Latest evidence for a late time vacuum -- geodesic CDM interaction
Natalie B. Hogg, Marco Bruni, Robert Crittenden, Matteo Martinelli, Simone Peirone
LLatest evidence for a late time vacuum–geodesic CDM interaction
Natalie B. Hogg a, ∗ , Marco Bruni a , b , Robert Crittenden a , Matteo Martinelli c , Simone Peirone d a Institute of Cosmology and Gravitation, University of Portsmouth, Burnaby Road, Portsmouth PO1 3FX, United Kingdom b INFN Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy c Instituto de F´ısica T´eorica UAM-CSIC, Campus de Cantoblanco, E-28049 Madrid, Spain d Institute Lorentz, Leiden University, PO Box 9506, Leiden 2300 RA, The Netherlands
Abstract
We perform a reconstruction of the coupling function between vacuum energy and geodesic cold dark matter usingthe latest observational data. We bin the interaction in seventeen redshift bins but use a correlation prior to preventrapid, unphysical oscillations in the coupling function. This prior also serves to eliminate any dependence of thereconstruction on the binning method. We use two di ff erent forms of the correlation prior, finding that both give similarresults for the reconstruction of the dark matter – dark energy interaction. Calculating the Bayes factor for each case,we find no meaningful evidence for deviation from the null interacting case, i.e. Λ CDM, in our reconstruction.
Keywords:
Cosmology, dark energy, dark matter
1. Introduction
One of the most pressing questions in modern cosmol-ogy concerns the true nature of dark energy: what is thephysical driver of the accelerated expansion of the Uni-verse? This phenomenon of accelerating expansion wasfirst established by [1] and [2] and is generally attributedto the existence of a positive cosmological constant, Λ ,in our Universe. While the cosmological constant is thesimplest proposed source of the accelerated expansion, itis widely agreed that it has long su ff ered from theoreticalproblems: namely, why its observed value and theoreti-cally predicted value di ff er so greatly [3, 4, 5], and whyit has only come to dominate over the other componentsin the Universe relatively recently [6]. However, someadvocate that these are not problems of cosmology butof particle physics or quantum field theory – or perhapsnot real problems at all [7].In addition to these theoretical and philosophical is-sues, the ever-increasing improvements in observationalcosmology have begun to reveal tensions in the valuesof cosmological parameters within the wider Λ CDMmodel, particularly between high and low redshift mea-surements of these quantities. The most striking of thesetensions is in the value of H , the Hubble parameter atredshift zero, which essentially informs us of the rate atwhich the Universe is expanding. ∗ Corresponding author: [email protected]
A value of H can be measured without assuming acosmological model, using low redshift probes such asType Ia supernovae which rely on the distance ladder tofix the distance–redshift relation. A recent example ofsuch a measurement is H = . ± .
66 kms − Mpc − [8]. The value of H can also be calculated using infor-mation coming from the cosmic microwave background(CMB) – the caveat being that a cosmological modelmust be adopted. In doing so, the Planck collaborationreports H = . ± . − Mpc − , when Λ CDM isassumed [9]. This signals a tension of more than 4 σ between the two measurements [10].Some other measurements of H that are indepen-dent of the distance ladder do not seem to relax thetension. For example, the H0LiCOW doubly imagedquasar measurement found H = . + . − . kms − Mpc − [11], the LIGO gravitational wave measurement found H = + . − . kms − Mpc − [12] and the very recentMegamaser Cosmology Project constraint was foundto be H = . ± . − Mpc − [13].The inverse distance ladder approach used in [14]found H = . ± .
30 kms − Mpc − . This methodentails anchoring the distance ladder using the baryonacoustic oscillation (BAO) signal combined with the sizeof the sound horizon at the drag epoch r s ( z d ), which is inturn obtained either through CMB measurements or BigBang nucleosynthesis constraints on the baryon density Ω b h . Anchoring the distance ladder in this way rather1 a r X i v : . [ a s t r o - ph . C O ] M a y han to Cepheid variable stars results in a value of H inagreement with the Planck 2018 result quoted above.Furthermore, another tension is becoming apparentin the value of σ . This parameter is a measure of thegrowth of cosmological perturbations and hence of thelarge scale structure formation. The tension in its mea-sured values is also between high and low redshift probes[15, 16].While it is entirely possible that these tensions arepresent simply due to systematic errors or noise in thedata, we must also consider the possibility – followingthe well-known aphorism in statistics that “all models arewrong” [17] – that the Λ CDM model is simply incorrect.In an attempt to extricate cosmology from this ratheralarming predicament, many alternatives to the Λ CDMmodel have been proposed. Some are as simple as al-lowing vacuum energy to interact with cold dark matter(CDM) (see e.g. [18, 19, 20, 21, 22, 23, 24]), others in-troduce an additional scalar field to drive the acceleratedexpansion (see [25] for a comprehensive review) andstill others eliminate general relativity entirely, explor-ing modified gravity theories in which self-accelerationcan be achieved (see e.g. [26, 27, 28, 29] for details ofvarious modified gravity models past and present).It is worth noting that some modified gravity modelsare motivated by the need to explain various features oflarge scale structure formation and thus do not necessar-ily alter the background cosmology. Since H is a probeof the background expansion, models that do have somee ff ect on the background cosmology are naturally moreattractive when the motivation is to relax the H tension.In this work we choose the first option, introducingan interaction between the vacuum and cold dark mat-ter, constructed in such a way that the cold dark matterremains geodesic, thus limiting any potentially patho-logical e ff ects on structure growth. In a previous work[22], we investigated whether a simple form of this in-teraction could relieve the tensions present in Λ CDM,testing the interaction acting in a single redshift bin andreconstructing the interaction using four redshift bins.We found that, while the interacting scenario does notmanage to relieve cosmological tensions, it is not ruledout by current observational data.In this work, we continue that investigation by increas-ing the number of redshift bins used in our reconstruction,thereby increasing the redshift range that the interactionacts over and ensuring a model-independent reconstruc-tion. We also use the up-to-date Planck 2018 likelihood[9], instead of the 2015 likelihood used in our previouswork. We study the constraining power of a theoreticalprior acting across the bins and reconstruct the final inter-action function. We then perform a Principal Component Analysis and calculate the Bayesian evidence for eachcase studied.This paper is organised as follows: in Section 2 werecapitulate the theory behind the interacting vacuumscenario that we test in this work. In Section 3 we de-scribe the implementation and numerical analysis done,explaining the role of the theoretical priors and the re-construction. We present our results and discussion inSection 4 and then conclude with Section 5.
2. The interacting vacuum
In this section, we limit ourselves to a basic discussionof the interaction in a spatially flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) background. We note thatthe interaction is constructed so that CDM remainsgeodesic; in brief, this is because the energy-momentumflow 4-vector between the vacuum and cold dark mat-ter, Q µ , can be projected in two parts, one parallel andone orthogonal to the CDM 4-velocity, Q µ = Qu µ + f µ ,where f µ is the momentum exchange, f µ = a µ ρ c .We set this momentum exchange to be equal to zero,implying that the 4-acceleration a µ must be zero andthus ensuring that CDM is always geodesic, as no ad-ditional acceleration due to the interaction acts on theCDM particles. With this choice, in the synchronouscomoving gauge that we use to describe perturbations,the interaction is unperturbed and fully encoded in thebackground Q .We refer the reader to [18] and [22] for a detailedtreatment of the covariant theory of the interaction, aswell as the behaviour of linear perturbations in this theoryand the e ff ect of the interaction on structure growth.In a spatially flat FLRW background, the interactionis introduced between CDM and the vacuum in the fol-lowing way: ˙ ρ c + H ρ c = − Q , (1)˙ V = Q , (2)where ρ c and V are the energy densities of CDM andthe vacuum respectively, H = ˙ a / a is the Hubble expan-sion, with a the cosmic scale factor, and Q is the energyexchange between the components. For Q = V isconstant and we recover a cosmological constant, i.e. Λ CDM.In order to reconstruct the behaviour of this interaction,we must choose a model for Q . We make the followingchoice, Q = − qHV , (3)2arameter Prior Ω b h [0 . , . Ω c h [0 . , . H [50 , A s [2 . , . n s [0 . , . q i [ − . , . Table 1: Prior ranges of the parameters sampled in our analysis. so that the coupled energy conservation equations (1),(2) become ˙ ρ c + H ρ c = qHV , (4)˙ V = − qHV , (5)where q = q ( a ) is a dimensionless function that encodesthe strength of the coupling between CDM and the vac-uum. A positive value of q indicates that the vacuum isdecaying and dark matter is growing, whereas a negativevalue of q indicates that dark matter is decaying and thevacuum is growing. We aim to reconstruct the couplingas a function of redshift z , i.e. q ( z ), using a cubic splineinterpolation and a Gaussian process.
3. Method
In this section, we describe the numerical codes usedand the modifications made to those codes, as well as thetheoretical priors and data considered in our analysis.
CAMB and
CosmoMC
The first step in our analysis is to constrain the cou-pling strength q ( a ) with cosmological data. To this endwe make use of modified versions of the CAMB [30, 31]and
CosmoMC codes [32, 33]. We bin the interactionfunction q ( a ) in terms of the cosmic scale factor, with q i being the constant parameter value within the i th bin.We choose to extend our previous four bin analysispresented in [22] to seventeen bins, with i = , ..., a = . a = .
14, plus a single large bin that extends to a ≈ . CosmoMC to produce MCMC sam-ples from the posterior distribution of the interactionparameter in each bin, plus the baryon and cold darkmatter densities Ω b h and Ω c h , the amplitude of theprimordial power spectrum and the spectral index A s and n s , and the value of the Hubble parameter today, H .We use flat priors on these parameters, with the rangesspecified in Table 1. Although we have no theoretically motivated modelfor the behaviour of the coupling as a function of scalefactor (or, equivalently, time) we do have one theoreticalprejudice: we do not expect the coupling function to os-cillate rapidly, as we consider very fast changes of signin the coupling function to be unphysical. We thereforetake the step of including a theoretical prior on the cou-pling parameter that actively suppresses high frequencyoscillations, thereby allowing the low frequency modesthat are potentially present in the data to dominate.The theoretical prior takes the form of a scale-factor-dependent correlation between the values of the couplingfunction in each bin. Values of the function in neigh-bouring bins are correlated, with the correlation growingweaker for bins of greater separation. This correlationprior was first proposed in [34] and the method has beensubsequently used in the reconstruction of the dark en-ergy equation of state function w ( z ) by [35, 36, 37]. Thecorrelation prior method was also used by [38] to re-construct the vacuum energy – CDM interaction at lowredshifts only, up to z = . ffi ciently large, as we will describe below.Following [34], we assume a correlation function thatdescribes fluctuations around some fiducial model, ξ ( | a − a (cid:48) | ) ≡ (cid:104) [ q ( a ) − ¯ q ( a )][ q ( a (cid:48) ) − ¯ q ( a (cid:48) )] (cid:105) , (6)and given a functional form for ξ , the correspondingcovariance matrix can be found: C i j = ∆ (cid:90) a i +∆ a i da (cid:90) a j +∆ a j da (cid:48) ξ ( | a − a (cid:48) | ) , (7)where ∆ is the bin width, ¯ q is the fiducial model and a is the cosmic scale factor. The fiducial model can beset to Λ CDM (i.e. ¯ q ( a i ) = fixed fiducial and mean fiducial respectively.We use the Crittenden-Pogosian-Zhao (CPZ) form forthe correlation function, as proposed in [34], ξ ( | a − a (cid:48) | ) = ξ (0) / [1 + ( | a − a (cid:48) | / a c ) ] , (8)3here a c is the correlation length. As previously stated,we want to ensure that our results are independent of thenumber of bins used. To ensure that we eliminate thispotential reconstruction bias, we require that N > N e ff , (9)where N is the number of bins, and N e ff = ( a max − a min ) / a c . (10)The parameters a max and a min are the limits of the scalefactor range used in our analysis, a = . a = . a c = .
06. This means that N e ff = .
7. Therefore, toensure that N > N e ff , we choose N = ξ (0), butfollowing [39], we use the variance of the mean instead,defined as σ q ≈ πξ (0) a c / ( a max − a min ). We set σ q = . ffi cient for the prior toprovide some constraining power, but not so much that itcompletely dominates over the constraints from data ineach bin. We discuss this point further in subsection 4.3. The data used in this work is a combination of thePlanck 2018 measurements of the CMB temperature andpolarisation [9], the BAO measurements from the 6dFGalaxy Survey [41] and the combined BAO and redshiftspace distortion (RSD) data from the SDSS DR12 con-sensus release [42], together with the Pantheon Type Iasupernovae sample [43].We note that some works in the literature that find aresolution to the H tension in an interacting dark energyscenario do so by omitting the BAO data from theiranalyses (see e.g. [44]). This is because, without usingBAO, the high redshift constraint on H becomes weaker,and a late time solution to the tension is possible. If BAOare used in combination with supernovae catalogues thenlate time solutions become disfavoured, and interactingdark energy models will therefore struggle to resolve thetension (see e.g. [45, 46]). However, this reasoning doesnot justify the exclusion of these datasets from modelconstraining analysis and we therefore make a point ofincluding multiple BAO measurements in this work.We also note that, due to the coupling between thevacuum and cold dark matter in this scenario, RSD donot directly constrain the growth factor f as they do in Λ CDM [46, 47]. Instead, the RSD constrain what wedenote as the interaction growth factor, f i , f i = f − QH ρ c , (11) where f is the usual growth factor for CDM, f = d ln Dd ln a , (12)with D being the amplitude of the linear growing mode.
4. Results and discussion
In this section, we describe and discuss the main re-sults of our investigation, beginning with the results ofthe MCMC analysis, then moving to the reconstructionof the coupling function, the Principal Component Anal-ysis performed and finally the findings of our Bayesianevidence calculation.
In Figure 1 we plot the 1D marginalised posteriorsfor the interaction parameter q i in each of the seventeenbins, where i = z =
0, up to i =
17 for the wide bin at high redshift. The posteriordistributions for q i are generally broader in the meanfiducial case compared to the fixed fiducial case. This isto be expected, as the mean fiducial case essentially hasone additional free parameter with respect to the fixedfiducial, this being ¯ q , the fiducial value for the correlationprior.We find that the null interacting case ( q = Λ CDM limit of the model, is always within1 σ of the achieved constraints. However, the boundsfound on the interaction parameter in every bin meansthe interacting scenario is still viable. It is clear froman Ockham’s razor standpoint that the Λ CDM scenarioshould be favoured over both the interacting cases. Wequantify this statement using the Bayes factor in subsec-tion 4.4.Table 2 shows the marginalised values of the standardcosmological parameters sampled in our MCMC anal-ysis, while in Figure 2 we show the 2D marginalisedjoint distributions for the cosmological parameters H , Ω m (the total matter density parameter) and σ . To pre-serve the readability of the plot, we choose to only showthe results of the mean fiducial case in this figure. Ascan be inferred from Figure 1, the constraints on thecosmological parameters in the fixed fiducial case are al-most identical to those in the mean fiducial case. In bothcases we found the value of H to be completely consis-tent with the Planck 2018 Λ CDM value of 67 . ± . − Mpc − [9]. The value of σ given by Planck is0 . ± . σ of thevalues for σ we find in both interacting cases.4 igure 1: The 1D marginalised posteriors of the interaction parameter in each bin. In each panel we report the best fit value of the interactionparameters and their 68% confidence level bounds for the fixed fiducial (red) and mean fiducial (blue) case.
5s discussed in the introduction, the tensions in thevalues of H and σ are commonly used as motivationsfor alternative models of dark energy. However, as wealso found in our previous work [22], the interactingvacuum fails to resolve the tensions when using the par-ticular datasets chosen here. This can clearly be seen inthe left panel of Figure 2, where the constraint on H in the interacting scenario is shown in conjunction withboth the Planck and local measurements. As we men-tioned in subsection 3.3, for the case of the H tension inparticular, this is attributable to the fact that by includingBAO and Type Ia supernovae in the same analysis thetension is shifted to a discrepancy in the sound horizonscale that cannot be resolved with a late time solution[48, 49, 50].The situation is slightly less clear with respect to the σ tension. In Λ CDM, the tension appears betweenCMB measurements coming from Planck and large scalestructure constraints on growth such as those from theDark Energy Survey (DES) [51]. This mild tension canbe seen in the right panel of Figure 2, with the Λ CDMconstraints plotted in black, the filled contour correspond-ing to Planck and the open contour to DES. The DESconstraint in the interacting scenario is plotted in theopen blue contour – again, in the interests of legibilitywe only show the mean fiducial case.From this plot, we can see that the tension is relaxedin the interacting case, but only due to the increased sizeof the contours, which in turn is due to the additionalfree parameters in the interacting model with respectto Λ CDM. This should not be regarded as a true relax-ation of the tension. Note that for the DES constraintspresented here we implemented an aggressive cut ofthe non-linear scales in the data. Since we have no un-derstanding of the non-linear regime in the interactingscenario we should not use this part of the data to obtainour constraints.
With the results of our MCMC analysis, we can recon-struct the coupling as a function of redshift. We show theresults of using two di ff erent methods for the reconstruc-tion: a simple cubic spline interpolation and a Gaussianprocess.A Gaussian process is defined as a collection of ran-dom variables, any finite number of which have a jointGaussian distribution [52]. It is completely specifiedby its mean and its covariance. In practice, the randomvariables represent the value of a given function f ( x ) ata location x . There are a wide range of choices for thecovariance function, or kernel, that is used to relate thefunction values at each point. In this work, we choose to use one of the simplest, the squared exponential kernel,given by k ( x , ˜ x ) = σ exp (cid:32) − ( x − ˜ x )2 (cid:96) (cid:33) . (13)The hyperparameters (cid:96) and σ that appear in this kernelcorrespond to the approximate length scale over whichthe function varies, and the variance of the function ateach point respectively. We optimise these by maximis-ing the log-likelihood of the functions they produce.In summary, the Gaussian process takes some giventraining data and constructs the best possible functionthat describes that data, given the kernel imposed. Thetraining data passed to the Gaussian process in our caseare the mean posterior values of the coupling parame-ter in each bin along with the corresponding 1 σ errorsgiven by our MCMC analysis, thereby allowing us toreconstruct the coupling function q ( z ).There are many packages and codes available to per-form Gaussian process regression. In this work, we usethe Gaussian process regressor available in the Pythonlibrary george [53].The results of our reconstructions for the cubic splineand the Gaussian process are shown in Figures 3 and 4respectively. It is clear to see that the Gaussian processresults in a smoother q ( z ) function, but that the highredshift part of the reconstruction is biased towards the Λ CDM value of q =
0, due to the baseline that theGaussian process is fixed to return to in the absence ofinformation.This is particularly obvious in the mean fiducial case,where the values of q themselves are very negative butthe combination of the Gaussian process baseline and thelarge 1 σ errors on q result in the reconstruction returningto zero. This is a problem that the cubic spline does notsu ff er from, hence the indication of a trend away from Λ CDM at high redshift in the mean fiducial case.The most interesting features of the reconstructionare the points where q ( z ) appears to peak or trough, forexample, the peak at around z =
1, which is clear inboth the spline and Gaussian process, or the trough ataround z =
3, more obvious in the Gaussian processreconstruction. A promising line of enquiry would beto focus on the behaviour of the interaction at thesepoints by using additional datasets in the analysis, butas z = ffi cult. https: // github.com / dfm / george Ω b h . ± . . ± . Ω c h . ± .
025 0 . ± . A . ± . . ± . n s . ± . . ± . H . ± .
74 68 . ± . σ . ± .
18 0 . ± . Table 2: Marginalised values of the cosmological parameters and their 68% confidence level bounds.Figure 2: 68% and 95% confidence levels in the H – Ω m plane (left panel) and Ω m – σ plane (right panel) for the mean fiducial case. The greybands in the left panel denote the 68% and 95% confidence levels of the Riess et al. local measurement of H = . ± .
66 kms − Mpc − [8]. α forest in the spectra of high-redshift quasars, which probes the matter distributionat redshifts of 2 to 3.5 [54]. Furthermore, the SquareKilometre Array is predicted to be able to probe red-shifts of between 3 and 25 using 21cm intensity mapping[55, 56]. Both of these new techniques could thereforebe used to constrain any interacting dark energy modelwhich a ff ects large scale structure growth or has otherhigh redshift e ff ects. In this work, we have aimed to be agnostic when itcomes to the reconstruction of the interaction functionand so used a larger number of bins than in [22], i.e.the minimum number to satisfy the criterion given byequation (9). However, it is also possible to investigatehow many modes in the result are informed by the obser-vational data used and whether any are informed by theprior alone, and thus understand how many e ff ective ad-ditional degrees of freedom our reconstruction has [57].To do this, we perform a principal component analysis.Principal component analysis, or PCA, can be thoughtof as finding the directions in the data that carry themost information. It also acts to decorrelate the errorson the interaction parameter in each bin. In practice, thisinvolves computing the eigenvalues and eigenvectors ofthe inverse covariance matrix (i.e. the Fisher matrix)of the data. In our case, the covariance matrix is oneof the products obtained after running GetDist [58] onour MCMC chains. We perform the PCA on the Fishermatrix for the q i alone, after marginalising over the othercosmological and nuisance parameters.The Fisher matrix is given by F = W T Λ W , (14)where W is the decorrelation matrix and its rows definethe eigenvectors; Λ is a diagonal matrix whose elementsare the eigenvalues λ i . The eigenvalues correspond to theamount of variance carried in each principal componentand therefore determine how well q i can be measured,i.e. σ ( q i ) = λ − / i .After finding the eigenvalues and eigenvectors of thecovariance matrix, the eigenvectors are sorted accordingto decreasing value of their corresponding eigenvalues.The first eigenvector after this sort is performed corre-sponds to the first principal component, the second eigen-vector corresponds to the second principal component https: // github.com / cmbant / getdist and so on, until the N th eigenvector for the N th princi-pal component is found (where the covariance matrix is N × N ).We show the results of our PCA in Figures 5 and 6.From these plots we can see that in the fixed fiducial casearound 15% of the total variance is in the first principalcomponent, we reach around 50% with four principalcomponents and 90% with 10. These results indicate thatit would be unwise to reduce the e ff ective degrees of free-dom by discarding some of the principal components, aseven the higher components contain a significant amountof information (above PC10 the remaining seven com-ponents together still contain approximately 11% of thevariance). This is less true in the case of the mean fidu-cial, in which around 25% of the total variance is con-tained in the first principal component, rising to nearly50% with just two principal components and reaching90% with seven. The final four principal componentstogether contain just 1% of the variance.To investigate whether the correlation prior dominatesover the data, we also ran an MCMC chain without anydata, using the prior alone to constrain the interaction.This prior alone case used ¯ q =
0, as in the case of thefixed fiducial. We plot the eigenvalues of the fixed fidu-cial case and the prior alone case as a function of princi-pal component number in Figure 7. This plot shows thatthe data permeates all the modes, meaning that the priordoes not completely dominate over the data at any pointand thus the selected prior strength was indeed su ffi cientto help constrain the interaction without washing out theinformation coming from the data. Note that we onlyshow the result for the case of the fixed fiducial prioralone and the fixed fiducial prior plus data, as the resultfor the mean fiducial is extremely similar.If we had found that the data dominated for say, thefirst three principal components and then the prior domi-nated over the rest, we would be able to conclude that ouranalysis e ff ectively only had an additional three degreesof freedom compared to the Λ CDM case. However, thisdoes not equate to doing an analysis using only threebins, as the principal components do not correspond tothe bins themselves, but to the eigenvectors of the covari-ance matrix of the interaction parameter in each bin. Wetherefore conclude that the best strategy for an analysissuch as this is to use as many bins as is computationallyfeasible, with the correlation prior being used to helpconstrain bins where data is scarce. The alternative isto increase the strength of the correlation prior, but thiscomes with its own pitfalls, as if the prior is too strong,it will completely wash out any contribution from thedata. A balance can be achieved, but to ensure that thereconstruction remains independent of the number of8 igure 3: The results of the cubic spline reconstruction of the coupling function q ( z ). Red and blue lines and areas refer to the fixed fiducial andmean fiducial cases respectively, and the shaded areas denote the 1 σ confidence interval.Figure 4: The results of the Gaussian process reconstruction of the coupling function q ( z ). Red and blue lines and areas refer to the fixed fiducial andmean fiducial cases respectively, and the shaded areas denote the 1 σ confidence interval. Principal component P e r c e n t a g e o f v a r i a n c e Cumulative varianceIndividual variance
Figure 5: Percentage variance explained by each principal componentin the fixed fiducial case.
Principal component P e r c e n t a g e o f v a r i a n c e Cumulative varianceIndividual variance
Figure 6: Percentage variance explained by each principal componentin the mean fiducial case. bins used, the correlation length and therefore the priorstrength should be determined by following equations(9) and (10). χ Finally, we want to compare the results for each casein a Bayesian way, which means making use of Bayes’theorem [59]: P ( θ | D , M ) = P ( D | θ, M ) P ( θ | M ) P ( D | M ) , (15)where θ is the parameter vector, D is the data vector and M is the model. The numerator contains the likelihoodand the prior, and the denominator is the evidence (some-times known as the marginal likelihood). These combineto form the posterior probability distribution P ( θ | D , M ),which is the distribution sampled in our MCMC analysis. Principal component / Prior and dataPrior alone
Figure 7: Showing that the data permeates all the modes.
As noted by [60], the use of model selection crite-ria such as the Bayesian Information Criterion (BIC),Akaike Information Criterion (AIC) and Deviance In-formation Criterion (DIC) are not strictly Bayesian asthey do not take into account the prior information. Wetherefore use the Bayes factor as our model comparisontool, defined in the following way:log B = log (cid:34) P ( D | M ) P ( D | M ) (cid:35) , (16) = log[ P ( D | M )] − log[ P ( D | M )] , (17)where D is the data vector, M and M are the models tobe compared, and P ( D | M ) is the Bayesian evidence, thenormalising factor in Bayes’ theorem.We calculate the Bayesian evidence from our MCMCchains for both of the two correlation prior cases studiedto determine the support for each case over Λ CDM. Thisanalysis was performed using the
MCEvidence code aspresented in [61]. In each case, we use Λ CDM as model1. We summarise our findings in Table 3.To interpret these values, we make use of the Je ff reysscale, as shown in Table 4. As pointed out in [62], thequalitative interpretations originally given by Je ff reys[63] are quite strong in the context of cosmology, wherechoosing suitable priors can often be an uncertain pro-cess. We therefore adopt the interpretations given in[62].We find that the Bayes factor for the fixed fiducialcase is 1.64. According to the Je ff reys scale, this reflectsa weak preference for Λ CDM over the interacting case.The Bayes factor for the mean fiducial case is -0.52. Inour evidence calculation, negative values indicate thatmodel 2 is preferred over model 1, where model 1 isalways Λ CDM. This result is therefore a slight indication10ase Bayes factor (log B ) ∆ χ Fixed fiducial 1.64 -2.5Mean fiducial -0.52 -2.2
Table 3: The Bayes factor and ∆ χ for each case. Bayes factor Interpretation (cid:12)(cid:12)(cid:12) log B (cid:12)(cid:12)(cid:12) < < (cid:12)(cid:12)(cid:12) log B (cid:12)(cid:12)(cid:12) < . . < (cid:12)(cid:12)(cid:12) log B (cid:12)(cid:12)(cid:12) < < (cid:12)(cid:12)(cid:12) log B (cid:12)(cid:12)(cid:12) Strong
Table 4: The Je ff reys scale, originally given in [63] and modified in [62]. for the mean fiducial case being favoured over Λ CDM.However, according to the Je ff reys scale, the very smallabsolute value of the Bayes factor means this is not worthmore than a bare mention.The fact that we find stronger evidence in favour of Λ CDM in the case where the fiducial is fixed as q = ffi ciently small forus to confidently say that the choice of fiducial modeldoes not drastically alter the result of a reconstruction.However, it has been argued that the Bayesian evi-dence is not a good model comparison tool when thereis uncertainty in the choice of priors [64]. We thereforealso compute the ∆ χ for each case, removing the con-tribution of the priors to the χ so that the values wecompare come from the data only. We find ∆ χ = − . ∆ χ = − . Λ CDM.In summary, it is clear that we cannot conclusivelystate that Λ CDM is preferred over the interacting case,but the hints given by the evidence indicate that an in-teresting future direction would be to repeat this type ofanalysis with the newest datasets as they are released, tosee if there is any strengthening in the evidence for oragainst Λ CDM. It is also worthwhile studying what pos-sible improvements on current constraints can be madeby future surveys.
5. Conclusions
In this work we have reconstructed a dark matter –vacuum energy interaction, using a correlation prior tocontrol the reconstruction bias. We implemented twodi ff erent versions of the prior: a fiducial value for the prior that is fixed in each bin and a fiducial value that iscomputed as the mean of the neighbouring bins.In our model comparison, we found evidence in favourof Λ CDM over the fixed fiducial model, but the Bayesfactor in that case was small enough to classify the ev-idence on the Je ff reys scale as weak. In contrast, wefound evidence for an interaction when comparing the Λ CDM case to the mean fiducial case, but the Je ff reysscale in that case classified the evidence as not worthmore than a bare mention.From our work, it is clear that a correlation prior, whene ff ectively tuned so as not to drown out the constrainingpower of the data, can improve the convergence speedof high-dimensionality MCMC sampling. The prior alsoeliminates any potential reconstruction bias, making it agood choice for any form of reconstructive analysis.Finally, we note that many recent works have foundevidence for an interaction in the dark sector or for dy-namical dark energy (see e.g. [65, 36, 66, 67, 68]), andthe attractive properties of such models combined withthe deficiencies of Λ CDM that we discussed in the intro-duction are su ffi cient to merit their continued study. It isclear that the large amounts of new data which upcomingsurveys are expected to yield will be a vital clue in thehunt for the true nature of dark energy, and robust fore-casting for the constraints these surveys are expected toprovide on alternative dark energy models will becomeever-more important. CRediT authorship contribution statementNatalie B. Hogg:
Software, Formal analysis, Investi-gation, Writing – original draft.
Marco Bruni:
Method-ology, Conceptualization, Writing – review & editing,Supervision.
Robert Crittenden:
Methodology, Con-ceptualization, Writing – review & editing, Supervi-sion.
Matteo Martinelli:
Methodology, Conceptualiza-tion, Writing – review & editing, Supervision.
Simone eirone: Software, Writing – review & editing, Visual-ization.
Declaration of competing interest
The authors declare that they have no known com-peting financial interests or personal relationships thatcould have appeared to influence the work reported inthis paper.
Acknowledgements
We thank Minas Karamanis, David Wands, Yut-ing Wang and Gong-Bo Zhao for enlightening discus-sions. This paper is based upon work from COSTaction CA15117 (CANTATA), supported by COST(European Cooperation in Science and Technology).Numerical computations were done on the SCIAMAHigh Performance Computer (HPC) cluster which issupported by the ICG, SEPNet and the University ofPortsmouth. NBH is supported by UK STFC stu-dentship ST / N504245 /
1. MB and RC are supportedby UK STFC Grant No. ST / S000550 /
1. MM hasreceived the support of a fellowship from “la Caixa”Foundation (ID 100010434), with fellowship codeLCF / BQ / PI19 / References [1] A. G. Riess, et al., Observational Evidence from Supernovae foran Accelerating Universe and a Cosmological Constant, TheAstronomical Journal 116 (1998) 1009–1038. doi: . arXiv:astro-ph/9805201 .[2] S. Perlmutter, et al., Measurements of Ω and Λ from 42 High-Redshift Supernovae, The Astrophysical Journal 517 (1999)565–586. doi: . arXiv:astro-ph/9812133 .[3] R. J. Adler, B. Casey, O. C. Jacob, Vacuum catastrophe: Anelementary exposition of the cosmological constant problem,American Journal of Physics 63 (1995) 620–626. doi: .[4] S. Weinberg, The cosmological constant problem, Reviews ofModern Physics 61 (1989) 1–23.[5] J. Martin, Everything you always wanted to know about thecosmological constant problem (but were afraid to ask), ComptesRendus Physique 13 (2012) 566–665. doi: . arXiv:1205.3365 .[6] H. E. S. Velten, R. F. vom Marttens, W. Zimdahl, As-pects of the cosmological coincidence problem, Eur. Phys. J.C74 (2014) 3160. doi: . arXiv:1410.2509 . [7] E. Bianchi, C. Rovelli, Why all these prejudices against a con-stant? (2010). arXiv:1002.3966 .[8] A. G. Riess, S. Casertano, W. Yuan, L. Macri, J. Anderson, J. W.MacKenty, J. B. Bowers, K. I. Clubb, A. V. Filippenko, D. O.Jones, B. E. Tucker, New Parallaxes of Galactic Cepheids fromSpatially Scanning the Hubble Space Telescope: Implications forthe Hubble Constant, The Astrophysical Journal 855 (2018) 136.doi: . arXiv:1801.01120 .[9] N. Aghanim, et al., Planck 2018 results. VI. Cosmologicalparameters, ArXiv e-prints (2018). arXiv:1807.06209 .[10] A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, D. Scol-nic, Large Magellanic Cloud Cepheid Standards Provide a 1%Foundation for the Determination of the Hubble Constant andStronger Evidence for Physics beyond Λ CDM, The Astrophysi-cal Journal 876 (2019) 85. doi: . arXiv:1903.07603 .[11] S. Birrer, T. Treu, C. E. Rusu, V. Bonvin, C. D. Fassnacht,J. H. H. Chan, A. Agnello, A. J. Shajib, G. C. F. Chen, M. Auger,H0LiCOW - IX. Cosmographic analysis of the doubly imagedquasar SDSS 1206 + . arXiv:1809.01274 .[12] B. P. Abbott, et al. (LIGO Scientific, VINROUGE, Las CumbresObservatory, DES, DLT40, Virgo, 1M2H, Dark Energy CameraGW-E, MASTER), A gravitational-wave standard siren mea-surement of the Hubble constant, Nature 551 (2017) 85–88.doi: . arXiv:1710.05835 .[13] D. W. Pesce, et al., The Megamaser CosmologyProject. XIII. Combined Hubble constant constraints (2020). arXiv:2001.09213 .[14] E. Macaulay, R. C. Nichol, D. Bacon, D. Brout, T. M.Davis, B. Zhang, B. A. Bassett, D. Scolnic, A. M¨oller, C. B.D’Andrea, First cosmological results using Type Ia super-novae from the Dark Energy Survey: measurement of the Hub-ble constant, Monthly Notices of the Royal AstronomicalSociety 486 (2019) 2184–2196. doi: . arXiv:1811.02376 .[15] R. A. Battye, T. Charnock, A. Moss, Tension between the powerspectrum of density perturbations measured on large and smallscales, Physical Review D 91 (2015) 103508. doi: . arXiv:1409.2769 .[16] M. Douspis, L. Salvati, N. Aghanim, On the Tension be-tween Large Scale Structures and Cosmic Microwave Back-ground, PoS EDSU2018 (2018) 037. doi: . arXiv:1901.05289 .[17] G. E. P. Box, Science and statistics, Journal of the Ameri-can Statistical Association 71 (1976) 791–799. doi: .[18] D. Wands, J. De-Santiago, Y. Wang, Inhomogeneous vacuum en-ergy, Classical and Quantum Gravity 29 (2012) 145017. doi: . arXiv:1203.6776 .[19] V. Salvatelli, N. Said, M. Bruni, A. Melchiorri, D. Wands, Indica-tions of a Late-Time Interaction in the Dark Sector, Physical Re-view Letters 113 (2014) 181301. doi: .[20] J. Sol`a, A. G´omez-Valent, J. de Cruz P´erez, Vacuum dynamicsin the Universe versus a rigid Λ = const, International Jour-nal of Modern Physics A32 (2017) 1730014. doi: . arXiv:1709.07451 .[21] S. Kumar, R. C. Nunes, Echo of interactions in the dark sector,Phys. Rev. D96 (2017) 103511. doi: . arXiv:1702.02143 .[22] M. Martinelli, N. B. Hogg, S. Peirone, M. Bruni, D. Wands,Constraints on the interacting vacuum-geodesic CDM sce- ario, Monthly Notices of the Royal Astronomical Soci-ety 488 (2019) 3423–3438. doi: . arXiv:1902.10694 .[23] S. Pan, W. Yang, E. Di Valentino, E. N. Saridakis, S. Chakraborty,Interacting scenarios with dynamical dark energy: observationalconstraints and alleviation of the H tension, ArXiv e-prints(2019). arXiv:1907.07540 .[24] S. Pan, G. S. Sharov, W. Yang, Field theoretic interpretations ofinteracting dark energy scenarios and recent observations (2020). arXiv:2001.03120 .[25] E. J. Copeland, M. Sami, S. Tsujikawa, Dynamics ofDark Energy, International Journal of Modern PhysicsD 15 (2006) 1753–1935. doi: . arXiv:hep-th/0603057 .[26] T. Clifton, P. G. Ferreira, A. Padilla, C. Skordis, Modified gravityand cosmology, Physics Reports 513 (2012) 1–189. doi: . arXiv:1106.2476 .[27] A. Joyce, L. Lombriser, F. Schmidt, Dark EnergyVersus Modified Gravity, Annual Review of Nuclearand Particle Science 66 (2016) 95–122. doi: . arXiv:1601.06133 .[28] J. M. Ezquiaga, M. Zumalac´arregui, Dark Energy AfterGW170817: Dead Ends and the Road Ahead, Physical Re-view Letters 119 (2017) 251304. doi: . arXiv:1710.05901 .[29] N. Frusciante, L. Perenon, E ff ective Field Theory of Dark Energy:a Review (2019). arXiv:1907.03150 .[30] A. Lewis, A. Challinor, A. Lasenby, E ffi cient computationof CMB anisotropies in closed FRW models, The Astro-physical Journal 538 (2000) 473–476. doi: . arXiv:astro-ph/9911177 .[31] C. Howlett, A. Lewis, A. Hall, A. Challinor, CMB powerspectrum parameter degeneracies in the era of precision cos-mology, Journal of Cosmology and Astroparticle Physics1204 (2012) 027. doi: . arXiv:1201.3654 .[32] A. Lewis, S. Bridle, Cosmological parameters from CMBand other data: A Monte Carlo approach, Physical ReviewD 66 (2002) 103511. doi: . arXiv:astro-ph/0205436 .[33] A. Lewis, E ffi cient sampling of fast and slow cosmologicalparameters, Physical Review D 87 (2013) 103529. doi: . arXiv:1304.4473 .[34] R. G. Crittenden, L. Pogosian, G.-B. Zhao, Investigating dark en-ergy experiments with principal components, Journal of Cosmol-ogy and Astroparticle Physics 0912 (2009) 025. doi: . arXiv:astro-ph/0510293 .[35] R. G. Crittenden, G.-B. Zhao, L. Pogosian, L. Samushia,X. Zhang, Fables of reconstruction: controlling bias in the darkenergy equation of state, Journal of Cosmology and Astroparti-cle Physics 2 (2012) 048. doi: . arXiv:1112.1693 .[36] Y. Wang, L. Pogosian, G.-B. Zhao, A. Zucca, Evolution ofdark energy reconstructed from the latest observations, As-trophys. J. 869 (2018) L8. doi: . arXiv:1807.03772 .[37] F. Gerardi, M. Martinelli, A. Silvestri, Reconstruction of the DarkEnergy equation of state from latest data: the impact of theoreti-cal priors, JCAP 1907 (2019) 042. doi: . arXiv:1902.09423 .[38] L. Dam, K. Bolejko, G. F. Lewis, Probing the independencewithin the dark sector in the fluid approximation, JCAP1912 (2019) 030. doi: . arXiv:1908.01953 .[39] Y. Wang, G.-B. Zhao, D. Wands, L. Pogosian, R. G. Crittenden, Reconstruction of the dark matter–vacuum energy interaction,Physical Review D 92 (2015) 103005. doi: . arXiv:1505.01373 .[40] Y. Wang, D. Wands, G.-B. Zhao, L. Xu, Post- Planck constraintson interacting vacuum energy, Phys. Rev. D90 (2014) 023502.doi: . arXiv:1404.5706 .[41] F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith,L. Campbell, Q. Parker, W. Saunders, F. Watson, The 6dF GalaxySurvey: baryon acoustic oscillations and the local Hubble con-stant, Monthly Notices of the Royal Astronomical Society 416(2011) 3017–3032. doi: . arXiv:1106.3366 .[42] S. Alam, et al. (BOSS), The clustering of galaxies in the com-pleted SDSS-III Baryon Oscillation Spectroscopic Survey: cos-mological analysis of the DR12 galaxy sample, Monthly No-tices of the Royal Astronomical Society 470 (2017) 2617–2652.doi: . arXiv:1607.03155 .[43] D. M. Scolnic, et al., The Complete Light-curve Sample of Spec-troscopically Confirmed SNe Ia from Pan-STARRS1 and Cos-mological Constraints from the Combined Pantheon Sample, As-trophys. J. 859 (2018) 101. doi: . arXiv:1710.00845 .[44] E. Di Valentino, A. Melchiorri, O. Mena, S. Vagnozzi, Interactingdark energy after the latest Planck, DES, and H measurements:an excellent solution to the H and cosmic shear tensions, arXive-prints (2019). arXiv:1908.04281 .[45] V. Poulin, K. K. Boddy, S. Bird, M. Kamionkowski, Implicationsof an extended dark energy cosmology with massive neutrinos forcosmological tensions, Phys. Rev. D97 (2018) 123504. doi: . arXiv:1803.02474 .[46] M. Martinelli, I. Tutusaus, CMB tensions with low-redshift H and S measurements: impact of a redshift-dependent type-Ia supernovae intrinsic luminosity, Symmetry 11 (2019) 986.doi: . arXiv:1906.09189 .[47] H. A. Borges, D. Wands, Growth of structure in interactingvacuum cosmologies (2017). arXiv:1709.08933 .[48] K. Aylor, M. Joy, L. Knox, M. Millea, S. Raghunathan, W. L. K.Wu, Sounds Discordant: Classical Distance Ladder & Λ CDM-based Determinations of the Cosmological Sound Horizon, As-trophys. J. 874 (2019) 4. doi: . arXiv:1811.00537 .[49] N. Arendse, et al., Cosmic dissonance: new physics or systemat-ics behind a short sound horizon? (2019). arXiv:1909.07986 .[50] L. Knox, M. Millea, The Hubble Hunter’s Guide (2019). arXiv:1908.03663 .[51] T. Abbott, et al. (DES), Dark Energy Survey year 1 results: Cos-mological constraints from galaxy clustering and weak lensing,Phys. Rev. D 98 (2018) 043526. doi: . arXiv:1708.01530 .[52] C. E. Rasmussen, C. K. I. Williams, Gaussian Processes forMachine Learning, MIT Press, 2006.[53] S. Ambikasaran, D. Foreman-Mackey, L. Greengard, D. W.Hogg, M. O’Neil, Fast Direct Methods for Gaussian Pro-cesses, IEEE Transactions on Pattern Analysis and MachineIntelligence 38 (2015). doi: . arXiv:1403.6015 .[54] R. A. C. Croft, A. Romeo, R. B. Metcalf, Weak lensing of theLyman α forest, Monthly Notices of the Royal AstronomicalSociety 477 (2018) 1814–1821. doi: . arXiv:1706.07870 .[55] R. Braun, T. Bourke, J. A. Green, E. Keane, J. Wagg, AdvancingAstrophysics with the Square Kilometre Array, Proceedings ofScience (AASKA14) (2015) 174.[56] D. J. Bacon, et al., Cosmology with Phase 1 of the SquareKilometre Array; Red Book 2018: Technical specifications and erformance forecasts, arXiv e-prints (2018) arXiv:1811.02743. arXiv:1811.02743 .[57] D. Huterer, G. Starkman, Parameterization of dark-energyproperties: A Principal-component approach, Phys. Rev. Lett.90 (2003) 031301. doi: . arXiv:astro-ph/0207517 .[58] A. Lewis, GetDist: a Python package for analysing Monte Carlosamples (2019). URL: https://getdist.readthedocs.io . arXiv:1910.13970 .[59] T. Bayes, LII. An essay towards solving a problem in the doctrineof chances. By the late Rev. Mr. Bayes, F. R. S. communicated byMr. Price, in a letter to John Canton, A. M. F. R. S, PhilosophicalTransactions of the Royal Society of London 53 (1763) 370–418.doi: .[60] F. X. Linares Cede˜no, A. Montiel, J. C. Hidalgo, G. Germ´an,Bayesian evidence for α -attractor dark energy models, JCAP1908 (2019) 002. doi: . arXiv:1905.00834 .[61] A. Heavens, Y. Fantaye, A. Mootoovaloo, H. Eggers, Z. Hosenie,S. Kroon, E. Sellentin, Marginal Likelihoods from Monte CarloMarkov Chains, ArXiv e-prints (2017). arXiv:1704.03472 .[62] M. P. Hobson, A. H. Ja ff e, A. R. Liddle, P. Mukherjee, D. Parkin-son, Bayesian Methods in Cosmology, Cambridge UniversityPress, 2010.[63] H. Je ff reys, Theory of Probability, Clarendon Press, Oxford,1961.[64] G. Efstathiou, Limitations of Bayesian Evidence applied to cos-mology, Monthly Notices of the Royal Astronomical Society 388(2008) 1314–1320. doi: . arXiv:0802.3185 .[65] G.-B. Zhao, et al., Dynamical dark energy in light of the latestobservations, Nature Astronomy 1 (2017) 627–632. doi: . arXiv:1701.08165 .[66] W. Yang, S. Pan, E. Di Valentino, E. N. Saridakis, S. Chakraborty,Observational constraints on one-parameter dynamical dark-energy parametrizations and the H tension, Phys. Rev.D99 (2019) 043543. doi: . arXiv:1810.05141 .[67] W. Yang, N. Banerjee, A. Paliathanasis, S. Pan, Recon-structing the dark matter and dark energy interaction scenar-ios from observations, Phys. Dark Univ. 26 (2019) 100383.doi: . arXiv:1812.06854 .[68] M. Lucca, D. C. Hooper, Tensions in the dark: shed-ding light on Dark Matter-Dark Energy Interactions (2020). arXiv:2002.06127 ..