Constraints on Multicomponent Dark Energy from Cosmological Observations
CConstraints on Multicomponent Dark Energy from Cosmological Observations
Ke Wang ∗ and Lu Chen † , ‡ Institute of Theoretical Physics & Research Center of Gravitation,Lanzhou University, Lanzhou 730000, China School of Physics and Electronics,Shandong Normal University, Jinan 250014, China (Dated: June 16, 2020)Dark energy (DE) plays an important role in the expansion history of our universe. In the earlierworks, universe including DE with a constant or time-varying equation of state (EOS) has been wellstudied. Different from predecessors, we investigate the cosmic expansion with multicomponentDE. Today’s total energy density of DE is separated into several parts equally and every part hasa constant EOS w i . We modify the Friedmann equation and the parameterized post-Friedmanndescription of dark energy, then put constraints on w i from Planck 2018 TT,TE,EE+lowE+lensing,BAO data, PANTHEON samples and SH0ES measurement. We find that the largest and thesmallest values of w i have no overlap at 95%C.L.. Our results indicate that DE is composed of morethan one candidate. Besides, the density evolution of multicomponent DE is tick-like and differentfrom the other DE models’. That is to say, multicomponent DE makes a contribution to late andearly universe. I. INTRODUCTION
The concept of dark energy (DE) has been widely accepted by many physicists since the discovery of cosmicaccelerating expansion in 1998 [1, 2]. Lots of methods are proposed to study the properties of DE. For example, thedistance measurements are used to detect DE, including the baryonic acoustic oscillation (BAO) observation [3–7],surveys on Type Ia supernovae (SNIa) [8–10] and the direct measurement of Hubble constant H (SH0ES) [11, 12].The formation of large scale structure [13–16] is also influenced by DE significantly due to its negative pressure. DE,as the largest proportion of the total energy density in today’s universe, leaves footprints on the cosmic microwavebackground (CMB) [17–20], too. However, the nature of DE is still a puzzle through decades of research.Theoretically, DE is considered as the cosmological constant firstly [21]. After that, fluids or fields, especiallyscalar fields, also become candidates of DE [22–25]. They are equivalent taking DE entropy into account to linearorder of perturbation. We determine DE with the following four quantities, the energy density ρ de , the equation ofstate (EOS) w ≡ p de /ρ de , the sound speed c s ≡ (cid:112) δp de /δρ de and the stress tensor Π de . Here p de is the pressureof DE, δρ de and δp de are the energy density fluctuation and the pressure fluctuation respectively. c s is set to 1following CAMB+CosmoMC packages [26–28] used in this work. It’s reasonable for classical scalar fields [29]. AndΠ de vanishes to the first order. In the base ΛCDM model, w is a constant -1. The simple cases of w is a constant or afunction of redshift z , (for instance, w CDM model and w w a CDM or CPL model [30, 31]) are well investigated in theprevious works [19, 20, 32–34], both theoretically and numerically. Besides, there’s possibilitiy that DE is composedof different components theoretically [20, 35]. However, there are few works aiming to study the constitution of DE inthe view of observation. In this work, we use the fluid descriptions of DE to detect the properties of multicomponentDE from cosmological observations. Assuming DE only has gravitational interaction or it is minimally coupled toother components of universe, we separate today’s total DE energy density into n parts equally (hereafter w n CDMmodel) and reconsider the cosmic expansion history. Then we put constraints on the constant EOSs of differentcomponents w i numerically. If the final results show no overlap between the EOSs of any two components, we willreach a conclusion that DE is composed of more than one candidate.This paper is organized as follows. In section II, we sketch out the parameterized post-Friedmann (PPF) descriptionof multicomponent DE. We utilize the CMB data [20], BAO data [4–6], SNIa measurement [10] and the SH0ESdata [12] to constrain the EOSs of multicomponent DE and show our results in section III. Finally, a brief summaryand discussion are included in section IV. ∗ [email protected] † [email protected] ‡ Corresponding author a r X i v : . [ a s t r o - ph . C O ] J un II. PARAMETERIZED POST-FRIEDMANN DESCRIPTION OF MULTICOMPONENT DE
The expansion of our universe is significantly influenced by DE. We divide today’s DE into n compositions equally.Then the total energy density of our universe is ρ tot ( a ) = ρ T ( a ) + ρ de ( a ) = ρ T ( a ) + n (cid:88) i =1 ρ de,i (1) a − − w i . (1)Here ρ de,i (1) is today’s energy density and w i is the EOS of the i th DE composition. ρ de is the total DE energydensity. The subscript T denotes all the other components excluding DE. a is scale factor of universe. Then, theFriedmann equation is modified with Eq. (1).To cross the phantom divide, the evolution of DE perturbation is described with PPF description [36–40]. In thesynchronous gauge, the perturbations of energy density and momentum of DE satisfy the following modified equations, ρ de δ de = − ρ wde v de k H − c K k H H πG Γ , (2) ρ wde v de = ρ wde v T − k H H πGF (cid:34) S − Γ − ˙Γ H (cid:35) . (3)Here ρ wde is defined as ρ wde ≡ n (cid:88) i =1 ρ de,i (1) a − − w i (1 + w i ) . (4) δ de = δρ de /ρ de = δ de,i = δρ de,i /ρ de,i is the density perturbation of total DE and δρ de = (cid:80) ni =1 δρ de,i . v denotesvelocity and v de = v de,i . G is Newton’s constant. H is Hubble parameter. c K is related to the background curvatureof our universe. For a spatial flat universe, we have c K = 1. k H = k/aH , where k is the wave number in fourier space.Overdot means the differentiation over cosmic time. Besides, F = 1 + 12 πGa k c K ( ρ T + p T ) , (5) S = 4 πGH ( v T + kα ) k H ρ wde , (6)where α = a ( ˙ h + 6 ˙ η ) / k , h and η is the metric perturbations in the synchronous gauge. Now, for any given evolutionof Γ, there is a specific evolution of perturbations of DE. PPF description provides a well approximation for minimallycoupled scalar field DE models and many smooth DE models. Therefore, DE ought to be relatively smoother thanmatter inside a transition scale c s k H = 1, ρ de δ de (cid:28) ρ T δ T . (7)To satisfy this condition, we have the differential equation for Γ(1 + c k H ) (cid:34) Γ + c k H Γ + ˙Γ H (cid:35) = S. (8)And c Γ = 0 . c s for the evolution of scalar field models. III. RESULTS
We refer to CAMB+CosmoMC packages and use the combination data of CMB, BAO, SNIa and SH0ES measure-ments to constrain the EOSs of multicomponent DE. Concretely, we use Planck2018 TT,TE.EE+lowE+lensing [20],the BAO measurements at z = 0 . , . , . , .
57, namely 6dFGS [4], MGS [5] and DR12 [6], as well as thePANTHEON samples [10] and SH0ES measurement, namely R19 [12].
TABLE I: Constraints on the cosmological parameters in different multicomponent DE models from the combination data ofPlanck 2018 TT,TE,EE+lowE+lensing, BAO data (6dFGS, MGS, DR12), PANTHEON samples and R19. The first set oferror bars indicates the 68% limits and the second set in parentheses reflects the 95% limits. w CDM model w CDM modelΩ b h . ± . ± . . ± . +0 . − . )Ω c h . ± . +0 . . ) 0 . ± . ± . θ MC . +0 . − . ( ± . . ± . +0 . − . ) τ . +0 . − . ( +0 . − . ) 0 . +0 . − . ( +0 . − . )ln(10 A s ) 3 . +0 . − . ( +0 . − . ) 3 . ± . +0 . − . ) n s . +0 . − . ( ± . . ± . ± . w − . +0 . − . ( +0 . − . ) − . +0 . − . ( +0 . − . ) w − . +0 . − . ( +0 . − . ) − . +0 . − . ( +0 . − . ) w − . +0 . − . ( +0 . − . ) − . +0 . − . ( +0 . − . ) w − . +0 . − . ( +0 . − . ) − . +0 . − . ( +0 . − . ) w − . +0 . − . ( +0 . − . ) − . +0 . − . ( +0 . − . ) w - − . +0 . − . ( +0 . − . ) w - − . +0 . − . ( +0 . − . ) w - − . +0 . − . ( +0 . − . ) w - − . +0 . − . ( +0 . − . ) w - − . +1 . − . ( +1 . − . ) Based on the previous discussion in Sec. II, we divide today’s total density of DE into five orten components averagely. In the case of w CDM model, there are 11 free parameters neededto be fitted, { Ω b h , Ω c h , θ MC , τ, ln(10 ) A s , n s , w , dw , dw , dw , dw } . Another five free parameters { dw , dw , dw , dw , dw } are added in the w CDM model. Six of them are parameters in the base ΛCDM model.Ω b h and Ω c h are today’s density of baryonic matter and cold dark matter respectively, 100 θ MC is 100 times the ratioof the angular diameter distance to the large scale structure sound horizon, τ is the optical depth, n s is the scalar spec-trum index, and A s is the amplitude of the power spectrum of primordial curvature perturbations. w is the largestone among w i . dw i ( i = 2 , , , , , , , ,
0) is the difference between the EOSs of two adjacent parts. In other words,EOSs of different DE components are listed from top to bottom as below: w , w = w − dw , w = w − dw − dw and so forth. The ranges of w is set to be [ − ,
10] and dw j ∈ [0 , w CDM model, the highest EOS is w = − . +0 . − . at 68% C.L. and w = − . +0 . − . at 95% C.L.. The lowest value reads w = − . +0 . − . at 68% C.L. and w = − . +0 . − . at 95% C.L.. Their allowed ranges of 2 σ don’t overlap with each other. As a result, DE iscomposed of constitutions with different EOSs and w i is allowed to deviate from -1. We reach similar conclusions inthe w CDM model. The probability densities of w i ( i = 1 , , , , w CDM model, or i = 1 , , , , , , , , , w CDM model) are illustrated in Fig. 1 and Fig. 2 vividly. The grey dashed lines donate w i = − w i = −
1. But there is a long left tail for w in the w CDMmodel (or w in the w CDM model). And much weaker limits are aquired when w i is much lower than -1. Thereason is that DE component with w i (cid:28) − w CDM model and the w w a CDM model are also shown besides the w CDM model and the w CDM model.The parameterizations of w CDM model and the w w a CDM model provide monotonous DE density evolutions withredshift obviously. However, the lines cross over the standard line from bottom to top in the multicomponent DEmodels. They have a tick-like density evolution. That means multicomponent DE can make a contribution to lateand early universe.
IV. SUMMARY AND DISCUSSION
In this paper, we investigate multicomponent DE with different EOSs from cosmological observations. The back-ground and perturbation evolutions of DE are modified assuming DE is composed of several equal parts with individualconstant EOS w i . With the modified CAMB+CosmoMC packages, we put constraints on parameters in the w CDM w i P r o b a b ili t y d e n s i t y w w w w w FIG. 1: The probability densities of w i ( i = 1 , , , ,
5) in the w CDM model. w i P r o b a b ili t y d e n s i t y w w w w w w w w w w FIG. 2: The probability densities of w i ( i = 1 , , , , , , , , ,
0) in the w CDM model. and w CDM model from Planck 2018 TT,TE,EE+lowE+lensing, BAO data, PANTHEON samples and SH0ES mea-surement. We find that the largest and the smallest values of w i have no overlap at 95%C.L., leading to a conclusionthat it’s probable that DE is composed of various constitutions. In addition, we show the plot of total DE energydensity evolving with redshift. It indicates that multicomponent DE models have a tick-like density evolution whichwill make a contribution to late and early universe. Acknowledgments
We acknowledge the use of HPC Cluster of Tianhe II in National Supercomputing Center in Guangzhou and HPC ρ d e ( z ) / ρ d e ( ) z0 2 4 6 8 10ΛCDMwCDMw w a CDMw CDMw CDM
FIG. 3: ρ de ( z ) /ρ de (0) as a function of redshift z . The black horizontal line with value of 1 indicates the values in the baseΛCDM model. The red and blue lines represent the values in the w CDM model and w w a CDM model. The green and purplelines denote those in the w CDM and w CDM model respectively. Here we use their mean values in Tab. I. For w CDM model,the mean value of EOS is w = − . w w a CDM model, the mean values are w = − . w a = − . Cluster of ITP-CAS. [1] A. G. Riess et al. [Supernova Search Team], Astron. J. , 1009-1038 (1998) doi:10.1086/300499 [arXiv:astro-ph/9805201[astro-ph]].[2] S. Perlmutter et al. [Supernova Cosmology Project], Astrophys. J. , 565-586 (1999) doi:10.1086/307221 [arXiv:astro-ph/9812133 [astro-ph]].[3] S. Cole et al. [2dFGRS], Mon. Not. Roy. Astron. Soc. , 505-534 (2005) doi:10.1111/j.1365-2966.2005.09318.x[arXiv:astro-ph/0501174 [astro-ph]].[4] F. Beutler et al. , Mon. Not. Roy. Astron. Soc. , 3017 (2011) [arXiv:1106.3366 [astro-ph.CO]].[5] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Burden and M. Manera, Mon. Not. Roy. Astron. Soc. , no. 1,835 (2015) [arXiv:1409.3242 [astro-ph.CO]].[6] S. Alam et al. [BOSS Collaboration], Mon. Not. Roy. Astron. Soc. , no. 3, 2617 (2017) [arXiv:1607.03155 [astro-ph.CO]].[7] M. Ata et al.
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