Cosmological perturbations in the interacting dark sector: Observational constraints and predictions
CCosmological perturbations in the interacting dark sector:Observational constraints and predictions
Joseph P. Johnson, ∗ Archana Sangwan, † and S. Shankaranarayanan ‡ Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India
Abstract
We consider an interacting field theory model that describes the interaction between dark en-ergy - dark matter interaction. Only for a specific interaction term, this interacting field theorydescription has an equivalent interacting fluid description. For inverse power law potentials andlinear interaction function, we show that the interacting dark sector model is consistent with fourcosmological data sets — Hubble parameter measurements (Hz), Baryonic Acoustic Oscillationdata (BAO), Supernova Type Ia data (SN), and High redshift HII galaxy measurements (HIIG).More specifically, these data sets prefer a negative value of interaction strength in the dark sec-tor and lead to the best-fit value of Hubble constant H = 69 . . . km s − Mpc − . Thus, theinteracting field theory model alleviates the Hubble tension between Planck and these four cos-mological probes. Having established that this interacting field theory model is consistent withcosmological observations, we obtain quantifying tools to distinguish between the interacting andnon-interacting dark sector scenarios. We focus on the variation of the scalar metric perturbedquantities as a function of redshift related to structure formation, weak gravitational lensing, andthe integrated Sachs-Wolfe effect. We show that the difference in the evolution becomes significantfor z <
20, for all length scales, and the difference peaks at smaller redshift values z <
5. We thendiscuss the implications of our results for the upcoming missions. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ a s t r o - ph . C O ] F e b . INTRODUCTION Cosmological observations suggest that the energy budget of the Universe is dominatedby dark energy and dark matter [1–6]. ΛCDM model provides the simplest descriptionof the Universe dominated by dark energy and dark matter while being highly successfulin describing various cosmological observations and phenomena like the cosmic microwavebackground (CMB) and nucleosynthesis [7]. But with the availability of high precisioncosmological observational data, there have been some inconsistencies in the values of cos-mological parameters estimated using the ΛCDM model, with the most prominent of thembeing the difference in the value of the Hubble’s constant estimated from the local dis-tance measurements and CMB observations [8–10]. These inconsistencies point towards thelimitations of the ΛCDM model and the need for modifications to the standard model ofcosmology.Apart from the gravitational interaction, we know very little about the properties ofdark matter and dark energy. ΛCDM model assumes that dark energy is constant in time.The quintessence model provides a more general time-varying dark energy represented bya scalar field [11, 12]. A quintessence dark energy model can be further generalized byintroducing a non-gravitational interaction between dark energy and dark matter, which isnot ruled out by cosmological observations [13–38]. Recently, it has been shown that thedark matter-dark energy interaction can reconcile the tensions in the estimated values ofHubble constant H [39–45]. Hence it is important to develop the analytical and numericaltools to detect the interaction between dark energy and dark matter. For this purpose, weneed a theoretical framework that provides a comprehensive description of the interactingdark sector.In Ref.[46], two of the current authors, have explicitly constructed such a frameworkstarting from a classical field theory action which describes interacting dark sector. Theauthors showed that: (i) A one-to-one mapping between the field theory description and thefluid description of the interacting dark sector exists for a unique interaction term. (ii) Thisclass of interacting dark sector models has an attractor solution describing the acceleratedexpansion of the Universe. The establishment of such a mapping enables us to analyzethe background and perturbed evolution of the Universe with dark energy - dark matterinteraction. 2o constrain the model parameters, especially the interaction strength, and to maketestable predictions, one needs to specify the scalar field potential and the interaction func-tion. In this work, we look at an inverse power law potential [47] U ( φ ) ∼ /φ n where( n = 1 ,
2) and a linear interaction function α ( φ ) ∼ Cφ where C ∈ [ − , C ) are allowed,observations show a preference for negative interaction strength ( C < H =69 . . . km s − Mpc − . Given that the value of Hubble constant reported by Planck is H = 67 . ± . − Mpc − which uses base ΛCDM cosmology [6] and the distanceladder estimates of Hubble constant is H = 73 . ± .
66 km s − Mpc − ( from SH0ESdata [69, 70]), and H = 74 . ± .
42 km s − Mpc − ( measurements of LMC Cepheids[71]), the interacting dark sector model considered in this work alleviate the H tension.3. The constraints on Ω m obtained by the model are consistent with Ω m = 0 . ± . δ m ), the Bardeen potential, and its derivative (Φ and Φ (cid:48) respectively) for theinverse power law potential U ( φ ) ∼ /φ n where ( n = 1 ,
2) and linear interaction functionwith negative interaction strength (
C < (cid:46) z <
0. We see a significant difference in the evolution of the relevantperturbed quantities in the interacting and non-interacting scenarios, at all length scales,for z <
20 and the maximum difference in the evolution is around z ∼
5. We thus explicitly3how that it is possible to detect and constrain the interaction between dark energy anddark matter from cosmological observations.In Sec. II, we introduce the interacting dark sector model we have used for the analysis. InSec. III we discuss the background evolution in the model and the numerical analysis usingvarious observational data sets to obtain the parameter constraints. The evolution of thecosmological perturbations and their observational consequences are discussed in Sec. IV. InSec. V, we briefly discuss the results and discuss the implications of our analysis. AppendicesA - C contain additional details.In this work, we use the natural units where m = G , and the metric signature (-,+,+,+). Greek letters denote the four-dimensional space-time coordinates, and Latin lettersdenote the three-dimensional spatial coordinates. Unless otherwise specified, dot representsderivative with respect to cosmic time and prime denotes derivative with respect to numberof e-foldings N ≡ ln a ( t ). II. INTERACTING DARK SECTOR: THE MODEL
In this work we consider the model described by the action [46], S = (cid:90) d x √− g (cid:18) πG R − g µν ∇ µ φ ∇ ν φ − U ( φ ) − e α ( φ ) g µν ∇ µ χ ∇ ν χ − e α ( φ ) V ( χ ) (cid:19) . (1)where φ corresponds to the dark energy and χ corresponds to the dark matter. The darkmatter fluid in a homogeneous and isotropic Universe can be described by defining the fourvelocity u µ u µ = − (cid:2) − g αβ ∇ α χ ∇ β χ (cid:3) − ∇ µ χ . (2)The energy density ( ρ m ) and pressure ( p m ) of the dark matter fluid are: p m = − e α (cid:2) g µν ∇ µ χ ∇ ν χ + e α V ( χ ) (cid:3) , ρ m = − e α (cid:2) g µν ∇ µ χ ∇ ν χ − e α V ( χ ) (cid:3) . (3)In this description, we can rewrite Einstein’s equation in terms of dark energy scalar fieldand dark matter fluid: G µν = 16 πG (cid:20) ∇ µ φ ∇ ν φ − g µν ∇ σ φ ∇ σ φ − g µν V ( φ ) + p m g µν + ( ρ m + p m ) u µ u ν (cid:21) , (4)where the energy-momentum tensor for the dark matter fluid is given by T ( m ) µν = p m g µν + ( ρ m + p m ) u µ u ν . (5)4he interaction between the dark energy and the dark matter fluid is described by: ∇ µ T ( m ) µν = Q (F) ν , (6)where the interaction term is given by Q (F) ν = − e α ( φ ) α ,φ ( φ ) ∇ ν φ (cid:2) ∇ σ χ ∇ σ χ + 4 e α ( φ ) V ( χ ) (cid:3) = − α ,φ ( φ ) ∇ ν φ ( ρ m − p m ) . (7)Identifying T ( m ) = T ( m ) µµ = − ( ρ m − p m ), we get Q (F) ν = T ( m ) ∇ ν α ( φ ) . (8)The time component of Q (F) ν represents the energy transfer between dark energy and darkmatter. For easy reading, we denote Q (F)0 as Q . Q will further be split into the backgroundand perturbed parts given by Q = Q + δQ .To study the cosmological evolution and obtain predictions and constraints, we need toconsider a specific form of scalar field potential U ( φ ) and the interaction function α ( φ ).In this work, we focus on the quintessence dark energy model with an inverse power lawpotential [47] and a linear interaction function U ( φ ) ∼ φ n , α ( φ ) ∼ φ , (9)where n = 1 and n = 2. III. BACKGROUND EVOLUTION AND OBSERVATIONAL CONSTRAINTS
We consider a spatially flat universe governed by Friedmann equations (cid:18) ˙ aa (cid:19) = 8 πG ρ tot , ¨ aa = − πG ρ tot + 3 P tot )where ρ tot and P tot denote the total energy density and pressure of the universe at a giventime. At late times, the contribution of the relativistic matter density ( ρ r ) is negligible ascompared to the dark (non-relativistic) matter ( ρ m ) and dark energy density ( ρ φ ). Hence,in this analysis, we neglect ρ r and total density is ρ tot = ρ m + ρ φ .The dynamics of the scalar field is governed by( ¨ φ + 3 H ˙ φ + U ,φ ) ˙ φ = Q, Q is the background interaction term. Here, φ is in the units of m Pl = G − / . Thescalar field potential is assumed to be U ( ˜ φ ) = κ m ˜ φ − n . (10)where κ is of the order of unity. To make the analysis simpler, we rescale the scalar field φ to ˜ φ = √ πG φ . Note that ˜ φ is dimensionless.The evolution of non-relativistic matter density is given by˙ ρ m + 3 Hρ m = − Q, where we have considered a pressureless dark matter fluid, p m = 0. For the interaction term, Q = − α ,φ ˙ φρ m , the above equation gives, ρ m = ρ m e − α ,φ φ a − .In terms of dimensionless scalar field variable ( ˜ φ ), the Friedmann equations and the fieldequation are: (cid:18) ˙ aa (cid:19) = H Ω m a − e − C ˜ φ + ˙˜ φ
12 + κm P l
12 ˜ φ − n (11)¨˜ φ + 3 H ˙˜ φ + U , ˜ φ ( ˜ φ ) = − H π C Ω m a − e − C ˜ φ , (12)where we have assumed α ( ˜ φ ) to be a linear function of ˜ φ , i. e. α ( ˜ φ ) = C ˜ φ , giving α , ˜ φ = C .The parameter C is dimensionless and defines the strength of interaction between darkenergy and dark matter. In our analysis, we obtain the constraint on C by keeping it as afree parameter with C ∈ [ − , A. Observational data
To constrain the model parameters in the interacting dark sector model, we analyze fourdifferent observational data sets. More specifically, we use Hubble parameter measurements(Hz) [48, 49, 51, 54, 81, 82], high redshift HII Galaxy (HIIG) data [56–61], Baryon acousticoscillation (BAO) data [62–67] and the joint lightcurve analysis (JLA) sample of Type Iasupernovae (SN) observations [68, 83–86].
Hubble Parameter Measurement (H(z)) data:
The Hubble parameter measruements(abbreviated as Hz) at different redshifts is an effective tool to constrain the cosmologicalparameters [48, 51]. Hz observations are useful in constraining the cosmological parameters6s it uses the model parameters directly without having an integral term that might obscureor cover valuable information. Broadly, different techniques employed to measure the Hubbleparameters can be classified into two methods: a) Differential age method [81] and b) RadialBAO method [82]. The Hubble rate as a function of redshift is evaluated by using H ( z ) = − z ) dzdt , (13)where t denotes the age of the universe when the observable photon is emitted. In thedifferential method, we can obtain a direct estimate of expansion rate by taking the derivativeof redshift with respect to time. Hubble parameter obtained through this method does notdepend on the cosmological model but on the age-redshift relation of cosmic chronometers.So very carefully the selection of passively evolving early galaxies as cosmic chronometers ismade depending upon a galaxy’s star formation history and its metallicity.In this work, we consider the Hz data points obtained through the cosmic chronometrictechnique and use the data points compiled in Ref. [48]. In this compilation, the authorsdropped older Hubble parameter estimates from SDSS galaxy clustering [87] and Ly- α forestmeasurement [88] and added new data sets. Out of the 38 data points reported in Ref. [48],in this analysis, we only use 31 independent measurements of the Hubble parameter (H(z)).More specifically, we use 9 data points from Ref. [49], 2 points from Ref. [50], 8 points fromRef. [51], 5 points from Ref. [52], 2 points from Ref. [53], 4 points from Ref. [54], and onepoint from Ref. [55]. Note that the three points reported in Ref. [89] and another threepoints in Ref. [62] are also used in the BAO observations, hence removed from this data set. BAO:
Baryon Acoustic Oscillations (BAO) are fluctuations in the correlation function oflarge scale structure that appears as overdense regions in the distribution of the visible,baryonic matter. This is the consequence of acoustic waves set up in the primordial plasmabecause of competing forces of radiation pressure and gravity. These acoustic waves travelwithin the plasma, however are frozen at the time of recombination when the plasma cooleddown enough to make the cosmos neutral. The distance where the waves stalls are imprintedas overdense regions and are used as a standard ruler to measure cosmological distances.The characteristic angular scale of the acoustic peak is given in terms of sound horizonat drag epoch, r s ( z d ), as θ A = r s ( z d ) /D V ( z ), where, D V is the effective distance ratio given7n terms of angular diameter distance D A : D V ( z ) = (cid:20) (1 + z ) D A ( z ) czH ( z ) (cid:21) / , r s ( z d ) = (cid:90) ∞ z d c s ( z (cid:48) ) dz (cid:48) H ( z (cid:48) ) . (14)In order to use the BAO data, the knowledge of the sound horizon scale at the z d (denoted by r s ) is required as the data is given in terms of H ( z ) r s /r s,fid , D M r s,fid /r s , D V r s,fid /r s , where r s,fid is 147.78 Mpc in [62] and [65], and 148.69 Mpc [64] and the comoving angular diameterdistance is given by D M ( z ) = (1 + z ) D A ( z ). The value of fiducial sound horizon, r s,fid whichwas calculated by assuming the ΛCDM model and the best fit values of parameters givenby Planck-2018 [6], is model dependent, but not to a significant degree. The quantities D V r s,fid /r s , D M r s,fid /r s , r s and r s,fid is given in units of M pc while H ( z ) r s /r s,fid is givenin units of km s − Mpc − . We compute r s is computed using the method given in Ref.[90].The BAO data in terms of Acoustic parameter A ( z ) is defined as [91]: A ( z ) = (cid:104) D v ( z ) (cid:112) (Ω NR h ) cz (cid:105) / . (15)Thus, the BAO data consists of A ( z ) and D v ( z ) (with associated errors) at different red-shifts. The measurement of these distances is a useful tool to constrain cosmological modelparameters. The BAO data we use in the analysis lie in the redshift span of 0 . − . HIIG:
The third data set we use is the high redshift HII galaxy (HIIG) observations [56–61].These observations are new independent cosmological observations that use the correlationbetween the Balmer emission line velocity dispersion ( σ ), and luminosity ( L ) in HIIG toobtain the distance estimator. This L - σ correlation is given by:log( L ) = β log( σ ) + γ , (16)where, γ and β are the intercept and slope, respectively and log = log . The tight correlationbetween the Balmer line luminosity ( L ) and velocity dispersion ( σ ) of the emission lines canbe used to constrain the cosmological model parameters.An extinction correction must be made to the observed fluxes to obtain the values ofthese parameters. We follow the method used in Ref. [56] and assume the extinction lawgiven in Ref. [92]. The resulting value of the intercept and slope are: β = 5 . ± . , (17) γ = 33 . ± . , (18)8espectively. In our analysis, we use these values of β and γ . Using these values in Eq.(16),we obtain the luminosity of a HII Galaxy. We then use the Luminosity to obtain the distancemodulus for that HII Galaxy: µ obs = 2 . L − . f − . , (19)where f denotes the measured flux of the HIIG, reported in the HIIG observational dataalong with error associated with it. We can predict the distance modulus for a given cos-mological model by using the theoretical definition: µ th ( z ) = 5 log D L ( z ) + 25 , (20)where the luminosity distance D L ( z ) (in the units of Mpc) is related to the angular sizedistance D A ( z ) via distance duality relation and the transverse comoving distance D M ( z )through D L ( z ) = (1 + z ) D A ( z ) = (1 + z ) D M ( z ). The HIIG data we use comprises 153measurements that span the redshift range of 0.0088 to 2.429, which covers a larger redshiftrange than the BAO data used in this analysis. SN data (JLA):
Type Ia supernovae, which are standardizable candles, is another usefultool to determine the expansion history of the Universe [68, 83–86]. The observable reportedin the sample is the distance modulus, which is extracted from light curves by assuming thatthe intrinsic luminosity on average is the same for Type Ia supernovae with the identicalcolor, shape, and environment, irrespective of the redshift measurement. The standardizeddistance modulus, obtained by using the following linear empirical relation: µ obs = m ∗ B + αx − βC − M B . (21)Here, m ∗ B is the peak magnitude observed in the B-band rest frame, α and β are nuisanceparameters, C is the color of supernovae at peak brightness, and x is ‘stretch’ of thelight curve. The values of the parameters ( m B , x , C ) are obtained by fitting supernovaespectral sequence to the photometric data. The parameter M B , which is the absolute B-bandmagnitude, depends on the host stellar mass. Theoretical value of the distance modulus µ th is given by Eq. (20), which depends on the cosmological model.By measuring the apparent brightness and comparing it to other candles, one can estimatethe distance the photons have traveled, and hence the rate of expansion of the Universe.Our analysis uses the full joint lightcurve analysis (JLA) sample comprising 740 type Ia9upernovae spanning a redshift range of z=0.01 to z= 1.4. We use the abbreviation ‘SN’ todenote these 740 sample points. B. Data analysis technique
For our analysis, we use the χ minimization technique. Any measurement data containsan observable quantity X o ( z i ) and its corresponding redshift z i , along with the error asso-ciated with each point σ i . Here, ‘ i ’ takes the values upto N (number of data points in eachobservation). We can also estimate these observable quantities theoretically [ X th ( z i )] for themodels considered in the analysis.For H ( z ) data, the observable is the expansion rate and we consider 31 points obtainedusing cosmic chronometer and the χ is defined as: χ ( p ) = (cid:88) i =1 [ H th ( p , z i ) − H o ( z i )] σ i , (22)where, σ i is the uncertainty of H o ( z i ). All these 31 points are independent of each otherand expansion rate depend on the specific model chosen represented by ‘ p ’ in the aboveexpression.The BAO data is a correlated data, so to calculate the χ , we need to use the covariancematrix in the definition, i. e., χ ( p ) = [ X th ( p ) − X o ( z i )] T C − [ X th ( p ) − X o ( z i )] , (23)where superscripts T and − C from Ref. [62].For HIIG data consisting of 153 measurements, the χ is given by χ ( p ) = (cid:88) i =1 [ µ th ( p , z i ) − µ o ( z i )] σ (cid:48) i , (24)where σ (cid:48) i is the uncertainty of the i th measurement (not to be confused with the velocitydispersion ( σ ) term in HIIG measurements) and is given by σ (cid:48) = (cid:113) σ (cid:48) + σ (cid:48) . (25) σ (cid:48) stat is the statistical uncertainties and is given by: σ (cid:48) stat = 6 . (cid:2) σ (cid:48) f + β σ (cid:48) σ + σ (cid:48) β (log σ ) + σ (cid:48) γ (cid:3) + (cid:18) ∂µ th ∂z (cid:19) σ (cid:48) z . (26)10ue to the distance modulus term in the expression, the systematic uncertainty calculatedthis way is model-dependent. However, when it comes to constraining the cosmologicalparameters, the model dependence is negligible [93]. In this analysis, we do not accountfor systematic uncertainties similar to the earlier work [56, 93] and take the statisticaluncertainty as total uncertainty, i. e., σ = σ stat .For the SN data with 740 joint light curves sample, the χ function is: χ ( p ) = (cid:88) i =1 [ µ th ( p , z i ) − µ o ( z i )] σ i , . (27)with σ i giving the total uncertainty of the i th measurement.For the joint analysis (Hz+BAO+HIIG+SN), we obtain the joint likelihood ( e − χ ) bymultiplying individual likelihoods such that χ = χ H + χ BAO + χ HIIG + χ SN . Here, themaximum likelihood corresponds to the minimum value of χ . C. Parameter constraints
Having discussed the data sets and the technique, we can now obtain parameter con-straints for the interacting dark sector model discussed at the starting of this section. Morespecifically, we use the χ technique (described in Sec. III B) to obtain the 1 σ , 2 σ , and 3 σ confidence regions corresponding to the four data sets for various cosmological parametersused in our dark energy- dark matter interaction model. For a given value of n in the scalar-field potential (10), we obtain the constraints on the standard model parameters H , Ω m , w and the parameter C , which describes the interaction strength in the dark sector.For the parameter fitting, we use priors that are consistent with the different constraintsobtained from various observations. For the Hubble constant, we take the range to be H = 60 −
80 km s − Mpc − . The present-day value of the dark-energy equation of stateparameter is set to be between − ≤ w ≤
1. The non-relativistic matter density is takento be in the range 0 . ≤ Ω m ≤ .
6, and the interaction strength between dark matter-darkenergy is taken to between − ≤ C ≤
1. These priors are listed in Table I.11 arameter Lower Limit Upper Limit H m w -1.0 1.0 C -1.0 1.0 TABLE I: Priors used in the analysis of parameter fitting.
Figure 1 contains the constraints on parameters H , interaction strength C , and Ω m forthe four observational data sets. The plots are for n = 1 (10). Analysis is also done for n = 2 and n = 3; however, there is no significant change in the parameter constraints. Forcompleteness, in Appendix A, we have presented the results for n = 2. The 1 σ , 2 σ , 3 σ contours corresponding to 67%, 95% and 99% confidence regions respectively, are shownin two-dimensional planes in Figure 1. The first, second, and third columns correspond to‘ H − Ω m ’, ‘ H − C ’ and ‘Ω m − C ’ planes, respectively. To show these two-dimensionalconfidence regions, we have marginalized over the other parameters. The two-dimensionalconfidence regions for standard parameters w and Ω m are shown in Figure 2. Table IIcontains the best fit values of the parameters, and Table III contains the allowed range ofparameters. In the first row, we show constraints from Hz measurements. In the second row,results from BAO+Hz observations are shown, and the third row represents the confidencecontours from HIIG data, while results in the fourth row are from SN+Hz observations. Thefifth row shows the constraints obtained from the combination of all the data sets mentionedin section III A. Observations H Ω m C w χ min Hz BAO+Hz
HIIG
SN+Hz
All combined
TABLE II: The best fit values of the parameters obtained for the dark-energy dark-matter interaction model. IG. 1: 1,2,3- σ likelihood contours for Hz data (I row), BAO+Hz data (II row), HIIG data (III row),SN+Hz data (IV row) and all four data sets (V row). The two-dimensional contours are obtained byperforming marginalization over other parameters. IG. 2: 1,2,3- σ likelihood contours in ‘ w -Ω m ’ plane. The top row shows constraints from Hz data (left)and BAO+Hz observations (right). The second row shows constraints from HIIG measurements(left) and SN+Hz observations (right). The key inferences from the Hz data are as follows: First, the minimum value of χ is 18.81which corresponds to the best fit values of the parameters H = 69 .
34 km s − Mpc − , Ω m =0 . , w = − .
98 and the interaction strength C = 0 .
98. Second, within the 2 σ region, the Hzdata allows H to take values between 61 . − .
12 km s − Mpc − which includes the valuesreported by Planck [6] and the local measurements [70, 94]. Hence, with Hz observations,the interacting dark sector model is consistent with both of these reported values. Third,the best fit value, as well as the allowed range of non-relativistic density parameter, is also14 ata set σ Confidence σ confidence σ confidenceHz ≤ H ≤ ≤ H ≤ ≤ H ≤ ≤ Ω m ≤ ≤ Ω m ≤ ≤ Ω m ≤ ≤ w ≤ -0.67 -1 ≤ w ≤ -0.24 -1 ≤ w ≤ ≤ C ≤ ≤ C ≤ ≤ C ≤ BAO+Hz ≤ H ≤ ≤ H ≤ ≤ H ≤ ≤ Ω m ≤ ≤ Ω m ≤ ≤ Ω m ≤ ≤ w ≤ -0.988 -0.999 ≤ w ≤ -0.983 -1 ≤ w ≤ -0.98-0.87 ≤ C ≤ -0.29 -1 ≤ C ≤ -0.12 -1 ≤ C ≤ HIIG ≤ H ≤ ≤ H ≤ ≤ H ≤ ≤ Ω m ≤ ≤ Ω m ≤ ≤ Ω m ≤ ≤ w ≤ -0.92 -1 ≤ w ≤ -0.9 -1 ≤ w ≤ -0.89-1 ≤ C ≤ ≤ C ≤ ≤ C ≤ SN+Hz ≤ H ≤ ≤ H ≤ ≤ H ≤ ≤ Ω m ≤ ≤ Ω m ≤ ≤ Ω m ≤ ≤ w ≤ -0.97 -1 ≤ w ≤ -0.93 -1 ≤ w ≤ -0.9-1 ≤ C ≤ -0.51 -1 ≤ C ≤ ≤ C ≤ Hz+BAO+HIIG+SN ≤ H ≤ ≤ H ≤ ≤ H ≤ ≤ Ω m ≤ ≤ Ω m ≤ ≤ Ω m ≤ ≤ w ≤ -0.988 -0.998 ≤ w ≤ -0.985 -0.999 ≤ w ≤ -0.983-0.68 ≤ C ≤ -0.26 -1 ≤ C ≤ -0.96 -1 ≤ C ≤ TABLE III: Confidence limits from various data sets for interacting dark sector cosmology. consistent with the constraints reported in the previous studies [6, 70, 94]. Fourth, aftermarginalizing over parameter w , the data allows the entire range of the coupling parameter( C ) considered in the analysis within the 1 σ region see Figure 1. However, we also find thatif we fix w at a particular value, say w = −
1, it does not constrain C at all, but if we moveaway from ΛCDM like scenarios at present, and consider w ≥ − C as well. As the value of w moves away from − C becomes tighter (cf. Figure 3). Fifth, from Figure 2, we see that the Hz data does not15rovide a lower limit on w ; however, an upper limit of -0.67 within 1 σ and w =0.04 within3 σ region is allowed showing that this particular model does not allow for a non-acceleratinguniverse within 1 σ region. Also, Hz is the only observation that allows for a non-acceleratinguniverse within the 3 σ region. The allowed range for w is the widest compared to the otherthree observations considered in the analysis. The Hz measurements constrain Ω m to takevalues within a range of 0 . − .
43 for 3 σ confidence level, which is very wide compared tothe ones obtained from BAO+Hz and SN+Hz data sets.The key inferences from BAO+Hz data are as follows: First, the minimum value of χ is 22.43 which corresponds to the best fit values of parameters giving H = 70 .
67 kms − Mpc − , Ω m = 0.3, w =-0.992 and the interaction strength is C =-0.63. Second, within1 σ region, BAO+Hz data allows H to take a very small range given by 69 . − . − Mpc − which lies between the value of H reported by Planck [6] and the local probes[70, 94]. Therefore, the interacting dark sector model can alleviate the H tension. Third,the best fit value of the non-relativistic matter density parameter is Ω m = 0 .
3. The allowedrange within the 3 σ region is very narrow and consistent with the constraints reportedin the previous studies [6, 70, 94]. Fourth, within 1 σ , BAO+Hz data also constrains theinteraction strength C within the range of -0.87 to -0.29 (cf. Figure 1) and between -1 to0.01 corresponding to 99% confidence region. Thus, BAO+Hz data prefers negative valuesof C . Here again, we find that if we fix w at a particular value, say w = −
1, the allowedrange is narrower than when w was a free parameter. And if we move away from ΛCDMlike scenarios at present, and consider w ≥ − C starts gettinglower as the contours start shifting to the negative regions on C . As we change w from − C becomes tighter as in Hz data, and we find that theBAO+Hz data prefers negative values of C .Fifth, from Figure 2, we see that the BAO+Hz data provide very small range on w for 1 σ , 2 σ region and within 3 σ region ΛCDM case is allowed. Therefore, the BAO+Hzobservational data do not allow for a non-accelerating universe and prefer a ΛCDM likescenario. It also provides the tightest constraints for the model parameters out of all theobservations considered.The key inferences from HIIG data are as follows: First, the minimum value of χ is410.12 which corresponds to the best fit parameters H = 71 . − Mpc − , Ω m = 0 . w = − .
98 and the interaction strength is C = − .
52. Second, HIIG data allows H to16ake values in the range 68 . − .
98 km s − Mpc − within 1 σ region. The best fit valuefor the model indicates the preference for the value of H reported by local measurements[70, 94]. However, the interacting dark sector model is also consistent with the H valuereported in Ref. [6] within 3 σ region. Third, the best fit value, as well as the allowed rangeof non-relativistic density parameter, is also consistent with the constraints reported in theprevious studies [6, 70, 94]. Fourth, similar to Hz data, HIIG data allows the entire rangeof coupling parameter ( C ) within 1 σ region, see Figure 1. Here again, we have marginalizedover parameter w . We also found that if we fix w at a particular value and consider w ≥ −
1, then we start getting a limit on C as well. As the value of w moves away from − C becomes tighter.Fifth, from Figure 2, we see that the HIIG data does not provide a lower limit on w . Still,an upper limit of − . σ region is allowed, showing that similar to BAO+Hz data,this particular model does not allow for a non-accelerating universe within 3 σ region. Theresults are consistent with the ΛCDM model. The constraints on Ω m by these observationsgive the widest range amongst all data sets considered in the analysis.The key inferences from SN+Hz data are as follows: First, the minimum value of χ is737.21 which corresponds to the best fit values of the parameters H = 69 .
51 km s − Mpc − ,Ω m = 0.31, w =-1.0 and C =-1. Second, the SN+Hz data allows H to take values between ∼ . − .
02 km s − Mpc − within 1 σ region, which lies between the values reported byPlanck-2018 [6] and the local H measurements [70, 94]. Interestingly, it provides a verynarrow range for H and, hence, the interacting dark sector model helps alleviate the H tension. Third, the best fit value, as well as the allowed range of non-relativistic densityparameter, is also consistent with the constraints reported in previous studies [6]. Fourth,like Hz data, SN+Hz data also allows the entire range of the interaction strength ( C ) withinthe 3 σ region. However, within 1 σ region, it constrains C to be less than 0.5, (cf. Figure 1).Here again, we have marginalized over parameter w . We find that, like other data sets,SN+Hz also prefers negative values of interaction strength.Fifth, from Figure 2, we see that the SN+Hz data does not provide a lower limit on w . However, within the 1 σ , there is an upper limit of − .
97, and w = − . σ region. Thus, the analysis shows that a non-accelerating universe is not allowed. Theallowed values of Ω m are very narrow and consistent with previous studies. This model isalso consistent with the ΛCDM model. 17he key inferences from the combined data are as follows: First, the minimum valueof χ is 1152.3, which corresponds to the best fit values of the parameters are H = 69 . − Mpc − , Ω m = 0.29, w =-0.99 and the interaction strength is C =-0.47. Second, theHz+BAO+HIIG+SN data allows H to take values between ∼ . − .
04 km s − Mpc − within 1 σ region, which lies between the values reported by Planck-2018 [6] and the localprobes [70, 94]. It provides a very narrow range for H within the 3 σ confidence region.Therefore, this dark sector interaction model puts very narrow constraints on model param-eters to alleviate the H tension with the joint analysis. Third, the best fit value, as wellas the allowed range of non-relativistic density parameter, is also consistent with a narrowrange of allowed values with 0 . ≥ Ω m ≥ .
26 within 3 σ region, and these constraints areconsistent with the ones reported in previous studies [6, 70]. Fourth, we get a very narrowrange for the coupling parameter for the joint analysis, C , which restricts it to take valuesonly within -0.68 to -0.26 for 1 σ and from -1 to 0.05 for 3 σ confidence regions, see Figure 1.In the joint analysis, the constraints are driven by the BAO observation, which has the mostconstraining capacity, followed by SN, Hz, and HIIG observations. Similar to the individualcases, if we fix w at a particular value, say w = −
1, the combination data gives slightlynarrower range, but if we move away from ΛCDM like scenarios at present. For w ≥ − C .Fifth, from Figure 2, we see that the combined data does not provide a lower limit on w .However, within 1 σ , we get the upper limit of − .
993 and w = − .
99 within 3 σ region. Thisagain shows that the model does not allow for a non-accelerating universe and constrains w to a value close to -1, and is consistent with the ΛCDM model.In Figure 3, instead of marginalizing w , we assume a value of w within 3 σ allowed rangereported in this work and see the change in the H − C plane. The first row is obtainedfor Hz data, and the value of w considered are -1, -0.6, and -0.1 (left, middle, and rightplots, respectively). In the second, third, fourth, and fifth rows, the results correspond toBAO+Hz, HIIG, SN+Hz, and combined analysis, respectively. For the left, middle andright plots respectively , we fix w at -1, -0.997 and -0.995. For Hz measurements, we see asignificant change in the constraints as w changes from − − .
1, and we start gettingconstraints on C . But for BAO+Hz, HIIG, SN+Hz, and combined case, there is a slightshift in contours in contours when w is varied from w =-1 to -0.995 (within 3 σ range).18 IG. 3: 1,2,3- σ likelihood contours in ’ H − C ’ plane for different values of w . The top row shows constraintsfrom Hz data (I row), BAO+Hz (II row), HIIG data (III row), SN+Hz (IV row) and all four data sets (Vrow). The left, middle and right plots correspond to different values of w . H tension. All the observational data sets consideredconstrain H to be close to 70 km s − Mpc − .2. The constraints on Ω m obtained from various data sets are consistent with each other.3. The constraints on w are consistent with ΛCDM model, and only Hz data allows fora non-accelerating universe.4. All data sets, except Hz, prefer negative value for the interaction strength ( C ).5. We have analyzed for n = 2 and the parameter constraints are roughly the same. (SeeAppendix A.) IV. EVOLUTION OF THE SCALAR PERTURBATIONS AND PREDICTIONSOF THE MODEL
In the previous sections, we have obtained the constraints on the various model param-eters based on the observational data related to the background evolution of the Universe.In this section, we look at the evolution of first-order perturbations for negative value forthe interaction strength ( C ).The perturbed perturbed FRW metric in the Newtonian gauge given by [7]: g = − (1 + 2Φ) , g i = 0 , g ij = a (1 − δ ij , (28)where Φ ≡ Φ( t, x, y, z ) and Ψ ≡ Ψ( t, x, y, z ) are the Bardeen Potentials.We obtain the evolution of three perturbed quantities, which are relevant to three differentcosmological observations:1. Stucture formation: δ m ( t, x, y, z ) ≡ δρ m ( t,x,y,z ) ρ m ( t )
2. Weak lensing : Φ + Ψ3. Integrated Sachs-Wolfe (ISW) effect: Φ (cid:48) + Ψ (cid:48) where δ m is the density perturbation of dark matter fluid. We study the evolution of theseperturbed quantities for various length scales specified by the wavenumber k .20o analyze the difference in the evolution of the scalar perturbations in dark sectorinteractions compared to standard cosmology, we study the following quantities:∆ δ m = δ m i − δ m ni , ∆ δ m rel = δ m i − δ m ni δ m ni = ∆ δ m δ m ni (29a)∆Φ = Φ i − Φ ni , ∆Φ rel = Φ i − Φ ni Φ ni = ∆ΦΦ ni (29b)∆Φ (cid:48) = Φ (cid:48) i − Φ (cid:48) ni , ∆Φ (cid:48) rel = Φ (cid:48) i − Φ (cid:48) ni Φ (cid:48) ni = ∆Φ (cid:48) Φ (cid:48) ni (29c)where the subscripts i and ni denote the interacting and non-interacting scenarios, respec-tively.The perturbed interaction term in the fluid description is given by δQ (F) = − ( δρ m − δp m ) α ,φ ( φ ) ˙ φ − ( ρ m − p m ) (cid:104) α ,φφ ( φ ) ˙ φδφ + α ,φ ( φ ) ˙ δφ (cid:105) (30)We use the evolution equations and dimensionless variables introduced in Ref. [46] to obtainthe evolution of the scalar perturbations. In the dimensionless variables, the evolutionequations are: δφ (cid:48)(cid:48) + (cid:20) (cid:0) y − x − ω Ω m + 1 (cid:1) − √ αβx (cid:18) c s − (cid:19)(cid:21) δφ (cid:48) + (cid:20) − β (cid:18) Ω m γ (cid:18) ω − (cid:19) β − y (cid:18) c s − (cid:19) λ (cid:19) α + 3Γ λ y + k a H (cid:21) δφ + (cid:20) − √ αβ (cid:18) c s − (cid:19) − √ x (cid:21) Φ (cid:48) − √ (cid:20) αβ (cid:18)(cid:18) c s − (cid:19) (cid:18) y + k a H (cid:19) + ( c s − ω ) (cid:19) + λy (cid:21) Φ =0 (31)Φ (cid:48)(cid:48) + 32 (cid:20) y − x − Ω m ω + 2 c s + 53 (cid:21) Φ (cid:48) + 3 (cid:20) c s (cid:18) k a H − x + 1 (cid:19) − Ω m ω + y (cid:21) Φ+ √ x c s − δφ (cid:48) − √ λy c s + 1) δφ =0 (32) δ (cid:48) + 3( ω − c s )( √ αβx − δ + 23 k a H Ω m Φ + (cid:18) − ω − k a H Ω m (cid:19) Φ (cid:48) − √ αβ (3 ω − δφ (cid:48) + √ (cid:20) αβ γ (3 ω − − k a H Ω m (cid:21) xδφ =0 , (33)where, ω and c s denote the equation of state and sound speed of the dark matter fluid,respectively. We solve these equations for the redshift range 0 ≤ z (cid:46) ω = c s = 0 (cf.Appendix C). As mentioned in Sec II, this analysis is done for U ( φ ) ∼ /φ and α ( φ ) ∼ φ .21nalysis is also done for n = 2, however, the results are not sensitive to n . For completeness,in Appendix B, we have presented the results for n = 2.To understand the effect of the interaction between dark energy and dark matter on theperturbed quantities, we define scaled interaction function δq : δq = δQH M P l . (34)Fig. 4 is the plot of δq as a function of number of e-foldings ( N ) for different k values. Since - - - - - - - × - × - × - × - × - × - - - - - - - - × - × - × - × - × - × - FIG. 4: Evolution of δq as a function of N for different values of k . this forms the basis of the rest of the analysis, we would like to stress the following points:First, we see that the interaction function peaks around N ∼ − z ∼ . − . k . Second, since the interaction in the darksector is a local interaction, the effect of the interaction should be least at the largest lengthscales (smallest k ), and this is what we see from the plots. In other words, the interactionstrength introduces a new length scale in the dynamics and leads to a preference for thegrowth of perturbations in certain length scales. We will see this feature for all the threequantities δ m , Φ + Ψ and Φ (cid:48) + Ψ (cid:48) .In the following subsections, we obtain the evolution of the perturbed quantities relevantto the upcoming cosmological observations and determine the constraints to distinguish theinteracting dark sector model from standard cosmology.22 . Structure formation Over the last few decades, the three-dimensional distribution of galaxies is available dueto many surveys. With the redshift measurement of millions of galaxies, there are two keyconclusions: First, if we smoothen the distribution on the largest scales, it approaches a ho-mogeneous distribution consistent with the FRW model. Second, in the smaller scales, thereare overdense regions (clusters) and underdense regions (voids); around 10 Mpc, the RMSdensity-fluctuation amplitude is of the order unity. Since the interaction function, δq in-creases with increasing values of k , we can expect that the cold matter density perturbationsin our model may have a different profile compared to standard cosmology.Hence, first we look at the evolution of the matter density perturbation δ m . More specifi-cally, δ m , ∆ δ m and ∆ δ m rel defined in Eq. (29a). Figures 5 and 6 [7 and 8] contain plots of δ m [∆ δ m , ∆ δ m rel ] as a function of N for different length scales in interacting and non-interactingscenarios. - - - - - - - × - - - - - - - - × - × - × - × - × - × - × - FIG. 5: Evolution of δ m as a function of N . Left: c = − .
6, Right: c = 0. From these plots, we infer the following: First, the difference in the evolution of δ m be-tween the interacting and non-interacting scenarios is significant after N ∼ −
3. Second, thisdifference increases with the increase in the value of the wavenumber k . This means thatthe interaction has a larger effect on the evolution of the scalar perturbations in the smallerlength scales (large values of k ) compared to the larger length scales (smaller values of k ).Third, these deviations become significant for z ∼ −
20 and lie in the epoch of reioniza-tion. During this epoch, a predominantly neutral intergalactic medium was ionized by theemergence of the first luminous sources. Before the reionization epoch, the formation and23 - - - - - - × - - - - - - - - × - × - × - × - × - × - × - FIG. 6: Evolution of δ m as a function of N . Left: c = − .
6, Right: c = 0. evolution of structure were dominated by dark matter alone. However, the interacting darksector leads to the exchange of density perturbations at smaller length scales. This indicatesthat it will be possible to detect the signatures of dark energy - dark matter interaction inthe large scale structure observations. This provides a possible way to detect the signaturesof dark sector interaction in the existing and upcoming cosmological observations like Euclidsatellite [95], GMRT, SKA [96] and LOFAR [97] . - - - - - - - × - - - - - - - - FIG. 7: Evolution of ∆ δ m (left), ∆ δ m /δ m ni (right) as a function of N . B. Weak gravitational lensing
The matter content of the Universe is dominated by dark matter. Most of the cosmologi-cal observations to study the matter distribution in the Universe depend on the observationsof the luminous matter, which gives us little information regarding the total mass distribu-24 - - - - - - × - - - - - - - - FIG. 8: Evolution of ∆ δ m (left), ∆ δ m /δ m ni (right) as a function of N . tion in the Universe. Gravitational lensing provides important information regarding thetotal mass distribution in the Universe, as it is independent of the nature of the matterand its interaction with electromagnetic radiation. Hence, weak gravitational lensing holdsenormous promise as it can reveal the distribution of dark matter independently of any as-sumptions about its nature. The quantity Φ + Ψ determines the geodesic of a photon, whichaffects the weak gravitational lensing [7]. Like the standard cosmology, for the dark-sectorinteracting model considered here, Φ( t, x, y, z ) = Ψ( t, x, y, z ). Hence, it is sufficient to studythe evolution of Φ to distinguish dark sector model with standard cosmology. - - - - - - - - × - - × - - × - × - - - - - - - - × - × - × - × - × - FIG. 9: Evolution of Φ as a function of N . Left: c = − .
6, Right: c = 0. To study the signatures of the interacting dark sector, we look at the evolution of scalarmetric perturbation Φ for different length scales starting from z ∼ rel . Figures9 and 10 contain plots of Φ as a function of N for different length scales in interacting and25 - - - - - - - × - - × - - × - - - - - - - - × - × - × - × - FIG. 10: Evolution of Φ as a function of N . Left: c = − .
6, Right: c = 0. non-interacting scenarios. Evolution of ∆Φ and ∆Φ rel as a function of N are plotted inFigures 11 and 12. - - - - - - - - × - - × - - × - - - - - - - - - - - - FIG. 11: Evolution of ∆Φ (left), ∆Φ / Φ ni (right) as a function of N . - - - - - - - - × - - × - - × - - - - - - - - - - - - FIG. 12: Evolution of ∆Φ (left), ∆Φ / Φ ni (right) as a function of N . z ∼ N ∼ −
3. This effect becomes even more prominent towards the lowerredshifts z <
5. By looking at the k − dependence of the evolution, this effect is enhancedat lower length scales. This means that the interaction has a larger effect on the evolutionof the scalar perturbations in the smaller length scales (large values of k ) compared to thelarger length scales (smaller values of k ). Thus, this indicates that observations of weaklensing can help us potentially distinguish between interacting and non-interacting scenar-ios and potentially provide a way to resolve the tension between Planck-2018 and KiDS-450in the σ − Ω m plane [98]. - - - - - - - - × - - × - × - × - - - - - - - - × - × - × - × - × - × - FIG. 13: Evolution of Φ (cid:48) as a function of N . Left: c = − .
6, Right: c = 0. - - - - - - - - × - - × - × - × - - - - - - - - × - × - × - × - × - FIG. 14: Evolution of Φ (cid:48) as a function of N . Left: c = − .
6, Right: c = 0. . Integrated Sachs-Wolfe effect The integrated Sachs-Wolfe (ISW) effect is a secondary anisotropy of the cosmic mi-crowave background (CMB), which arises because of the variation in the cosmic gravitationalpotential between local observers and the surface of the last scattering [99]. The ISW effectis related to the rate of change of (Φ + Ψ) w.r.t. conformal time ( η ) [7]. While weak grav-itational lensing is determined by the spatial dependence of the metric scalar perturbationΦ, the ISW effect provides valuable information about the time evolution of the same, espe-cially in the late accelerating Universe. Even though its detectability is weaker than weaklensing, it is a powerful tool to study the underlying cosmology. It can be detected usingthe cross-correlation between the observational data on CMB and large scale structures. Inthe flat Λ-CDM model, detection of the ISW signal provides direct detection of dark energy[100]. - - - - - - - - × - - × - × - - - - - - - - - - - FIG. 15: Evolution of ∆Φ (cid:48) (left), ∆Φ (cid:48) / Φ (cid:48) ni (right) as a function of N . Since the Bardeen potential Φ evolve differently in the interacting and non-interactingscenarios, this change should potentially change the temperature fluctuations of the CMBphotons. Figures 13 and 14 contain plots of Φ (cid:48) as a function of N for different length scalesin interacting and non-interacting scenarios. Evolution of ∆Φ (cid:48) and ∆Φ (cid:48) rel as a function of N are plotted in Figures 15 and 16.Like δ m and Φ, we see that the difference in the evolution of Φ (cid:48) in these two scenariosbecomes significant at N ∼ −
3. Consistent with the fact that the first-order interactionterm is larger at the smaller length scales, the difference in the evolution of Φ (cid:48) in theinteracting and non-interacting scenarios is enhanced for larger values of k . This indicates28 - - - - - - - × - - × - × - - - - - - - - - - - FIG. 16: Evolution of ∆Φ (cid:48) (left), ∆Φ (cid:48) / Φ (cid:48) ni (right) as a function of N . that observations on the ISW effect can detect or constrain dark energy and dark matterinteraction. V. CONCLUSIONS
In Ref. [46] two of the current authors found a mapping between phenomenological modelsof the dark-energy dark matter coupling functions Q from a consistent classical field theory.It was shown that the mapping holds both at the background and first-order perturbationslevel. In this work we used this interacting field theory framework for a specific scalarfield potential U ( φ ) ∼ /φ n and linear interaction function α ( φ ) ∼ φ . We analyzed thebackground cosmological evolution in this model and obtained the model parameters fromcosmological observations. We evolved the perturbed equations in the redshift range 1500 (cid:46) z ≤ Constraints from observations:
We obtained constraints for the model parameters fromfour observational data sets — Hubble parameter measurements, baryon acoustic oscillationobservation, high- z HII galaxy measurements, and Type Ia supernovae observations. Fornumerical analysis, we rewrote the evolution equations in terms of dimensionless variables.Using the χ minimization technique, we obtained the constraints on H , Ω m , w , and theinteraction strength C .The key conclusions of the analysis for n = 1 ( U ( φ ) ∼ φ − ) case are: (i) All the fourdata sets constrain the value of H to be close to 70 km s − Mpc − . BAO+Hz and SN+Hzobservations provide the tightest constraints, followed by HIIG and Hz measurements. (ii)29hen a combined analysis of all four data sets is performed, the constraints are impacted byBAO and SN observations the most, and the allowed range for H becomes even narrower.(iii) The constraints on Ω m obtained from various data sets are consistent with each other,and BAO+Hz provides the smallest allowed range, which drives the limit for combinedanalysis, followed by SN + Hz, Hz, and then HIIG data. (iv) When it comes to constraining w , all the observations are consistent with the ΛCDM model, and only Hz data allows for anon-accelerating universe. (v) As for the constraints on C , we find that only BAO+Hz dataconstrains C within 3 σ confidence region, and hence, when analysis with a combination ofthe data sets is performed, the allowed values of C is influenced by BAO+Hz data the most.We also find that, except for Hz measurements, all the three data sets show a preference fora negative value C (cf. Table II). The Hz data is nearly insensitive to the sign and value of C within the considered range.The key conclusions of the analysis for n = 2 ( U ( φ ) ∼ φ − ) case are: (i) Constraintsfrom Hz data set does not change significantly, for other data sets, there is a slight shift ofthe contours. (ii) The observations prefer slightly higher values of Ω m , the contours fromHz data shifts towards higher values of Ω m . HIIG data allows a significantly larger rangeof Ω m compared to n = 1. For SN+Hz observations, there is no significant change in thelower range but the upper limit on Ω m shifts slightly higher. For BAO+Hz data, the changein the allowed range of Ω m is insignificant. (iii) For H , the change is not noticeable whenwe go from U ( φ ) ∼ φ − to φ − . For w , from Hz data, there is no noticeable change, butthe allowed ranges increase when the n = 1 is changed to n = 2 for BAO+Hz, HIIG, andSN+Hz observations. (iv) The constraints on coupling parameter C changes significantlywhen n changes. For n = 2, constraints on C from Hz do not show much change. Still,for BAO+Hz, HIIG, and SN+Hz data, we get un upper limits on C , and the contours shifttowards negative values of C , showing their preference for a negative value of interactionstrength. We can also see this in constraints obtained from the combination of data sets. Allthe observations are consistent with C = −
1, but BAO+Hz, HIIG, and SN+Hz observationsdo not agree with C = 1 within 1 σ confidence regions for n = 2 case. Distinguishing dark sector interacting model from standard cosmology:
To dis-tinguish the interacting dark sector from the non-interacting dark sector, we looked at theevolution of the scalar perturbations in the interacting dark sector model. We considereda inverse potential U ( φ ) ∼ /φ and a linear interaction function α ( φ ) ∼ φ with negative30alues of interaction strength C . We evolved three perturbed quantities ( δ m , Φ, Φ (cid:48) ) from lastscattering surface to present epoch (1500 (cid:46) z ≤ δ m grows faster in the interacting scenarios, especially at thelower length scales. The difference in the evolution becomes significant for z <
20, for alllength scales, and the difference peaks at smaller redshift values z <
5. This means thatcosmological observations related to the formation of large scale structures can potentiallydetect the signatures of dark matter - dark energy interaction. We see a similar trend in theevolution of Φ and Φ (cid:48) as well, which indicates an interaction between dark energy and darkmatter will be reflected on the observational data on weak gravitational lensing and ISWeffect. We get a similar behaviour for inverse-square potential U ( φ ) ∝ /φ . The evolutionof the perturbations in the interacting dark sector also differ from the ones inmodified gravitymodels like f ( R ) gravity, which descibes the late time acceleration of the Universe[101].It is interesting to note that all the perturbed quantities significant for z ∼ − σ − Ω m plane between Planck and cosmic shearexperiments [98]. We plan to address this in future work.Currently, we are looking to obtain the constraints on the model from the evolution ofthe perturbations using the relevant observational data sets. It will also be interesting tolook at the observational consequences of the difference in the evolution of the density per-turbation. Since interaction is higher for the smaller length scales, it can significantly affectthe evolution of the mass distribution of the binary black holes detected by the gravitationalwave observations [102]. 31 I. ACKNOWLEDGEMENTS
We thank T. Padmanabhan for fruitful discussions. We thank Ana Luisa Gonz´alez-Mor´an for providing Gordon extinction corrected HIIG measurements and useful informationrelated to the measurements. JPJ is supported by CSIR Senior Research Fellowship, India.The work is partially supported by the ISRO-Respond grant.
Appendix A: Parameter constraints for U ( φ ) ∼ /φ For completeness, in this Appendix, we present the constraints for n = 2 in thequintessence potential (10). Note that in Sec. III C, we presented the detailed analysisfor n = 1. As mentioned earlier, the parameter constraints are roughly the same for n = 1and n = 2. Figures 17 and 18 contain the constraints on parameters H , interaction strength C , and Ω m for the four observational data sets — Hz, BAO, HIIG, and SN.Here are the key inferences from Figures 17 and 18: (i) For n = 2, the constrains on H , w and C obtained from the data sets are almost same as for n = 1. (ii) From Hz data, theminimum value of χ is 18.77 which corresponds to the best fit values of the parameters are H = 69 .
37 km s − Mpc − , Ω m = 0.29, w =-0.98 and the interaction strength is C =0.98. (iii)For BAO+Hz data, when it comes to the interaction strength C the preference for negativevalue is more evident here than for n = 1. Although for 1 /φ potential, the data doesnot allow for a non-accelerating universe, a larger range for w is obtained. (iii) For HIIGobservations, n = 2 provides a larger range of parameters than n = 1. (iv) For SN+Hzdata, the H and Ω m constraints are as narrow as in n = 1 case, but the observations prefernegative value for C .From Figure 18 we see that the four data sets do not provide a lower limit on w . Hzdata provides an upper limit of -0.68 within 1 σ and w =0.03 within 3 σ region, showing thatthis particular model does not allow for a non-accelerating universe within 1 σ region. Theallowed ranges are almost the same as in the case n = 1.32 IG. 17: 1,2,3- σ likelihood contours for Hz data (I row), BAO+Hz data (II row), HIIG data (III row),SN+Hz data (IV row) and all four data sets (V row). The two-dimensional contours are obtained byperforming marginalization over other parameters. σ region,and the allowed range for w is wider as compared to the n = 1 case. The HIIG data alsoallows a wider range for w of − .
83 within the 3 σ region and allows the entire range of Ω m considered in the analysis. The SN+Hz data also allows a wider range for w and Ω m ascompared to n = 1 case. Apart from Hz data, the three remaining observational data setsconsidered in the analysis do not allow for a non-accelerating universe for both n = 1 and2. For w , Hz observations provide the widest allowed range within 3 σ confidence level. FIG. 18: 1,2,3- σ likelihood contours in ‘ w -Ω m ’ plane. The top row shows constraints from Hz data (left)and BAO+Hz observations (right). The second row shows constraints from HIIG measurements (left) andSN+Hz observations (right). ppendix B: Evolution of scalar perturbations for φ − potential For completeness, in this Appendix, we present the evolution of the matter density per-turbation δ m and related quantities fo n = 2 in the quintessence potential (10). Note thatin Sec. IV, we presented the detailed analysis for n = 1. As mentioned earlier, the evolutionof the perturbed quantities is not sensitive to n .
1. Evolution of the scaled interaction function δq Fig. 19 is the plot of δq (cf. Eq. 34) as a function of N for different k values. Comparingthis plot with the plots in Fig. 4, we see that evolution of the interaction function is roughlythe same for the both the cases. Hence, the evolution of scaled interaction function δq isnot sensitive to n . - - - - - - - × - × - × - - - - - - - - × - × - × - FIG. 19: Evolution of δq as a function of N for different values of k ;
2. Structure formation
Figures 20 and 21 contain plots of δ m as a function of N for different length scales ininteracting and non-interacting scenarios. Figures 22 and 23 contain the plots of ∆ δ m and∆ δ m rel as a function of N for different length scales, respectively. Thus, we see that evolutionof δ m is roughly the same for n = 1 and n = 2.35 - - - - - - × - × - × - × - - - - - - - - × - × - × - FIG. 20: Evolution of δ m as a function of N . Left: c = − .
6, Right: c = 0. - - - - - - - × - × - × - × - - - - - - - - × - × - × - × - × - × - × - FIG. 21: Evolution of δ m as a function of N . Left: c = − .
6, Right: c = 0. - - - - - - - × - × - × - × - - - - - - - - FIG. 22: Evolution of ∆ δ m (left), ∆ δ m /δ m ni (right) as a function of N . - - - - - - × - × - × - × - - - - - - - - FIG. 23: Evolution of ∆ δ m (left), ∆ δ m /δ m ni (right) as a function of N .
3. Weak gravitational lensing
Figures 24 and 25 contain plots of Φ as a function of N for different length scales ininteracting and non-interacting Figures 26 and 27 contain the plots of ∆Φ and ∆Φ / Φ ni asa function of N for different length scales, respectively. Thus, we see that evolution of Φ isroughly the same for the both the cases and is not sensitive to n . - - - - - - - - × - - × - × - - - - - - - - × - × - × - × - × - FIG. 24: Evolution of Φ as a function of N . Left: c = − .
6, Right: c = 0.
4. ISW effect
Figures 28 and 29 contain plots of Φ (cid:48) as a function of N for different length scales ininteracting and non-interacting Figures 30 and 31 contain the plots of ∆Φ (cid:48) and ∆Φ (cid:48) / Φ (cid:48) ni asa function of N for different length scales, respectively. Thus, we see that evolution of Φ (cid:48) is37 - - - - - - - × - - × - × - - - - - - - - × - × - × - × - × - FIG. 25: Evolution of Φ as a function of N . Left: c = − .
6, Right: c = 0. - - - - - - - - × - - × - - × - - × - - × - - × - - - - - - - - - - - - - FIG. 26: Evolution of ∆Φ (left), ∆Φ / Φ ni (right) as a function of N . - - - - - - - - × - - × - - × - - × - - × - - × - - - - - - - - - - - - - FIG. 27: Evolution of ∆Φ (left), ∆Φ / Φ ni (right) as a function of N . roughly the same for the both the cases and is not sensitive to n .We thus conclude that the evolution of δ m , Φ and Φ (cid:48) for the inverse square potential follow38 - - - - - - - × - × - - - - - - - - × - × - × - × - × - × - FIG. 28: Evolution of Φ (cid:48) as a function of N . Left: c = − .
6, Right: c = 0. - - - - - - - - × - × - - - - - - - - × - × - × - × - × - × - FIG. 29: Evolution of Φ (cid:48) as a function of N . Left: c = − .
6, Right: c = 0. - - - - - - - - × - - × - × - - - - - - - - - - - - - FIG. 30: Evolution of ∆Φ (cid:48) (left), ∆Φ (cid:48) / Φ (cid:48) ni (right) as a function of N . a similar trend as compared to the U ( φ ) ∼ /φ case. The difference in the evolution becomessignificant for z <
20, for all length scales. This means that cosmological observations related39 - - - - - - - × - - × - × - - - - - - - - - - - - - FIG. 31: Evolution of ∆Φ (cid:48) (left), ∆Φ (cid:48) / Φ (cid:48) ni (right) as a function of N . to the formation of large scale structures can potentially detect the signatures of dark matter- dark energy interaction. Appendix C: Sound speed of the scalar field
Sound speed and adiabatic sound speed of the dark energy scalar field ( φ ) is given by[103] c s = δp φ δρ φ , c s ad = ˙¯ p φ ˙¯ ρ φ = − − φ H ˙¯ φ + 2 α φ ¯ ρ m M P l (C1)In terms of the dimensionless variables, these quantities can be expressed as c s ad = h (cid:48) ( x − y ) + h ( xx (cid:48) − yy (cid:48) ) h (cid:48) ( x + y ) + h ( xx (cid:48) + yy (cid:48) ) = − − (cid:0) x (cid:48) + h (cid:48) h (cid:1) √ (cid:0) √ x − √ αβ Ω m (cid:1) (C2) c s = 12Φ x − √ xδφ (cid:48) − √ λy δφ x − √ xδφ (cid:48) + 3 √ λy δφ (C3)For a quintessence model, c s = 1 in the rest frame of φ [104]. In this work, the perturbedquantities are evaluated in the dark matter rest frame. [1] A. G. Riess and Others, Astron. J. , 1009 (1998), arXiv:astro-ph/9805201 [astro-ph].[2] S. Perlmutter and Others, Astrophys. J. , 565 (1999), arXiv:astro-ph/9812133 [astro-ph].[3] D. N. Spergel and Others, Astrophys. J. Suppl. , 377 (2007), arXiv:astro-ph/0603449[astro-ph].
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