MG-MAMPOSSt

where σz(R, r) is the velocity dispersion along the line of sight defined as:

σz(R, r) = σr(r)

r

1 − β(r)R

2

r2 . (3.40)

steps through which the MAMPOSSt procedure works can be summarized as follows:

(i) For a given choice of the parameter vector θ = {rs, r200, rν, β} (see below) σr(r) is

computed from eq. (3.37) over a logarithmic grid of radii riand then performs a cubic-

spline interpolation to evaluate at intermediate radii.

(ii) The solution of the Jeans’ equation is then used to calculate g(R, vz) for each (Ri, vz,i).

The integral of eq. (3.39) is numerically solved by assuming a cutoff for the upper limit of ∼ 15r200, where the velocity of the matter pushed by the Hubble flow is roughly 3σ

above the mean value of the cluster. Variations of the viral radius by up to a factor of 2 do not change significantly the result of the integration (ref. [32]).

(iii) The Likelihood is computed from eq. (3.38) as: − log L = −

n

X

i=1

log q(Ri, vz,i|θ). (3.41)

The minimum value of − log L is found by searching over a grid in the parameter space. In its original version, the MAMPOSSt code works with four free parameters, namely the scale radius of the tracers density profile rν, the parameter of the velocity anisotropy profile, the

scale radius rsand the radius r200of the chosen parametric mass profile.

3.3.1 MG-MAMPOSSt

I have modified the MAMPOSSt procedure by introducing the generic form for the gravitational potential Φ of eq. (2.23): Φ(r) = h1  G Z r r0 dx x2MN F W(x, rs, r200) + 2Q 2_φ mg(x, rs, r200, m, S)  , (3.42)

which accounts for a broad range of viable modified gravity models, including the f (R) case. I have expanded the parameter array by introducing 4 additional parameters, which are the interaction range λ = 1/m, the conformal rescaling h1, the coupling factor Q and the screening

radius S, under the assumption of an instantaneous transition between the screened and non- screened regime (see Sec. 7.2).

For the purposes of this Thesis, I will focus on f (R) models as a case study; in this framework, as already discussed in Chapter 2, Q is a constant fixed such that 2Q2 = 1/3. Furthermore, h1 = 1/(1 + f,R); I will restrict the analysis to the range λ ≤ 100Mpc, which, for example,

in the Hu & Sawicki model of f (R) (ref. [5]) roughly corresponds to |f,R| < 10−3, so one

can safely set h1 = 1. It is also worth to notice that in general h1 is totally degenerate with

the observed mass of the overdensity, whose information is already encoded in the virial radius r200.

The modified gravity module of the code (MG-MAMPOSSt hereafter) is implemented only for the case of a NFW mass density profile, which provides the highest Likelihood in the GR analysis of both clusters used in this work (see Chapter 5); an extension of the code to work with other choices for the parametrization of the mass profile in generic modified gravity models is relatively straightforward and will be implemented for future analyses.

I tested the MG-MAMPOSSt code in linear f (R) models (i.e. λ = const) to reproduce the results of ref. [25] for MACS 1206 in the limit of standard gravity λ → 0; as shown in Figure 3.5, for λ = 0.1 Mpc (black solid line) the contribution given by the modification of gravity is very small and, except for the innermost region, the effective mass profile is very close to the GR profile obtained by ref. [25] in the standard MAMPOSSt analysis (red dotted line and red shaded area). In the opposite situation (λ  1) I checked that the modification in the effective mass profile reaches the maximum enhancement of 1/3 as expected for f (R) gravity. It is worth to notice that the contribution of the additional degree of freedom satu- rates for λ& 10 Mpc: the overall relative difference in the effective mass profile computed for λ = 10 Mpc (blue line) and λ = 1000 Mpc (brown dotted line) is less than 1%. This is not surprising since λ determines the typical length at which the fifth force decreases; for scales much larger than the cluster size it is impossible to distinguish the effect induced by different values of the interaction range. In Fig. 3.6 I also show the enhancement in the radial velocity dispersion profile σrgiven by the solution of the Jeans’ equation (3.27) for different values of

the interaction range λ. Since σris proportional to the square root of the effective mass profile,

for λ → ∞ I have σ_rM G→ 1.16σGR r .

I will apply our MG-MAMPOSSt code on the data of two galaxy clusters in the framework of f (R) gravity for the case of a constant value of m = 1/λ, and for environment-dependent function:

λ(r) = ¯λ × f (r; S), (3.43)

where ¯λ is the background value and f (r; S) is a step function equal to 0 for r ≤ S and equal to 1 otherwise. The screening radius S at which the transition happens will be fixed by assuming the H&S functional form of f (R) (Sec. 2.2.1). The results will be presented and discussed in Chapter 7.

Figure 3.5: Upper panel: mass profiles in f (R) gravity, expressed in unit of M200 =

200H2_(z)r3

200/2G, for different values of the interaction range and r200 = 1.96 Mpc, rs= 0.27 Mpc

(best fit values of the GR analysis of ref [25] for MACS 1206). Brown dotted line: λ = 1000 Mpc. Blue solid line: λ = 10 Mpc. Purple solid curve: λ = 1 Mpc. Black solid curve: λ = 0.1 Mpc. The red shaded area shows the GR profile within the 68% C.L. in the NFW parameters with the best fit indicated by the red dashed curve. Bottom panel: relative enhancement of the profiles with respect of the GR one, for the same value of λ as in the upper plot. For λ  1 Mpc the profile is enhanced by a factor 1/3 with respect the GR value, while for λ = 0.1 Mpc the modifications of gravity are suppressed with the exception of the central regions.

Figure 3.6: Upper panel: radial velocity dispersions in f (R) gravity for different values of the inter- action range and r200 = 1.96 Mpc, rs = 0.27 Mpc (best fit values of the GR analysis of ref [25] for

MACS 1206). The velocity anisotropy is parametrized as a "O" model with normalization parameter β∞ = 0.5 and the scale radius of the number density profile rνis set to be equal to rs. Brown dotted

line: λ = 1000 Mpc. Blue solid line: λ = 10 Mpc. Purple curve: λ = 1 Mpc. Black solid curve: λ = 0.1 Mpc. the GR profile is indicated by the red dashed curve. Bottom panel: relative enhancement of the profiles with respect of the GR one, for the same value of λ as in the upper plot For λ  1 Mpc the profile is enhanced by a factor 0.16 with respect the GR value, while for λ = 0.1 Mpc the result is very close to standard gravity except in the cluster central regions.

Gravitational Lensing

with galaxy clusters

In this Chapter I briefly discuss the theory of photon propagation in curved spacetime. I will derive the lensing potential Φlens from the linearly perturbed FLRW metric, showing how it

depends on both Bardeen potentials, which is a crucial point of the method for constraining modified gravity presented in this Thesis. Section 4.2 then provides an overview on galaxy cluster mass profiles reconstruction with gravitational lensing in the two regimes of Strong Lensing and Weak Lensing, highlighting what kind of data are required to perform these anal- yses.

For a more detailed description, I refer to the lecture notes of ref. [165] and to refs. [166, 167]

4.1

Propagation of light rays in a gravitational field

The idea that the trajectory of a light ray could be bended by the presence of a gravitational field dates back to the Eighteenth century. In 1804 Soldner, embracing the current view of corpuscular nature of light, applied the Newton’s gravitational law to compute the deflection angle of a light "particle" under the influence of the gravitational potential at the surface of the Sun:

˜

α = 2GM

c2_R ' 0.83 arcsec.

This results was then derived and corrected by a factor 2 by Einstein using the General Rela- tivity principles; the value of 1.7 arcsec was observationally proved during the total eclipse in

1919 by measuring the apparent displacement of a star near the Sun surface (Dyson, Edding- ton, Davidson, 1920) and it is considered one of the most important fundamental tests of GR in literature. The light deflection induced by the presence of mass distributions in spacetime is called gravitational lensing and it can be used to investigate properties and composition of both the region in which those bending effects are generated (the lens) and the background "lensed" sources whose light rays are distorted by the lens. In particular, gravitational lensing effects in galaxy clusters are an exceptional tool to study the structure of Dark Matter within the cluster and the nature of gravity, as well as to probe cosmology from the analysis of lensed images (e.g. ref [168] and references therein).

As already mentioned, the geometry of a galaxy cluster is described by the linearly perturbed FLRW metric. Since possible departures from GR are expected to be small, as confirmed by recent observations (see Chapter 2), the Bardeen potential Ψ should be of the same order of Φ ∼ 10−4c2_{. I am now going to define the effective refractive index n(x) for light propaga-}

tion in a FLRW spacetime which is linearly perturbed by the presence of a matter distribution, showing the connection between n and the two Bardeen potentials.