New constraints on Coupled Dark Energy from Planck

New constraints on Coupled Dark Energy from Planck

Valentina Salvatelli Physics Department and INFN, Università di Roma “La Sapienza”, Ple Aldo Moro 2, 00185, Rome, Italy    Andrea Marchini Physics Department and INFN, Università di Roma “La Sapienza”, Ple Aldo Moro 2, 00185, Rome, Italy    Laura Lopez-Honorez Theoretische Natuurkunde, Vrije Universiteit Brussel and The International Solvay Institutes Pleinlaan 2, B-1050 Brussels, Belgium    Olga Mena IFIC, Universidad de Valencia-CSIC, 46071, Valencia, Spain
Abstract

We present new constraints on coupled dark energy from the recent measurements of the Cosmic Microwave Background Anisotropies from the Planck satellite mission. We found that a coupled dark energy model is fully compatible with the Planck measurements, deriving a weak bound on the dark matter-dark energy coupling parameter at c.l.. Moreover if Planck data are fitted to a coupled dark energy scenario, the constraint on the Hubble constant is relaxed to km/s/Mpc, solving the tension with the Hubble Space Telescope value. We show that a combined Planck+HST analysis provides significant evidence for coupled dark energy finding a non-zero value for the coupling parameter , with at c.l.. We also consider the combined constraints from the Planck data plus the BAO measurements of the 6dF Galaxy Survey, the Sloan Digital Sky Survey and the Baron Oscillation Spectroscopic Survey.

pacs:
98.80.Es, 98.80.Jk, 95.30.Sf

I Introduction

The Planck satellite experiment has recently provided new and precise measurements of the Cosmic Microwave Background (CMB) anisotropy, covering a wide range of angular scales up to multipole Ade:2013xsa (); Ade:2013lta (); Planck:2013kta (). The new data are in full agreement with the expectations of the so-called standard CDM scenario.

There exist however some tensions when the value of a number of cosmological parameters, as measured by Planck data, are compared with the values of the same parameters as measured by independent cosmological probes.

The most notable case concerns the constraint on the Hubble constant . The value measured by the Planck team , km/s/Mpc (at c.l.), is significantly lower than the previous measurement of km/s/Mpc (at c.l.) from the Hubble Space Telescope (HST) arising from optical and infrared observations of Cepheid variables  riess2011 (). While systematics can certainly be present in both measurements, it is timely to investigate if this discrepancy can be explained by the inclusion of new physical phenomena.

Very recently, Marra et al. marra (), have pointed out that the HST value could be affected by the cosmic variance of the local expansion rate and inhomogeneities. In this approach, the HST value is therefore biased and the correct value is given by Planck measurements.

On the other hand, one has to consider that the CMB determination of the Hubble parameter is not a direct measurement but it is based on the assumption of an underlying theoretical model.

For example, the inclusion in the Planck data analysis of an extra relativistic energy component, parameterized via the effective number of relativistic degrees of freedom , changes the constraint on the Hubble constant to km/s/Mpc at c.l. Ade:2013lta (), totally compatible with the HST determination. Several physical candidates have been recently examined to account for the extra dark radiation component (see e.g. eleonora () and references therein). A combined Planck and HST data analysis yields evidence for dark radiation at more than c.l. (Ade:2013lta (),eleonora2 ()).

In this paper we investigate the possibility that the solution of the Planck-HST tension could arise from an interaction between dark energy and dark matter. The assumption of a coupled dark energy model can change significantly the CMB constraints on the Hubble parameter (see e.g. Olivares:2005tb (); Valiviita:2009nu (); Martinelli:2010rt (); DeBernardis:2011iw ()). In particular, if coupled models are fitted to a (wrong) CDM cosmology, the reconstructed from CMB will be shifted from its real value Honorez:2010rr (). A mismatch between low and high redshift measurements could therefore be a smoking gun for an interaction in the dark sector.

Moreover, even if a cosmological constant is a good fit to the current data, the dark matter-dark energy interaction alleviates the well-known coincidence problem that plagues the CDM scenario (see e.g. Peebles:2002gy ()). Several candidates for coupled dark energy models have been proposed and investigated (see e.g. models ()). Since coupled quintessence models can resemble to scalar-tensor or brans dicke gravitational theories, it has been suggested that coupled dark energy can also hint to a modification of general relativity on cosmological scales (see e.g. Amendola:1999er (); Farrar:2003uw (); Pettorino:2008ez ()).

The paper is organized as follows. In Section II we briefly present the coupled dark energy model we consider for our analysis, in Sec. III we describe the analysis method, in Sec. IV we show our results and in SectionV we draw our conclusions.

Ii Dark interaction parametrization

We assume a flat universe described by the Friedmann-Robertson-Walker metric. We parametrize the interaction as follows:

(1)
(2)

where and are the energy-momentum tensors for the dark matter and dark energy components respectively, is the dark matter four-velocity and the coefficient encodes the interaction rate between the two dark components.

In particular we restrict the parametrization to the case where the interaction rate is proportional to the dark energy density :

(3)

where is a dimensionless parameter and (the dot indicates derivative respect to conformal time ). Such an interacting model is in agreement with cosmological constraints and it does not suffer from early time instabilities if the coupling is negative and the dark energy equation of state satisfies  He:2008si (); Gavela:2009cy (). We shall follow here the former stability conditions.

The background evolution equations in the coupled model considered here read Gavela:2010tm ()

(4)
(5)

In the synchronous gauge, the evolution of the dark matter and dark energy perturbations in the linear regime reads Gavela:2010tm ()

(8)
(9)

where and are the density perturbations and velocities of the dark matter and dark energy fluids, respectively, is the center of mass velocity for the total fluid and is the usual synchronous gauge metric perturbation. Equations (II)-(9) include the contributions of the perturbation in the expansion rate , the dark energy speed of sound has been fixed to 1, i.e. , and the equation of state for dark energy has been taken to be constant.

Also notice that in our numerical analysis, we have considered adiabatic initial conditions for all components, see appendix A.

The dark interaction we consider affects the CMB temperature spectrum in several ways. In Fig. 1, we illustrate the impact of up to multipole for assuming a cold dark matter density and km/s/Mpc. Notice that the presence of a coupling among the dark matter and the dark energy fluids shifts the position of the peaks to larger multipoles. At low multipoles, a value of different from zero contributes to the late integrated Sachs-Wolfe (ISW) effect, while at high multipoles changes the amplitude of the gravitational lensing.

Figure 1: CMB temperature power spectrum in the CDM case and in the coupled cases for , , km/s/Mpc. The main effects of the coupling are shifting the position of the acoustic peaks and varying their amplitude.

Iii Data Analysis Method

The theoretical CMB angular spectra are computed with a modified version of the CAMB code Lewis:1999bs (), in which we have included the background and linear density perturbation equations in the presence of a coupling between the dark matter and the dark energy sectors.

Parameters Prior
[0.005, 0.100]
[0.005, 0.100]
[0.5, 10]
[0.01, 0.80]
[0.9, 1.1]
[2.7, 4.0]
[-1,0]
Table 1: Ranges for the priors of different cosmological parameters considered in the analysis.

For the Planck data set111Planck data set is publicly available at http://pla.esac.esa.int/pla/aio/planckProducts.html, we consider the high- TT likelihood, including measurements up to a maximum multipole number of , combined with the low- TT likelihood which includes measurements up to . We also consider low- () TE,EE,BB likelihood which includes polarization measurements from nine years of observation of the Wilkinson Microwave Anisotropy Probe (WMAP) Bennett:2012fp (). We refer to this data combination as the PLANCK data set.

We also consider the effect of a gaussian prior on the Hubble constant consistent with the measurements of the Hubble Space Telescope as in riess2011 (). We refer to this prior as HST.

Finally we analyze the impact of baryon acoustic oscillations (BAO) measurements, using the data of three surveys at different redshifts: the 6dF Galaxy Survey measurement at provided in Beutler:2011hx (), the SDSS DR7 measurement at from Padmanabhan:2012hf () and the BOSS DR9 measurement at discussed in Anderson:2012sa (). We refer to this combination as the BAO data set.

We sample a seven-dimensional set of cosmological parameters, adopting flat priors on them (see Tab. 1): the coupling parameter, the baryon and cold dark matter densities and , the ratio of the sound horizon to the angular diameter distance at decoupling , the optical depth to reionization , the scalar spectral index and the amplitude of the primordial scalar perturbation spectrum, at . We fix the relativistic number of degrees of freedom parameter to , the helium abundance to , the total neutrino mass to , the spectrum lensing normalization to and the dark energy equation of state to . We marginalize over all foregrounds parameters as described in Ade:2013lta (). We also consider the effect of letting free to vary under the condition .

Our analysis follows a Markov chains approach and is based on a modified version of the public available code CosmoMC Lewis:2002ah (); Lewis:2013hha (), setting the statistical convergence for Gelman and Rubin values below .

Iv Results

We have initially performed four runs, fixing the dark energy equation of state to : a first run with no coupling, , which corresponds to the CDM case (as in Ade:2013lta ()). We have then let the coupling parameter to vary freely, performing separately an analysis with Planck data alone, Planck data plus the HST prior on the Hubble constant and Planck data plus BAO data.

The constraints on the parameters and the best fit values are reported in Tab. 2 while we plot the 1-D posteriors for the parameters in Fig. 2 and the 2-D posterior for the main parameter degeneracies in Fig. 34, 5. Note that the coupling is only between cold dark matter and dark energy, therefore the bounds on the baryon density remain unchanged when is allowed to freely vary.

The presence of a dark coupling is perfectly compatible with the PLANCK data set. Coupled cosmologies provide even a slightly better best fit than the CDM model, see Tab. 2. Looking at the marginalized value of in Tab. 2 it seems to exist also a preference for but this is mainly due to the large number of coupled dark models compatible with the data. The PLANCK data set alone does not exclude 2.

As expected (see e.g Honorez:2010rr ()), there exists a strong degeneracy between and the cold dark matter density , see Fig. 3. Negative values of the coupling implies a larger matter density in the past. Since the PLANCK data are sensitive to the amount of cold dark matter density at recombination, leads to a lower value of the cold dark matter density today. The degeneracy is nearly perfect and for large negative values of the PLANCK data set is compatible with even negligible values of .

This degeneracy affects the geometrical parameters values and, in particular, the Hubble constant value, which in this case assume values significantly larger than in the standard CDM case. The introduction of a coupling changes the constraint of km/s/Mpc (at c.l.), obtained in the CDM case, to km/s/Mpc (at c.l.).

It is therefore not surprising that, when the HST prior is included, the PLANCK+HST data combination suggests a value for the coupling different from zero. The best fit values and the c.l. constraints we have obtained in this case are reported in the third column of table 2 while 2-D posteriors are shown in Fig. 4. The combined PLANCK+HST constraint excludes a zero value of the coupling parameter at more than 2 sigma ( at c.l.), while the value of the Hubble constant is km/s/Mpc at c.l..

If we add instead the BAO low-redshift measurements to the Planck data, we can observe that a zero coupling it is admitted but not favoured and that the tension between the Hubble constant measurements is still alleviated, as in the Planck alone case. The results of this run are shown in the fourth column of table 2 and in Fig. 5.

We have verified that none of the foreground parameters is sensitive to the coupling component, confirming that our analysis is not biased by the foreground marginalization.

We have also explored the effect of a freely varying dark energy equation of state in our results. If is added in the MonteCarlo analyses, the parameters constraints are comparable and the same degeneracies appear, even if in this case the constraint on is weaker when we consider PLANCK alone. This is due to the fact that CMB measurements alone can weakly constrain in general, also in the CDM case, and can not break the degeneracy between the coupling parameter and .

The degeneracy between the coupling parameter , and when we consider a varying for PLANCK data set is shown in Figure 6. Note that the values of obtained for reasonable values of are in a much better agreement with low redshift measurements of the Hubble constant, if compared to the values of obtained in the CDM PLANCK case.

The best fit values and the c.l. constraints from the three dataset combinations when is marginalized over are shown in the table 3. While the 1D and 2D posterior distributions are presented in Fig.7 and 8910.

PLANCK + CDM PLANCK+ PLANCK+ HST+ PLANCK+ BAO+
Parameters Best fit limit Best fit limit Best fit limit Best fit limit
0.02203 0.02205 0.00028 0.02204 0.02200 0.00027 0.02201 0.02203 0.00027 0.02195 0.02192 0.00025
0.1204 0.1199 0.0027 0.0664 0.0450 0.096 0.
1.04119 1.04131 0.00063 1.0445 1.0456 0.0026 1.0461 1.0466 0.0021 1.0427 1.0445 0.0021
0.093 0.089 0.0867 0.087 0.080 0.088 0.090
0.9619 0.9603 0.0073 0.9543 0.9580 0.0071 0.9586 0.9589 0.0070 0.9596 0.9556 0.0060
3.098 3.089 3.090 3.083 3.070 3.084 3.090
—— —— -0.46 -0.58 -0.58 -0.22
0.318 0.315 0.177 0.155 0.127 0.122 0.246
0.682 0.823 0.845 0.873 0.878 0.754
11.4 10.9 10.9 1.1 10.2 10.9 1.1 11.2 10.7 1.1
67.0 67.3 1.2 71.0 72.1 73.0 73.3 69.3
13.824 13.817 0.048 13.747 13.720 13.711 13.785 13.765 0.044
4902.95 4902.45 4902.52 4902.71
Table 2: Best fit values and 68 c.l. constraints for the CDM model (first column) and the dark coupled model described in the text from PLANCK data set (second column). We can see that the effect of the coupling, encoded by the parameter , is to increase and to decrease respect to the standard model values. If we consider the combined constraint on the dark coupled model from PLANCK and HST (third column) we see similar bounds but now a negative coupling is significantly favoured. If we instead combine PLANCK and BAO measurements (fourth column) we observe that a lower value of is favoured, compared to coupled model in the PLANCK alone case, but the value is still in agreement with the Hubble Space Telescope measurement.
PLANCK + wCDM PLANCK+ PLANCK+ HST+ PLANCK+ BAO+
Parameters Best fit limit Best fit limit Best fit limit Best fit limit
0.02220 0.02206 0.00028 0.02207 0.02197 0.00028 0.02207623 0.02205 0.00027 0.02192 0.02197 0.00025
0.1200 0.1199 0.0026 0.0607 0.013 0.012
1.04125 1.04132 0.00063 1.0445 1.0465 0.0026 1.0492 1.0479 0.0019 1.0487 1.0467 0.0023
0.090 0.089 0.093 0.087 0.094 0.089 0.085
0.9598 0.9602 0.0071 0.9614 0.9568 0.0071 0.9582 0.9590 0.0070 0.9563 0.9570 0.0064
3.093 3.089 3.095 3.084 3.098 3.086 0.024 3.086
—— —— -0.48 -0.76 -0.67 -0.82
w -1.92 -0.870 -0.952 -0.887
0.147 0.218 0.179 0.148 0.065 0.131
0.853 0.821 0.852 0.935 0.902 0.869
11.1 11.4 10.9 1.1 11.5 11.0 1.1 10.8 10.9 1.1
98.6 83 68.2 68.5 73.7 72.2 70.8 69.6 2.0
13.449 13.817 0.048 13.789 13.696 13.726 0.048 13.770 13.783 0.046
4903.14 4902.02 4901.90 4902.59
Table 3: Best fit values and 68 c.l. constraints for the CDM model (first column) and the dark coupled model described in the text from PLANCK data set alone (second column), PLANCK+HST (second column) and PLANCK+BAO (third column) when we marginalize over the dark energy equation of state parameter . It is important to note that the prior on in the coupled model () is not the same of the CDM case. As we can expect the constraints are weaker in this case compared to Tab. 2, since both and depends on , but the same trends are evident. Also note that the constraints on the background evolution parameters, as or , from CMB measurements alone are very large also in the CDM model.
Figure 2: Posterior distributions for the cosmological parameters presented in Tab.2 from PLANCK data set alone (solid black line), PLANCK plus HST prior (red dashed line) and PLANCK plus BAO measurements (blue dot-dashed line). The effect of HST prior is to provide narrower distributions and stronger constraints, especially for the coupling parameter .
Figure 3: 2-D posterior distributions of parameters most degenerate with the coupling . A strong correlation is evident with the cold dark matter density parameter. A larger absolute value of the coupling implies a decrement of the dark cold matter and a consequent decrease of the dark matter density. Since the assumption of a flat universe, it also implies a larger dark energy amount that brings to an increment of and consequently an increase of .
Figure 4: 2-D posterior distributions in presence of the HST prior for the same parameters shown in Fig.3.
Figure 5: 2-D posterior distributions in presence of the HST prior for the same parameters shown in Fig.3.
Figure 6: 3D degeneracy between the coupling parameter , and from a run for the PLANCK data set with varying . Note that the values of obtained for reasonable values of are in a much better agreement with low redshift measurements of the Hubble constant, if compared to the values of obtained in the pure CDM PLANCK case.
Figure 7: Posterior distributions for the cosmological parameters presented in Tab.1 from PLANCK data set alone (solid black line), PLANCK plus HST prior (red dashed line) and PLANCK plus BAO measurements (blue dot-dashed line).
Figure 8: 2-D posterior distributions of parameters most degenerate with the coupling . A strong correlation is evident with the cold dark matter density parameter. A larger absolute value of the coupling implies a decrement of the dark cold matter and a consequent decrease of the dark matter density. Since the assumption of a flat universe, it also implies a larger dark energy amount that brings to an increment of and consequently an increase of .
Figure 9: 2-D posterior distributions in presence of the HST prior for the same parameters shown in Fig.3.
Figure 10: 2-D posterior distributions in presence of the HST prior for the same parameters shown in Fig.3.

V Conclusions

In this paper we have presented novel cosmological constraints on a dark matter-dark energy interaction from the new CMB measurements provided by the Planck experiment. We have found that a dark coupling interaction is compatible with Planck data and that the coupling parameter is weakly constrained by Planck measurements . However, the inclusion of a dark coupling opens up new degeneracies and affects strongly the constraints on the remaining cosmological parameters. The model with the dark interaction gives a lower matter density and a larger Hubble parameter . Since the value of the Hubble constant is compatible with the HST value, we have combined the Planck and HST data sets, finding that, in this case, a non-zero value of the dark coupling is suggested by the data, with at c.l.. The analysis presented here points out that an interaction in the dark sector is not only allowed by current CMB data but can even resolve the tension between the Planck and the HST measurements of the Hubble parameter. The results we have found are in agreement with the results obtained in former analyses for similar models using previous cosmological data Honorez:2010rr (); Pettorino:2012ts (); Valiviita:2009nu (); Clemson:2011an ().

Acknowledgements

It is a pleasure to thank Eleonora Di Valentino, Martina Gerbino and Najla Said for useful discussions and help.

Appendix A Adiabatic initial conditions

In ref. Gavela:2010tm (), the evolution equations for the coupled dark matter dark energy model considered here were studied in a gauge invariant way. It was demonstrated that, assuming adiabatic initial conditions for all the standard cosmological fluids (photon, baryons,…), the coupled dark energy fluid also obeys adiabatic initial conditions, see also Ref. Doran:2003xq () in the uncoupled case and Ref. Majerotto:2009np () for another coupled example. At leading order in , in the synchronous gauge, the initial conditions read:

(10)

where are the initial conditions for the photon density perturbations and is the equation of state of the photon. The latter reduce to the adiabatic initial conditions for dark energy perturbations in the synchronous gauge obtained in Ref. Ballesteros:2010ks () in the uncoupled case.

References

  • (1) P. A. R. Ade et al. [ Planck Collaboration], arXiv:1303.5062 [astro-ph.CO].
  • (2) P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5076 [astro-ph.CO].
  • (3) P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5075 [astro-ph.CO].
  • (4) A. G. Riess, L. Macri, S. Casertano, H. Lampeitl, H. C. Ferguson, A. V. Filippenko, S. W. Jha and W. Li et al., Astrophys. J. 730 (2011) 119 [Erratum-ibid. 732 (2011) 129] [arXiv:1103.2976 [astro-ph.CO]].
  • (5) V. Marra, L. Amendola, I. Sawicki and W. Valkenburg, arXiv:1303.3121 [astro-ph.CO].
  • (6) E. Di Valentino, A. Melchiorri and O. Mena, arXiv:1304.5981 [astro-ph.CO].
  • (7) N. Said, E. Di Valentino and M. Gerbino, arXiv:1304.6217 [astro-ph.CO].
  • (8) G. Olivares, F. Atrio-Barandela and D. Pavon, Phys. Rev. D 71, 063523 (2005) [astro-ph/0503242].
  • (9) J. Valiviita, R. Maartens and E. Majerotto, Mon. Not. Roy. Astron. Soc. 402, 2355 (2010) [arXiv:0907.4987 [astro-ph.CO]].
  • (10) M. Martinelli, L. Lopez Honorez, A. Melchiorri and O. Mena, Phys. Rev. D 81, 103534 (2010) [arXiv:1004.2410 [astro-ph.CO]].
  • (11) F. De Bernardis, M. Martinelli, A. Melchiorri, O. Mena and A. Cooray, Phys. Rev. D 84, 023504 (2011) [arXiv:1104.0652 [astro-ph.CO]].
  • (12) L. L. Honorez, B. A. Reid, O. Mena, L. Verde and R. Jimenez, JCAP 1009, 029 (2010) [arXiv:1006.0877 [astro-ph.CO]].
  • (13) P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003) [astro-ph/0207347].
  • (14) L. Amendola, G. Camargo Campos and R. Rosenfeld, Phys. Rev. D 75 (2007) 083506 [arXiv:astro-ph/0610806]; J. Valiviita, E. Majerotto and R. Maartens, JCAP 0807, 020 (2008) [arXiv:0804.0232 [astro-ph]]; G. Caldera-Cabral, R. Maartens and B. M. Schaefer, JCAP 0907, 027 (2009) [arXiv:0905.0492 [astro-ph.CO]]. J. H. He, B. Wang and E. Abdalla, Phys. Lett. B 671, 139 (2009) [arXiv:0807.3471 [gr-qc]]; B. M. Jackson, A. Taylor and A. Berera, Phys. Rev. D 79, 043526 (2009) [arXiv:0901.3272 [astro-ph.CO]].
  • (15) L. Amendola, Phys. Rev. D 62, 043511 (2000) [astro-ph/9908023]
  • (16) G. R. Farrar and P. J. E. Peebles, Astrophys. J. 604, 1 (2004) [astro-ph/0307316].
  • (17) V. Pettorino and C. Baccigalupi, Phys. Rev. D 77 (2008) 103003 [arXiv:0802.1086 [astro-ph]].
  • (18) J. -H. He, B. Wang and E. Abdalla, Phys. Lett. B 671, 139 (2009) [arXiv:0807.3471 [gr-qc]].
  • (19) M. B. Gavela, D. Hernandez, L. Lopez Honorez, O. Mena and S. Rigolin, JCAP 0907 (2009) 034 [Erratum-ibid. 1005 (2010) E01] [arXiv:0901.1611 [astro-ph.CO]].
  • (20) M. B. Gavela, L. Lopez Honorez, O. Mena and S. Rigolin, JCAP 1011, 044 (2010) [arXiv:1005.0295 [astro-ph.CO]].
  • (21) A. Lewis, A. Challinor and A. Lasenby, Astrophys. J. 538, 473 (2000) [astro-ph/9911177].
  • (22) C. L. Bennett, D. Larson, J. L. Weiland, N. Jarosik, G. Hinshaw, N. Odegard, K. M. Smith and R. S. Hill et al., arXiv:1212.5225 [astro-ph.CO].
  • (23) F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell, Q. Parker and W. Saunders et al., Mon. Not. Roy. Astron. Soc. 416, 3017 (2011) [arXiv:1106.3366 [astro-ph.CO]].
  • (24) N. Padmanabhan, X. Xu, D. J. Eisenstein, R. Scalzo, A. J. Cuesta, K. T. Mehta and E. Kazin, arXiv:1202.0090 [astro-ph.CO].
  • (25) L. Anderson, E. Aubourg, S. Bailey, D. Bizyaev, M. Blanton, A. S. Bolton, J. Brinkmann and J. R. Brownstein et al., Mon. Not. Roy. Astron. Soc. 427, no. 4, 3435 (2013) [arXiv:1203.6594 [astro-ph.CO]].
  • (26) A. Lewis and S. Bridle, Phys. Rev. D 66, 103511 (2002) [astro-ph/0205436].
  • (27) A. Lewis, arXiv:1304.4473 [astro-ph.CO].
  • (28) V. Pettorino, L. Amendola, C. Baccigalupi and C. Quercellini, Phys. Rev. D 86, 103507 (2012) [arXiv:1207.3293 [astro-ph.CO]].
  • (29) T. Clemson, K. Koyama, G. -B. Zhao, R. Maartens and J. Valiviita, Phys. Rev. D 85, 043007 (2012) [arXiv:1109.6234 [astro-ph.CO]].
  • (30) M. Doran, C. M. Muller, G. Schafer and C. Wetterich, Phys. Rev. D 68, 063505 (2003) [astro-ph/0304212].
  • (31) E. Majerotto, J. Valiviita and R. Maartens, Mon. Not. Roy. Astron. Soc. 402, 2344 (2010) [arXiv:0907.4981 [astro-ph.CO]].
  • (32) G. Ballesteros and J. Lesgourgues, JCAP 1010, 014 (2010) [arXiv:1004.5509 [astro-ph.CO]].
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