12.3 f(𝒢 ) gravity

Let us consider the theory (12.7View Equation) in the presence of matter, i.e.
∫ [ ] S = -1- d4x√ −-g 1R + f(𝒢) + SM , (12.9 ) κ2 2
where we have recovered 2 κ. For the matter action SM we consider perfect fluids with an equation of state w. The variation of the action (12.9View Equation) leads to the following field equations [178Jump To The Next Citation Point383Jump To The Next Citation Point]
[ ] G μν + 8 R μρνσ + R ρνgσμ − R ρσg νμ − R μνgσρ + Rμσg νρ + (R ∕2)(gμνgσρ − gμσgνρ)∇ ρ∇ σf,𝒢 + (𝒢f − f)g = κ2T , (12.10 ) ,𝒢 μν μν
where Tμν is the energy-momentum tensor of matter. If f ∝ 𝒢, then it is clear that the theory reduces to GR.

12.3.1 Cosmology at the background level and viable f (𝒢 ) models

In the flat FLRW background the (00) component of Eq. (12.10View Equation) leads to

3H2 = 𝒢f,𝒢 − f − 24H3 f˙,𝒢 + κ2(ρm + ρr), (12.11 )
where ρm and ρr are the energy densities of non-relativistic matter and radiation, respectively. The cosmological dynamics in f(𝒢) dark energy models have been discussed in [458Jump To The Next Citation Point165383Jump To The Next Citation Point188Jump To The Next Citation Point633Jump To The Next Citation Point430]. We can realize the late-time cosmic acceleration by the existence of a de Sitter point satisfying the condition [458]
3H21 = 𝒢1f,𝒢(𝒢1 ) − f (𝒢1 ), (12.12 )
where H1 and 𝒢1 are the Hubble parameter and the GB term at the de Sitter point, respectively. From the stability of the de Sitter point we require the following condition [188Jump To The Next Citation Point]
0 < H6 f (H ) < 1∕384. (12.13 ) 1 ,𝒢𝒢 1

The GB term is given by

2 2 ˙ 4 𝒢 = 24H (H + H ) = − 12H (1 + 3weff), (12.14 )
where weff = − 1 − 2H˙∕(3H2 ) is the effective equation of state. We have 𝒢 < 0 and 𝒢˙> 0 during both radiation and matter domination. The GB term changes its sign from negative to positive during the transition from the matter era (4 𝒢 = − 12H) to the de Sitter epoch (4 𝒢 = 24H). Perturbing Eq. (12.11View Equation) about the background radiation and matter dominated solutions, the perturbations in the Hubble parameter involve the mass squared given by M 2 ≡ 1∕(96H4f, 𝒢𝒢) [188Jump To The Next Citation Point]. For the stability of background solutions we require that M 2 > 0, i.e., f,𝒢𝒢 > 0. Since the term 24H3 f˙𝒢 in Eq. (12.11View Equation) is of the order of 8 H f,𝒢𝒢, this is suppressed relative to 2 3H for 6 H f,𝒢𝒢 ≪ 1 during the radiation and matter dominated epochs. In order to satisfy this condition we require that f,𝒢𝒢 approaches 0 in the limit |𝒢 | → ∞. The deviation from the ΛCDM model can be quantified by the following quantity [182Jump To The Next Citation Point]
˙ ˙ 6 ′ μ ≡ H f,𝒢 = H 𝒢f,𝒢𝒢 = 72H f,𝒢𝒢 [(1 + weff )(1 + 3we ff) − w eff∕2], (12.15 )
where a prime represents a derivative with respect to N = ln a. During the radiation and matter eras we have μ = 192H6f, 𝒢𝒢 and μ = 72H6f,𝒢𝒢, respectively, whereas at the de Sitter attractor μ = 0.

The GB term inside and outside a spherically symmetric body (mass M ⊙ and radius r⊙) with a homogeneous density is given by 𝒢 = − 48(GM )2∕r6 ⊙ ⊙ and 𝒢 = 48 (GM )2∕r6 ⊙, respectively (r is a distance from the center of symmetry). In the vicinity of Sun or Earth, |𝒢 | is much larger than the present cosmological GB term, 𝒢0. As we move from the interior to the exterior of the star, the GB term crosses 0 from negative to positive. This means that f(𝒢) and its derivatives with respect to 𝒢 need to be regular for both negative and positive values of 𝒢 whose amplitudes are much larger than 𝒢0.

The above discussions show that viable f(𝒢 ) models need to obey the following conditions:

1.
f(𝒢 ) and its derivatives f,𝒢, f,𝒢𝒢, …are regular.
2.
f > 0 ,𝒢𝒢 for all 𝒢 and f ,𝒢𝒢 approaches +0 in the limit |𝒢| → ∞.
3.
6 0 < H 1f,𝒢 𝒢(H1) < 1∕384 at the de Sitter point.

A couple of representative models that can satisfy these conditions are [188Jump To The Next Citation Point]

( ) ( ) -𝒢--- 𝒢-- 1- ∘ --- 𝒢2- ∘ --- (A ) f(𝒢 ) = λ √ 𝒢-arctan 𝒢 − 2λ 𝒢∗ln 1 + 𝒢2 − αλ 𝒢∗, (12.16 ) ∗ ( ∗) ∗ (B ) f(𝒢 ) = λ √𝒢--arctan 𝒢-- − αλ∘ 𝒢-, (12.17 ) 𝒢∗ 𝒢∗ ∗
where α, λ and 4 𝒢 ∗ ∼ H0 are positive constants. The second derivatives of f in terms of 𝒢 for the models (A) and (B) are f,𝒢𝒢 = λ∕[𝒢3∗∕2(1 + 𝒢2∕𝒢2∗ )] and f,𝒢𝒢 = 2λ∕ [𝒢3∗∕2(1 + 𝒢2∕ 𝒢2∗)2], respectively. They are constructed to give rise to the positive f,𝒢𝒢 for all 𝒢. Of course other models can be introduced by following the same prescription. These models can pass the constraint of successful expansion history that allows the smooth transition from radiation and matter eras to the accelerated epoch [188Jump To The Next Citation Point633]. Although it is possible to have a viable expansion history at the background level, the study of matter density perturbations places tight constraints on these models. We shall address this issue in Section 12.3.4.

12.3.2 Numerical analysis

In order to discuss cosmological solutions in the low-redshift regime numerically for the models (12.16View Equation) and (12.17View Equation), it is convenient to introduce the following dimensionless quantities

H˙- H-- κ2ρm- κ2-ρr x ≡ H2 , y ≡ H∗ , Ωm ≡ 3H2 , Ωr ≡ 3H2 , (12.18 )
where 1∕4 H ∗ = G ∗. We then obtain the following equations of motion [188Jump To The Next Citation Point]
1 [ 𝒢f − f ] x′ = − 4x2 − 4x + ---------- --,𝒢-----− 3(1 − Ωm − Ωr) , (12.19 ) 242H6f,𝒢 𝒢 H2 y′ = xy, (12.20 ) ′ Ω m = − (3 + 2x )Ωm, (12.21 ) Ω ′r = − (4 + 2x )Ωr, (12.22 )
where a prime represents a derivative with respect to N = ln a. The quantities 6 H f,𝒢𝒢 and 2 (𝒢f,𝒢 − f)∕H can be expressed by x and y once the model is specified.
View Image

Figure 12: The evolution of μ (multiplied by 104) and w eff versus the redshift z = a ∕a − 1 0 for the model (12.16View Equation) with parameters α = 100 and − 4 λ = 3 × 10. The initial conditions are chosen to be x = − 1.499985, y = 20, and Ωm = 0.99999. We do not take into account radiation in this simulation. From [182Jump To The Next Citation Point].

Figure 12View Image shows the evolution of μ and we ff without radiation for the model (12.16View Equation) with parameters α = 100 and λ = 3 × 10− 4. The quantity μ is much smaller than unity in the deep matter era (weff ≃ 0) and it reaches a maximum value prior to the accelerated epoch. This is followed by the decrease of μ toward 0, as the solution approaches the de Sitter attractor with w = − 1 eff. While the maximum value of μ in this case is of the order of −4 10, it is also possible to realize larger maximum values of μ such as μmax ≳ 0.1.

For high redshifts the equations become too stiff to be integrated directly. This comes from the fact that, as we go back to the past, the quantity f,𝒢𝒢 (or μ) becomes smaller and smaller. In fact, this also occurs for viable f (R) dark energy models in which f ,RR decreases rapidly for higher z. Here we show an iterative method (known as the “fixed-point” method) [420188Jump To The Next Citation Point] that can be used in these cases, provided no singularity is present in the high redshift regime [188Jump To The Next Citation Point]. We define 2 H¯ and ¯ð’¢ to be H¯2 ≡ H2 ∕H20 and ¯ð’¢ ≡ 𝒢 ∕H40, where the subscript “0” represents present values. The models (A) and (B) can be written in the form

2 2 f (𝒢) = ¯f(𝒢 )H 0 − ¯ΛH 0, (12.23 )
where ¯Λ = α λ√G---∕H2 ∗ 0 and ¯f(𝒢 ) is a function of 𝒢. The modified Friedmann equation reduces to
¯2 ¯ 2 1- ¯ ¯ ¯ df¯,¯ð’¢ ¯4 H − H Λ = 3 (f,¯ð’¢ð’¢ − f ) − 8 dN H , (12.24 )
where ¯ 2 (0) 3 (0) 4 ¯ H Λ = Ωm ∕a + Ω r ∕a + Λ âˆ•3 (which represents the Hubble parameter in the ΛCDM model). In the following we omit the tilde for simplicity.

In Eq. (12.24View Equation) there are derivatives of H in terms of N up to second-order. Then we write Eq. (12.24View Equation) in the form

2 2 ( 2 2′ 2′′) H − HΛ = C H ,H ,H , (12.25 )
where 4 C = (f,𝒢𝒢 − f )∕3 − 8H (df,𝒢∕dN ). At high redshifts (a ≲ 0.01) the models (A) and (B) are close to the ΛCDM model, i.e., H2 ≃ H2Λ. As a starting guess we set the solution to be H2(0) = H2Λ. The first iteration is then H2 = H2 + C(0) (1) Λ, where C (0) ≡ C (H2 ,H2 ′,H2 ′′) (0) (0) (0). The second iteration is 2 2 H (2) = H Λ + C (1), where ( 2 2 ′ 2 ′′) C (1) ≡ C H(1),H(1),H (1).
View Image

Figure 13: Plot of the absolute errors 2 2 log10(|H i − H Λ − Ci|) (left) and [|H2i−H2Λ−Ci|] log10 |H2i−H2Λ+Ci| (right) versus N = lna for the model (12.16View Equation) with i = 0,1, ⋅⋅⋅,6. The model parameters are α = 10 and λ = 0.075. The iterative method provides the solutions with high accuracy in the regime N ≲ − 4. From [188].

If the starting guess is in the basin of a fixed point, 2 H (i) will converge to the solution of the equation after the i-th iteration. For the convergence we need the following condition

2 2 2 2 H-i+1-−-H-i < H-i-−-H-i−-1, (12.26 ) H2i+1 + H2i H2i + H2i− 1
which means that each correction decreases for larger i. The following relation is also required to be satisfied:
H2 − H2 − C H2 − H2 − C --i2+1-----Λ2----i+1 < --i2-----Λ2----i. (12.27 ) H i+1 − H Λ + Ci+1 H i − H Λ + Ci
Once the solution begins to converge, one can stop the iteration up to the required/available level of precision. In Figure 13View Image we plot absolute errors for the model (12.16View Equation), which shows that the iterative method can produce solutions accurately in the high-redshift regime. Typically this method stops working when the initial guess is outside the basin of convergence. This happen for low redshifts in which the modifications of gravity come into play. In this regime we just need to integrate Eqs. (12.19View Equation) – (12.22View Equation) directly.

12.3.3 Solar system constraints

We study local gravity constraints on cosmologically viable f(𝒢 ) models. First of all there is a big difference between f(𝒢 ) and f (R) theories. The vacuum GR solution of a spherically symmetric manifold, the Schwarzschild metric, corresponds to a vanishing Ricci scalar (R = 0) outside the star. In the presence of non-relativistic matter, R approximately equals to the matter density κ2 ρm for viable f (R) models.

On the other hand, even for the vacuum exterior of the Schwarzschild metric, the GB term has a non-vanishing value αβγδ 2 6 𝒢 = RαβγδR = 12rs∕r [178185Jump To The Next Citation Point], where rs = 2GM ⊙∕r⊙ is the Schwarzschild radius of the object. In the regime |𝒢 | ≫ 𝒢∗ the models (A) and (B) have a correction term of the order √--- λ 𝒢 ∗𝒢2∗∕𝒢2 plus a cosmological constant term √--- − (α + 1)λ 𝒢 ∗. Since 𝒢 does not vanish even in the vacuum, the correction term 𝒢2∗∕𝒢2 can be much smaller than 1 even in the absence of non-relativistic matter. If matter is present, this gives rise to the contribution of the order of R2 ≈ (κ2ρ )2 m to the GB term. The ratio of the matter contribution to the vacuum value (0) 2 6 𝒢 = 12rs∕r is estimated as

-R2- (8π-)2ρ2mr6- Rm ≡ 𝒢 (0) ≈ 48 M 2⊙ . (12.28 )
At the surface of Sun (radius 10 24 −1 r⊙ = 6.96 × 10 cm = 3.53 × 10 GeV and mass M ⊙ = 1.99 × 1033 g = 1.12 × 1057 GeV), the density ρm drops down rapidly from the order ρm ≈ 10−2 g∕cm3 to the order ρm ≈ 10 −16 g∕cm3. If we take the value ρm = 10−2 g∕cm3 we have R ≈ 4 × 10−5 m (where we have used 1 g∕cm3 = 4.31 × 10−18 GeV4). Taking the value −16 3 ρm = 10 g∕cm leads to a much smaller ratio: −33 Rm ≈ 4 × 10. The matter density approaches a constant value 3 ρm ≈ 10−24 g∕cm around the distance r = 103r⊙ from the center of Sun. Even at this distance we have Rm ≈ 4 × 10−31, which means that the matter contribution to the GB term can be neglected in the solar system we are interested in.

In order to discuss the effect of the correction term 2 2 𝒢∗∕ 𝒢 on the Schwarzschild metric, we introduce a dimensionless parameter

∘ ------ 𝜀 = 𝒢∗∕𝒢s, (12.29 )
where 𝒢s = 12∕r4s is the scale of the GB term in the solar system. Since √ --- 𝒢∗ is of the order of the Hubble parameter H ≈ 70 kmsec− 1 Mpc −1 0, the parameter for the Sun is approximately given by −46 𝜀 ≈ 10. We can then decompose the vacuum equations in the form
G μν + ðœ€Σ μν = 0, (12.30 )
where G μν is the Einstein tensor and
ρ σ &tidle; Σ μν = 8[R μρνσ + R ρνgσμ − Rρσgνμ − R μνgσρ + Rμσgνρ + R (gμνgσρ − gμσg νρ)∕2]∇ ∇ f,𝒢 + (𝒢 &tidle;f,𝒢 − &tidle;f)gμν. (12.31 )
Here &tidle; f is defined by &tidle; f = 𝜀f.

We introduce the following ansatz for the metric

ds2 = − A (r,𝜀)dt2 + B −1(r,𝜀)dr2 + r2(d𝜃2 + sin2 𝜃dϕ2), (12.32 )
where the functions A and B are expanded as power series in 𝜀, as
A = A0 (r) + A1 (r)𝜀 + O (𝜀2), B = B0 (r) + B1 (r)𝜀 + O(𝜀2). (12.33 )
Then we can solve Eq. (12.30View Equation) as follows. At zero-th order the equations read
G μν(0)(A0, B0 ) = 0, (12.34 )
which leads to the usual Schwarzschild solution, A0 = B0 = 1 − rs∕r. At linear order one has
𝜀[G μν(1)(A1, B1,A0, B0 ) + Σ μν(0)(A0, B0)] = 0. (12.35 )
Since A0 and B0 are known, one can solve the differential equations for A1 and B1. This process can be iterated order by order in 𝜀.

For the model (A) introduced in (12.16View Equation), we obtain the following differential equations for A1 and B1 [185Jump To The Next Citation Point]:

√ -- √ -- √-- ρdB1- + B1 = 32 3λρ3 + 12 3λ ρ2ln(ρ) + (4ln𝜀 − 2α − 28 ) 3 λρ2, (12.36 ) dρ 2 dA1 √ -- 4 √ -- 3 (ρ − ρ )-dρ- + A1 = 8 3λρ − 2 3(10 + 6 ln ρ + 2ln𝜀 − α )λρ √ -- − 2 3(α − 6lnρ − 2 ln𝜀 − 6)λρ2 + ρB1, (12.37 )
where ρ ≡ r∕rs. The solutions to these equations are
√ -- 3 √ -- 2 2√ -- 2 B1 = 8 3λρ + 4 3λρ ln ρ + 3 3 (2ln𝜀 − α − 16)λρ , (12.38 ) 16√ -- 2 √ -- A1 = − --- 3λρ3 + -- 3(4 − α + 6 ln ρ + 2ln 𝜀)λρ2. (12.39 ) 3 3

Here we have neglected the contribution coming from the homogeneous solution, as this would correspond to an order 𝜀 renormalization contribution to the mass of the system. Although 𝜀 ≪ 1, the term in ln𝜀 only contributes by a factor of order 2 10. Since ρ ≫ 1 the largest contributions to B1 and A1 correspond to those proportional to ρ3, which are different from the Schwarzschild–de Sitter contribution (which grows as ρ2). Hence the model (12.16View Equation) gives rise to the corrections larger than that in the cosmological constant case by a factor of ρ. Since 𝜀 is very small, the contributions to the solar-system experiments due to this modification are too weak to be detected. The strongest bound comes from the shift of the perihelion of Mercury, which gives the very mild bound λ < 2 × 105 [185Jump To The Next Citation Point]. For the model (12.17View Equation) the constraint is even weaker, λ(1 + α) < 1014. In other words, the corrections look similar to the Schwarzschild–de Sitter metric on which only very weak bounds can be placed.

12.3.4 Ghost conditions in the FLRW background

In the following we shall discuss ghost conditions for the action (12.9View Equation). For simplicity let us consider the vacuum case (SM = 0) in the FLRW background. The action (12.9View Equation) can be expanded at second order in perturbations for the perturbed metric (6.1View Equation), as we have done for the action (6.2View Equation) in Section 7.4. Before doing so, we introduce the gauge-invariant perturbed quantity

H ℛ = ψ − --δ ξ, where ξ ≡ f,𝒢. (12.40 ) ˙ξ
This quantity completely describes the dynamics of all the scalar perturbations. Note that for the gauge choice δξ = 0 one has ℛ = ψ. Integrating out all the auxiliary fields, we obtain the second-order perturbed action [186Jump To The Next Citation Point]
∫ [ 2 ] (2) 3 3 1-˙2 1-cs- 2 δS = dtd xa Qs 2ℛ − 2 a2(∇ ℛ ) , (12.41 )
where we have defined
24(1 + 4μ)μ2 Qs ≡ --2--------2-, (12.42 ) κ (1 + 6 μ) 2 2H˙ cs ≡ 1 + --2-= − 2 − 3weff. (12.43 ) H
Recall that μ has been introduced in Eq. (12.15View Equation).

In order to avoid that the scalar mode becomes a ghost, one requires that Qs > 0, i.e.

μ > − 1∕4. (12.44 )
This relation is dynamical, as one requires to know how H and its derivatives change in time. Therefore whatever f(𝒢 ) is, the propagating scalar mode can still become a ghost. If ˙ f,𝒢 > 0 and H > 0, then μ > 0 and hence the ghost does not appear. The quantity cs characterizes the speed of propagation for the scalar mode, which is again dependent on the dynamics. For any GB theory, one can give initial conditions of H and H˙ such that c2s becomes negative. This instability, if present, governs the high momentum modes in Fourier space, which corresponds to an Ultra-Violet (UV) instability. In order to avoid this UV instability in the vacuum, we require that the effective equation of state satisfies weff < − 2∕3. At the de Sitter point (weff = − 1) the speed cs is time-independent and reduces to the speed of light (cs = 1).

Suppose that the scalar mode does not have a ghost mode, i.e., Qs > 0. Making the field redefinition u = zsℛ and √ --- zs = a Qs, the action (12.41View Equation) can be written as

∫ [ 1 1 1 ] δS(2) = dηd3x -u′2 − -c2s(∇u )2 − -a2M s2u2 , (12.45 ) 2 2 2
where a prime represents a derivative with respect to ∫ η = a−1dt and M 2 ≡ − z′′∕(a2zs) s s. In order to realize the positive mass squared (2 M s > 0), the condition f,𝒢𝒢 > 0 needs to be satisfied in the regime μ ≪ 1 (analogous to the condition f,RR > 0 in metric f (R) gravity).

12.3.5 Viability of f (𝒢) gravity in the presence of matter

In the presence of matter, other degrees of freedom appear in the action. Let us take into account a perfect fluid with the barotropic equation of state w = P ∕ρ M M M. It can be proved that, for small scales (i.e., for large momenta k) in Fourier space, there are two different propagation speeds given by [182Jump To The Next Citation Point]

c21 = wM , (12.46 ) ˙ 2 c2 = 1 + 2H--+ 1-+-wM--κ-ρM--. (12.47 ) 2 H2 1 + 4μ 3H2

The first result is expected, as it corresponds to the matter propagation speed. Meanwhile the presence of matter gives rise to a correction term to c22 in Eq. (12.43View Equation). This latter result is due to the fact that the background equations of motion are different between the two cases. Recall that for viable f (𝒢 ) models one has |μ | ≪ 1 at high redshifts. Since the background evolution is approximately given by 2 3H ≃ 8πG ρM and ˙ 2 H ∕H ≃ − (3∕2)(1 + wM ), it follows that

2 c2 ≃ − 1 − 2wM . (12.48 )
Hence the UV instability can be avoided for wM < − 1∕2. During the radiation era (wM = 1∕3) and the matter era (wM = 0), the large momentum modes are unstable. In particular this leads to the violent growth of matter density perturbations incompatible with the observations of large-scale structure [383182Jump To The Next Citation Point]. The onset of the negative instability can be characterized by the condition [182]
μ ≈ (aH ∕k)2. (12.49 )
As long as μ ⁄= 0 we can always find a wavenumber k (≫ aH ) satisfying the condition (12.49View Equation). For those scales linear perturbation theory breaks down, and in principle one should look for all higher-order contributions. Hence the background solutions cannot be trusted any longer, at least for small scales, which makes the theory unpredictable. In the same regime, one can easily see that the scalar mode is not a ghost, as Eq. (12.44View Equation) is satisfied (see Figure 12View Image). Therefore the instability is purely classical. This kind of UV instability sets serious problems for any theory, including f(𝒢) gravity.

12.3.6 The speed of propagation in more general modifications of gravity

We shall also discuss more general theories given by Eq. (12.8View Equation), i.e.

∫ 4 √ --- S = d x − gf (R, 𝒢), (12.50 )
where we do not take into account the matter term here. It is clear that this function allows more freedom with respect to the background cosmological evolution8, as now one needs a two-parameter function to choose. However, once more the behavior of perturbations proves to be a strong tool in order to have a deep insight into the theory.

The second-order action for perturbations is given by

∫ [ ] 3 3 1-˙2 1-B1- 2 1-B2- 2 2 S = dtd xa Qs 2ℛ − 2 a2 (∇ ℛ ) − 2 a4 (∇ ℛ ) , (12.51 )
where we have introduced the gauge-invariant field
2 ℛ = ψ − H-(δF--+-4H--δξ), (12.52 ) F˙ + 4H2 ˙ξ
with F ≡ f,R and ξ ≡ f,𝒢. The forms of Qs (t), B1 (t) and B2(t) are given explicitly in [186Jump To The Next Citation Point].

The quantity B2 vanishes either on the de Sitter solution or for those theories satisfying

( ) ∂2f ∂2f ∂2f 2 Δ ≡ ---2 ---2 − ------ = 0. (12.53 ) ∂R ∂𝒢 ∂R ∂ 𝒢
If Δ â„= 0, then the modes with high momenta k have a very different propagation. Indeed the frequency ω becomes k-dependent, that is [186]
4 ω2 = B2 k-. (12.54 ) a4
If B2 < 0, then a violent instability arises. If B2 > 0, then these modes propagate with a group velocity
∘ ---k- vg = 2 B2 a. (12.55 )
This result implies that the superluminal propagation is always present in these theories, and the speed is scale-dependent. On the other hand, when Δ = 0, this behavior is not present at all. Therefore, there is a physical property by which different modifications of gravity can be distinguished. The presence of an extra matter scalar field does not change this regime at high k [185], because the Laplacian of the gravitational field is not modified by the field coupled to gravity in the form f (ϕ, R, 𝒢).
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