13.1 Weak lensing

Weak gravitational lensing is sensitive to the growth of large scale structure as well as the relation between matter and gravitational potentials. Since the evolution of matter perturbations and gravitational potentials is different from that of GR, the observations of weak lensing can provide us an important test for probing modified gravity on galactic scales (see [252727595Jump To The Next Citation Point528Jump To The Next Citation Point548] for theoretical aspects and [546348322Jump To The Next Citation Point362937332617773Jump To The Next Citation Point] for observational aspects). In particular a number of wide-field galaxy surveys are planned to measure galaxy counts and weak lensing shear with high accuracy, so these will be useful to distinguish between modified gravity and the ΛCDM model in future observations.

Let us consider BD theory with the action (10.10View Equation), which includes f (R) gravity as a specific case. Note that the method explained below can be applied to other modified gravity models as well. The equations of matter perturbations δm and gravitational potentials Φ,Ψ in BD theory have been already derived under the quasi-static approximation on sub-horizon scales (k ≫ aH), see Eqs. (10.38View Equation), (10.39View Equation), and (10.40View Equation). In order to discuss weak lensing observables, we define the lensing deflecting potential

Φwl ≡ Φ + Ψ, (13.1 )
and the effective density field
a δeff ≡ − -----(0)k2Φwl, (13.2 ) 3H20Ωm
where the subscript “0” represents present values with a = 1 0. Using the relation ρ = 3F H2 Ω (0)∕a3 m 0 0 m with Eqs. (13.1View Equation) and (13.2View Equation), it follows that
a2ρm- F0- Φwl = − k2 F δm, δeff = F δm. (13.3 )

Writing the angular position of a source and the direction of weak lensing observation to be ⃗𝜃 S and ⃗𝜃 I, respectively, the deformation of the shape of galaxies can be quantified by the amplification matrix ⃗ ⃗ 𝒜 = d𝜃S∕d 𝜃I. The components of the matrix 𝒜 are given by [66Jump To The Next Citation Point]

∫ χ ′ ′ 𝒜 μν = Iμν − χ-(χ-−-χ-)∂ μνΦwl[χ ′⃗𝜃, χ′]dχ ′, (13.4 ) 0 χ
where ∫ z ′ ′ χ = 0 dz ∕H (z ) is a comoving radial distance (z is a redshift). The convergence κwl and the shear ⃗γ = (γ1,γ2) can be derived from the components of the 2 × 2 matrix 𝒜, as κwl = 1 − (1∕2)Tr 𝒜 and ⃗γ = ([𝒜22 − 𝒜11]∕2,𝒜12 ). For a redshift distribution p(χ )dχ of the source, the convergence can be expressed as ∫ κwl(⃗𝜃) = p(χ)κwl(⃗𝜃,χ )dχ. Using Eqs. (13.2View Equation) and (13.4View Equation) it follows that
3 ∫ χH δ [χ ⃗𝜃,χ] κwl(⃗𝜃) = --H20Ω(m0) g(χ)χ -eff------dχ, (13.5 ) 2 0 a
where χH is the maximum distance to the source and ∫ g(χ) ≡ χHp(χ ′)(χ′ − χ)∕χ ′dχ ′ χ.

The convergence is a function on the 2-sphere and hence it can be expanded in the form ∫ ⃗⃗ 2⃗ κwl(⃗𝜃) = ˆκwl(⃗ℓ)eiℓ⋅𝜃 d2ℓπ-, where ⃗ℓ = (ℓ1,ℓ2) with ℓ1 and ℓ2 integers. We define the power spectrum of the shear to be ⟨ˆκwl(⃗ℓ)ˆκ∗ (⃗ℓ′)⟩ = Pκ(ℓ)δ(2)(⃗ℓ − ⃗ℓ′) wl. Then the convergence has the same power spectrum as Pκ, which is given by [66601]

∫ [ ] [ ] 9H40(Ω(m0))2- χH g(χ-) 2 ℓ- P κ(ℓ) = 4 a(χ ) P δeff χ ,χ dχ. (13.6 ) 0
We assume that the sources are located at the distance χs (corresponding to the redshift zs), giving the relations p(χ ) = δ(χ − χs) and g(χ ) = (χs − χ )∕χs. From Eq. (13.3View Equation) P δeff can be expressed as P = (F ∕F )2P δeff 0 δm, where P δm is the matter power spectrum. Hence the convergence spectrum (13.6View Equation) reads
4 (0) 2∫ χs ( )2 [ ] Pκ(ℓ) = 9H-0(Ω-m-)- χs −-χ-F0- Pδm -ℓ,χ dχ. (13.7 ) 4 0 χsa F χ

We recall that, during the matter era, the transition from the GR regime (2∕3 δm ∝ t and Φwl = constant) to the scalar-tensor regime (√ ------- δm ∝ t( 25+48Q2−1)∕6 and √ ------- Φwl ∝ t( 25+48Q2−5)∕6) occurs at the redshift z k characterized by the condition (10.45View Equation). Since the early evolution of perturbations is similar to that in the ΛCDM model, the weak lensing potential at late times is given by the formula [214Jump To The Next Citation Point]

9 D (k,a) Φwl(k,a) = 10Φwl(k,ai)T (k)---a---, (13.8 )
where Φwl(k,ai) ≃ 2Φ (k,ai) is the initial potential generated during inflation, T (k) is a transfer function describing the epochs of horizon crossing and radiation/matter transition (50 ≲ z ≲ 106), and D (k,a) is the growth function at late times defined by D (k,a)∕a = Φwl(a)∕Φwl(aI) (aI corresponds to the scale factor at a redshift 1 ≪ z < 50 I). Our interest is the case in which the transition redshift z k is smaller than 50, so that we can use the standard transfer function of Bardeen–Bond–Kaiser–Szalay [58]:
[ ] −0.25 T (x) = ln(1 +-0.171x-) 1.0 + 0.284x + (1.18x)2 + (0.399x )3 + (0.490x )4 , (13.9 ) 0.171x
where x ≡ k∕kEQ and (0) 2 −1 kEQ = 0.073Ω m h Mpc. In the ΛCDM model the growth function D (k,a) during the matter dominance is scale-independent (D (a ) = a), but in BD theory with the action (10.10View Equation) the growth of perturbations is generally scale-dependent.

From Eqs. (13.2View Equation) and (13.8View Equation) we obtain the matter perturbation δm for z < zI:

-3-F----k2--- δm(k,a ) = − 10 F (0) 2Φwl(k,ai)T (k)D (k, a). (13.10 ) 0Ω m H 0
The initial power spectrum generated during inflation is PΦwl ≡ 4|Φ|2 = (200π2∕9k3 )(k ∕H0 )nΦ −1δ2H, where n Φ is the spectral index and δ2 H is the amplitude of Φwl [71214]. Therefore we obtain the power spectrum of matter perturbations, as
( F )2 kn Φ Pδm(k,a ) ≡ |δm |2 = 2π2 --- --(0)--------δ2HT 2(k)D2 (k, a). (13.11 ) F0 (Ωm )2Hn0Φ+3
Plugging Eq. (13.11View Equation) into Eq. (13.7View Equation), we find that the convergence spectrum is given by
9π2 ∫ zs ( X )2 1 ( ℓ)n Φ ( Φ (z) )2 Pκ(ℓ) = ---- 1 − --- -----δ2H --- T2(x) --wl---- dz, (13.12 ) 2 0 Xs E (z) X Φwl(zI)
H-(z-) -H0--ℓ- E(z) = H0 , X = H0 χ, x = kEQ X . (13.13 )
Note that X satisfies the differential equation dX ∕dz = 1∕E (z).
View Image

Figure 14: The convergence power spectrum P κ(ℓ) in f (R) gravity (√ -- Q = − 1∕ 6) for the model (5.19View Equation). This model corresponds to the field potential (10.23View Equation). Each case corresponds to (a) p = 0.5, C = 0.9, (b) p = 0.7, C = 0.9, and (c) the ΛCDM model. The model parameters are chosen to be (0) Ω m = 0.28, n Φ = 1, and 2 − 10 δH = 3.2 × 10. From [595Jump To The Next Citation Point].

In Figure 14View Image we plot the convergence spectrum in f (R) gravity with the potential (10.23View Equation) for two different values of p together with the ΛCDM spectrum. Recall that this model corresponds to the f (R) model f(R ) = R − μRc [1 − (R∕Rc )−2n] with the correspondence p = 2n ∕(2n + 1). Figure 14View Image shows the convergence spectrum in the linear regime characterized by ℓ ≲ 200. The ΛCDM model corresponds to the limit n → ∞, i.e., p → 1. The deviation from the ΛCDM model becomes more significant for smaller p away from 1. Since the evolution of Φwl changes from Φwl = constant to √ ------- Φwl ∝ t( 25+48Q2− 5)∕6 at the transition time tℓ characterized by the condition 2 2 2 M ∕F = (ℓ∕χ) ∕a, this leads to a difference of the spectral index of the convergence spectrum compared to that of the ΛCDM model [595]:

∘ ---------2 --Pκ(ℓ)---∝ ℓΔns, where Δns = (1-−-p)(--25-+-48Q--−--5). (13.14 ) PΛκCDM (ℓ) 4 − p
This estimation is reliable for the transition redshift zℓ much larger than 1. In the simulation of Figure 14View Image the numerical value of Δns for p = 0.7 at ℓ = 200 is 0.056 (with zℓ = 3.26), which is slightly smaller than the analytic value Δns = 0.068 estimated by Eq. (13.14View Equation). The deviation of the spectral index of Pκ from the ΛCDM model will be useful to probe modified gravity in future high-precision observations. Note that the galaxy-shear correlation spectrum will be also useful to constrain modified gravity models [528].

Recent data analysis of the weak lensing shear field from the Hubble Space Telescope’s COSMOS survey along with the ISW effect of CMB and the cross-correlation between the ISW and galaxy distributions from 2MASS and SDSS surveys shows that the anisotropic parameter η ≡ Ψ âˆ•Φ is constrained to be η < 1 at the 98% confidence level [73]. For BD theory with the action (10.10View Equation) the quasi-static results (10.38View Equation) and (10.39View Equation) of the gravitational potentials give

2 2 2 2 η ≃ (k-∕a-)(1 −-2Q-)F-+--M--. (13.15 ) (k2∕a2)(1 + 2Q2 )F + M 2
Since η ≃ (1 − 2Q2 )∕(1 + 2Q2) in the scalar-tensor regime (k2∕a2 ≫ M 2∕F), one can realize η < 1 in BD theory. Of course we need to wait for further observational data to reach the conclusion that modified gravity is favored over the ΛCDM model.

To conclude this session we would like to point out the possibility of using the method of gravitational lensing tomography [574]. This method consists of considering lensing on different redshift data-bins. In order to use this method, one needs to know the evolution of both the linear growth rate and the non-linear one (typically found through a standard linear-to-non-linear mapping). Afterward, from observational data, one can separate different bins in order to make fits to the models. Having good data sets, this procedure is strong enough to further constrain the models, especially together with other probes such as CMB [322320632292].

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