + 2Φ)( dx + dy + dz )] 2 δ k = P( k, z Density fluctuation in space ) Matter power spectrum P matter ( k, z) Galaxy power spectrum b 2 ( k, z) P matter ( k, z) Galaxy power spectrum in redshift space (1+ f (k,z) b(k,z) cos2 θ) 2 b 2 (k,z)g 2 (k,z)p matter initial (k,z in ) L. Amendola, Vitoria,

40 Deconstructing the galaxy power spectrum P galaxy (k,z,µ = cosθ)= (1+ f (k,z) b(k,z) cos2 θ) 2 b 2 (k,z)g 2 (k,z)p matter initial (k,z in ) b(k,z) depend on local, non-cosmological physics P matter initial (k,z in ) G(k,z) f (k,z) depend on initial conditions (inflation) depend on dark energy/gravity L. Amendola, Vitoria,

41 First way to go: Parametric method P galaxy (k,z,µ = cosθ)= (1+ f (k,z) b(k,z) cos2 θ) 2 b 2 (k,z)g 2 (k,z)p matter initial (k,z in ) b(k,z) depend on local, non-cosmological physics P matter initial (k,z in ) G(k,z) f (k,z) Given a model, eg. inflation+λcdm, the power spectrum can be parametrized by a small number of parameters: n s,λ,ω m,h,w, etc L. Amendola, Vitoria,

42 Euclid parameter constraints A compilation of all available datasets of lensing and growth P 3 P 2 E E 42

43 Second way to go: non-parametric method P galaxy (k,z,µ = cosθ)= (1+ f (k,z) b(k,z) cos2 θ) 2 b 2 (k,z)g 2 (k,z)p matter initial (k,z in ) b(k,z) depend on local, non-cosmological physics P matter initial (k,z in ) G(k,z) f (k,z) depend on initial conditions (inflation) depend on dark energy/gravity L. Amendola, Vitoria,

44 Deconstructing the galaxy power spectrum Line-of-sight decomposition µ cosθ δ galaxy (k,z,µ)= G(k,z)(1+ f (k,z) b(k,z) µ2 ) b(k,z)δ matter initial (k,z in ) A + Rµ 2 δ lensing (k,z) = 3 2 G(k,z)Ω δ matter (k,z m initial in ) L L. Amendola, Vitoria,

45 Three linear observables: A, R, L galaxy clustering A = Gbδ m,0 (k) R = Gf δ m,0 (k) weak gravitational lensing L = 3 2 GΩ m δ m,0 (k) L. Amendola, Vitoria,

46 The model-independent observables Redshift distortion/amplitude P 1 = R A = f b Lensing/Redshift distortion Redshift distortion rate Expansion rate P 2 = L R = Ω m0 f P 3 = R' f ' R = f + E = H H 0 f L. Amendola, Vitoria,

47 All we know about present-day cosmology A compilation of all available datasets of lensing and growth P 3 P 2 E E Collab. with Santiago Casas and Ana M. Pinho arxiv: L. Amendola, Vitoria,

48 Properties of Dark Energy L. Amendola, Vitoria,

49 Properties of Dark Energy Isotropy Abundance Observational requirements acceleration Weak clustering L. Amendola, Vitoria,

50 Properties of Dark Energy Observations: Isotropy Large abundance Acceleration Weak clustering Theory: Scalar field? ΩDE Ωm weff -1 cs 1 L. Amendola, Vitoria,

51 The magic of Lagrangians dx 4 g # $ R + L matter % & variation R µν 1 2 Rg µν = 8πT µν The two laws of Lagrangian-cooking: a) form a scalar: Eqs are covariant b) no explicit functions of coords: Energymomentum tensor is conserved Random example: dx 4 g # f (φ)r + R µν φ,ν φ,µ +V (φ) + R µνστ φ,ν φ,µ φ,σ φ,τ + R µνστ R,στ φ,νµ L % $ matter & L. Amendola, Vitoria,

52 The magic of Fourier space Fourier space φ(x, y, z,t) φ(t)exp(i! k! x) therefore x φ(x, y, z,t) φ(t)ik x exp(i! k! x) φ(x, y, z,t) φ(t)i! k exp(i! k! x) φ(x, y, z,t) φ(t)k 2 exp(i! k! x) Very useful for linear equations because the space part drops out! 52

53 The past ten years of DE research 1 2 4, µ dx g R+ φφ, µ + V( φ) + L matter 1 2 4, dx g f ( φ) R+ φ µ, µ φ + V( φ) + L matter 1 2 4, dx g f ( φ) R+ K( φ µ, µ φ ) + V( φ) + L matter ,,,, dx g f ( φ, φ µ, ) R G ν µ K( µ µ φ + µν φ φ + φ, µ φ ) + V( φ) + L matter Cosmological constant, Dark energy w=const, Dark energy w=w(z),quintessence, scalar-tensor model, coupled quintessence, k-essence, f(r), Gauss-Bonnet, Galileons, KGB, L. Amendola, Vitoria,

54 The Horndeski Lagrangian The most general 4D scalar field theory with second order equations of motion 4 dx g L i +Lmatter i X = 1 2 φ,µ φ,µ define X 1 2 φ ;µ φ ;ν ü First found by Horndeski in 1975 ü rediscovered by Deffayet et al. in 2011 ü no ghosts, no classical instabilities ü it modifies gravity! ü it includes f(r), Brans-Dicke, k-essence, Galileons, etc etc etc ü Invariant under conformal and disformal transformations

55 Horndeski Second-order scalar-tensor field equations in a four-dimensional space Gregory Walter Horndeski (Waterloo U.) Int.J.Theor.Phys. 10 (1974) Horndeski L. Amendola, Vitoria,

56 Limits of the Horndeski Lagrangian 4 dx g L i +Lmatter i define X 1 2 φ ;µ φ ;µ L. Amendola, Vitoria,

57 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 L. Amendola, Vitoria,

58 The Great Horndeski Hunt Let us assume we have only 1) a perturbed FRW metric 2) pressureless matter 3) the Horndeski field L. Amendola, Vitoria,

59 Standard rulers D(z) = R (1+ z) 1 dz = sinh(h θ 0 Ω k 0 ) H 0 Ω H(z) k 0 R θ L. Amendola, Vitoria,

60 Standard rulers H( z) = dz R R L. Amendola, Vitoria,

61 BAO ruler Charles L. Bennett Nature 440, (27 April 2006) L. Amendola, Vitoria,

62 Background: SNIa, BAO, Then we can measure H(z) and D(z) = 1 dz sinh(h 0 Ω k 0 ) H 0 Ω H(z) k 0 and therefore we can reconstruct the full FRW metric ds 2 = dt 2 a(t) 2! 1 Ω k 0 # " 4 r2 2 $ (dx2 + dy 2 + dz 2 ) & % L. Amendola, Vitoria,

63 Two free functions The most general linear, scalar metric ds = a [(1 + 2Ψ) dt (1 + 2Φ)( dx + dy + At linear order we can write: dz 2 )] Poisson s equation 2 Ψ= 4πGρmδm anisotropic stress 1 Ψ = Φ Warning: all the perturbation variables in this talk are root mean squares! L. Amendola, Vitoria,

64 The most general way Two free functions The most general linear, scalar metric The most general parametrization of gravity at linear level Poisson s equation η( ka, = ) anisotropic stress 0 = Φ+Ψ Φ Ψ Ψ L. Amendola, Vitoria,

65 Modified Gravity at the linear level standard gravity Yka (, ) = 1 η( ka, ) = 1 scalar-tensor models f(r) DGP * 2 G 2( F + F' ) Ya ( ) = 2 FG 2F + 3 F ' cav,0 2 F ' η( a) = 1+ 2 F + F' 2 2 k k * 1+ 4m m G 2 2 Ya ( ) = ar, η( a) = 1+ ar 2 2 FGcav,0 k k 1+ 3m 1+ 2m 2 2 ar ar 1 Y ( a) = 1 ; β = 1+ 2Hrc w 3β 2 η( a) = 1+ 3 β 1 Ya ( ) =... DE Bean et al Hu et al Tsujikawa 2007 Lue et al. 2004; Koyama et al massive bi-gravity F. Koennig and L. A. 2014, η( a) =... Boisseau et al Acquaviva et al Schimd et al L.A., Kunz &Sapone 2007 Y. Akrami et al L. Amendola, Vitoria,

66 Modified Gravity at the linear level In the quasi-static limit, every Horndeski model is characterized at linear scales by the two functions η( ka, ) Yka (, ) 2 1+ kh 4 = h kh kh 5 = h kh3 De Felice et al. 2011; L.A. et al.prd, arxiv: , 2012 L. Amendola, Vitoria,

67 Modified Gravity at the linear level De Felice et al. 2011; L.A. et al.,prd, arxiv: , 2012 L. Amendola, Vitoria,

68 Yukawa Potential 2 1+ kh 4 η( ka, ) = h kh kh 5 Yka (, ) = h kh3 Momentum space k 2 Ψ = 4πGY(a,k)ρ m (k) k 2 Φ = 4πGY(a,k)η(a,k)ρ m (k) Ψ = GM r h 2 (1+ h 4 h 5 h 5 e r/ h 5 ) = GM r (1+ Qe mr ) Real space Φ = GM r h 1 (1+ h 3 h 5 h 5 e r/ h 5 ) = ĜM r (1+ ˆQe mr ) De Felice et al. 2011; L.A. et al.prd, arxiv: , 2012 L. Amendola, Vitoria,

69 2 nd order perturbed Lagrangians An elegant way to derive the perturbation equations is to write down the perturbed Lagrangian at second order: when you differentiate it, you get first order perturbation equations! For instance standard EH Lagrangian gives in Minkowski Definition Propagating degree of freedom: a field with two time derivatives in the 2 nd order Lagrangian, once all the constraints have been taken into account L. Amendola, Vitoria,

70 Identifying the degrees of freedom Classifying theories by their dof Standard gravity: no scalar degree of freedom, 2 tensor dofs Horndeski: one scalar dof, 2 tensor dofs Massive grav: one scalar dof, 2+2 tensor dofs Bimetric models: one scalar dof, 2 vector dofs, 2+2 tensor dofs 2 nd order pert. Lagrangian shows explicitly the dofs For Horndeski: 2 De Felice & Tsujikawa 2011 L. Amendola, Vitoria,

71 Identifying the degrees of freedom 2 nd order pert. Lagrangian shows explicitly the dofs 2 scalar tensor and the minimal number of free functions: two for each dof (plus a function for the background, i.e. 5 functions for Horndeski) L. Amendola, Vitoria,

72 The magic of 2 nd order perturbed Lagrangians 2 nd order pert. Lagrangian also clarifies the stability conditions 2 Equation of motion in Fourier space!! φ + c s 2 k 2 φ = 0 Positive squared sound speeds c 2 s,t 0 Positive kinetic terms Q s,t 0 L. Amendola, Vitoria,

73 Horndeski Lagrangian L. Amendola, Vitoria,

74 Linearized Horndeski: Bellini-Sawicki parametrization L. Amendola, Vitoria,

75 sound speeds 2 L. Amendola, Vitoria,

76 Bellini-Sawicki parametrization L. Amendola, Vitoria,

77 Beyond Horndeski, I Look now at the matter equations of motion 4 dx g L i +Lmatter i Standard sub-horizon matter equations θ = ik j v j continuity Euler Poisson Modifying gravity Yη Modifying continuity equation Gleyzes et al 2014; Lombriser et al 2015 L. Amendola, Vitoria,

78 Beyond Horndeski, II Or, we can add several Horndeski fields! dx 4 g # $ L H1 + L H L matter % & A. Silvestri et al T. Baker et al V. Vardanyan & L.A., 2015 L. Amendola, Vitoria,

79 Beyond Horndeski, III Torsion Non-metricity Non-universal coupling Palatini Vectors Tensors L. Amendola, Vitoria,

80 Beyond Horndeski, IV Pauli-Fierz (1939) action: the only ghost-free quadratic action for a massive spin two field The three deadly sins of Pauli-Fierz theory: It does not reduce to massless gravity for m 0 (vdvz disc.) It violates diffeomorphism invariance It contains a ghost when extended to non-linear level (Boulware-Deser ghost) L. Amendola, Vitoria,

81 Ghost-free Bigravity The first problem was solved by Vainshtein (1972): there exists a radius below which the linear theory cannot be applied; For the Sun, this radius is larger than the solar system! The second and third problems can be solved introducing a second metric: The only ghost-free local non-linear massive gravity theory! derham, Gabadadze, Tolley 2010 Hassan & Rosen, 2011 L. Amendola, Vitoria,

82 Modified Gravity in bimetric gravity In the quasi-static limit, every Horndeski model and bimetric gravity model is characterized at linear scales by the two functions η( ka, ) Yka (, ) 2 1+ kh 4 = h kh kh 5 = h kh3 De Felice et al. 2011; L.A. et al.prd, arxiv: , 2012 L. Amendola, Vitoria,

83 The speed of gravitational waves 2017: GW and gamma-ray simultaneous detection LIGO, Integral, Fermi coll. 83

84 The speed of gravitational waves L. Amendola, Vitoria,

85 The speed of gravitational waves c 2 T = 1+ α T Only way to have c T =1 is G 4,X = 0 G 5 = 0 Conformal coupling! L. Amendola, Vitoria,

86 Caveats The speed of GW has been measured only at z = 0! L. Amendola, Vitoria,

87 Observing η η( ka 0, = ) = Φ+Ψ Φ Ψ Ψ L. Amendola, Vitoria,

88 Deconstructing the galaxy power spectrum P galaxy (k,z,µ = cosθ)= (1+ f (k,z) b(k,z) cos2 θ) 2 b 2 (k,z)g 2 (k,z)p matter initial (k,z in ) b(k,z) P matter initial (k,z in ) do not depend on gravity G(k,z) f (k,z) depend on gravity L. Amendola, Vitoria,

89 Deconstructing the galaxy power spectrum Line-of-sight decomposition µ cosθ δ galaxy (k,z,µ)= G(k,z)(1+ f (k,z) b(k,z) µ2 ) b(k,z)δ matter initial (k,z in ) A + Rµ 2 δ lensing (k,z) = 3 2 Y(1+ η)g(k,z)ω δ matter (k,z m initial in ) L L. Amendola, Vitoria,

90 Three linear observables: A, R, L galaxy clustering A = Gbδ m,0 (k) R = Gf δ m,0 (k) weak gravitational lensing L = 3 2 Y(1+ η)gω m δ m,0 (k) L. Amendola, Vitoria,

91 The only model-independent ratios Redshift distortion/amplitude P 1 = R A = f b Lensing/Redshift distortion Redshift distortion rate Expansion rate P 2 P 3 Ω +η = L = R f = R' f ' f R = f + E = H H 0 Y(1 ) m0 How to combine them to test the theory? L. Amendola, Vitoria,

92 Summarizing. Matter conservation equation independent of gravity theory δ ''+(1+ H' H )δ ' 3 2 Ω m Yδ = 0 Observables P 2 L Ω Y(1 ) m0 +η = = P3 R f = R' f ' R = f + f E = H H 0 L. Amendola, Vitoria,

93 Testing the entire Horndeski Lagrangian A unique combination of observables Independent of the bias, of the initial conditions, of the specific model P2(1 + z) 1+ k h 4 1 = η = h2 2 2 E ' 2 E ( P ) + kh E Observables Theory L.A., M. Motta, I. Sawicki, M. Kunz, I. Saltas, L. Amendola, Vitoria,

94 Testing the entire Horndeski Lagrangian η obs 3P 2 (1+ z) 3 2E 2 (P E ' E ) 1 η obs 1 gravity is modified η obs (k) η(horndeski) The entire Horndeski model is falsified L.A., M. Motta, I. Sawicki, M. Kunz, I. Saltas, L. Amendola, Vitoria,

95 Can we measure it? L. Amendola, Vitoria,

96 The first measurement ever A compilation of all available datasets of lensing and growth P 3 P 2 E E L. Amendola, Vitoria,

97 The first model-independent measurement ever ΛCDM Result compressed in a single central bin η = 0.40 ± 0.60 Collab. with Santiago Casas and Ana M. Pinho arxiv: L. Amendola, Vitoria,

98 Results...! 1+ k 2 H η(k,a) = H 4 2 # " 1+ k 2 H 5 $ & % Model 1: η constant for all z, k Error on η around 2% Model 2: η varies in z Error on η η=1/2 for f(r)! (at small scales) L. Amendola, Vitoria, 2018 L.A, M. Kunz, A. Vollmer, A. Guarnizo, S. Fogli, ,

99 Conclusions Euclid will combine weak lensing, galaxy clustering, galaxy clusters, to determine the cosmic expansion and the growth of fluctuations There is a way to systematically classify all models of gravity plus a single scalar field, the Horndeski Lagrangian The GW speed already kills a sector of this mod. grav. theory The surviving freedom can be constrained in a modelindependent way