Addressing H 0 tension by means of VCDM
AAddressing H tension by means of VCDM Antonio De Felice, ∗ Shinji Mukohyama,
1, 2, † and Masroor C. Pookkillath ‡ Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502, Kyoto, Japan Kavli Institute for the Physics and Mathematics of the Universe (WPI),The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
Tensions in quantifying the present expansion rate of the universe, H , from the high-redshiftobservation and low-redshift observations have been growing during these past few years. This isone of the most surprising and hardest challenges that the present day cosmology needs to face.These experimental results are difficult to accept as they immediately challenge the standard modelof cosmology known as the Λ Cold Dark Matter ( Λ CDM) model. On the other hand, once theexperimental results are accepted, we need to face the possibility that the choice of this model isnothing but an arbitrarily chosen theoretical prior in the estimation of the cosmological parametersof the universe (including H ). Here we show that the H -tensions can be resolved by changing thetheoretical prior on using a new theory, dubbed VCDM. This is essentially a low-redshift resolutionof the Hubble tension. The value of today’s rate of expansion of the universe, H , has been measured, as a direct measurement, fromlow-redshift observations, namely, Hubble Space Tele-scope [1] (HST), H0LiCOW [2], Megamaser Project [3](MCP) and Carnegie-Chicago Hubble Program (CCHP)Collaboration [4]. Among these observations HST inparticular has achieved a remarkable precision provid-ing H = ± H can also be deduced from themeasurement of temperature power spectra in the Cos-mic Microwave Background Radiation (CMB) which isproduced at the recombination time. The recent PlanckLegacy 2018 release gives H = ± Λ CDM model (non-flat versions are knownto be strongly disfavored by other data, e.g. BaryonAcoustic Oscillations (BAO)) [5]. Hence, the tensionbetween this theoretical model and experimental resultsadds up to more than 4 σ ’s [6, 7].However, the flat- Λ CDM could be representing only afirst approximation to another, improved model of ouruniverse. The VCDM theory, described in the following,was originally introduced for the purpose of seeking min-imal theoretical deviations from the standard model ofgravity and cosmology, i.e. General Relativity (GR) and Λ CDM, as it does not introduce any new physical degreesof freedom, but on the other hand, one can have, as wewill show in the following, a non-trivial and interestingphenomenology.The VCDM theory [8], in which the cosmological con-stant Λ in the standard Λ CDM is promoted to a function V ( φ ) of a non-dynamical, auxiliary field φ without intro-ducing extra physical degrees of freedom. This theory ofmodified gravity breaks four dimensional diffeomorphisminvariance at cosmological scales but keeps the three di-mensional spatial diffeomorphism invariance. On doingso, the theory modifies gravity at cosmological scaleswhile it only possess two gravitational degrees of freedomas in GR. In general this allows a spectrum of possibili-ties typically much larger than the case of a scalar-tensor theory. For the latter, the extra scalar degree of free-dom is not only strongly bound on solar system scales(for which one needs the scalar to be very massive or tobe shielded by some non-trivial dynamical mechanisms,e.g. chameleon or Vainshtein), but also on cosmologicalscales (for which one needs to constrain the backgrounddynamics as to avoid ghost and gradient instabilities).The equations of motion for the VCDM theory on ahomogeneous and isotropic background can be writtenas V = 13 φ − ρM , dφd N = 32 ρ + PM H , dρ i d N − ρ i + P i ) = 0 , (1)where N = ln( a/a ) ( a being the scale factor and a being its present value), H = ˙ a/a is the Hubble expan-sion rate (a dot denotes differentiation with respect tothe conformal time), ρ = (cid:80) i ρ i and P = (cid:80) i P i (the sumis over the standard matter species). Unless ρ + P = 0 ,the following equation follows from the above equations: φ = V ,φ − H . When V is a linear function of φ , as in V = λ φ + λ , then the equations of motion (1) reduceto H = ρM +Λ , H dHd N = − ρ + P M , dρ i d N − ρ i + P i ) = 0 , (2)where Λ ≡ λ + 3 λ / . These are nothing butthe equations of motion in the standard Λ CDM model.Hence, the VCDM theory extends the Λ CDM model byreplacing the constant Λ with a free function V ( φ ) . Yet,the VCDM theory does not introduce extra degrees offreedom in the sense that the number of independent ini-tial conditions is the same as Λ CDM. The “V” of VCDMtherefore stands for the free function V ( φ ) introduced inthis theory.In the following we want to be able to use the freefunction V ( φ ) in order to give any wanted backgroundevolution for H which can be given as H = H ( N ) . Fromthe 2nd of (1), having given H as a function of N , then a r X i v : . [ a s t r o - ph . C O ] S e p one obtains φ ( N ) = φ + ˆ NN ρ ( N (cid:48) ) + P ( N (cid:48) ) M H ( N (cid:48) ) d N (cid:48) , (3)where φ = φ ( N ) . Assuming that ρ + P > , and H > ,the right hand side of (3) is an increasing function of N and thus the function φ ( N ) has a unique inverse function, N = N ( φ ) . Obviously, N is an increasing function of φ .By combining this with the 1st of (1), one obtains V ( φ ) = 13 φ − ρ ( N ( φ )) M . (4)Therefore we have obtained a simple and powerful recon-struction mechanism for V for a given and wanted evo-lution of H . Once V ( φ ) is specified in this way, we knowhow to evolve not only the homogeneous and isotropicbackground but also perturbations around it.We now introduce two choices of H ( z ) to address cur-rent cosmological tensions: one in the flat- Λ CDM and the other in the VCDM. The former is H ≡ H [ ˜Ω Λ +˜Ω m (1 + z ) + ˜Ω r (1 + z ) ] , where ˜Ω Λ ≡ − ˜Ω m − ˜Ω r ,and the latter is H = H + A H (cid:20) − tanh (cid:18) z − A A (cid:19)(cid:21) , (5)with the idea that < A < . In this case wehave at early times, for z (cid:29) | A | , that the systemwill tend to be the standard flat- Λ CDM evolution, i.e. H ≈ H . Today, i.e. for z = 0 , we have H = H + A H (cid:104) (cid:16) A A (cid:17)(cid:105) , which can be solved todayfor A , to give A = H − H H (cid:2) tanh (cid:0) A A (cid:1) +1 (cid:3) .Let us further consider the following parameter redefi-nitions ˜Ω m = Ω m H H , ˜Ω r = Ω r H H , and β H = H Λ0 H ,then we find H H = Ω m0 (1 + z ) + Ω r0 (1 + z ) + (1 − β H ) 1 + tanh (cid:16) A − zA (cid:17) (cid:16) A A (cid:17) + β H (cid:18) − Ω m0 β H − Ω r0 β H (cid:19) . (6)So in total we have six background parameters (threemore than Λ CDM). However, we can reduce them to five(two more than Λ CDM) by fixing A as we expect to havea large degeneracy (after assuming A ∼ ). Accordingto Akaike Information Criterion (AIC), we can accept themodel if we can have an improvement of χ larger thanfour in comparison with Λ CDM [9]. In fact, we will showlater on that the χ has improved remarkably by 20.60with respect to Λ CDM. In particular, we will fix, lateron, A to the value of 10 − .Two things need to be noticed. First, having giventhe expression for H = H ( z ) , one automatically can de-duce all the needed background expressions. Second, thefact we have an MMG-component does not mean we areadding a physical dark-component degree of freedom. Infact, for this theory, there is no additional physical de-gree of freedom, beside the tensorial gravitational wavesand the standard ones related to the presence of matter fields [8].After having introduced the behavior of the VCDMmodel on a FLRW background, we will test it againstseveral cosmological data to see how well it can ad-dress the H tension. Here we use Planck Legacy 2018data with planck_highl_TTTEEE , planck_lowl_EE , and planck_lowl_TT [10], baryon acoustic oscillation (BAO)from 6dF Galaxy Survey [11] and the Sloan Digital SkySurvey [12, 13], Joint light Analysis (JLA) comprised of740 type Ia supernovae [14], and SH0ES consisting of asingle data point [1] H = ± S SSL = − ˆ d x √− g [ ρ ( n, s ) + J µ ( ∂ µ (cid:96) + θ∂ µ s + A ∂ µ B + A ∂ µ B )] , (7)with n = (cid:112) − J µ J µ g µν , and 4-velocity u α = J α /n .We consider several copies of the previous action each describing a different fluid, labeled with an index I .Then we can expand the previous SSL up to secondorder in the perturbation fields, and to this one, wethen add a correction aimed to describe an anisotropicfluid as follows S (2)m = S (2)SSL + S (2)corr , where S (2)corr = ´ dtd xN a (cid:80) I σ I Θ I and Θ I stands for a linear com-bination of perturbation fields. Since for each matterspecies ρ I = ρ I ( n I ) , and n I = (cid:112) − J µI J νI g µν , one canfind a relation among δρ I and the other fields as δJ I = ρ I n I ρ I.n δρ I ρ I − α , where δN = N ( t ) α , which can be used asa field redefinition to replace δJ I in terms of δρ I ρ I . We alsodefine gauge invariant combinations v I = − ak θ I + χ − a N ∂ t ( E/a ) , α = ψ − ˙ χN + N − ∂ t [ a N − ∂ t ( E/a )] , and ζ = − φ − H χ + a HN ∂ t ( E/a ) , where δγ ij = 2[ a ζδ ij + ∂ i ∂ j E ] , and δu Ii = ∂ i v I . We find finally that S (2)corr = ˆ dtd x N a (cid:88) I σ I [ δρ I + 3 ( ρ I + P I ) ζ ] . (8) For vector transverse modes, we can define T iI j ≡ p I δ ij + p I δ ik a π Ikj , and π Iij ≡ ( ∂ i π I,Tj + ∂ j π I,Ti ) . Thenwe can introduce the 1+3 decompositions for the 4-velocity of the fluid u V,IIi = δu IIi , the shift N i = a N G i ,and the 3D metric δγ ij = a ( ∂ i C j + ∂ j C i ) . We canalso introduce the following gauge invariant variables V i = G i − aN ddt (cid:0) C i a (cid:1) , and F Ii = C i a − b I i (cid:126)b I · (cid:126)b I δB I − b I i (cid:126)b I · (cid:126)b I δB I ,where b I i b I i = 0 = b I i k i = b I i k i . Then, on following asimilar approach one finds the total Lagrangian densityfor the vector perturbations, in VCDM, can be writtenas L = N a δ ij (cid:40)(cid:88) I n I ρ I,n ˙ F Ii N δu Ij + 1 a (cid:88) I n I ρ I,n (cid:20) aδu Ii V j − δu Ii δu Ij (cid:21) − M a V i ( δ lm ∂ l ∂ m V j ) − a (cid:88) I p I π I,Ti ( δ lm ∂ l ∂ m F Ij ) (cid:41) , (9)which reduces to the same result as in GR.Finally, for the tensor modes, let us define δγ ij = a ( h + ε + ij + h × ε × ij ) , where ε + , × ij = ε + , × ji , δ ij ε + , × ij =0 , ε + ij ε × mn δ im δ jn = 0 , and ε + ij ε + mn δ im δ jn = 1 = ε × ij ε × mn δ im δ jn . As for the energy-momentum tensor wehave instead for the perturbations δT iI j ≡ p I δ ik a π I,T Tkj ,so that the total Lagrangian density in MMG becomes L = M a N ( ˙ h + ˙ h × ) − N a M ∂ i h + ) δ ij ( ∂ j h + ) + ( ∂ i h × ) δ ij ( ∂ j h × )] + N a (cid:88) I p I ( h + π I + + h × π I × ) , (10)which reduces to the same form of GR.All the equations of motion (including the ones for thematter fields) for the perturbations are, in form, exactlythe same as for Λ CDM (the only difference being theexplicit dependence of H on the redshift), except thefollowing one, which is written in terms of the Newtonian-gauge invariant fields φ and ψ : ˙ φ + aHψ = 3 [ k − a ( ˙ H/a )] k [2 k /a + 9 (cid:80) j ( (cid:37) j + p j )] (cid:88) i ( (cid:37) i + p i ) θ i , (11)which is used to find the evolution of the curvature per-turbation φ , and where a dot represents a derivative withrespect the conformal time.The parameter estimation is made via Markovchain Monte Carlo (MCMC) sampling by using MontePython [17, 18] against the above mentioned data sets. The analysis of the MCMC chains is performed using achain analyzer package, GetDist [19].We have considered the prior for the parameters ofVCDM such that Λ CDM is well inside the region. Inparticular, we give 0.6 < β H < − < A < A = − as it has large degeneracy. Devitationsof β H from 1 imply that the cosmological data sets preferthe VCDM model over Λ CDM.After doing the chain analysis we found a remarkableimprovement in the fitness parameter with respect tothat of Λ CDM, ∆ χ = H from VCDM after the parameter estima-tion is in full agreement with the low redshift measure-ment, H = +2 . − . . Hence, the H tension is unravelledwithin this model.In order to have a better picture of the χ for cosmo-logical data sets, in Table I we compare effective χ of Experiments VCDM Λ CDM
Planck_highl_TTTEEE .
36 2351 . Planck_lowl_EE .
01 396 . Planck_lowl_TT .
00 22 . JLA .
00 683 . bao_boss_dr12 .
81 3 . bao_smallz_2014 .
48 2 . hst . × − . Total .
67 3473 . Table I. Comparison of effective χ between VCDM and Λ CDM for individual data sets. H m H n s r e i o b A A reio n s
70 72 74 76 78 H m Figure 1. 2-dimensional marginalised likelihoods for theVCDM model fitting against the cosmological data sets. each experiment between VCDM and Λ CDM.Fig. 1 shows 2-dimensional marginalised likelihoods forthe cosmological parameters of interest in VCDM model.The Table II gives the values of the parameters within σ ’s.From Table II, it is interesting to notice that the valueof β H does not reach 1 even at 2 σ . It means that thedata prefer VCDM over Λ CDM. Moreover, the value of H estimated is in agreement with the low redshift mea-surement, and the tension is resolved. On top of that,it is to be noticed that even high redshift data (PlanckLegacy 2018 data) tend to prefer the VCDM model (SeeTable I).Let us look at the evolution of the background and per-turbation variables to see the behaviour of the minimumof VCDM. Fig. 2 shows the behaviour of the function V ( φ ) with respect to the auxiliary field φ for the bestfitVCDM model. The behavior of H ( z ) in VCDM, with a Parameters limits β H . +0 . − . A . +0 . − . ω b . +0 . − . τ reio . +0 . − . n s . +0 . − . H . +2 . − . Ω m . +0 . − . σ . +0 . − . Table II. One-dimensional 2 σ constraints for the cosmologicalparameters of interest after the estimation with the cosmo-logical data sets considered. V / H - φ /H Figure 2. Behavior of the function V ( φ ) with respect to theauxiliary field φ in VCDM model. For high values of φ (i.e.for high redshifts) the theory reduces to Λ CDM as V ( φ ) ap-proaches a straight line. very small transition in the low redshift region which isvisible in Fig.3, between the redshift 10 − and 10 − . Itis clear from the choice of H ( z ) that this is a low-redshiftresolution for Hubble tension.Let us now look at the behaviour of the perturbationvariables. First of all, Fig. 4 compares the CMB temper-ature correlation given the bestfit values for both VCDMand Λ CDM, with planck error bars. The low-redshift − − z × − × − H Figure 3. Zoomed version of H vs z plot. Here we can seethe transition in the H ( z ) at very low redshift between 10 − and 10 − . VCDM transition does not affect much the perturbationevolution. For example, the evolution of the energy den-sity contrast δ γ of photons at low redshifts, in Fig. 5. Wecan see that the transition is smooth even at the order of O ( − ) . ℓ [ ℓ ( ℓ + ) / π ] C TT ℓ [ µ K ] VCDM Λ CDMPlanck data with error bar
Figure 4. CMB TT correlation with bestfit for VCDM and Λ CDM including planck error bars.
Let us understand why the perturbation variables arealways finite as long as the transition of H is finite. Wehave two perturbation equations of motion to be solved.Among the two, ˙ φ has to be integrated, which dependon ˙ H . Consider integration of this equation between t ∗ − ∆ t/ and t ∗ + ∆ t/ , where t ∗ is the time at whichtransition happens and ∆ t is the time width of the tran-sition, ∆ φ = ˆ t ∗ +∆ t/ t ∗ − ∆ t/ ˙ φ dt = ˆ t ∗ +∆ t/ t ∗ − ∆ t/ A (cid:124)(cid:123)(cid:122)(cid:125) finite + B (cid:124)(cid:123)(cid:122)(cid:125) finite ˙ H dt = B ∆ H (cid:124) (cid:123)(cid:122) (cid:125) at t=t ∗ + O (∆ t ) . (12)Since A and B are finite and as far as ∆ H is finite, thenthe perturbation φ should also be finite even in the limit ∆ t → . Hence all the perturbation variables affected -3.0e-05-2.9e-05-2.8e-05-2.7e-05-2.6e-05-2.5e-0510 -3 -2 δ γ z Figure 5. Zoomed version of δ γ , in the redshift range, −
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