Testing Gravity Cosmologically

Transcription

1 Testing Gravity Cosmologically Philippe Brax IPhT Saclay Asphon Toulouse March 2013

2 The Big Puzzle

3 How do we know? measuring distances! Absolute luminosity. Received flux: what we see in the telescope Hubble parameter acceleration parameter: we need large red-shift z

4 Evidence: The Hubble Diagram The explosion of high red-shift SN Ia (standard candles): Within General Relativity, link to matter and dark energy Dark Energy must exist!

5 The Cosmic Microwave Background Fluctuations of the CMB temperature across the sky lead to acoustic peaks and troughs, snapshot of the plasma oscillations at the last scattering surface when the universe became transparent The position of the first peak: The universe is spatially flat WMAP data

6 Cosmological constant Dark Energy? Dark energy is an unknown form of matter whose energy density and pressure are almost opposite. ρ=λ/8πg Constant energy density ρ: de= ρdv=-pdv Newton s constant p=-ρ The tiny value of ρ has prompted the search for alternative explanations.

7 Dark Energy? A scalar field is a good candidate as its energy density can be dominated by its potential implying that its pressure is almost opposite to ρ A field rolling down a runaway potential, reaching large values now?

8 It could also be that what we interpret as acceleration is in fact a manifestation of something more subtle: A modification of the laws of gravity on large scales Gravity is described by the general theory of relativity (Einstein 1915) and encompasses several aspects which needs to be carefully understood when one tries to modify gravity. The equivalence principle The Einstein equation of General Relativity

9 General relativity is a metric theory relating energy and geometry of the Universe: Ricci tensor Energy momentum tensor

10 The equivalence principle is at the heart of general relativity. There are several versions which must be distinguished: The weak equivalence principle (or universality of free fall) states that to massive tests bodies fall in the same way independently of their composition. The equality of the inertial and gravitational masses for test bodies has been tested at the level of thirteen decimal places! General relativity satisfies a stronger version of the equivalence principle.

11 The Einstein equivalence principle is at the heart of General Relativity The weak equivalence principle is valid. Result of any local non-gravitational experiment independent of the velocity of free falling frame (local Lorentz invariance) Result of any non-gravitational experiment independent of where and when performed. In the local free falling frame: the constants of nature are constant Violated if the fine structure constant or particle masses vary on cosmological scales.

12 The Einstein equivalence principle is at the heart of General Relativity The weak equivalence principle is valid. Result of any local non-gravitational experiment independent of the velocity of free falling frame (local Lorentz invariance) Result of any non-gravitational experiment independent of where and when performed. In the local free falling frame: the constants of nature are constant the laws of physics are the ones of special relativity

13 The Einstein equivalence principle is at the heart of General Relativity The weak equivalence principle is valid. Result of any local non-gravitational experiment independent of the velocity of free falling frame (local Lorentz invariance) Result of any non-gravitational experiment independent of where and when performed. In the local free falling frame: the constants of nature are constant the laws of physics are the ones of special relativity the effect of the gravitational field can be effaced

14 The Einstein equivalence principle is at the heart of General Relativity The weak equivalence principle is valid. Result of any local non-gravitational experiment independent of the velocity of free falling frame (local Lorentz invariance) Result of any non-gravitational experiment independent of where and when performed. In the local free falling frame: the constants of nature are constant the laws of physics are the ones of special relativity the effect of the gravitational field can be effaced test bodies follow straight lines (geodesics globally) Not true in modified gravity: fifth force

15 General relativity even satisfies a strong equivalence principle The weak equivalence principle is valid even for bodies with self-gravity. Result of any local experiment independent of the velocity of free falling frame. Result of any experiment independent of where and when performed. This is violated in the theories which modify gravity on large scales.

16 The acceleration of the expansion of the Universe could have (at least) two dynamical explanations: Dark Energy: a new form of matter Modified law of gravity on large scales Einstein-Hilbert action describing General Relativity. Einstein s equations are simply obtained using the least action principle Lagrangian density of dark energy Arbitrary corrections to the Einstein-Hilbert Lagrangian involving the Riemann and Ricci tensors. Surprisingly, both types of models involve scalar fields.

17 Modifying Gravity Mass term for a graviton The simplest modification is massive gravity (Pauli-Fierz): Pauli-Fierz gravity is ghost free (negative kinetic energy terms). Unfortunately, a massive graviton carries 5 polarisations when a massless one has only two polarisations. In the presence of matter, the graviton wave function takes the form: The massless limit does not gives GR! (van Dam-Velman-Zakharov discontinuity). The extra polarization is lethal.

18 Pauli- Fierz massive gravity has a ghost in curved backgrounds (cosmology). An infinite class of modified gravity models can be considered: These Lagrangian field theories fall within the category of higher derivative theories. Ostrosgradski s theorem states that these theories are generically plagued with ghosts. Quantum mechanically, this implies an explosive behaviour with particles popping out of the vacuum continuously. In particular an excess in the gamma ray background.

19 A large class is ghost-free though, the f(r) models:

20 As promised, f(r) is totally equivalent to an effective field theory with gravity and scalars! The potential V is directly related to f(r) Crucial coupling between matter and the scalar field

21 The acceleration of the Universe could be due to either: ϕ Modified Gravity Dark Energy In both cases, current models use scalar fields. In modified gravity models, this is due to the scalar polarisation of a massive graviton (5=2+2+1). In dark energy, it is by analogy with inflation. The fact that the scalar field acts on cosmological scales implies that its mass must be large compared to solar system scales.

22 F(R) gravity Deviations from Newton s law are parametrised by: For large range forces with large λ, the tightest constraint on the coupling β comes from the Cassini probe measuring the Shapiro effect (time delay): Bertotti et al. (2004) The effect of a long range scalar field must be screened to comply with this bound and preserve effects on cosmological scales.

23 Around a background configuration and in the presence of matter, the Lagrangian can be linearised and the main screening mechanisms can be schematically distinguished : The chameleon mechanism makes the range become smaller in a dense environment by increasing m The Damour-Polyakov mechanism reduces β in a dense environment The Vainshtein mechanism reduces the coupling in a dense environment by increasing Z

24 The chameleon mechanism requires a non-linear potential V: f(r) gravity, chameleons The Damour-Polyakov mechanism requires a non linear coupling β symmetrons, dilatons Unified description of screening in a single formalism: Unique coupling, no violation of the weak equivalence principle

25 The Vainshtein mechanism requires non linear kinetic terms: DGP, Galileons, massive gravity Vainshtein 1972 Dvali-Gabadatze-Porrati 2000 Nicolis-Rattazzi-Trincherini 2009 derham-gabadadze-tolley 2010 Unified treatment for Horndeski models (most general second order eom): Horndeski 1974 Deffayet-Gao-Steer-Zahariade 2011

26 The effect of the environment (excluding Vainshtein) When coupled to matter, scalar fields have a matter dependent effective potential Environment dependent minimum Chameleon The field generated from deep inside is Yukawa suppressed. Only a thin shell radiates outside the body. Hence suppressed scalar contribution to the fifth force.

27 Symmetron

28 Dilaton

29 These theories violate the strong equivalence principle. Thin shell When objects are big enough/dense enough, or if they are surrounded by big objects, the field is screened. Inside it is nearly constant apart from inside thin shell whose size is inversely proportional to Newton s potential at the surface.

30 Due to the scalar interaction, within the Compton wavelength of the scalar field, the inertial and gravitational masses differ for unscreened objects: A Gravity + scalar force B If only unscreened objects existed, one could rescale Newton s constant and recover the equality between the gravitational and inertial masses. This cannot be done as light rays deviate in the presence of matter according to the unrescaled Newton constant! This corresponds to the fact that matter particles do not follow geodesics of the geometrical metric g (one of the consequence of Einstein s equivalence principle).

31 Due to the scalar interaction, within the Compton wavelength of the scalar field, the inertial and gravitational masses differ for screened objects: A Gravity + scalar force B Value of the field far away Massive bodies with differ scalar charges fall differently. Hence a violation of the strong equivalence principle. Newtonian potential at the surface of the body.

32 Screening effective when: or equivalently: The solar system tests impose that the sun must be screened: If this ratio is such that: Small galaxies are unscreened while the largest ones are screened.

33 Astrophysical tests: Motion of screened stars different from unscreened HI gas in unscreened dwarf galaxies. screened Distance indicators for cepheids and TRGB stars in screened and unscreened dwarf galaxies are different as their luminosities vary SDSS catalogue, within 200 Mpc, scalar range 1 Mpc No effects measured so far: bound on the range of the scalar interaction.

34 All these models can be entirely characterised by 2 time dependent functions. The non-linear potential and coupling of the model can be reconstructed using: m(a) cosmological mass β(a) cosmological coupling Works for chamelons, dilatons, symmetrons etc Screening in the Milky Way: Models with cosmological effects on scales smaller than 1 Mpc. Similar constraint from binary pulsars and Cepheid distance indicator.

35 These models break also the invariance of constants. Particle masses: The Universe behaves like the concordance model in the recent past: Big Bang Nucleosynthesis imposes that masses cannot vary by more than 1% during BBN. Late time acceleration If the field has not the minimum of the potential by BBN, large variation of particle masses unless m>>h. Lurking cosmological constant

36 There is no intrinsic variation of the fine structure constant. On the other hand, the fact that the field rolls slowly with the minimum of the effective potential implies that the electron mass varies a little bit in the late time Universe. Assuming that the bulk of the proton mass comes from the gluons and is therefore time independent, the electron to proton mass ratio evolves in time: Weak constraints from variation of constants, e.g. the electron to proton mass ratio μ: Which gives a constraint on the coupling to matter now (essentially): The coupling is bounded by: which is essentially of the same order as the solar system constraint.

37 The cosmological background evolves like in the concordance model. The main difference coming from the modification of gravity arises at the perturbation level where the Cold Dark Matter density contrast evolves like: Modified gravity The new factor in the bracket is due to a modification of gravity depending on the comoving scale k.

38 The growth of structures depends on the comoving Compton length: Gravity acts in an usual way for scales larger than the Compton length (matter era) Gravity is modified inside the Compton length with a growth (matter era):

39 Large scale structures in the non-linear regime determined by m(a) and β(a). Non-linear effects simulated with ECOSMOG. Power Spectrum deviation (symmetron) Range of the scalar field.

40 Screening on large scales where k<<am(a) in the linear regime Large scale screening Power Spectrum Deviation (symmetron) Range of the scalar field.

41 Vainshtein Mechanism The Vainshtein mechanism differs greatly from the chameleon and Damour-Polyakov ones. We can pick up the gist of Galileon-Horndeski using a simple example with higher derivatives: m graviton mass Non-linearity screened Well inside the Vainshtein radius, Newtonian gravity is restored. Well outside gravity is modified. The Vainshtein radius is very large for stars, typically 0.1 kpc for the sun, and a mass for the graviton of the order of the Hubble rate.

42 The most general class of theories with the Vainshtein property is described by the Horndeski Lagrangian:

43 The quasi-static profile depends on a quintic equation: Mass scale The Vainshtein radius is obtained for A=1. The solution interpolates between: Modified gravity Vainshtein radius inside the Vainshtein radius where GR is recovered and GR outside where GR is modified.

44 The motion of astrophysical objects in Vainshtein theories does not violate the strong equivalence principle: Trajectories due to gravity + scalar Universal scalar charge, no violation of the strong equivalence principle. Unlike chameleons et al., Vainshtein does not affect the charge Q: Outside the Vainshtein radius, the object feels the scalar force The scalar force is screened because the gradient of the external scalar field is suppressed. outside the Vainshtein radius

45 On very large scales where structure formation takes place, there is a gradient for the scalar field coming from the large scale structures. It penetrates inside the Vainshtein radius of galaxies. Stars inside the Vainshtein radius feel this gradient and have a motion affected by the large scale structures of the Universe. This is the same for all stars, no violation of the strong equivalence principle. The only objects which carry no scalar charge are black holes which have Q=0. BH do not feel the gradient hence the BH at the centre of the galaxy lags behind. The offset from the centre is such that gravitational pull from the centre equates the scalar gradient (a fraction of kpc) The central BH does not follow the bulk flow followed by stars. Hence a maximal violation of the strong equivalence principle.

46 Conclusions Modified gravity on very large scales is constrained: Astrophysical tests of the strong equivalence principle Effects on the growth of structures in the quasi-linear regime The different models are classified according to the mechanism preserving gravity in the solar system: Chameleon/Damour-Polyakov Vainshtein These tests are complementary to lab/solar system experiments.