Indications of a late-time interaction in the dark sector
We show that a general late-time interaction between cold dark matter and vacuum energy is favoured by current cosmological datasets. We characterize the strength of the coupling by a dimensionless parameter that is free to take different values in four redshift bins from the primordial epoch up to today. This interacting scenario is in agreement with measurements of cosmic microwave background temperature anisotropies from the Planck satellite, supernovae Ia from Union 2.1 and redshift space distortions from a number of surveys, as well as with combinations of these different datasets. Our analysis of the 4-bin interaction shows that a non-zero interaction is likely at late times. We then focus on the case in a single low-redshift bin, obtaining a nested one parameter extension of the standard CDM model. We study the Bayesian evidence, with respect to CDM, of this late-time interaction model, finding moderate evidence for an interaction starting at , dependent upon the prior range chosen for the interaction strength parameter . For this case the null interaction (, i.e. CDM) is excluded at c.l..
Introduction: Measurements of anisotropies of the cosmic microwave background (CMB) from experiments including the WMAP Bennett:2012zja () and Planck Ade:2013zuv () satellites, combined with independent measurements of the cosmic expansion history, such as baryon acoustic oscillations Anderson:2013zyy (), have provided strong support for the standard model of cosmology with dark energy (specifically a cosmological constant, ) and cold dark matter (CDM). However the latest CMB data are in tension with local measurements of the Hubble expansion rate from supernovae Ia Riess:2011yx () and other cosmological observables which point towards a lower growth rate of large-scale structure (LSS), including cluster counts Ade:2013lmv (); Vikhlinin:2008ym () and redshift-space distortions (RSD) from galaxy peculiar velocities Samushia:2012iq ().
At the present time it remains unclear whether these discrepancies may be due to systematic effects in the different methods used for measurements, or whether they could instead be evidence for deviations from CDM. Massive neutrinos have been proposed to reconcile CMB with LSS observations Battye:2013xqa (), but they increase the tension with local measurements of the Hubble rate Giusarma:2013pmn (). Dynamical dark energy can help reconcile CMB and local Hubble expansion measurements, but does not ease the tension with LSS Ade:2013zuv (). However, a coupling between the components of the dark sector can strongly influence the evolution of both the background and perturbations. Models with a constant interaction between cold dark matter and dark energy have already been proposed as one possible solution to solve the tension in the measurements of the Hubble constant from CMB and Supernovae Salvatelli:2013wra ().
In this Letter we investigate a minimal extension of the CDM model where dark matter is allowed to interact with vacuum energy, without introducing any additional degrees of freedom. We allow the interaction strength to vary with redshift and show that energy transfer from dark matter to the vacuum can resolve the tension between the CMB and RSD measurements of the growth of LSS, making it consistent to combine these two datasets. We consider only RSD measurements as these probe the gravitational potential in the linear regime and do not depend on non-linear evolution and the formation of collapsed halos and clusters. Our main result is that a model where an interaction in the dark sector switches on at late times is particularly favoured with respect to CDM. Assuming an interaction starting at redshift the null interaction case (i.e. CDM) is excluded at c.l..
Model: Interacting vacuum models (iVCDM) allow energy-momentum transfer between CDM and the vacuum Bertolami:1986bg (); Freese:1986dd (); Carvalho:1991ut (); Berman:1991zz (); Pavon:1991uc (); Wands:2012vg (). The background evolution is encoded in the coupled energy conservation equations
for the CDM and vacuum densities, and , the standard conservation equations for baryons, photons and neutrinos, and the Friedmann equation
where is the total matter and radiation energy density, is the expansion rate of the universe, is the interaction term and we assume a spatially flat universe. When there is no interaction () we have , the cosmological constant, and we recover the standard CDM model.
In general the interaction is covariantly represented by a 4-vector ; if we assume that this is proportional to the 4-velocity of CDM () then the matter flow remains geodesic () and in the comoving-synchronous gauge the vacuum energy is spatially homogeneous Wands:2012vg (). Hence the perturbation equations in this gauge are the same as in CDM, with zero effective sound speed Wang:2013qy ().
Recent studies of interacting vacuum cosmologies have focused on specific models for the interaction, DeSantiago:2012xh (); Wang:2013qy (); Borges:2013bya (); Sola:2014tta (); Wang:2014xca (). In this Letter we want to consider a general interaction in different redshift bins. Thus we take an interaction of the form , where is a dimensionless parameter that encodes the strength of the coupling Quercellini:2008vh (). We require to ensure that the matter density remains non-negative. Note that in our notation a negative implies dark matter decaying into vacuum.
We first consider a model in which is a binned (stepwise-defined) function. We have subdivided the redshift range from last scattering until today into four bins, with (), i.e. parametrizing our iVCDM model with four parameters. We have chosen to include all the redshifts from the primordial epoch to in a single bin (bin 1), as we have few measurements in that range after CMB last scattering. The other three bins have been chosen with the aim to be mainly sensitive to supernovae (bin 4, ), to RSD (bin 3, ) and to the farthest supernova observations available (bin 2, ).
In the light of our results for , we then focus on the case of a late-time interaction, with in a single low-redshift bin.
Analysis: We have performed a Bayesian analysis with the Monte Carlo Markov chain code CosmoMC Lewis:2002ah (); Lewis:2013hha () and a modified version of the Boltzmann code CAMB Lewis:1999bs (). The datasets we have considered to assess the likelihood of the model are CMB measurements from Planck Ade:2013kta () including polarization from WMAP Bennett:2012zja (), SNIa from the compilation Union2.1 Suzuki:2011hu () and RSD measurements from a number of surveys Beutler:2012px (); Percival:2004fs (); Blake:2011rj (); Samushia:2011cs (); Reid:2012sw (); delaTorre:2013rpa (), see Fig. 2. We also considered baryon acoustic oscillations Beutler:2012px (); Percival:2009xn (); Anderson:2013oza (); Blake:2011rj () and radio galaxies data Daly:2007pp (), finding that the constraints from these datasets are equivalent to those from SN; therefore their addition to our analysis doesn’t change our results. A comprehensive analysis including the effects of these datasets will be presented in a forthcoming paper Inprep ().
In this analysis we have chosen a flat prior [-10,0] for the parameters since the parameters’ magnitude is assumed to be of order one. (We will consider later the effect of a wider logarithmic prior, see Fig. 5 and Inprep ().)
In the 4-bin interaction case, when considering CMB only or CMB+SN measurements, the presence of an interaction is allowed but a null interaction is not excluded in any bin (see column 1 and 2 in Table 1). This is due to the unbroken degeneracy between the strength of the interaction parameters, , and the present-day CDM density (), shown in Fig. 1.
The degeneracy between and the interaction parameters can be broken by the addition of RSD measurements. This imposes a lower limit on the present cold dark matter density and leads to a shift in the posterior distributions for the interaction, as clearly shown in Fig. 1. A null interaction is then excluded at 99% c.l. in bin 3 and at 95 % in bin 4, showing that a late-time interaction is preferred by observations (see column 3 in Table 1). This result is also supported by a principal component analysis Inprep ().
We note that the iVCDM model can also alleviate the tension that arises in CDM between the Hubble constant measurements from Planck () and the Hubble Space Telescope Riess:2011yx () (). The constraint from Planck in the iVCDM case is (see also Wang:2014xca ()). The combination with RSD measurements breaks the degeneracy between and , leading to .
In the light of this analysis we have also explored the viability of a simpler model with an interaction that switches on at low redshift whose strength is encoded in a constant for . In particular, based on the preceding results, we have selected as the interaction starting point , i.e., the upper limit of redshift bin 3. For this reason we will refer to it as the -model. In this case a null interaction is excluded at 99% c.l. Results are shown in Table 2 and Fig. 2.
As shown in Table 3, when we introduce four parameters to determine the interaction strength in 4 redshift bins, we obtain a much better fit to the data with respect to CDM. Remarkably, the -model with a single interaction parameter can match the best fit of the more complex model with 4 independent redshift bins. It reproduces the same best-fit with three fewer parameters.
An alternative way to compare different models is to compute the Bayesian evidence. Since the model is a one-parameter nested extension of CDM, we can simply compute the Bayes factor , that represents the ratio of the models’ probability, using the Savage-Dickey Density Ratio formula
where is the additional parameter and is the value of the parameter for which model 0 (CDM) is recovered.
The Bayesian evidence for the extended model is ; thus less than one means that the model is preferred over CDM.
The Bayesian evidence inevitably depends on the prior distribution of model parameters, decreasing with the prior width. When dealing with phenomenological parameters, such as , it is not clear what range for the prior should be considered when computing of the evidence Trotta:2008qt (). For this reason we have explored in Fig. 3 how the evidence changes with the width of the prior. For comparison we have computed the evidence for three late-time interaction models with different choices for . We have also evaluated the Bayes factor between CDM and another one parameter extension that alleviates the tension between CMB and RSD measurements, namely CDM with massive neutrinos Battye:2013xqa (), see Fig. 4. In this case the nested extra parameter is the sum of the neutrino masses, for which we obtain eV. In the rest of our analysis we use the standard fixed value eV. We see that the evidence for the model with is always higher than the other one-parameter models we study for a given prior width relative to the standard deviation from the mean. The evidence remains moderate even when allowing a prior range for equal to standard deviations from the mean.
Possible biases: In order to check the robustness of our results we have performed some further analysis Inprep (). First we have explored a model where is free to vary. Table 3 shows that is a good approximation of the best-fit point of this extended model, and hence maximises the Bayesian evidence computed above. Marginalizing over slightly broadens and shifts the posterior distribution for , as shown in Fig. 5. We also show in Fig. 5 that choosing a wider logarithmic prior, for fixed , has a small effect on the posterior.
Moreover we have tested our results against variation of the lensing amplitude parameter of the CMB temperature, . In CDM Planck measurements point towards an value that is higher than the standard value, , Ade:2013zuv (); Said:2013hta () used in the preceding analysis. In our iVCDM model a degeneracy exists between and that reduces the strength of the interaction when increases. However the indication for an interaction is maintained at 95% c.l..
Finally, given the recent results from the BICEP2 experiment that claims a detection for a tensor to scalar ratio different from zero Ade:2014xna (), we have investigated if our results may be affected. The interaction parameter is actually very poorly degenerate with and the inclusion of the BICEP2 dataset changes the Bayes evidence by only 1%.
Conclusions: We have shown that an interacting vacuum cosmology, where the strength of the coupling with CDM varies with redshift, is a possible solution to the tension that arises in the standard CDM model between CMB data and LSS linear growth measured by RSD, see Fig. 4. In particular we have found that an interaction which switches on at late times () is particularly favoured. In this context we have obtained a very tight constraint on the interaction strength parameter, excluding the CDM model (i.e., a null interaction) at 99% c.l.. We have also verified that the probability of late-time interaction is only weakly affected by changes in the value of the tensor-to-scalar ratio or the lensing amplitude parameters.
We have only considered here constraints on the linear growth of LSS, as the non-linear coupled evolution of interacting vacuum and dark matter has yet to be studied in detail. It will be important to examine the predictions of iVCDM for cluster number counts; this provides tight constraints, e.g. on CDM, but requires non-linear modelling. Interacting vacuum models can be recast Wands:2012vg () as clustering quintessence with vanishing sound speed Creminelli:2009mu () and/or irrotational dark matter Sawicki:2013wja (), either of which could have distinctive predictions for non-linear collapse.
Acknowledgements.Acknowledgments We would like to thank Rob Crittenden, Karen Masters, Lado Samushia and Gong-Bo Zhao for helpful comments. The work of MB and DW was supported by STFC grants ST/K00090X/1 and ST/L005573/1. VS and NS are grateful to the ICG for its hospitality. We thank an anonymous referee for suggesting the use of radio galaxies data Daly:2007pp ().
- (1) C. L. Bennett et al. [WMAP Collaboration], Astrophys. J. Suppl. 208, 20 (2013) [arXiv:1212.5225 [astro-ph.CO]].
- (2) P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5076 [astro-ph.CO].
- (3) L. Anderson et al. [BOSS Collaboration], arXiv:1312.4877 [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, 119 (2011) [Erratum-ibid. 732, 129 (2011)] [arXiv:1103.2976 [astro-ph.CO]].
- (5) P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5080 [astro-ph.CO].
- (6) A. Vikhlinin, A. V. Kravtsov, R. A. Burenin, H. Ebeling, W. R. Forman, A. Hornstrup, C. Jones and S. S. Murray et al., Astrophys. J. 692 (2009) 1060 [arXiv:0812.2720 [astro-ph]].
- (7) L. Samushia, B. A. Reid, M. White, W. J. Percival, A. J. Cuesta, L. Lombriser, M. Manera and R. C. Nichol et al., Mon. Not. Roy. Astron. Soc. 429, 1514 (2013) [arXiv:1206.5309 [astro-ph.CO]].
- (8) R. A. Battye and A. Moss, Phys. Rev. Lett. 112, 051303 (2014) [arXiv:1308.5870 [astro-ph.CO]].
- (9) E. Giusarma, R. de Putter, S. Ho and O. Mena, Phys. Rev. D 88, no. 6, 063515 (2013) [arXiv:1306.5544 [astro-ph.CO]].
- (10) V. Salvatelli, A. Marchini, L. Lopez-Honorez and O. Mena, Phys. Rev. D 88, no. 2, 023531 (2013) [arXiv:1304.7119 [astro-ph.CO]].
- (11) O. Bertolami, Nuovo Cim. B 93, 36 (1986).
- (12) K. Freese, F. C. Adams, J. A. Frieman and E. Mottola, Nucl. Phys. B 287, 797 (1987).
- (13) J. C. Carvalho, J. A. S. Lima and I. Waga, Phys. Rev. D 46, 2404 (1992).
- (14) M. S. Berman, Phys. Rev. D 43, 1075 (1991).
- (15) D. Pavon, Phys. Rev. D 43, 375 (1991).
- (16) D. Wands, J. De-Santiago and Y. Wang, Class. Quant. Grav. 29, 145017 (2012) [arXiv:1203.6776 [astro-ph.CO]].
- (17) J. De-Santiago, D. Wands and Y. Wang, arXiv:1209.0563 [astro-ph.CO].
- (18) H. A. Borges, S. Carneiro, J. C. Fabris and W. Zimdahl, Phys. Lett. B 727, 37 (2013) [arXiv:1306.0917 [astro-ph.CO]].
- (19) J. Sola, arXiv:1402.7049 [gr-qc].
- (20) Y. Wang, D. Wands, L. Xu, J. De-Santiago and A. Hojjati, Phys. Rev. D 87, 083503 (2013) [arXiv:1301.5315 [astro-ph.CO]].
- (21) Y. Wang, D. Wands, G. -B. Zhao and L. Xu, arXiv:1404.5706 [astro-ph.CO].
- (22) C. Quercellini, M. Bruni, A. Balbi and D. Pietrobon, Phys. Rev. D 78, 063527 (2008) [arXiv:0803.1976 [astro-ph]].
- (23) A. Lewis and S. Bridle, Phys. Rev. D 66, 103511 (2002) [astro-ph/0205436].
- (24) A. Lewis, Phys. Rev. D 87, no. 10, 103529 (2013) [arXiv:1304.4473 [astro-ph.CO]].
- (25) A. Lewis, A. Challinor and A. Lasenby, Astrophys. J. 538, 473 (2000) [astro-ph/9911177].
- (26) P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5075 [astro-ph.CO].
- (27) N. Suzuki, D. Rubin, C. Lidman, G. Aldering, R. Amanullah, K. Barbary, L. F. Barrientos and J. Botyanszki et al., Astrophys. J. 746, 85 (2012) [arXiv:1105.3470 [astro-ph.CO]].
- (28) F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, G. B. Poole, L. Campbell and Q. Parker et al., Mon. Not. Roy. Astron. Soc. 423, 3430 (2012) [arXiv:1204.4725 [astro-ph.CO]].
- (29) W. J. Percival et al. [2dFGRS Collaboration], Mon. Not. Roy. Astron. Soc. 353, 1201 (2004) [astro-ph/0406513].
- (30) C. Blake, S. Brough, M. Colless, C. Contreras, W. Couch, S. Croom, T. Davis and M. J. Drinkwater et al., Mon. Not. Roy. Astron. Soc. 415, 2876 (2011) [arXiv:1104.2948 [astro-ph.CO]].
- (31) L. Samushia, W. J. Percival and A. Raccanelli, Mon. Not. Roy. Astron. Soc. 420, 2102 (2012) [arXiv:1102.1014 [astro-ph.CO]].
- (32) B. A. Reid, L. Samushia, M. White, W. J. Percival, M. Manera, N. Padmanabhan, A. J. Ross and A. G. Sanchez et al., arXiv:1203.6641 [astro-ph.CO].
- (33) S. de la Torre, L. Guzzo, J. A. Peacock, E. Branchini, A. Iovino, B. R. Granett, U. Abbas and C. Adami et al., arXiv:1303.2622 [astro-ph.CO].
- (34) W. J. Percival et al. [SDSS Collaboration], Mon. Not. Roy. Astron. Soc. 401, 2148 (2010) [arXiv:0907.1660 [astro-ph.CO]].
- (35) L. Anderson, E. Aubourg, S. Bailey, F. Beutler, A. S. Bolton, J. Brinkmann, J. R. Brownstein and C. H. Chuang et al., arXiv:1303.4666 [astro-ph.CO].
- (36) R. A. Daly, M. P. Mory, C. P. O’Dea, P. Kharb, S. Baum, E. J. Guerra and S. G. Djorgovski, Astrophys. J. 691, 1058 (2009) [arXiv:0710.5112 [astro-ph]].
- (37) D. Pietrobon, A. Balbi, M. Bruni and C. Quercellini, Phys. Rev. D 78 (2008) 083510 [arXiv:0807.5077 [astro-ph]].
- (38) N. Said, V. Salvatelli, M. Bruni and D. Wands [In prep.].
- (39) H. Jeffreys, Theory of Probability, Oxford: Clarendon Press, (1961)
- (40) R. Trotta, Contemp. Phys. 49 (2008) 71 [arXiv:0803.4089 [astro-ph]].
- (41) B. Leistedt, H. V. Peiris and L. Verde, arXiv:1404.5950 [astro-ph.CO].
- (42) N. Said, E. Di Valentino and M. Gerbino, Phys. Rev. D 88, no. 2, 023513 (2013) [arXiv:1304.6217 [astro-ph.CO]].
- (43) P. A. R. Ade et al. [BICEP2 Collaboration], arXiv:1403.3985 [astro-ph.CO].
- (44) P. Creminelli, G. D’Amico, J. Norena, L. Senatore and F. Vernizzi, JCAP 1003, 027 (2010) [arXiv:0911.2701 [astro-ph.CO]].
- (45) I. Sawicki, V. Marra and W. Valkenburg, Phys. Rev. D 88, 083520 (2013) [arXiv:1307.6150 [astro-ph.CO]].