Overall observational constraints on the running parameter\lambda of ...

Overall observational constraints on the running parameter λ of Hoˇ rava-Lifshitz gravity Sourish Dutta1, ∗ and Emmanuel N. Saridakis2, † 1

arXiv:1002.3373v2 [hep-th] 4 May 2010

Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235 2 College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing, 400065, P.R. China We use observational data from Type Ia Supernovae (SNIa), Baryon Acoustic Oscillations (BAO), and Cosmic Microwave Background (CMB), along with requirements of Big Bang Nucleosynthesis (BBN), to constrain the running parameter λ of Hoˇrava-Lifshitz gravity, which determines the flow between the Ultra-Violet and the Infra-Red. We consider both the detailed and non-detailed balance versions of the gravitational sector, and we include the matter and radiation sectors. Allowing for variation of all the parameters of the theory, we construct the likelihood contours and we conclude that in 1σ confidence λ is restricted to |λ − 1| . 0.02, while its best fit value is |λb.f − 1| ≈ 0.002. Although this observational analysis restricts the running parameter λ very close to its IR value 1, it does not enlighten the discussion about the theory’s possible conceptual and theoretical problems. PACS numbers: 98.80.-k, 04.60.Bc, 04.50.Kd

I.

INTRODUCTION

Hoˇrava recently proposed a power-counting renormalizable, Ultra-Violet (UV) complete theory of gravity [1– 4]. Although presenting an Infra-Red (IR) fixed point, namely General Relativity, in the UV the theory possesses a fixed point with an anisotropic, Lifshitz scaling between time and space. Since then there has been a significant progress in examining the properties of the theory itself, including various extensions of the original basic version [5–30]. Additionally, application of HoˇravaLifshitz gravity as a cosmological framework gives rise to Hoˇrava-Lifshitz cosmology [31, 32], and in particular one can study specific solution subclasses [33–47], the phasespace behavior [48–52], the gravitational wave production [53–58], the perturbation spectrum [59–68], the matter bounce [69–74], the black hole properties [75–89], the dark energy phenomenology [90–95], the astrophysical phenomenology [96–100], the thermodynamic properties [101–103] etc. However, despite this extended research, there are still many ambiguities if Hoˇrava-Lifshitz gravity is reliable and capable of a successful description of the gravitational background of our world, as well as of the cosmological behavior of the universe [11, 13, 15, 26, 104– 108]. Although the discussion about the foundations and the possible conceptual and phenomenological problems of Hoˇrava-Lifshitz gravity and cosmology is still open in the literature, it is worthy to examine the possible constraints that observational and cosmological data could impose on the parameters of the scenario (of the basic as well as of the various extended versions). In such an investigation a reasonable assumption is to set the running parameter λ to its IR value 1, since all observa-

∗ Electronic † Electronic

address: [email protected] address: [email protected]

tions lie deep inside the IR. Thus, under this assumption in [96–98] the authors used Solar System observations in order to constrain some of the remaining model parameters of the basic version of Hoˇrava-Lifshitz cosmology under detailed balance, while in [100] the analysis was performed including a soft detailed-balance breaking. Similarly, in [109] we used cosmological observations (Type Ia Supernovae (SNIa), Baryon Acoustic Oscillations (BAO) and Cosmic Microwave Background (CMB) ones, together with Big Bang Nucleosynthesis conditions) in order to impose complete constraints on all the parameters of the basic version of Hoˇrava-Lifshitz cosmology and construct the corresponding contour plots, with or without the detailed-balance condition. Although setting λ to its IR value is a first and reasonable assumption, one could go beyond it, and examine the observational constraints that the data could impose on λ itself. However, allowing λ varying, that is preserving the Lorentz invariance breaking (which is restored in the exact IR value λ = 1), one must take into account that in theories with Lorentz invariance breaking the “gravitational” Newton’s constant Ggrav , that is the one that is present in the gravitational action, does not coincide with the “cosmological” Newton’s constant Gcosmo , that is the one that is present in Friedmann equations [110]. Thus, in the case of Hoˇrava-Lifshitz gravity one could use the deviation between Ggrav and Gcosmo , as it is constrained by measurements of the primordial abundance of He4 [110], in order to extract an upper bound on |λ − 1|. This approach was followed in [20, 106] with the result 0 < |λ − 1| . 0.1. However, it is obvious that such an approach can only provide a crude upper bound on |λ − 1|, since it considers that all the other parameters of the theory remain constant. The correct approach should be to perform a systematic investigation, allowing for simultaneous variations of all model parameters, and constrain all of them using observations. In the present work we are interested in performing such an holistic observational constraining of all the pa-

2 rameters of the basic version of Hoˇrava-Lifshitz cosmology, and especially of the running parameter λ, using SNIa, BAO and CMB cosmological observations in order to construct the corresponding probability contourplots. Furthermore, in order to be general and modelindependent, we perform our analysis with and without the detailed-balance condition. The plan of the work is the following: In section II we present the basic ingredients of Hoˇrava-Lifshitz cosmology, extracting the Friedmann equations, and describing the dark matter and dark energy dynamics. In section III we constrain both the detailed-balance and the beyond-detailed-balance formulations using cosmological observations, and we present the corresponding likelihood contours. Finally, in section IV we summarize the obtained results. II.

ˇ HORAVA-LIFSHITZ COSMOLOGY

In this section we briefly review the scenario where the cosmological evolution is governed by Hoˇrava-Lifshitz gravity [31, 32]. The dynamical variables are the lapse and shift functions, N and Ni respectively, and the spatial metric gij (roman letters indicate spatial indices). In terms of these fields the full metric is written as: ds2 = −N 2 dt2 + gij (dxi + N i dt)(dxj + N j dt),

(1)

where indices are raised and lowered using gij . The scaling transformation of the coordinates reads: t → l3 t and xi → lxi . A.

Detailed Balance

The gravitational action isR decomposed into a kinetic √ and a potential part as Sg = dtd3 x gN (LK +LV ). The assumption of detailed balance [3] reduces the possible terms in the Lagrangian, and it allows for a quantum inheritance principle [1], since the (D + 1)-dimensional theory acquires the renormalization properties of the Ddimensional one. Under the detailed balance condition the full action of Hoˇrava-Lifshitz gravity is given by  Z √ 2 Sg = dtd3 x gN (Kij K ij − λK 2 ) κ2 κ2 κ2 µ ǫijk κ 2 µ2 ij l + C C − R ∇ R + Rij Rij √ ij il j k 2w4 2w2 g 8   1 − 4λ 2 κ 2 µ2 2 R + ΛR − 3Λ , (2) + 8(3λ − 1) 4 where Kij =

1 (g˙ij − ∇i Nj − ∇j Ni ) 2N

(3)

is the extrinsic curvature and
ǫijk 1 C ij = √ ∇k Rij − Rδij g 4

(4)

the Cotton tensor, and the covariant derivatives are defined with respect to the spatial metric gij . ǫijk is the totally antisymmetric unit tensor, λ is a dimensionless constant and the variables κ, w and µ are constants with mass dimensions −1, 0 and 1, respectively. Finally, we mention that in action (2) we have already performed the usual analytic continuation of the parameters µ and w of the original version of Hoˇrava-Lifshitz gravity, since such a procedure is required in order to obtain a realistic cosmology [33, 37, 81, 101] (although it could fatally affect the gravitational theory itself). Therefore, in the present work Λ is a positive constant, which as usual is related to the cosmological constant in the IR limit. In order to add the matter component (including both dark and baryonic matter) in the theory one can follow two equivalent approaches. The first is to introduce a scalar field [31, 32] and thus attribute to dark matter a dynamical behavior, with its energy density ρm and pressure pm defined through the field kinetic and potential energy. Although such an approach is theoretically robust, it is not suitable from the phenomenological point of view since it requires specially-designed matter-potentials in order to acquire an almost constant matter equation-ofstate parameter (wm = pm /ρm ) as it is suggested by observations. In the second approach one adds a cosmological stress-energy tensor to the gravitational field equations, by demanding to recover the usual general relativity formulation in the low-energy limit [13, 46, 48]. Thus, this matter-tensor is a hydrodynamical approximation with ρm and pm (or ρm and wm ) as parameters. Similarly, one can additionally include the standard-modelradiation component (corresponding to photons and neutrinos), with the additional parameters ρr and pr (or ρr and wr ). Such an approach, although not fundamental, is better for a phenomenological analysis, such the one performed in this work. In order to investigate cosmological frameworks, we impose the projectability condition [11] and we use an FRW metric N = 1 , gij = a2 (t)γij , N i = 0 ,

(5)

with γij dxi dxj =

dr2 + r2 dΩ22 , 1 − Kr2

(6)

where K 0 corresponding to open, flat, and closed universe respectively (we have adopted the convention of taking the scale factor a(t) to be dimensionless and the curvature constant K to have mass dimension 2). By varying N and gij , we extract the Friedmann equations:   κ2 H2 = ρm + ρr + 6(3λ − 1)   κ2 3κ2 µ2 K 2 3κ2 µ2 Λ2 + − + 6(3λ − 1) 8(3λ − 1)a4 8(3λ − 1) κ4 µ2 ΛK − , (7) 8(3λ − 1)2 a2

3   κ2 3 wm ρm + wr ρr − H˙ + H 2 = − 2 4(3λ − 1)   κ2 κ 2 µ2 K 2 3κ2 µ2 Λ2 − − − 4(3λ − 1) 8(3λ − 1)a4 8(3λ − 1) κ4 µ2 ΛK − , (8) 16(3λ − 1)2 a2 where H ≡ aa˙ is the Hubble parameter. As usual, ρm (dark plus baryonic matter) follows the standard evolution equation ρ˙ m + 3H(ρm + pm ) = 0,

(9)

while ρr (standard-model radiation) follows ρ˙ r + 3H(ρr + pr ) = 0.

(10)

ρDE ≡

3κ2 µ2 K 2 3κ2 µ2 Λ2 + 8(3λ − 1)a4 8(3λ − 1)

(11)

pDE ≡

κ 2 µ2 K 2 3κ2 µ2 Λ2 − . 8(3λ − 1)a4 8(3λ − 1)

(12)

The term proportional to a−4 is the usual “dark radiation term”, present in Hoˇrava-Lifshitz cosmology [31, 32], while the constant term is just the explicit cosmological constant. Therefore, in expressions (11),(12) we have defined the energy density and pressure for the effective dark energy, which incorporates the aforementioned contributions. Finally, note that using (11),(12) it is straightforward to show that these dark energy quantities satisfy the standard evolution equation: (13)

Using the above definitions, we can re-write the Friedmann equations (7),(8) in the standard form: H2 =

i h κ4 µ2 ΛK κ2 ρm + ρr + ρDE − (14) 6(3λ − 1) 8(3λ − 1)2 a2

i h κ4 µ2 ΛK κ2 3 pm +pr +pDE − . H˙ + H 2 = − 2 4(3λ − 1) 16(3λ − 1)2 a2 (15) Therefore, if we require these expressions to coincide with the standard Friedmann equations, in units where c = 1 we set [31, 32]: Gcosmo =

κ2 16π(3λ − 1)

κ 4 µ2 Λ = 1, 8(3λ − 1)2

Ggrav =

(16)

where Gcosmo is the “cosmological” Newton’s constant. Note that as we said in the Introduction, in theories

κ2 , 32π

(17)

as it can be straightforwardly read from the action (2) (our definitions of Gcosmo , Ggrav coincide with those of [20, 106]). Thus, it becomes obvious that in the IR (λ = 1), where Lorentz invariance is restored, Gcosmo and Ggrav coincide. B.

Lastly, concerning the dark-energy sector we can define

ρ˙ DE + 3H(ρDE + pDE ) = 0.

with Lorentz invariance breaking Gcosmo does not coincide with the “gravitational” Newton’s constant Ggrav , unless Lorentz invariance is restored [110]. For completeness we mention that in our case

Beyond Detailed Balance

The above formulation of Hoˇrava-Lifshitz cosmology has been performed under the imposition of the detailedbalance condition. However, in the literature there is a discussion whether this condition leads to reliable results or if it is able to reveal the full information of HoˇravaLifshitz gravity [31, 32]. Thus, one should study also the Friedmann equations in the case where detailed balance is relaxed. In such a case one can in general write [11, 13, 15, 48, 49]:  2σ0  ρm + ρr + H2 = (3λ − 1)   2 σ1 σ4 K σ3 K 2 + + + + (3λ − 1) 6 6a4 6a6 K σ2 + (18) 3(3λ − 1) a2  3 3σ0  wm ρm + wr ρr − H˙ + H 2 = − 2 (3λ − 1)   σ1 3 σ4 K σ3 K 2 − + + − + (3λ − 1) 6 18a4 6a6 σ2 K + , (19) 6(3λ − 1) a2 where σ0 ≡ κ2 /12, and the constants σi are arbitrary (with σ2 being negative). Note that one could absorb the factor of 6 in redefined parameters, but we prefer to keep it in order to coincide with the notation of [13, 48]. As we observe, the effect of the detailed-balance relaxation is the decoupling of the coefficients, together with the appearance of a term proportional to a−6 . In this case the corresponding quantities for dark energy are generalized to σ4 K σ3 K 2 σ1 + + 4 6 6a 6a6 2 σ4 K σ3 K σ1 + . ≡− + 6 18a4 6a6

ρDE |non-db ≡

(20)

pDE |non-db

(21)

Again, it is easy to show that ρ˙ DE |non-db + 3H(ρDE |non-db + pDE |non-db ) = 0.

(22)

4 Finally, if we force (18),(19) to coincide with the standard Friedmann equations, we result to: 6σ0 8π(3λ − 1) σ2 = −3(3λ − 1),

Gcosmo =

(23)

while in this case the “gravitational” Newton’s constant Ggrav reads [13]: Ggrav =

III.

6σ0 . 16π

(24)

OBSERVATIONAL CONSTRAINTS

Having presented the cosmological equations of a universe governed by Hoˇrava-Lifshitz gravity, both with and without the detailed-balance condition, we now proceed to study the observational constraints on the model parameters. This is performed in the following two subsections, for the detailed and non-detailed balance scenarios separately. We mention that, contrary to [109], in this work we allow the running parameter λ to vary too, and in order to be general enough we do not use any theoretical argument to restrict it in any specific interval, handling it as completely free.

A.

Constraints on Detailed-Balance scenario

We work in the usual units suitable for observational comparisons, namely setting 8πGgrav = 1 (we have already set c = 1 in order to obtain (16)). This allows us to reduce the parameter space, since in this case (17) gives κ2 = 4,

(25)

and thus (16) lead to: 1 4π(3λ − 1) (3λ − 1)2 . µ2 Λ = 2

Gcosmo =

(26)

Inserting these relations into Friedmann equation (7) we obtain    1  3K 2 K 3Λ 2 − 2 . (27) + ρm +ρr + H2 = 3(3λ − 1) 3 2Λa4 2 a In terms of the usual density parameters (Ωm ≡ ρm /(3H 2 ), ΩK ≡ −K/(H 2 a2 ), Ωr ≡ ρr /(3H 2 )) this expression becomes:   2   1 2 K Λ 1− Ωm +Ωr −ΩK = 2 . (28) + (3λ − 1) H 2Λa4 2

Applying this relation at present time and setting the current scale factor a0 = 1 we obtain:   2   2 Λ K 1 1− + Ωm0 + Ωr0 − ΩK0 = 2 , (3λ − 1) H0 2Λ 2 (29) where a 0-subscript denotes the present value of the corresponding quantity. Note that Ωm0 includes contributions from both baryons Ωb0 as well as dark matter ΩDM0 . In order to proceed to the elaboration of observational data, we consider as usual the matter (dark plus baryonic) component to be dust, that is wm ≈ 0, and similarly for the standard-model radiation we consider wr = 1/3, where both assumptions are valid in the epochs in which observations focus. Therefore, the corresponding evolution equations (9),(10) give ρm = ρm0 /a3 and ρr = ρr0 /a4 respectively. Finally, it proves convenient to use the redshift z as the independent variable instead of the scale factor (1 + z ≡ a0 /a = 1/a). Inserting these into Friedmann equation (27) we obtain i h 2 Ωm0 (1 + z)3 + Ωr0 (1 + z)4 + (3λ − 1) h io Ω2 +ΩK0 (1 + z)2 + ω + K0 (1 + z)4 , (30) 4ω

H 2 = H02

n

where we have also introduced the dimensionless parameter ω≡

Λ . 2H02

(31)

Thus, the constraint (29) can be rewritten as:   Ω2 2 Ωm0 + Ωr0 + ΩK0 + ω + K0 = 1. (3λ − 1) 4ω

(32)

We remind that the term Ω2K0 /(4ω) is the coefficient of the dark radiation term, which is a characteristic feature of the Hoˇrava-Lifshitz gravitational background. Since this dark radiation component has been present also during the time of nucleosynthesis, it is subject to bounds from Big Bang Nucleosynthesis (BBN). As discussed in more details in the Appendix of [109], if the upper limit on the total amount of dark radiation allowed during BBN is expressed through the parameter ∆Nν of the effective neutrino species [111–114], then we obtain the following constraint : Ω2K0 = 0.135∆Nν Ωr0 . 4ω

(33)

In this work, in order to ensure consistency with BBN, we adopt an upper limit of ∆Nν ≤ 2.0 following [113].

In most studies of dark energy models it is customary to ignore curvature (e.g.[116–123]), especially concerning observational constraints. This practice is well motivated

5
K κ2 /(8πGgrav ) 1/H02 Λ (8πGgrav H0 ) µ λ ∆Nν >0 4 (0, 1.46) (1.37, ∞) (0.98, 1.01) (0, 0.32) 0

1.06

1.02 λ

TABLE I: 1σ limits on the parameter values for the detailedbalance scenario, for positive and negative curvature.The cosmological parameters Ωm0 , Ωb0 , Ωr0 and H0 have been marginalized over.

1.1

0.98

0.94

since most inflationary scenarios predict a high degree of spatial flatness, and furthermore the CMB data impose stringent constraints on spatial flatness in the context of constant-w models (for example a combination of WMAP+BAO+SNIa data [115] provides the tight simultaneous constraints −0.0179 ≤ ΩK0 ≤ 0.0081 and −0.12 ≤ 1 + w ≤ 0.14, both at 95% confidence).

In summary, the scenario at hand involves seven model parameters namely the cosmological parameters Ωm0 , Ωb0 , Ωk0 , Ωr0 and H0 and the model parameters λ, ω and ∆Nν , subject to constraint equations (32) and (33). We marginalize over the cosmological parameters Ωm0 , Ωb0 , Ωr0 and H0 . Of the four remaining parameters, only two are independent. We choose λ and ∆Nν as our free parameters. Once these are chosen, and for a given choice of curvature, ΩK0 and ω are immediately fixed from the constraint equations. In particular, ω can be determined

0.1

0.2 ∆ Nν

0.3

0.4

1.1 K 0 is required also for the stability of the gravitational perturbations of the theory [13, 15]. For convenience we moreover assume σ3 ≥ 0, that is ω3 ≥ 0. In summary, the present scenario involves the following parameters: the cosmological parameters H0 , Ωm0 , ΩK0 , Ωb0 , Ωr0 , and the model parameters λ, ω1 , ω3 and ω4 . Similarly to the detailed-balance section these are subject to two constraints. The first one arises from the Friedman equation at z = 0, which leads to i h 2 Ωm0 + Ωr0 + ω1 + ω3 + ω4 + ΩK0 = 1. (38) (3λ − 1) This constraint eliminates the parameter w1 . The second constraint arises from BBN considerations. The term involving ω3 represents the usual dark-radiation component. In addition, the ω4 -term represents a kination-like component (a quintessence field dominated by kinetic energy [135, 136]). If ∆Nν represents the BBN upper limit on the total energy density of the universe beyond standard model constituents, then as we show in the Appendix of [109] we acquire the following constraint at the time of BBN (z = zBBN ) [111–114]: 2

ω3 + ω4 (1 + zBBN ) = ω3max ≡ 0.135∆Nν Ωr0 .

(39)

ω3 ω3max

,

(40)

which has the interesting physical meaning of denoting the ratio of the energy density of the Hoˇrava-Lifshitz dark radiation to the total energy density of Hoˇrava-Lifshitz dark radiation and kination-like components at the time of BBN. We use relation (39) to eliminate ω4 in favor of α and ∆Nν , and treat λ, α, ΩK0 and ∆Nν as our free parameters, marginalizing over H0 , Ωm0 , Ωb0 and Ωr0 . Using the combined SNIa+CMB+BAO data, we construct likelihood contours for different combinations of the above parameters, which are presented in Fig. 2. In each case, the free parameters not included in the plot have been marginalized over. The details and the techniques of the construction have been described in the Appendix of [109], with the only difference being that in the present work WMAP 7-year data [133] and the more recent Constitution Supernovae dataset [134] have been used. In Table II we summarize the 1σ limits on the parameter values for the beyond-detailed-balance scenario. ΩK0 ∆Nν α λ (−0.01, 0.01) (0, 2) (0, 1) (0.98, 1.01)

TABLE II: 1σ limits on the free parameters of the beyonddetailed-balance scenario. The cosmological parameters Ωm0 , Ωb0 , Ωr0 and H0 have been marginalized over.

Furthermore, as we observe, in 1σ confidence the running parameter λ of Hoˇrava-Lifshitz gravity is restricted to the interval |λ − 1| . 0.02, for the entire allowed range of ω3 (that is of σ3 ). Finally, the best fit value for λ restricts |λ − 1| to much more smaller values, namely |λb.f − 1| ≈ 0.002. IV.

CONCLUSIONS

In this work we have constrained the running parameter λ of Hoˇrava-Lifshitz gravity using observational SNIa, BAO and CMB data as well as considerations from BBN. In order to be general enough we have not used any theoretical argument to a priori restrict λ in any specific interval, handling it as a completely free parameter. Additionally, we have performed our investigation under the detailed-balance condition, as well as under its relaxation. Finally, we have included the matter and radiation sectors following the usual effective fluid approach. We

7 1.1

1.06

1.06

1.02

1.02

λ

λ

1.1

0.98

0.98

0.94

0.94

0.9 0

0.2

0.4

α

0.6

0.8

0.9 0

1

2

1.06

1.6

1.02

1.2

0.8

∆ Nν

1.2

1.6

2

λ

∆N

ν

1.1

0.4

0.98

0.8

0.94

0.4

0.9 −8

−6

−4

−2 ΩK0

0

2

4

0 −8

−6

−3

x 10

−4

−2 ΩK0

0

2

4 −3

x 10

FIG. 2: (Color online) Contour plots of different pairs of free parameters in the beyond-detailed-balance scenario, under SNIa, BAO and CMB observational data. In each case the parameters not included in the plots have been marginalized over. Color scheme as in Fig. 1.

stress that we have let all the parameters of the theory to vary, performing an overall and holistic investigation, and we have not followed the naive approach, that is to keep everything fixed and vary only λ in order to match observations. As we showed, Hoˇrava-Lifshitz cosmology, either with or without the detailed-balance condition, can be compatible with observations. We constructed the likelihoodcontours for the involved free parameters, and we found that in 1σ level λ is restricted to |λ − 1| . 0.02. As expected, these bounds are one order of magnitude tighter than the corresponding ones arising from the consideration of primordial He4 -abundance measurements in order to extract a crude upper bound [20, 106]. Finally, concerning the best fit value for λ we obtain |λb.f −1| ≈ 0.006 for the detailed balance and |λb.f − 1| ≈ 0.002 for the beyond-detailed-balance cases. The aforementioned features act as additional arguments in favor of the present holistic investigation, where all the parameters of the theory are allowed to vary, comparing to the naive or partial ones in which only λ varies. Lastly, it is interesting to note that these results, arising from cosmological observations, are relatively close to the preliminary parametrized post-Newtonian (PPN) estimations for Hoˇrava-Lifshitz gravity (0 < |λ − 1| . 4 × 10−7

[137]), despite the fact that the Solar System measurements (that lie at the basis of PPN parameters [138–141]) are much more accurate than the cosmological ones. In summary, as expected, we found that the value of the running parameter λ of Hoˇrava-Lifshitz gravity is restricted to a very tight window around its IR value. Finally, we should mention that although the present analysis provides the bounds for the running parameter, it does not enlighten the discussion about the possible conceptual problems and instabilities of Hoˇrava-Lifshitz gravity, nor it can address the questions concerning the validity of its theoretical background, which is the subject of interest of other studies. The present work just faces the problem from the phenomenological point of view, which is necessary but not sufficient, and thus its results can been taken into account only if HoˇravaLifshitz gravity passes successfully the aforementioned theoretical tests.

V.

ACKNOWLEDGEMENTS

The authors wish to thank Y. F. Cai for useful discussions.

8

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