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Cosmic acceleration caused by the extra-dimensional evolution in a generalized Randall-Sundrum model

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1. Salem, S.. The cosmological constant problem and the extra dimensions of N=2 D=5 supergravity[J]. Physics of the Dark Universe, 2024. doi: 10.1016/j.dark.2024.101499

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Guang-Zhen KANG, De-Sheng ZHANG, Chao SHI, Long DU, Dan SHAN and Hong-Sh ZONG. Cosmic Acceleration Caused by the Extra-Dimensional Evolution in a Generalized Randall-Sundrum Model[J]. Chinese Physics C. doi: 10.1088/1674-1137/abadec
Guang-Zhen KANG, De-Sheng ZHANG, Chao SHI, Long DU, Dan SHAN and Hong-Sh ZONG. Cosmic Acceleration Caused by the Extra-Dimensional Evolution in a Generalized Randall-Sundrum Model[J]. Chinese Physics C.  doi: 10.1088/1674-1137/abadec shu
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Cosmic acceleration caused by the extra-dimensional evolution in a generalized Randall-Sundrum model

    Corresponding author: Guang-Zhen Kang, gzkang@nju.edu.cn
  • 1. School of Science, Yangzhou Polytechnic Institute, Yangzhou 225127, China
  • 2. Department of Physics, Nanjing University, Nanjing 210093, China
  • 3. School of Science, Changzhou Institute of Technology, Changzhou 213032, China
  • 4. Department of Nuclear Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
  • 5. School of Science, Guangxi University of Science and Technology, Liuzhou 545006, China
  • 6. Department of Physics, Anhui Normal University, Wuhu 241000, China
  • 7. Nanjing Proton Research and Design Center, Nanjing 210093, China

Abstract: We investigate an (n+1)-dimensional generalized Randall-Sundrum model with an anisotropic metric which has three different scale factors. One obtains a positive effective cosmological constant Ωeff10124(in Planck units), which only needs a solution kr5080 without fine tuning. Both the visible and hidden brane tensions are positive, which renders the two branes stable. Then, we find that the Hubble parameter is close to a constant in a large region near its minimum, thus causing the acceleration of the universe. Meanwhile, the scale of extra dimensions is smaller than the observed scale but greater than the Planck length. This may suggest that the observed present acceleration of the universe is caused by the extra-dimensional evolution.

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    1.   Introduction
    • The current cosmic acceleration is an unexpected picture of the universe, revealed by the data sets of the last two decades from astrophysics and cosmology [1-15]. These data, which come from the cosmic microwave background radiation, supernovae surveys, baryon acoustic oscillations, etc, indicate that the universe consists of 5% ordinary baryonic matter, 27% dark matter, and 68% dark energy [6-16]. Dark energy not only has an unknown form of energy but also has not been detected directly. Additionally, dark energy is very similar to the cosmological constant which was proposed by Einstein. In Planck units, the observed value of the cosmological constant is an extravagantly tiny positive value of order 10124. This is the well-known cosmological constant fine tuning problem [17, 18]. There have been numerous attempts in order to solve this problem, such as quintessence, the anthropic principle, the f(R) model, etc. [19-26]. But none of these theories are problem-free. In astrophysics and cosmology, it is still an open question.

      Another perspective for resolving the problem described above, which seems to be more radical, is the following: must the dimensions of our universe be four? Are there any extra dimensions which are too small to be observed? Does the evolution of these extra dimensions contribute to the current cosmic acceleration? If so, would this help in solving the cosmological constant fine tuning problem? Therefore, we have investigated some higher-dimensional theories [27-40]. Among them, the Randall-Sundrum (RS) two-brane model [30], which has a natural solution to the hierarchy problem with warped extra dimension, has attracted our attention. The hierarchy problem is essentially a fine tuning problem that can be described as: why is there such a large discrepancy between the electroweak scale/Higgs mass MEW1 TeV and the Planck mass Mpl1016 TeV? In the RS two-brane scenario, our universe is described by a five dimensional line element [30]

      ds2=e2σ(y)ημνdxμdxν+r2cdy2,

      (1)

      where y is the extra dimensional coordinate, rc is the extra dimensional compactification radius, e2σ is the well-known warp factor with the term σ=krc|ϕ|, k=Λ/24M3, and M is the five dimensional Planck mass. Then a large hierarchy is generated by the warp factor e2krcπ, meanwhile one requires only kr10. The cosmological constant fine tuning problem is similar to the hierarchy problem. In the RS model, the visible brane is unstable, which is caused by the negative brane tension. Furthermore, the cosmological constant on the visible brane is zero, which is not consistent with our data sets of the last two decades [31, 41].

      The above problems can be solved in a generalized RS braneworld scenario in which gμν replaces ημν in the RS model [31]. In this scenario, the tension of the visible brane and the hidden brane can both be positive with a negative induced cosmological constant. It is very interesting because both branes are stable [42-47]. In order to be consistent with the current constraints, the negative induced cosmological constant Ω should be transformed into the positive effective cosmological constant Ωeff. This positive Ωeff can be obtained in an (n+1)-dimensional (-d) generalized RS model with two (n1)-branes instead of two 3-branes [48]. In this model, adopting an anisotropic metric ansatz with two different scale factors, one obtains the positive effective cosmological constant Ωeff10124 (in Planck units), which only needs a solution kr5080 without fine tuning. The cosmological constant fine tuning problem can be solved quite well [48].

      But there is no reason to exclude the possibility of the anisotropic metric ansatz with the form of scale factors more than two. In this paper, we investigate an (n+1)-dimensional generalized Randall-Sundrum model with an anisotropic metric that has three different scale factors. We obtain that H1 has a lower bound, H1min. Near this minimum value, the Hubble parameter is seen to be a constant in a large region, thus causing the acceleration of the universe. Meanwhile, the scale of extra dimension is smaller than the observed scale but greater than the Planck length. This may suggest that the observed present acceleration of the universe is caused by the extra-dimensional evolution rather than by dark energy. Our work is organized as follows: In Sec. 2, by considering the two (n1)-branes with the matter field Lagrangian in the (n+1)-d generalized RS model, the n-d Einstein field equations are obtained. In Sec. 3, we focus on the evolution of a (n+1)-brane solved from the above field equation with an anisotropic metric ansatz that has three different scale factors. Finally, the summary and conclusion are presented in Sec. 4.

    2.   (n+1)-d generalized Randall-Sundrum model
    • We consider an (n+1)-d generalized RS braneworld model that is consistent with Ref. [48]. The action Sn+1 is:

      Sn+1=Sbulk+Svis+Shid,

      (2)

      where Sbulk is the bulk action and Svis and Shid are the (n1)-brane visible action and hidden action, respectively:

      Sbulk=dnxdyG(Mn1n+1RΛ),

      (3)

      Svis=dnxgvis(LvisVvis),

      (4)

      Shid=dnxghid(LhidVhid),

      (5)

      where Λ denotes a bulk cosmological constant, Mn+1 is the (n+1)-d fundamental mass scale, GAB and R are the (n+1)-d metric tensor and Ricci scalar, respectively, Li is the matter field Lagrangian of the visible and hidden branes, and Vi is the tension of the visible and hidden branes, here with i=hid or vis. In this (n+1)-d generalized RS scenario, the metric takes the form:

      ds2=GABdxAdxB=e2A(y)gabdxadxb+r2dy2,

      (6)

      where e2A(y) is known as the warp factor, capital letter A,B,... indices run over all spacetime coordinate labels, y is the extra dimensional coordinate of length r, lowercase letter a,b=0,1,2,,n1 does not include the coordinate y, and gab is the n-d metric tensor. Variation with respect to the metric GAB and after some easy manipulations then modulo surface terms, one obtains

      RAB12GABR=12Mn1n+1{GABΛ+i[TiABδ(yyi)GabδaAδbBViδ(yyi)]},

      (7)

      where RAB is the (n+1) -d Ricci tensor and TiAB is the (n+1)-d energy-momentum tensors. Note that here the energy-momentum tensor is given by Tiab=diag[ci,ci,,ci] [47, 48]. A solution to Eq. (7) with the metric tensor in Eq. (6) has been derived in Ref. [48], and reads

      A=ln[ωcosh(k|y|+c)],

      (8)

      where the constant kΛ/[Mn1n+1n(n1)] Planck mass. ω is given by

      ω2Ω(n1)(n2)k2,

      (9)

      and the term c takes the form

      cln11ω2ω.

      (10)

      Meanwhile, an n-d Einstein field equation can be obtained:

       ˜Rab12gab˜R=Ωgab,

      (11)

      where Ω is the induced cosmological constant on the visible brane, and ˜R and ˜Rab are the n-d Ricci scalar and Ricci tensor, respectively,

      Note that the solution derived above has the negative induced cosmological constant Ω. Here we do not consider the situation where Ω>0, since the tension on the visible brane is negative, which results in instability [31, 41, 47, 48]. Thus, an anisotropic metric is assumed to be of the following form [47-49]:

      gab=diag[1,a21(t),a22(t),a23(t),,a2n1(t)],

      (12)

      where ai is the scale factor. The case where the scale factors on the visible brane evolve with two different rates has been studied recently [48]. In this case, we can obtain the positive effective cosmological constant Ωeff10124 and only requiring kr5080, where for convenience, the Planck mass has been set to unity. Thus, the cosmological constant fine tuning problem can be solved quite well. Furthermore, the three dimensional (3D) Hubble parameter H(z) is consistent with the cosmic chronometers dataset extracted from [6-15]. The observed 3D universe naturally shifts from deceleration expansion to accelerated expansion. This shows that the accelerated expansion of the observed universe is intrinsically an extra-dimensional phenomenon. But there is no reason to make the scale factor evolve with only two kinds of rates. Therefore, we investigate the case where the scale factors on the visible brane evolve with three different rates.

    3.   Anisotropic evolution of (n1)-brane
    • For the anisotropic metric Eq. (12) with three kinds of scale factors and the negative induced cosmological constant Ω10124, the field equations (11) can be written:

      ini(ni1)H2i+ijninjHiHj=2Ω,

      (13)

      ini˙Hi˙H1+(iniHi)2H1iniHi=2Ω,

      (14)

      ini˙Hi˙H2+(iniHi)2H2iniHi=2Ω,

      (15)

      ini˙Hi˙H3+(iniHi)2H3iniHi=2Ω,

      (16)

      where i=1,2,3, and the terms n1, n2, and n3 are the number of dimensions which evolve with three kinds of rates, respectively, the Hubble parameter H˙a/a, and ˙Hi is the first time derivative of Hi. Computing the sum of Eqs. (14), (15), and (16) yields a simplified expression for iniHi:

      iniHi=χ1tanβ,

      (17)

      where the term β=χ1t+θ0, θ0 is the initial phase angle which is determined by the scale of the formation of the brane, and the term χ1 takes the form

      χ1=2(n1)Ωn2.

      (18)

      It is convenient to redefine the sum of the Hubble parameters in the following manner:

      iniHif.

      (19)

      Replacing Eqs. (17) and (19) into Eq. (14), then performing some manipulations, one obtains

      ˙H1+H1f=˙f+f22Ω.

      (20)

      The solution of the above equation is

      H1=efdt[(˙f+f22Ω)efdtdt+c],

      (21)

      where c is an integration constant. Combining Eqs. (17) and (19), H1 is given by

      H1=χ1n1tanβ+csecβ.

      (22)

      Using Eqs. (19) and (20), Eq. (13) can be rewritten as

      n2H22+n3H23=n1H21+f22Ω.

      (23)

      Eqs. (22) and (23) could be combined to give the following equation, eliminating H2 completely:

      (n3+n23n2)H23+2n3n2(n1H1f)H3+[(1n21)f2+n21n2H212n1n2fH1+n1H21+2Ω]=0.

      (24)

      Then the Hubble parameter H3 can be obtained:

      H3=fn1H1n2+n31n2+n3[n2n3(n2+n31)f2n1n2n3(n1)H21+2n1n2n3fH12n2n3(n2+n3)Ω]1/2.

      (25)

      Combining Eqs. (17), (19), and (22), H3 is given by

      H3=χ1n1tanβχ3+n1cn2+n3secβ,

      (26)

      where the term χ3 is

      χ3=2n2n3(n2+n3)Ωn1n2n3(n1)c2.

      (27)

      Note here that we have chosen H2 to always be greater than H3. Finally, using Eqs. (17) and (26), one can also obtain the Hubble parameter H2:

      H2=χ1n1tanβ+χ2n1cn2+n3secβ,

      (28)

      where the term χ2 is

      χ2=2n3n2(n2+n3)Ωn1n3n2(n1)c2.

      (29)

      It is easy to see that the integration constant c is constrained by Eqs. (27) and (29). The constant c must be set to a value less than cmax to guarantee that the value in the root is greater than zero, which yields

      c2Ω(n2+n3)n1(n1)cmax,

      (30)

      where cmax is the maximum value of c. It is evident from Eqs. (26), (27), (28), and (29) that, if c=cmax, H2 is equal to H3. For convenience, we define a parameter d satisfying

      c=dcmax,

      (31)

      where the value of d is between 0 and 1. First, we investigate the effect of parameter d on the Hubble parameters H. We choose n1=3, which is most in line with the presently observed three dimensional (3D) space. Further setting n2=1 and n3=1, we plot the Hubble parameters H1, H2, and H3 as a function of β in Fig. 1 with d=0, d=0.5, and d=1 respectively. In Fig. 1(a)-(c), the three curves having same type (color) correspond to three different values of d, respectively. In Fig. 1(a), we plot the Hubble parameter H1 as a function of β. When the parameter is d=0 or d=0.5, the Hubble parameter H1 monotonically decreases with β. And when

      Figure 1.  (color online) The Hubble parameters H1 (n1=3), H2 (n2=1), and H3 (n3=1), with three different parameter values d. (a) The Hubble parameters H1 with d=0,0.5,1 correspond to the solid (blue) curve, the dashed (black) curve, and the dotted curve (red) respectively (n2=1). (b)-(c) The case for the Hubble parameters H2 and H3 , respectively.

      d>n1/[(n2+n3)(n2)]dmin=6/40.612

      for n1=3, n2=1, and n3=1, the Hubble parameter H1 has a minimum

      H1min=c2(n1)2χ21n1,

      (32)

      when the term β in Eq. (22) takes the form

      βmin=arcsin[χ1c(n1)].

      (33)

      If ββmin, the Hubble parameter H1 is a monotonic function decreasing with time, which is not in accordance with cosmological observations and experiments. As can be seen easily from Fig. 1(a), the the Hubble parameter H1 has a minimum when d=dmin0.612. This case may be consistent with the present observations because H1 tends to a constant near the minimum, which can lead to accelerated expansion without the contribution of dark energy (or an inflaton field).

      Combining Eqs. (18), (22), (28) and (29), we obtain H1=H2 if c satisfies

      c=2Ωn3(n1)(n1+n2)ceq.

      (34)

      It is shown that H2 tends toward H1 when cceq. In the case c=ceq, the parameter deq is defined as:

      deq=ceqcmax=n1n3(n1+n2)(n2+n3).

      (35)

      For n1=3, n2=1, and n3=1, deq=6/40.612. Since the extra dimension in our universe is not observed presently, it cannot be too large, and the current scale of extra dimensions is still outside the observable range. When d is equal to deq, the extra dimension Hubble parameter H2 is converted to H1. This case is inconsistent with the presently observed 3D space. The Hubble parameter H2 of the extra dimensions is plotted in Fig. 1(b) for d=0,0.5,1. It can be easily seen that the Hubble parameter H2 has a minimum with d=0 and 0.5, which can lead to accelerated expansion of the extra dimensions. We are not interested in this situation because it is not in line with observation. However, the Hubble parameter H2 is always negative with d=1, which ensures that the extra dimensions always exceed the observable range. Note here that the ratio of dmin to deq is

      dmindeq=(n1+n2)n3(n1+n2+n31)1,

      (36)

      where dmin/deq=1 if and only if n3=1. So we only consider the case ddeq because then we obtain ddmin when ddeq.

      In Fig. 2, we have plotted the parameter deq versus the number of extra dimensions n2 and n3, respectively. The figure on the left is the curve of the parameter deq with n3 when n2=1, 5, and 30. It is shown that deq monotonically increases with n3. The parameter deq tends toward n1/(n1+n2) in the n3 limit. In particular, deq3/20.866 for n2=1. The figure on the right depicts the evolution of deq with n2 when n3=1, 5, and 30. In this case, deq monotonically decreases with n2. The parameter deq0 in the n2 limit. To be consistent with observation, we should set d to be greater than 0.866 and closer to 1. Furthermore, we set the constant d=0.98 in the following.

      Figure 2.  (color online) The parameter deq versus the number of extra dimensions n2 and n3, respectively. The figure on the left is the curve of the parameter deq with n3 when n2=1,5,30, and the figure on the right depicts the evolution of deq with n2 when n3=1,5,30.

      In Fig. 3(a)-(c), we plot the Hubble parameters H1, H2 and H3 as a function of β with n1=3 when d=0.98. From top to bottom, the three curves having same type (color) corresponds to H1, H2, and H3, respectively, for fixed values of n2 and n3. In Fig. 3(a), we plot the Hubble parameter as a function of β at n2=1. From top to bottom, the three solid curves correspond to H1, H2, and H3 at n3=1. The three dashed curves (n3=3) and the three dotted curves (n3=5) are similar to the above case. The Hubble parameter of the extra dimensions is always negative at n3=1. With the increase of n3, the coordinate of the minimum value of H1 tends toward β=0. Meanwhile, H2 changes from positive to negative in the region near β1.5 and H3 is closer to zero. In Fig. 3(b) and (c), the Hubble parameters as a function of β are shown with n2=3 and n2=5. H2 is always negative with n2=5, which is similar to the situation in Fig. (1) of Ref. [48]. H1 has a lower bound H1min=0.982c2maxχ21/(n1)2 when βmin=arcsin{η1/[0.98(n1)cmax]}. In the region near βmin, we find that Ωeff>0 is of order Ω. This situation is similar to the case with two different scale factors, and the negative induced cosmological constant Ω can be transformed into the positive effective cosmological constant Ωeff. This tells us that the observed current cosmic acceleration is intrinsically an extra-dimensional phenomenon rather than dark energy. The cosmological constant fine tuning problem can be solved by this extra-dimensional evolution.

      Figure 3.  (color online) The Hubble parameters H1 (n1=3), H2, and H3 when d=0.98. (a) The Hubble parameters with n3=1,3,5 correspond to the solid (blue) curve, the dashed (black) curve and the dotted curve (red) respectively (n2=1). (b)-(c) The cases for n2=3 and n2=5, respectively.

      From Eqs. (22), (26), and (28) we can get the scale factors a1, a2, and a3 of the form

      a1=a10|cosβ|1n1|secβ+tanβ|cχ1,

      (37)

      a2=a20|cosβ|1n1|secβ+tanβ|χ2n1cχ1(n2+n3),

      (38)

      a2=a30|cosβ|1n1|secβ+tanβ|χ3+n1cχ1(n2+n3),

      (39)

      where a10, a20, and a30 are the scale factors when the brane forms. Further, since the volume of the visible brane is obtained by

      Vb=an11an22an33=an110an220an330cosβ,

      (40)

      when the brane is just forming, there is no particular reason to make the scale factor different, so we choose a10=a20=a30. If the initial scale of the brane is of order 1035 in Planck units, and considering the presently observed scale of our universe (of approximate order 1061), we obtain the scale of extra dimension which is at least of order 1022 with n2=n3=3. It can be shown that the scale of the extra dimension should be much larger than the Planck length. This ensures that physics is still valid in the evolution of the visible brane. The θ0 we obtain is close to π/2 if one wants to have a sufficiently small initial scale. So in the region of θ0+π/2η1tπ/2, the Hubble parameter H1 is of the form:

      H1[cχ1+1n1]1t=1n1[1+d(n2+n3)(n2)n1]1t.

      (41)

      When n2=n3=1 and d=0.98, H1 is about 0.52t, which is as similar to the radiation dominant era. In the limit n2 (or n3), H13d/3t.

    4.   Summary and conclusion
    • In conclusion, we investigate an (n+1)-d generalized Randall-Sundrum model with an anisotropic metric that has three different scale factors. In this model, we obtain the positive effective cosmological constant Ωeff10124 (in Planck unit), which only needs a solution kr5080 without fine tuning. This is consistent with the case of two different scale factors.

      In this model, the Hubble parameter H2 tends toward H1 when the integration constant d tends toward deq. This indicates that the Hubble parameter of observable dimensions is related to the value of the integral constant d. For convenience, here we have selected the Hubble parameter H1 to show the observable dimensions. To be consistent with observation, we should set d to be greater than 0.866 and closer to 1. Further setting the constant d=0.98, we obtain that H1 has a lower bound H1min=0.982c2maxχ21/(n1)2 when βmin=arcsin{η1/[0.98(n1)cmax]}. Meanwhile, the scale of extra dimension is smaller than the observed scale but greater than the Planck length. This may suggest that the observed current cosmic acceleration is caused by extra-dimensional evolution rather than dark energy (or an inflaton field). Of course, there are still many problems to be solved in this model. This includes the question of how the quantum fluctuations contribute to the amount of the expected value of the cosmological constant.

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