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Resolving Hubble tension with quintom dark energy model

  • Recent low-redshift observations have yielded the present-time Hubble parameter value H074kms1Mpc1. This value is approximately 10% higher than the predicted value of H0=67.4kms1Mpc1, based on Planck's observations of the Cosmic Microwave Background radiation (CMB) and the ΛCDM model. Phenomenologically, we show that, by adding an extra component, X, with negative density to the Friedmann equation, it can address the Hubble tension without changing the Planck's constraint on the matter and dark energy densities. To achieve a sufficiently small extra negative density, its equation-of-state parameter must satisfy 1/3wX1. We propose a quintom model of two scalar fields that realizes this condition and potentially alleviate the Hubble tension. One scalar field acts as a quintessence, while another “phantom” scalar conformally couples to matter such that a viable cosmological scenario is achieved. The model only depends on two parameters, λϕ and δ , which represent the rolling tendency of the self-interacting potential of the quintessence and the strength of the conformal phantom-matter coupling, respectively. The toy quintom model with H0=73.4kms1Mpc1 (Quintom I) yields a good Supernovae-Ia luminosity fit and acceptable rBAO fit but slightly small acoustic multipole A=285.54. A full parameter scan revealed that the quintom model was superior to the ΛCDM model in certain regions of the parameter space, 0.02<δ<0.10,Ω(0)m<0.31, while significantly alleviating the Hubble tension, although it is not completely resolved. A benchmark quintom model, Quintom II, is presented as an example.
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  • [1] A. G. Riess et al., Astron. J. 116, 1009 (1998), arXiv:astro-ph/9805201 doi: 10.1086/300499
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    [54] W. Hu and N. Sugiyama, Astrophys. J. 471, 542 (1996), arXiv:astro-ph/9510117 doi: 10.1086/177989
    [55] D. J. Eisenstein and W. Hu, Astrophys. J. 496, 605 (1998), arXiv:astro-ph/9709112 doi: 10.1086/305424
    [56] P. A. R. Ade et al., Astron. Astrophys. 571, A16 (2014), arXiv:1303.5076[astro-ph.CO doi: 10.1051/0004-6361/201321591
    [57] W. J. Percival, S. Cole, D. J. Eisenstein et al., Mon. Not. Roy. Astron. Soc. 381, 1053 (2007), arXiv:0705.3323[astro-ph doi: 10.1111/j.1365-2966.2007.12268.x
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    [61] F. Arevalo, A. Cid, and J. Moya, Eur. Phys. J. C 77(8), 565 (2017), arXiv:1610.09330[astro-ph.CO doi: 10.1140/epjc/s10052-017-5128-7
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    [63] M. Doran and M. Lilley, Mon. Not. Roy. Astron. Soc. 330, 965-970 (2002), arXiv:astro-ph/0104486[astro-ph doi: 10.1046/j.1365-8711.2002.05144.x
    [64] N. Aghanim et al., Astron. Astrophys. 594, A11 (2016), arXiv:1507.02704[astro-ph.CO doi: 10.1051/0004-6361/201526926
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Sirachak Panpanich, Piyabut Burikham, Supakchai Ponglertsakul and Lunchakorn Tannukij. Resolving Hubble Tension with Quintom Dark Energy Model[J]. Chinese Physics C. doi: 10.1088/1674-1137/abc537
Sirachak Panpanich, Piyabut Burikham, Supakchai Ponglertsakul and Lunchakorn Tannukij. Resolving Hubble Tension with Quintom Dark Energy Model[J]. Chinese Physics C.  doi: 10.1088/1674-1137/abc537 shu
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Resolving Hubble tension with quintom dark energy model

  • 1. High Energy Physics Theory Group, Department of Physics, Faculty of Science, Chulalongkorn University, Phayathai Rd., Bangkok 10330, Thailand
  • 2. Department of Physics and Astronomy, Sejong University, Seoul 05006, Republic of Korea
  • 3. Theoretical and Computational Physics Group, Theoretical and Computational Science Center(TaCS), Faculty of Science, King Mongkut’s University of Technology Thonburi, Prachautid Rd., Bangkok 10140, Thailand

Abstract: Recent low-redshift observations have yielded the present-time Hubble parameter value H074kms1Mpc1. This value is approximately 10% higher than the predicted value of H0=67.4kms1Mpc1, based on Planck's observations of the Cosmic Microwave Background radiation (CMB) and the ΛCDM model. Phenomenologically, we show that, by adding an extra component, X, with negative density to the Friedmann equation, it can address the Hubble tension without changing the Planck's constraint on the matter and dark energy densities. To achieve a sufficiently small extra negative density, its equation-of-state parameter must satisfy 1/3wX1. We propose a quintom model of two scalar fields that realizes this condition and potentially alleviate the Hubble tension. One scalar field acts as a quintessence, while another “phantom” scalar conformally couples to matter such that a viable cosmological scenario is achieved. The model only depends on two parameters, λϕ and δ , which represent the rolling tendency of the self-interacting potential of the quintessence and the strength of the conformal phantom-matter coupling, respectively. The toy quintom model with H0=73.4kms1Mpc1 (Quintom I) yields a good Supernovae-Ia luminosity fit and acceptable rBAO fit but slightly small acoustic multipole A=285.54. A full parameter scan revealed that the quintom model was superior to the ΛCDM model in certain regions of the parameter space, 0.02<δ<0.10,Ω(0)m<0.31, while significantly alleviating the Hubble tension, although it is not completely resolved. A benchmark quintom model, Quintom II, is presented as an example.

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    I.   INTRODUCTION
    • After the discovery of the accelerated expansion of the universe [1, 2], several hypotheses have been proposed to address the dark energy problem, such as Horndeski theories [3-5], generalized proca theories [6, 7], and the ghost-free massive gravity [8, 9]. However, the discovery of gravitational waves GW170817 [10] severely constrains these modified gravity models [11-13]. The simplest standard model of cosmology, without the introduction of any new gravitational degrees of freedom, is the ΛCDM. Based on the proposal of “minimal” dark matter and dark energy components, it could reasonably explain the accelerated expansion of the universe, as well as other observational data, until recently [14]. A number of low-redshift observations have revealed that there are discrepancies between the values of the Hubble parameter at the present time, H0, based on observations of Cepheids in the Large Magellanic Cloud (LMC) [15], the gravitational lensing of quasar measurements [16, 17], and the predicted value obtained from the Planck CMB data within the ΛCDM (Note that an intermediate value was originally found using Red Giants as the distance ladder [18]. These values were subsequently corrected to a value consistent with the low-redshift measurements [19]). Since the difference between H0 is approximately 5σ, this means that the standard model of cosmology, the ΛCDM, may not be correct. There is a tension between the H0 predicted from the early universe and the values obtained by direct measurements at low redshifts.

      Many ideas have been proposed to resolve the Hubble tension, such as the modified gravity and brane model [20-23], the gravitational and vacuum [24-26] phase transition, early dark energy [27], dark matter decay [28], dark sector interaction [29, 30], neutrino self-interaction [31], phenomenological dark energy [32], and the negative cosmological constant [33] (moreover, see [34]). In this work, we demonstrate that the usual Friedmann equation allows for a higher value of H0=74.03kms1Mpc1 while keeping the matter contribution to 31% and the dark energy contribution to 69%, provided that an extra component with a very small negative density is introduced. The negative component must be a very small fraction to the total density of the universe; otherwise, it would have been previously detected (see Ref. [35], except for the possible galactic effects of a small negative density on the rotation curves). As a theoretical model that considers such a possibility, we propose a modified quintom model [36] to realize the negative-density component that is phenomenologically required by the Friedmann equation. The quintom model consists of a quintessence scalar field and a phantom scalar field. The model introduces dark energy with phantom divide crossing and a stable late-time solution. The use of two scalar fields for dark energy is not novel (see e.g. [37-39]. It is shown in Ref. [40] that a model with only one quintessence scalar and a cosmological constant exacerbates the Hubble tension. A model called the gravitational scalar-tensor theory also incudes two scalar fields [41, 42]. It is interesting to examine whether the phantom scalar field of the quintom model matches the required negative density and is capable of addressing the Hubble tension.

      This work is organized as follows. Section II discusses the physical requirement of the extra component X for coexistence within the standard Friedmann model to alleviate the Hubble tension. In Section III, we propose a modified quintom model with scalar-matter coupling that realizes the negative density requirement, giving the correct value of H0 while keeping the density parameters Ω(0)m0.31,Ω(0)DE0.69 consistent with Planck's early universe constraints. We use a dynamical system approach to find cosmological solutions of the modified quintom model in Section IV. Section V presents the numerical analysis results of the quintom model, yielding a realistic cosmological solution. Section VI compares the theoretical prediction of our model to the observational data, and Section VII summarizes our main results.

    II.   GENERAL PHENOMENOLOGY
    • In this section, a general physical condition is discussed based on the Friedmann equation with one extra component in addition to the standard ΛCDM model. In this approach, it is assumed that Planck's constraints from the early universe on H0 are valid, i.e., Ω(0)m=0.31,Ω(0)Λ=0.69, with very small contributions from other components at present.

      Using a density parameter, Ωiρi/ρc, where ρc3H2/8πG is the critical density of the universe, the generalized Friedmann equation for the spatially flat universe is given by:

      H2(z)=H20[Ω(0)r(1+z)4+Ω(0)m(1+z)3+Ω(0)DEexp(3z01+wDE(z)1+zdz)+Ω(0)Xexp(3z01+wX(z)1+zdz)],

      (1)

      where the subscripts r,m,DE,X represent radiation, matter, dark energy, and the extra unknown component X, respectively. The notation “(0)” denotes the present value at zero redshift. In the ΛCDM model with wDE=1, observational data from the CMB and high-redshifts prefer Ω(0)m=0.308, Ω(0)r=9.2×105, and Ω(0)Λ=DE=0.692, and H0=67.4kms1Mpc1 [14, 43]. We can thus calculate H(z) at the last-scattering surface (z1100) to be approximately 1.57537×106kms1Mpc1. To address the Hubble tension wherein the value of H at small z is relatively large compared to the Planck value H0=67.4 kms1Mpc1, we use H(z=1100)=1.57537×106kms1Mpc1 and H0=74.03kms1Mpc1 to determine the physical constraint on the extra unknown component X from Eq. (1). The allowed values of wX,Ω(0)X are shown in Fig. 1, assuming the simplest case where wX is constant.

      Figure 1.  (color online) The relation between Ω(0)X and wX from Eq. (1) for Ω(0)m=0.308, Ω(0)r=9.2×105, and Ω(0)Λ=DE=1Ω(0)rΩ(0)mΩ(0)X,wDE=1.

      Interestingly, a negative energy density Ω(0)X<0 is required. According to Fig. 1, the negative density cannot be a negative mass (wX0); otherwise, Ω(0)X is too large, i.e., Ω(0)X0.07, and it should have been observed. For 1/3wX1, however, the amount of the extra component X is very small, i.e., 0.0000636042Ω(0)X5.247×1011. Thus, to address the Hubble tension problem, a negative density is required with 1/3wX1 without modification of the CMB observational data.

      This particular phenomenological model fails when extrapolated back to early times due to overpopulation of the negative energy. At z=1100, the density parameter becomes ΩX=0.23 , which should be excluded by the constraints on the power spectrum of the CMB on the early DE. We need a more realistic model with cosmological evolution that suppresses the early DE population and satisfies the early-time constraints from the CMB and the late-time constraints from Baryon Acoustic Oscillations (BAO) and the integrated Sach-Wolfe (iSW) effect.

      In Section III, a quintom model with two scalars is proposed as a realization of the extra component X. A “phantom” (with a negative kinetic energy term) scalar is assumed to couple to matter, while the other scalar serves as the dark energy that is responsible for the accelerated expansion of the universe. Matter-phantom coupling stabilizes the phantom value by allowing the matter to drag it along, together with the cosmological expansion. The quintessence scalar serves as the DE with negligible contribution in the early times and only increases to dominate in the late epoch. By tuning the model parameters and initial condition, a viable realistic quintom model is achieved that passes some of the early and late time constraints.

    III.   QUINTOM DARK ENERGY MODEL
    • We consider the quintom action with 2 scalar fields and an interaction between matter fields and one of the scalar fields as follows:

      S=d4xg[12κ2R12(ϕ)2+12(σ)2V(ϕ)]+SM(gμν,σ,ψM),

      (2)

      where κ2=8πG is the inverse of the reduced Planck mass squared. R is the Ricci scalar, ϕ is a quintessence scalar field, σ is a “phantom” scalar field, and ψM is a matter field. We assume that there is only one self-interacting potential of the quintessence scalar field, whereas the “phantom” scalar field is rolling on an effective potential arising from the phantom-matter interaction, as will be explained. Strictly speaking, this σ is not exactly the standard phantom but rather a ghost field since its equation of state is Pσ=ρσ<0. However, here and henceforth, we will refer to it as the phantom field for convenience and also in accordance with the original name quintom (quintessence + phantom). The extra component X is identified using the phantom field σ in this model. It shoud be noted that the quintom model a has negative kinetic energy term in the Lagrangian. Thus, the phantom scalar field encounters a quantum instability problem of its own. In our model, however, the total energy density of all components in the universe is always positive, and the evolution of the universe always obeys the positive energy condition.

      By varying the action with respect to gμν, we obtain the equation of motion

      Rμν12gμνR=κ2(T(M)μν+T(ϕ)μν+T(σ)μν),

      (3)

      where T(M)μν is an energy-momentum tensor of non-relativistic matter and radiation. The energy-momentum tensors of the quintessence scalar field and the phantom scalar field are given by:

      T(ϕ)μν=μϕνϕ+gμν(12(ϕ)2V(ϕ)),

      (4)

      T(σ)μν=μσνσ+gμν(12(σ)2),

      (5)

      respectively. Using the flat Friedmann-Lemaître-Robertson-Walker (FLRW) metric, ds2=dt2+a2(t)dx2, and assuming ϕ=ϕ(t) and σ=σ(t), we obtain the Friedmann equations,

      3H2=κ2(ρm+ρr+ρϕ+ρσ),

      (6)

      3H2+2˙H=κ2(PmPrPϕPσ).

      (7)

      ρm, ρr, Pm, and Pr are the energy densities and pressures of non-relativistic matter and radiation, respectively. The notation “.” denotes a derivative with respect to time. The energy densities and pressures of the scalar fields are

      ρϕ=12˙ϕ2+V(ϕ),ρσ=12˙σ2,

      (8)

      Pϕ=12˙ϕ2V(ϕ),Pσ=12˙σ2.

      (9)

      We can then define an equation of the state parameter of the dark energy and an effective equation of state parameter as

      wDEPϕ+Pσρϕ+ρσ=12˙ϕ212˙σ2V(ϕ)12˙ϕ212˙σ2+V(ϕ),

      (10)

      weffPϕ+Pσ+Pm+Prρϕ+ρσ+ρm+ρr=123˙HH2.

      (11)

      It should be noted thatwDE in this section and henceforth is contributed by both scalar fields, which is different from the wDE in the Section II. For each scalar field, the equation of state parameters are

      wϕ=Pϕρϕ=12˙ϕ2V(ϕ)12˙ϕ2+V(ϕ),wσ=Pσρσ=+1.

      (12)

      It should be noted that ρσ is negative, and wσ is always equal to +1. This is crucial in resolving the Hubble tension problem, which will be shown in the Section IV.

      We assume that there is only an interaction between the matter field and the phantom field, i.e., μTμ(ϕ)ν=0. In this work we consider an interaction of the form

      μTμ(M)ν=κδTMνσ,

      (13)

      μTμ(σ)ν=κδTMνσ,

      (14)

      where TM=ρM+3PM, and δ is a dimensionless constant. This is a conformal interaction form that arises in many scalar-tensor theories after performing a conformal transformation to the Einstein frame [44, 45]. Hence, the continuity equations are

      ˙ρm+3Hρm=κδρm˙σ,

      (15)

      ˙ρσ+3H(ρσ+Pσ)=κδρm˙σ,

      (16)

      ˙ρϕ+3H(ρϕ+Pϕ)=0,

      (17)

      ˙ρr+4Hρr=0.

      (18)

      Substituting for the energy density and pressure of each scalar field, we obtain the equations of motion

      ¨σ+3H˙σ=κδρm,

      (19)

      ¨ϕ+3H˙ϕ+V,ϕ=0.

      (20)

      The right-hand-side (RHS) of the equation of motion of the phantom scalar field acts as an effective potential. This is similar to the effective potential in chameleon or symmetron gravity [46-48]

    IV.   DYNAMICAL SYSTEM

      A.   Autonomous equations and fixed points

    • We will use a dynamical system approach to study the cosmological scenarios of the quintom dark energy model based on the behaviour of their fixed points. First, the dimensionless dynamic variables are defined as follows;

      x1κ˙ϕ6H,x2κV(ϕ)3H,x3κ˙σ6H,x4κρr3H.

      (21)

      According to the Friedmann Eq. (6), the density parameters in terms of the dynamic variables are

      Ωm=1x21x22+x23x24,

      (22)

      Ωr=x24,

      (23)

      ΩDE=x21+x22x23,

      (24)

      Ωϕ=x21+x22,

      (25)

      Ωσ=x23,

      (26)

      where Ωmκ2ρm/3H2. In addition, the equation of states are

      wDE=x21x22x23x21+x22x23,

      (27)

      wϕ=x21x22x21+x22,

      (28)

      wσ=1,

      (29)

      weff=x21x22x23+x243.

      (30)

      In the last equation , we used the second Friedmann Eq. (7), which leads to

      ˙HH2=12(3+3x213x223x23+x24).

      (31)

      Differentiating the dynamic variables with respect to N, where N=lna is an e-folding number, we obtain a set of autonomous equations:

      dx1dN=62λϕx223x1x1˙HH2,

      (32)

      dx2dN=62λϕx1x2x2˙HH2,

      (33)

      dx3dN=62δ(1x21x22+x23x24)3x3x3˙HH2,

      (34)

      dx4dN=2x4x4˙HH2,

      (35)

      where λϕV,ϕ/κV. For an exponential form of a potential, i.e., V(ϕ)=V0eκλϕϕ, λϕ is a constant, and the autonomous equations are closed.

      The fixed points of the system can be obtained by setting dx1/dN=dx2/dN=dx3/dN=dx4/dN=0. They are shown in Appendix A. The dynamic variables x2 and x4 are always positive, whereas x1 and x3 can be positive or negative depending on the signs of ˙ϕ or ˙σ. We are only interested in the case where λϕ>0 (an exponential decay) and δ>0. For these fixed points, the density parameters and equation of state parameters are presented in Table 1.

      Ωm Ωr ΩDE wDE weff
      (a) 0 0 1 1 1
      (b) 0 1 0 13
      (c) 13δ2 1+12δ2 16δ2 1 13
      (d) 1+2δ23 0 2δ23 1 2δ23
      (e) 0 14λ2ϕ 4λ2ϕ 13 13
      (f) 13δ2 1+12δ24λ2ϕ 16δ2+4λ2ϕ 8δ2λ2ϕ24δ2λ2ϕ 13
      (g) 0 0 1 1+λ2ϕ3 1+λ2ϕ3
      (h) (λ2ϕ3)(3λ2ϕ+2δ2(λ2ϕ6))3(λ2ϕ2δ2)2 0 12δ4+9λ2ϕ2δ2(183λ2ϕ+λ4ϕ)3(λ2ϕ2δ2)2 2δ2(2δ2λ2ϕ)(λ2ϕ3)12δ4+9λ2ϕ2δ2(183λ2ϕ+λ4ϕ) 2δ2(λ2ϕ3)6δ23λ2ϕ

      Table 1.  Density and equation of state parameters of each fixed point.

      In the next section, the stability of each fixed point will beexamined by considering its corresponding eigenvalues.

    • B.   Eigenvalues of fixed points

    • The eigenvalues of each fixed point in Table A1 are as follows (the definition of eigenvalues is presented in Appendix B).

      x1 x2 x3 x4 existence
      (a) x21x23=1 0 0 x21x23=1
      (b) 0 0 0 1 All
      (c) 0 0 16δ 1+12δ2 All
      (d) 0 0 23δ 0 All
      (e) 263λϕ 233λϕ 0 14λ2ϕ λϕ2
      (f) 263λϕ 233λϕ 16δ λ2ϕ+2δ2(λ2ϕ4)2δλϕ 0<λϕ<2,0<δλ2ϕ82λ2ϕ or λϕ2,δ>0
      (g) λϕ6 1λ2ϕ6 0 0 0<λϕ6
      (h) (32δ2)λϕ6(λ2ϕ2δ2) 36δ2+9λ2ϕ4δ4(λ2ϕ6)6(λ2ϕ2δ2)2 23δ(λ2ϕ3)2δ2λ2ϕ 0 δ>0,λϕ=3 or δ>0,0<λϕ3,6λ2ϕ6λ2ϕ2δ or 0<δ32,λϕ>3 or 0<λϕ<3,2δ6 or 3<λϕ<6,6λ2ϕ6λ2ϕ2δ

      Table A1.  Fixed points of the autonomous equations (32)-(35).

    • 1.   Fixed Point (a)
    • The eigenvalues of the fixed point are

      μ(a)=1,0,3±6δx211,332λϕx1.

      (36)

      Although the sign ± depends on the roots of the condition x21x23=1, the fixed point is either a saddle or an unstable point. Since this fixed point does not match with any known cosmological era, it is no longer considered.

    • 2.   Radiation Dominated Solutions
    • The eigenvalues of the fixed point (b), (c), (e), and (f) are given by

      μ(b)=2,1,1,1.

      (37)

      μ(c)=1,2,12±12(2δ2+3),

      (38)

      μ(e)=1,1,12±16λ2ϕ154,

      (39)

      μ(f)=12±12δ2λ2ϕδ2λ4ϕ+δ4(32λ2ϕ9λ4ϕ)+δ4λ4ϕ(λ4ϕ+4δ4(163λ2ϕ)24δ2λ2ϕ(16+3λ2ϕ)),12±12δ2λ2ϕδ2λ4ϕ+δ4(32λ2ϕ9λ4ϕ)δ4λ4ϕ(λ4ϕ+4δ4(163λ2ϕ)24δ2λ2ϕ(16+3λ2ϕ)).

      (40)

      Therefore, the fixed point (b), (c), and (e) are saddle points. For the point (f), we can understand the behaviour of the fixed point when we set the value of λϕ and δ.

    • 3.   Matter Dominated Solutions
    • For the fixed points (d) and (h), their corresponding eigenvalues are

      μ(d)=32δ2,32δ2,12δ2,32δ2,

      (41)

      μ(h)=λ2ϕ+2δ2(λ2ϕ4)4δ22λ2ϕ,3λ2ϕ+2δ2(λ2ϕ6)4δ22λ2ϕ,14(λ2ϕ2δ2)2(3λ4ϕ2δ2λ2ϕ(λ2ϕ9)+4δ4(λ2ϕ6)±3(λ2ϕ2δ2)2(72λ2ϕ21λ4ϕ+4δ2(72+18λ2ϕ+λ4ϕ)+4δ4(6028λ2ϕ+3λ4ϕ))).

      (42)

      The fixed point (d) is stable when δ2>32, whereas it is a saddle point when δ2<32. For the fixed point (h), the eigenvalues can be understood once we set the values of λϕ and δ.

    • 4.   Accelerated Expansion Solutions
    • The eigenvalues of the fixed point (g) are

      μ(g)=12(λ2ϕ6),12(λ2ϕ6),12(λ2ϕ4),λ2ϕ3.

      (43)

      Thus, the fixed point is stable when λ2ϕ<3. For the point (h), it is the same as the previous case.

    V.   NUMERICAL SOLUTIONS
    • In this section, the autonomous Eqs. (32)-(35) are solved numerically, wherein we set λϕ=0.1 and δ=0.113. This choice lies within the range of values that satisfy the fixed-points scenario λϕ<3,δ<3/2 discussed previously and the condition weff<1/3 at the fixed point (g) which gives λϕ<2 for accelerated expansion. By tuning the model parameters and the initial condition, we can obtain the Hubble parameter in the desired range of values, namely H074 km s1 Mpc1, to address the Hubble tension. Since a viable cosmology requires a sufficiently long period of a radiation dominated era, we choose the initial point of the numerical solution to be deep into the radiation epoch at z=9.74811×105 corresponding to N=ln(1+z)=13.79, to guarantee a long radiation epoch. The cosmological evolution is insensitive to the choice of the initial value of N, e.g., setting N=13.81 would yield indistinguishable results provided that we define the present-day N0 such that Ω(0)m has the same value. When we change the initial value of N, the present-day value N0 at some fixed choice of (Ω(0)m,Ω(0)r) will shift accordingly. The overall cosmological evolution remains the same with the physical redshift defined as NN0=ln(1+z). Evolutions of the density parameters and equation of state parameters are shown in Fig. 2.

      Figure 2.  (color online) Evolutions of the density and equation of state parameters according to Eqs. (22)-(30), where initial conditions are x1=1×105, x2=1×1010, x3=1×105, and x4=0.9983 at z=9.74811×105(N=13.79).

      Figure 2 demonstrates a viable cosmological scenario wherein the universe evolves from the radiation dominated era to the matter dominated epoch, followed by the late-time accelerated expansion. Since λϕ=0.1 and δ=0.113, the fixed points (e), (f), and (h) do not exist, whereas the fixed point (c) yields Ωr40, which is too large. Since we are interested in Ωr1, we choose the initial condition to be close to the fixed point (b). The matter dominated era is automatically point (d), and the accelerated expansion is the point (g). Therefore, the cosmological viable evolution is

      (b)(d)(g).

      (44)

      According to the middle figure of Fig. 2, the density of the phantom scalar field is negative, and wσ=1 as desired, whereas the density of dark energy (quintessence + phantom) increases at late-time. We can obtain the evolution of the Hubble parameter by integrating Eq. (31),

      H(N)=Cexp[12(3N3x21dN+3x22dN+3x23dNx24dN)],

      (45)

      where C is a constant of integration, which can be obtained by comparing the preceding equation to the Hubble parameter from the ΛCDM at the last-scattering surface. x1, x2, x3, and x4 are obtained from numerical solutions with the same initial conditions used in Fig. 2. We set H0=67.4kms1Mpc1 to find the Hubble parameter at z=1100 using the ΛCDM model; then, we start the evolution in the quintom model using this value of H(1100). The evolution of the Hubble parameter of the quintom compared to that of the ΛCDM is shown in Fig. 3.

      Figure 3.  (color online) Top figure represents the evolution of the Hubble parameters at late-time, while the bottom figure represents the percent deviation from ΛCDM from the last-scattering surface to the present.

      In Fig. 3, the Hubble parameter of the quintom model decreases at a slightly different rate compared to the ΛCDM, where we obtain H0=73.356kms1Mpc1 at the present time. The data of H(z) are from Ref. [49]. Remarkably, the Hubble tension is alleviated. It should be noted that the value of H0 depends on the initial conditions, and the values of λϕ and δ can be tuned to yield superior precision compared to the observational results, as will be shown.

      The cosmological parameters at the present time that were obtained based on numerical simulations are presented in Table 2. These parameters correspond to the redshift at z=0 in Figs. 2 and 3.

      Ω(0)m Ω(0)DE Ω(0)r Ω(0)σ Ω(0)ϕ
      0.3078 0.69211 7.66×105 −0.00164 0.69376
      wDE weff wσ wϕ
      −1.003 −0.694 1 −0.9985

      Table 2.  Cosmological parameters at the present time from the quintom model

      The motivation of this work is to modify the standard ΛCDM model such that the early-time parameters are mostly unchanged. In contrast, an additional phantom field only exhibits small effects, but the accumulative change in the value of H0 is significant. We thus explore the parameter space of the coupled quintom model with respect to H0, where values of λϕ<2,δ<3/2 are required for a late-time accelerated expansion. In Fig. 4, the contour of constant H0 is shown with respect to phantom-matter coupling δ and the present-day matter density parameter Ω(0)m, where the superscript is suppressed. The value of H0 depends not only on Ω(0)m but also on the coupling δ. Notably, the present-day Hubble parameter depends weakly on λϕ, which governs the quintessence evolution.

      Figure 4.  (color online) Parameter space of the coupled quintom model for fixed λϕ, the number on each pair of contour is the value of the corresponding H0. The solid (dashed) line represents contours with λϕ=0.1(1) respectively. The parameter space changes only slightly with λϕ.

      To examine the viability of the coupled quintom model, we plot the contour of each value of H0 in the parameter space with respect to the constraints Ω(0)m=0.308,Ω(0)γ=5.38×105,Neff=3.13 in Fig. 5. The value of H0 varies significantly with δ but is relatively insensitive to the parameter λϕ. For a given value of Ω(0)m, the quintom model presents a range of possibilities for H0, starting from the value corresponding to the ΛCDM model, to higher values. Interestingly, this unique property of the coupled quintom model makes it a good candidate to resolve the Hubble tension problem. For H074 km s1 Mpc1, it corresponds to the range δ=0.110.12.

      Figure 5.  (color online) Parameter space of the coupled quintom model under the condition Ω(0)m=0.308,Ω(0)γ=5.38×105,Neff=3.13; the number on each contour is the value of the corresponding H0. Region with small (λϕ,δ) gives H06768kms1Mpc1, close to the value of the ΛCDM model.

    VI.   COMPARISON WITH DATA AND OBSERVATIONAL CONSTRAINTS FROM THE EARLY AND LATE TIMES
    • In this section, we compare the cosmology data obtained using the quintom model with observational data, particularly for the CMB constraints from the early time zzdec, BAO constraints that originate from z=zdrag, and Type Ia Supernovae (SN) from the late time z2.3. The Planck constraints are determined from the base ΛCDM model. Therefore, certain constraints are not necessarily valid for other models. Since the Hubble tension arises due to the more accurately measured luminosity of SN Ia resulting in a larger value of H0 compared to the value implied from the Planck CMB measurement based on the assumption of a ΛCDM model, some of the constraints that depend on the cosmological model could be relaxed, e.g., the constraint Ω(0)mh2=0.14170±0.00097 [14]. The benchmark quintom model we are considering is based on the choice of parameters that would suppress the difference in the iSW effects in the late time for the Λ CDM, by tuning the initial conditions and model parameters so that Ω(0)DE,Ω(0)m are as close to the best-fit values of the ΛCDM model as possible. The cosmological parameters are given in Table 2 and were obtained from the initial conditions x1=1×105, x2=1×1010, x3=1×105, and x4=0.9983 at N=13.79, and the model parameters λϕ=0.10,δ=0.113. Another example of evidence for this benchmark is the excellent fit with the SN Ia data. Subsequent analysis reveals that Ω(0)m0.3080.315 is prefered for the quintom model with H073.4 km s1 Mpc1. However, this yields Ω(0)mh2=0.1660.170. Here and henceforth, we refer to this quintom model as Quintom I.

      Starting with the SN Ia observations, we use observational data between the magnitude mB and redshift parameter of Type Ia Supernovae from Ref. [50] (Pantheon analysis) and take the absolute magnitude M to be a fitting parameter that is unique for the entire set of data. To focus only on the essential differences between the quintom and ΛCDM models, the statistical analysis is simplified to contain only one parameter, M, which is assumed to include not only the absolute magnitude but also the combined effects of other nuisance parameters such as the stretch and color measure of the SN Ia data (for more careful analyses including stretch and color measurements, see e.g. Refs. [50-52]). The distance modulus μL is related to the observable mB and the luminosity distance dL by

      μL=mBM=5log10(dLMpc)+25.

      (46)

      The luminosity distance contains information about the evolution of the universe via the Hubble parameter,

      dL=c(1+z)z0dzH(z),

      (47)

      where H(z) can be calculated from Eq. (1) and Eq. (45) depending on the model. For the ΛCDM and other non-coupled phenomenological models, Eq. (1) will suffice. However, our quintom model with phantom-matter coupling can be more accurately calculated using Eq. (45). Figure 6 shows a comparison between the theoretical models and the Supernovae observational data. The ΛCDM parameters were chosen to be h=67.4,Ω(0)m=0.308. The fits of all 1048 data points for z<2.3 and the 211 data points for a small redshift z<0.1 are presented in Fig. 6, for which the two models appear to be degenerate on a single line. We define

      Figure 6.  (color online) Comparison between Type Ia Supernovae data from Ref. [50] and theoretical models: ΛCDM and quintom.

      χ2ΔmTC1Δm,

      (48)

      where Δmi=mobsB,imthB,i(i=1 to number the data points), C=Dstat+Csys. The uncertainty matrix C contains the diagonal statistical matrix Dstat and the off-diagonal covariance matrix Csys; see Ref. [50]. For all 1048 data points, the chi-square values of the fitting are χ2ΛCDM=1027.07 for M=19.429 and χ2quin=1026.74 for M=19.247. For the 211 low-redshift data points, χ2ΛCDM=217.804 for M=19.442 and χ2quin=217.784 for M=19.257 . The quintom model clearly yields an equally good fit to the ΛCDM.

      The degenerate plots of both models are the result of the same values for matter and dark energy densities for late times between the two models (since we tune the benchmark quintom model such that this is the case), while the difference in H0 is compensated by the different fitting values of the absolute magnitude M. The quintom model prefers the absolute magnitude to be approximately -19.25 to -19.26, whereas ΛCDM prefers |M|19.4319.44. The quintom model gives a value for the best-fit M, which is very close to the central value ¯M=19.25±0.20 given in Ref. [53]. However, the error bar is sufficiently large to accommodate the best-fit M of ΛCDM. More precise measurement of the absolute magnitude of the SN Ia could potentially facilitate the identification of the superior model .

      Next, we consider the acoustic peaks of the CMB in the quintom model. Generically, the multipole A of the acoustic peaks in the CMB is given by

      A=πθA=π(1+zdec)dA(zdec)rs(zdec),

      (49)

      where θA is the acoustic angular scale, dA(z) is the angular diameter distance, rs(z) is the comoving sound horizon, and zdec is the redshift parameter at the matter-radiation decoupling. dA(z),rs(z) can be calculated from

      dA(z)=c1+zz0dzH(z),

      (50)

      rs(z)=c3zdzH(z)1+Rs(z),

      (51)

      where Rs=3ρb4ργ=3Ω(0)b4Ω(0)γ(1+z). From the lower figure of Fig. 2, the phantom contribution Ωσ is smaller than 0.01 or 1 percent throughout the history of the universe, and consequently, its effect appears only in H(z) at the leading order. The phantom-matter coupling only reduces the matter density very slowly without interfering with the physics of matter-radiation during the transition epoch. Therefore, during the radiation-matter transition era, zdec can be approximated using Hu and the Sugiyama formula [54]

      zdec=1048(1+0.00124w0.738b)(1+g1wg2m),

      (52)

      where wm,b=Ω(0)m,bh2,

      g1=0.0783w0.238b/(1+39.5w0.763b),g2=0.560/(1+21.1w1.81b).

      In our case, we assume the baryon density to be given by Ω(0)bh2=0.02226 [14] and the photon density by

      Ω(0)γ=Ω(0)r/(1+0.2271Neff),

      (53)

      where Neff=3.13 is the effective number of relativistic neutrinos. This gives zdec=1093.98. We then numerically calculate the multipole to be A=285.54, in disagreement with the CMB result from Planck Collaboration that prefers A300.

      Another evaluation of the model is via the baryon acoustic oscillations. The relative BAO distance rBAO can be calculated from

      rBAO(z)=rs(zdrag)[(1+z)2d2A(z)cz/H(z)]1/3.

      (54)

      Again, since the fraction of phantom is less than 1 percent and its effect is only to reduce the matter density very slowly, the value of zdrag can be approximated using the usual Eisenstein and Hu formula [55]

      zdrag=1291w0.251m1+0.659w0.828m(1+b1wb2b),

      (55)

      where

      b1=0.313w0.419m(1+0.607w0.674m),b2=0.238w0.223m.

      In our quintom model, we adjust the value of rs(zdrag) by a factor of 1.0275 to compensate for the discrepancy between the Eisenstein&Hu formula and the numerical result [56]. For zdrag=1065.71, the plot of rBAO is shown in Fig. 7. The observational data are obtained from Refs. [57-60]. To be consistent with the quintom evolution, the cutoff in the integration limit in Eq. (51) is set to zcut=exp(N0Ni)1, where Ni=13.79 for the quintom models. Both the quintom model and the ΛCDM fit the BAO observations (ΛCDM is superior except for the z=0.20 point).

      Figure 7.  (color online) rBAO versus redshift plots in comparison with observational data. Plot of quintom (ΛCDM) model is represented by the solid (dashed) line, and that of Quintom I (II) is in red (blue)

      In summary, the benchmark model Quintom I yields an equally good fit for the SN Ia data, an acceptable fit for rBAO, but a slightly small value of A=285.54. Although the coupled quintom model resolves the Hubble tension, it is in disagreement with the acoustic peak and BAO measurements. However, the complete scan of the parameter space (δ,Ω(0)m) of the coupled quintom model shown in Fig. 8 reveals that there are regions that yield more satisfactory fits to the BAO data and the CMB's first acoustic peak, while significantly alleviating the tension in H0 even though it is not completely resolved. Using the minimization of chi-square for BAO fitting as the anchor, the benchmark quintom model, Quintom II, is defined with λϕ=0.10,δ=0.06,Ω(0)m=0.308. Quintom II fits the SN Ia data with χ2quin=1026.81 for M=19.3925 for all 1048 data points. All chi-square values of Quintom II are actually smaller than those of the ΛCDM, as will be shown below.

      Figure 8.  (color online) χ2 contours of the quintom model (λϕ=0.10). Dashed lines represent χ2l1 of the first acoustic peak l1 of the CMB spectrum; non-shaded solid lines represent H0 of the quintom model.

      In addition to Quintom II, Fig. 8 shows that the models in the middle region of the parameter space, δ=0.020.08, can effectively fit both the BAO data and the first CMB peak, while giving the present-time Hubble parameter in the range H0=6869.5kms1Mpc1. Fig. 9 shows the chi-square contours of the SN Ia fit of the quintom model. The model prefers Ω(0)m0.300.31 for δ=00.10.

      Figure 9.  (color online) χ2 contours of the quintom model (λϕ=0.10) fitting to the SNIa data. Best-fit M contours are depicted as horizontal lines.

      To identify the best model, we have to consider the total chi-square of each model. The total chi-square is defined as

      χ2=χ2SN+χ2BAO+χ2l1,

      (56)

      where the chi-square of the BAO is given by

      χ2BAO=NBAOi=1(rBAOi,thrBAOi,obs)2σ2rBAOi.

      (57)

      The rBAOi,th term is defined by Eq. (54), whereas rBAOi,obs and σrBAOi are observational data and error bars as shown in Fig. 7. For the CMB, we consider only the first acoustic peak of the CMB anisotropy, and the chi-square of the first peak is [61]

      χ2l1=(lth1lobs1σl)2.

      (58)

      l1 is the position of the first peak given by [62, 63]

      l1=lA(1δ1),

      (59)

      where δ1=0.267(r/0.3)0.1 and r=ρr/ρm at zdec. From Ref. [64], the first peak of the TT power spectrum is lobs1=220.0, where σl=0.5. We thus find

      l1(QuintomI)=210.093,l1(QuintomII)=220.357,l1(ΛCDM)=221.757.

      The values of chi-square of both models are presented in Table 3.

      H0/(kms1Mpc1) χ2SN χ2BAO χ2l1 χ2
      Quintom I (δ=0.113) 73.356 1026.74 161.83 392.626 1581.20
      Quintom II (δ=0.06) 68.55 1026.81 13.711 0.5104 1041.03
      ΛCDM 67.4 1027.07 21.2862 12.3425 1060.70

      Table 3.  Chi-square values of the quintom models (λϕ=0.10) and the ΛCDM model.

      Since the quintom model has 2 more parameters (λϕ and δ) compared to the ΛCDM model, according to the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), we must take into account the model's parameters and number of data points as [61]

      AIC=χ2min+2d,

      (60)

      BIC=χ2min+dlnN,

      (61)

      where d is the number of the free parameters in the model, N is the number of data points, and χ2min is the minimum value of the chi-square total. The preferred model is the model that has a small value for AIC and BIC. The ΛCDM model has 4 free parameters (Ω(0)m,Ω(0)b,H0,M), and the data points used in this work are 1048(SN)+7(BAO)+1(CMB)=1056. We thus obtain

      AIC(QuintomI,II)=1593.20,1053.03,AIC(ΛCDM)=1068.70,BIC(QuintomI,II)=1622.97,1082.80,BIC(ΛCDM)=1088.55.

      Comparison of the three observations (the Type Ia supernovae, the baryons acoustic oscillation, and the first acoustic peak of the CMB anisotropy) indicates that the ΛCDM model is more (less) preferred compared to Quintom I (II). Quintom II alleviates the Hubble tension and even results in better fits to the BAO and CMB l1 data.

    VII.   CONCLUSIONS AND DISCUSSIONS
    • In this work, the Hubble tension is alleviated via the addition of a very small negative density component to the universe. Such a small contribution does not change the values Ω(0)m0.31,Ω(0)DE0.69, which are constrained by the Planck's CMB observations based on the early universe. To realize this idea, we consider a quintom model with conformal phantom-matter coupling and self-interacting quintessence that gives a viable cosmological scenario with the correct density parameters. The model satisfies the general phenomenological conditions, i.e., starting with a radiation dominated era and continuing with matter and dark energy dominated era. It also contains the phantom divide crossing and effectively alleviates the Hubble tension, giving H0=73.356kms1Mpc1 and Ω(0)m=0.308,Ω(0)ϕ=0.692,Ω(0)σ=0.00164, as shown in Table 2.

      Phenomenologically, as discussed in Section II, the required negative density of the extra component X for wX=1 is Ω(0)X=5.247×1011. This is based on the assumption of the non-coupling of X to normal matter. In our quintom model, the conformal phantom-matter coupling is introduced to control the size of the negative density of the phantom field, σ. In this coupled model, the negative density of the phantom field becomes Ω(0)σ=0.00164 for wσ=1.

      For the benchmark quintom model that mimicked the late-time densities of the ΛCDM, the small redshift (z<10) iSW effect originated from the dark energy should be closely similar to that of the ΛCDM. We found that the SN Ia fits of the benchmark quintom were just as good as the fiducial ΛCDM but with a different best-fit absolute magnitude M. More precise determination of the absolute magnitude of SN Ia in future observations could potentially identify the superior model. The BAO distance fit with the observation is acceptable, but the ΛCDM fit is better, except for one point (z=0.20). However, the acoustic peak multipole A=285.54 is approximately 5% smaller than that of the observation. The benchmark model Quintom I completely resolves the tension in the Hubble parameter but is in obvious tension with the peak position of the CMB and the BAO measurements. A parameter scan of the quintom model revealed the region 0.02<δ<0.10,Ω(0)m<0.31 of the parameter space, which yielded good to excellent fits for the BAO, first acoustic peak of CMB anisotropy, and SN Ia data. An example of a Quintom II model was presented and demonstrated using AIC and BIC, that it is a better-fit model compared to the ΛCDM, and yet, significantly alleviated the Hubble tension.

    ACKNOWLEDGEMENTS
    • We appreciate the very helpful suggestions from D.M. Scolnic on the SN Ia data.

    APPENDIX A: FIXED POINTS
    • From Table A1, fixed point (a) is a kinetic-dominated point. Radiation dominated epoch can be realized by the fixed point (b), (c), (e), or (f) because weff=1/3. Fixed point (b) is a standard radiation dominated era, whereas the other points are a mixture of radiation and other components. Point (d) or (h) can possibly be a matter dominated point, wherein both of them also have a dark energy component in the matter dominated epoch. The accelerated expansion era can be realized by point (g) or (h). Fixed point (g) is an accelerating expansion fixed point that arises in the quintessence model, whereas point (h) is a scaling solution (i.e., the ratio of matter and dark energy is not equal to zero at late-time). Fixed point (d) cannot be an accelerating solution because the dark energy density is not negative.

    APPENDIX B: STABILITY ANALYSIS
    • The autonomous equations can be rewritten as

      dx1dN=F(x1,x2,x3,x4),dx2dN=G(x1,x2,x3,x4),dx3dN=H(x1,x2,x3,x4),dx4dN=I(x1,x2,x3,x4).

      The stability of the fixed points will be investigated using linear perturbation analysis around each fixed point, (x(c)1,x(c)2,x(c)3,x(c)4), by setting

      xi(N)=x(c)i+δxi(N),

      where i=1,2,3,4. The perturbation equations then take the form

      ddN(δx1δx2δx3δx4)=M(δx1δx2δx3δx4),

      where the matrix M is given by

      M=(Fx1Fx2Fx3Fx4Gx1Gx2Gx3Gx4Hx1Hx2Hx3Hx4Ix1Ix2Ix3Ix4)|x(c)1,x(c)2,x(c)3,x(c)4.

      The first order coupled differential equation (B1) has a general solution

      δxieμN,

      where μ is an eigenvalue of the matrix M. Thus, if all eigenvalues are negative (or their real parts are negative for complex eigenvalues), the fixed point is stable. If at least one eigenvalue is positive, the fixed point is a saddle point. When all of the eigenvalues are positive, the fixed point is unstable. The components of the matrix M are as follows:

      Fx1=12(3+9x213x223x23+x34),Fx2=3x1x2+6x2λϕ,Fx3=3x1x3,Fx4=x1x4,Gx1=3x1x232x2λϕ,Gx2=12(3+3x219x223x23+x246x1λϕ),Gx3=3x2x3,Gx4=x2x4,Hx1=x1(3x36δ),Hx2=x2(3x3+6δ),Hx3=12(3+3x213x229x23+x24+26x3δ),Hx4=x4(x36δ),Ix1=3x1x4,Ix2=3x2x4,Ix3=3x3x4,Ix4=12(1+3x213x223x23+3x24).

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