Cosmological model from the holographic equipartition law with a modified Rényi entropy
Eur. Phys. J. C
Cosmological model from the holographic equipartition law with a modified R?nyi entropy
Nobuyoshi Komatsu 0
0 Department of Mechanical Systems Engineering, Kanazawa University , Kakumamachi, Kanazawa, Ishikawa 9201192 , Japan
Cosmological equations were recently derived by Padmanabhan from the expansion of cosmic space due to the difference between the degrees of freedom on the surface and in the bulk in a region of space. In this study, a modified R?nyi entropy is applied to Padmanabhan's 'holographic equipartition law', by regarding the BekensteinHawking entropy as a nonextensive Tsallis entropy and using a logarithmic formula of the original R?nyi entropy. Consequently, the acceleration equation including an extra driving term (such as a timevarying cosmological term) can be derived in a homogeneous, isotropic, and spatially flat universe. When a specific condition is mathematically satisfied, the extra driving term is found to be constantlike as if it is a cosmological constant. Interestingly, the order of the constantlike term is naturally consistent with the order of the cosmological constant measured by observations, because the specific condition constrains the value of the constantlike term.

To explain the accelerated expansion of the late universe,
CDM (lambda cold dark matter) models assume a
cosmological constant related to an additional energy
component called dark energy [1?5]. However, measured by
observations is many orders of magnitude smaller than the
theoretical value estimated by quantum field theory [6?13].
This discrepancy is the socalled cosmological constant
problem. To resolve this theoretical difficulty, numerous
cosmological models have been proposed [14,15], such as CCDM
(creation of CDM) models [16?27] and (t )CDM models,
which assume a timevarying cosmological term (t ) [28?
53]. In CCDM models, a constant term is obtained from a
dissipation process based on gravitationally induced particle
creation [16?27], while in (t )CDM models, a constant term
is obtained from an integral constant of the renormalization
group equation for the vacuum energy density [39].
For these models, thermodynamic scenarios have attracted
attention, in which the Bekenstein?Hawking entropy (which
is proportional to the surface area of the event horizon) [54?
60] and the holographic principle (which refers to the
information of the bulk stored on the horizon) [61?63] play
important roles. For example, using the holographic principle it
has been proposed that gravity is itself an entropic force
derived from changes in the Bekenstein?Hawking entropy
[64?67]. Based on this concept, the cosmological
equations have been extensively examined in a homogeneous
and isotropic universe [68?72], although the cosmological
constant has not been discussed. In an alternative treatment,
Easson et al. [73,74] proposed an entropic cosmology that
assumes the usually neglected surface terms on the
horizon of the universe [75?90]. In entropic cosmology, an extra
driving term to explain the accelerated expansion is derived
from entropic forces on the horizon of the universe. An
area entropy (the Bekenstein?Hawking entropy), a volume
entropy (the Tsallis?Cirto entropy) [91], a quartic entropy
[84,85], and a general form of entropy [88?90] have been
applied to entropic cosmology. However, the entropicforce
term (related to dark energy and ) is generally considered
to be a tuning parameter, which makes it difficult to include
in discussion of the cosmological constant problem.
Padmanabhan recently provided a new insight into the
origin of spacetime dynamics using another thermodynamic
scenario called the ?holographic equipartition law? [92].
Based on the holographic equipartition law with Bekenstein?
Hawking entropy, cosmological equations in a flat universe
can be derived from the expansion of cosmic space due to the
difference between the degrees of freedom on the surface and
in the bulk [92]. The emergence of the cosmological
equations (i.e., the Friedmann and acceleration equations) has
been examined from various viewpoints, such as a nonflat
universe and quantumcorrected entropy [93?108]. However,
dark energy and have not yet been discussed
fundamentally, though they have been considered in several studies
[105?108]. This is likely because dark energy and (which
are related to an extra driving term) have been assumed to
explain the accelerating universe. If the extra driving term is
naturally derived from the holographic equipartition law, it
is possible to study various cosmological models based on
holographic equipartition law. However, such a driving term
should not be derived using Bekenstein?Hawking entropy.
Selfgravitating systems exhibit peculiar features due
to longrange interacting potentials. Therefore, the R?nyi
entropy [109] and the Tsallis entropy [110] can also be used
for astrophysical problems [91,111?132]. Bir? and Czinner
[112] have recently suggested a novel type of R?nyi entropy
on a blackhole horizon, in which the Bekenstein?Hawking
entropy is considered to be a nonextensive Tsallis entropy
[112,113].
In this paper, the modified R?nyi entropy is applied to the
holographic equipartition law in a homogeneous, isotropic,
and spatially flat universe. It is expected that an extra driving
term can be derived from the holographic equipartition law
with the modified R?nyi entropy. Therefore, the present study
should help in developing cosmological models based on the
holographic equipartition law. In addition, the extra driving
term is expected to be constantlike under specific conditions.
The specific condition and constantlike term may provide
new insights into the cosmological constant problem. (The
present study focuses on the derivation of the extra driving
term and the specific condition for the constantlike term.
Accordingly, the inflation of the early universe and density
perturbations related to structure formations are not
examined.)
The remainder of the article is organized as follows. In
Sect. 2, (t )CDM models are briefly reviewed for a
typical formulation of the cosmological equations. In Sect. 3,
entropies on the horizon are discussed. The Bekenstein?
Hawking entropy is reviewed in Sect. 3.1, while a modified
R?nyi is introduced in Sect. 3.2. The holographic
equipartition law is discussed in Sect. 4. In Sect. 5, the modified R?nyi
entropy is applied to the holographic equipartition law, to
derive the acceleration equation that includes an extra
driving term. The extra driving term is then discussed in Sect. 6,
focusing on the specific condition required for obtaining a
constantlike term. Finally, in Sect. 7, the conclusions of the
study are presented.
(t)CDM models
Cosmological equations derived from the holographic
equipartition law are expected to be similar to those for (t )CDM
models [28?53] in a nondissipative universe. Therefore, in
this section, the (t )CDM model is reviewed, to discuss a
typical formulation of the cosmological equations.
A homogeneous, isotropic, and spatially flat universe is
considered, and the scale factor a(t ) is examined at time t in
the Friedmann?Lema?tre?Robertson?Walker metric [85,88].
In the (t )CDM model, the Friedmann equation is given as
= H (t )2 =
and the acceleration equation is
= ?
where the Hubble parameter H (t ) is defined by
H (t ) ?
G, c, ?(t ), and p(t ) are the gravitational constant, the speed
of light, the mass density of cosmological fluids, and the
pressure of cosmological fluids, respectively [85,88], and
(t ) is a timevarying cosmological term. Based on Eqs. (1)
and (2), the continuity equation [85] is given by
The righthand side of this continuity equation is usually
nonzero, except for the case (t ) = . Accordingly, the
(t )CDM model can be interpreted as a kind of energy
exchange cosmology in which the transfer of energy between
two fluids is assumed [133?136]. When (t ) = , the
Friedmann, acceleration, and continuity equations are identical to
those for the standard CDM model. In this paper,
cosmological models based on the holographic equipartition law
are assumed to be a particular case of (t )CDM models,
although the theoretical backgrounds are different.
3 Entropy on the horizon
In the holographic equipartition law, the horizon of the
universe is assumed to have an associated entropy [92]. The
Bekenstein?Hawking entropy is generally used, replacing
the horizon of a black hole by the horizon of the universe.
In Sect. 3.1, the Bekenstein?Hawking entropy is briefly
reviewed. In Sect. 3.2, a novel type of R?nyi entropy
proposed by Bir? and Czinner [112] is introduced for the entropy
on the horizon of the universe.
3.1 The Bekenstein?Hawking entropy
The Bekenstein?Hawking entropy SBH [54?56] is given as
SBH =
Substituting AH = 4?r H2 into Eq. (5) and using Eq. (6), we
obtain [82?84]
SBH =
4 =
where K is a positive constant given by
K =
L p =
As shown in Eqs. (5) and (8), the Bekenstein?Hawking
entropy SBH on the Hubble horizon is proportional to H ?2
(and AH ) and is related to the Planck length.
In this paper, a spatially flat universe, k = 0, is
considered, where k is a curvature constant. For a spatially
nonflat universe (k = 0), the apparent horizon given by
r A = c/ H 2 + (k/a2) is used instead of the Hubble
horizon; see, e.g., Refs. [94?104].
3.2 Modified R?nyi entropy
where pi is a probability distribution and q is the socalled
nonextensive parameter [91,111]. In addition, the original
R?nyi entropy [91,109] is defined as
When q = 1, both ST and SoRrg recover the Boltzmann?Gibbs
entropy given by
SBG = ?kB
i=1
The original R?nyi entropy [91,109] can be written as
SoRrg = 1 ?1 q ln[1 + (1 ? q)ST ] = ?1 ln[1 + ?ST ],
In this paper, ? is considered to be nonnegative, as discussed
later.
A novel type of R?nyi entropy has been proposed and
examined in Refs. [112,113], in which not only is the
logarithmic formula of the original R?nyi entropy used but
the Bekenstein?Hawking entropy SBH is considered to be a
nonextensive Tsallis entropy ST . Using Eq. (14) and
replacing ST by SBH, we obtain a modified R?nyi entropy SR [113]
given by
When ? = 0 (i.e., q = 1), SR becomes SBH. We call SR the
modified R?nyi entropy. In the present study, the modified
R?nyi entropy is applied to the holographic equipartition law.
To this end, the holographic equipartition law is reviewed in
the next section.
Note that the physical origin of the modified R?nyi entropy
is likely unclear at the present time. However, it is worthwhile
to examine cosmological models based on the holographic
equipartition law from various viewpoints. Therefore, in this
paper, the modified R?nyi entropy is applied to the
holographic equipartition law.
4 Holographic equipartition law
Padmanabhan [92] derived the Friedmann and acceleration
equations in a flat universe from the expansion of cosmic
space due to the difference between the degrees of freedom on
the surface and in the bulk in a region of space. In this section,
the Padmanabhan idea called the ?holographic equipartition
law? is reviewed, based on his work [92] and related work
[93?108]. (The surface terms assumed in entropic cosmology
[73,74] are not considered in the holographic equipartition
law. In Ref. [108], the holographic equipartition law is
examined, including the surface terms.)
In this calculation, r has been set to be rH before the
time derivative is calculated [92]. Next, we calculate Nbulk
included in the righthand side of Eq. (17). According to Ref.
[92], ?c2 + 3 p < 0 is selected and, therefore, = +1 from
Eq. (21). Note that the following discussion is not affected
even if ?c2 + 3 p > 0 is selected [93]. Substituting Eqs. (20)
and (22) into Eq. (19), using Eq. (18), and rearranging, we
obtain
= ?
= +1 and Eqs. (10), (23), (24),
In an infinitesimal interval dt of cosmic time, the increase
dV of the cosmic volume can be expressed as
and solving this with regard to H? , we have
= ?
= ?
where K is given by Eq. (9). As shown in Eq. (2), a? /a is
written as a? /a = H? + H 2. Substituting Eq. (27) into this
equation, we obtain
where Nsur is the number of degrees of freedom on the
spherical surface of Hubble radius rH and Nbulk is the number
of degrees of freedom in the bulk [92]. L p is the Planck
length given by Eq. (10) and is a parameter discussed later.
Equation (17) represents the socalled holographic
equipartition law. Note that the righthand side of Eq. (17) includes
c, because c is not set to be 1 in the present paper. Using
rH = c/H given by Eq. (6), the Hubble volume V can be
written as
where the Komar energy E  contained inside the Hubble
volume V is given by
E  = (?c2 + 3 p)V = ? (?c2 + 3 p)V ,
and is a parameter defined as [92,93]
From this definition, E  is confirmed to be nonnegative.
Note that ?c2 + 3 p < 0 corresponds to an
accelerating universe (e.g., dark energy dominated universe), while
?c2 + 3 p > 0 corresponds to a decelerating universe (e.g.,
matter and radiation dominated universe). The temperature
T on the horizon is written as
This temperature is used for calculating Nbulk from Eq. (19)
[137]. In contrast, the number of degrees of freedom on the
spherical surface is given by
Nsur =
where SH is the entropy on the Hubble horizon. When SH =
SBH, Eq. (23) is equivalent to that in Ref. [92]. In the next
section, SH is replaced by the modified R?nyi entropy SR
given by Eq. (16).
We now derive the cosmological equations from the
holographic equipartition law. To this end, we first calculate the
lefthand side of Eq. (17). Substituting Eq. (18) into Eq. (17),
the lefthand side of Eq. (17) is given by
The above equation is the acceleration equation derived from
the holographic equipartition law, where SH is the entropy
on the Hubble horizon. Interestingly, the second term on the
righthand side, i.e., H 2(1 ? SH H 2/K ), appears to be an
extra driving term.
When the Bekenstein?Hawking entropy is used, i.e.,
SH = SBH, the second term H 2(1 ? SH H 2/K ) is zero
because SBH = K /H 2 given by Eq. (7). Consequently, the
acceleration equation can be written as
The obtained cosmological equation is considered to be a
particular case of (t )CDM models, as mentioned in Sect. 2.
Accordingly, Eq. (29) indicates that (t ) in Eq. (2) is zero.
Substituting (t ) = 0 into Eqs. (1) and (4), we have the
Friedmann and continuity equations, given by
X X
H 2 =
p
? + c2
= 0.
The three equations (i.e., the Friedmann, acceleration, and
continuity equations) agree with those in Ref. [92].
In the above, the second term on the righthand side of
Eq. (28) is zero because the Bekenstein?Hawking entropy
SBH is used for the entropy SH on the horizon. However, if a
different SH is assumed, the second term (corresponding to
an extra driving term) is expected to be nonzero. In the next
section, we examine this expectation, applying the modified
R?nyi entropy SR to the holographic equipartition law. Note
that a similar driving term has been studied, e.g., in Ref.
[108], using quantumcorrected entropy.
5 Acceleration equation from the holographic equipartition law with a modified R?nyi entropy
In this section, the modified R?nyi entropy SR is applied to
the holographic equipartition law, instead of the Bekenstein?
Hawking entropy. The modified R?nyi entropy given by
Eq. (16) can be written as
where SR becomes SBH when ? = 0. The acceleration
equation derived from the holographic equipartition law given by
Eq. (28) is
+ H 2 1 ?
where SH is the entropy on the Hubble horizon and K given
by Eq. (9) is written as
? kBc5 ? kBc2
K = h?G = L2p . (34)
Regarding SR as SH and substituting Eq. (32) into
Eq. (33), we have
= ?
+ H 2 1 ?
f (H ) = H 2 1 ?
Fig. 1 Properties of the normalized extra driving term f (H )/(?K ).
The horizontal axis is X ? ?K /H 2. The logarithmic formula given
by Eq. (39) is plotted. The two approximation curves for 1/2 and 1/ X
given by Eq. (42) are also shown
where, using SBH = K /H 2 given by Eq. (7), the extra driving
term f (H ) is written as
f (H ) = H 2 1 ?
and ? is a nonnegative constant, i.e., ? ? 0. The
acceleration equation given by Eq. (35) can be derived from the
holographic equipartition law with the modified R?nyi entropy.
When ? = 0 (i.e., SR = SBH), f (H ) is zero, as discussed in
the previous section. However, f (H ) is expected to be
nonzero when ? > 0. Consequently, f (H ) given by Eq. (36) is
not a simple power series of H but a logarithmic formula.
From Eq. (36), f (H ) can be written as
1 ?
The positive dimensionless parameter X is defined as
where ? is considered to be positive, i.e., ? > 0. Dividing
Eq. (37) by ?K , and applying Eq. (38), we have the
normalized extra driving term given by
To observe the properties of f (H ), the normalized term
is plotted in Fig. 1. Consequently, f (H )/(?K ) gradually
approaches 1/2 with decreasing X , while it approaches a
common curve with increasing X . Therefore, we focus on
two specific conditions, X ? 1 and X 1. Under the two
conditions, the approximate formulas are given
mathematically by
(t )/3 = f (H ) into Eqs. (1), (2), and (4), we obtain the
Friedmann, acceleration, and continuity equations written as
? 0 (X
Highorder terms have been neglected in Eq. (40).
Substituting Eqs. (40) and (41), respectively, into Eq. (39), we have
From Eqs. (38) and (42), f (H ) can be written as
The constantlike and H 2like terms are, respectively,
obtained from each condition, as shown in Eq. (43). The
constantlike term, ?K /2, can be interpreted as a kind
of cosmological constant, although the specific condition
?K /H 2 ? 1 is required. In the next section, we discuss
the constantlike term and the specific condition.
As a matter of fact, the extra driving term (which behaves
like effective dark energy) can be derived even if dark energy
is not assumed, i.e., even when p = 0. Alternatively, the
modified R?nyi entropy is assumed in the present paper. In
this sense, the original holographic equipartition law is
modified. This modification is expected to provide new insights
into cosmological models based on the holographic
equipartition law.
As shown in Eq. (43), H 2like terms are derived under
a specific condition. However, the driving term is not likely
to be related to the inflation of the early universe, because
higher exponents terms such as H 4 terms should be required
for the inflation [88]. In (t )CDM models, the higher
exponents terms have been closely examined in Refs. [40?
42]. In entropic cosmology, a quantumcorrected entropy is
introduced for the higher exponents terms; see, e.g., Refs.
[74,108].
6 Constantlike term under a specific condition
The acceleration equation can be derived from the
holographic equipartition law with the modified R?nyi entropy,
as examined in the previous section. Interestingly, an extra
driving term included in the equation is found to be
constantlike under a specific condition. In this section, we discuss
the constantlike term under the specific condition. To this
end, the present cosmological model is reviewed, assuming
that it is a particular case of (t )CDM models. Substituting
H 2 =
where the extra driving term f (H ) is given by
f (H ) = H 2 1 ?
and K is
K =
As shown in Eq. (46), the righthand side of the continuity
equation is not zero generally. This nonzero righthand side
may be interpreted as the interchange of energy between the
bulk (the universe) and the surface (the horizon of the
universe) [79,88]. (The previous work of Sol? et al. [43,44]
imply that the value of the nonzero righthand side of the
continuity equation should be small in (t )CDM models
[88].) In the following, the righthand side of the continuity
equation is zero, because f (H ) is considered to be constant
under a specific condition. General solutions for the present
model, which includes a logarithmic term, are separately
discussed in Appendix A.
This paper focuses on the constantlike term under a
specific condition. From Eq. (43), the constantlike term and the
specific condition can be written as
f (H ) ?
where H0 is the Hubble parameter at the present time and
?K /(2H02) is a dimensionless constant (corresponding to ?4
in Ref. [85]). Hereafter, we call the constantlike term the
constant term. Under the specific condition given by Eq. (49),
the Friedmann, acceleration, and continuity equations are
written as
H 2 =
= 0.
The three equations are essentially the same as those in the
standard CDM model. Setting p = 0 for nonrelativistic
matter, the background evolution of the universe is
analytically given by
= (1 ?
?3
?3
where a0 represents the scale factor at the present time and
h is a holographic parameter defined by
In the present model, the constant term ?K /2 and the
holographic parameter h have been restricted by the inequality
given in Eq. (49). Accordingly, the constraint on both ?K /2
and h can be discussed, without tuning.
In the history of an expanding universe, H0 is expected to
be the minimum value of H , because rH increases with time
and H is given by H = c/rH from Eq. (6). Therefore, when
H = H0, the most severe constraint on ?K can be obtained
from the inequality of Eq. (49). The constraint is written as
In the present model, Eq. (56) is mathematically required
to obtain the constant term. From Eq. (56), the order of the
constant term, ?K /2, is approximately given by
or equivalently, from Eqs. (54) and (56), the order of h can
be approximately written as
O( h ) = O
We now compare the constant term ?K /2 in the present
model and the cosmological constant term /3 in the
standard CDM model. In the CDM model, the density
parameter for the cosmological constant is given by
? 3H02
Numerous cosmological observations indicate that the order
of is expected to be 1, e.g., = 0.692 from the Planck
2015 results [5]. Accordingly, the order of can be
approximately written as
) ? O(1).
Using Eqs. (59) and (60), we have
= O(
H02) ? O(H02).
Equations (57) and (61) imply that the order of the
constant term ?K /2 is consistent with the order of /3. That
is, interestingly, the constant term in the present model is
naturally consistent with the order of measured by
cosmological observations as if it is . Similarly, from Eqs. (58)
and (60), the order of h is likely consistent with the order of
. Of course, in the present model, ?K H02 is required
to obtain the constant term, assuming the modified R?nyi
entropy instead of the Bekenstein?Hawking entropy. This
extension may be beyond the original holographic
equipartition law. However, the constant term and the specific
condition considered here may provide new insights into the
cosmological constant problem.
Equation (56) can be written as ? L2p/(? kBr H20)
using Eq. (48) and H0 = c/rH0, where rH0 is the
Hubble horizon at the present time. This constraint indicates
that ? is an extremely small positive value. In other words,
the modified R?nyi entropy is approximately equivalent to
the Bekenstein?Hawking entropy although it slightly
deviates from the Bekenstein?Hawking entropy. Accordingly, the
present model may imply that the cosmological constant is
related to a small deviation from the Bekenstein?Hawking
entropy. As time passes, the constraint is expected to be
more severe because rH0 increases with time. Consequently,
the extra driving term in the present model should gradually
deviate from the constant value, even if the severe condition
is satisfied at the present time. That is, the present model
can be distinguished from the CDM model, as discussed in
Appendix A.
In this paper, we have focused on the derivation of an
extra driving term from the holographic equipartition law
with the modified R?nyi entropy. The obtained term f (H )
given by Eq. (47) is a logarithmic formula. However, under
two specific conditions, it can be systematically written as
f (H ) = C0 H 2
0 + C1 H 2, where C0 and C1 are
dimensionless constants. In the (t )CDM model, the above formula
has been examined in detail [44?48,80,81]. For example,
G?mezValent et al. [45?47] and Basilakos et al. [80,81] have
shown that it is not the H 2, H? , and H terms, but rather the
constant term that plays an important role in cosmological
fluctuations related to structure formations [88]. In addition,
recently, (t )CDM models, which include power series of
H , have been found to be more suitable than the standard
CDM model [43]. In particular, the simple combination of
the constant and H 2 terms, f (H ) = C0 H 2
0 + C1 H 2, is likely
favored; see, e.g., the work of Sol? et al. [44], G?mezValent
et al. [45?47], and Lima et al. [48], which indicate that C0 is
dominant and C1 is expected to be small [43,44]. The
smallness of C1 has not yet been explained by the holographic
approach [88], though it may be explained by a deeper
understanding of the present model. For example, the logarithmic
formula discussed in the present model slightly deviates from
the constant value when ?K /H 2 ? 1, as shown in Fig. 1. Of
course, this small deviation is not proportional to H 2.
However, the small deviation may be related to the smallness of
C1, if the smallness can be interpreted as a deviation from a
constant term. This task is left for future research.
We keep in mind that the present result depends on the
choice of entropy. In this paper, the modified R?nyi entropy
[112,113] is employed, in which the Bekenstein?Hawking
entropy is considered to be a nonextensive Tsallis entropy.
That is, the Bekenstein?Hawking entropy is assumed to
satisfy the Tsallis composition rule. In addition, a logarithmic
formula of the original R?nyi entropy is used for the
modified R?nyi entropy [112,113], where the logarithmic formula
is obtained from the Tsallis composition rule. For details of
the modified R?nyi entropy, see Refs. [112,113]. The
physical origin of the modified R?nyi entropy is likely unclear
at the present time. However, it is worthwhile to examine
cosmological models based on the holographic equipartition
law from various viewpoints. Note that the choice of
temperature does not affect main results in the present study,
because the extra driving term discussed here is related to
Nsur = 4SH / kB from Eq. (23) and is independent of the
temperature [137].
7 Conclusions
Recently, a novel type of R?nyi entropy on a blackhole
horizon was proposed by Bir? and Czinner [112], in which a
logarithmic formula of the original R?nyi entropy was used and
the Bekenstein?Hawking entropy was regarded as a
nonextensive Tsallis entropy. In this study, the modified R?nyi
entropy has been applied to the holographic equipartition law
proposed by Padmanabhan [92], to investigate cosmological
models based on the holographic equipartition law.
Consequently, the acceleration equation, which includes an extra
driving term, can be derived in a homogeneous, isotropic, and
spatially flat universe. The extra driving term is a logarithmic
formula which can explain the accelerated expansion of the
late universe, as for effective dark energy.
Under two specific conditions, the extra driving term is
found to be the constantlike and H 2like terms, respectively.
In particular, when a specific condition is satisfied, the extra
driving term is found to be constantlike, i.e., ?K /2, as if it is a
cosmological constant . In the present model, ?K H02 is
required to obtain the constantlike term from the most severe
constraint. In other words, the constantlike term must be
extremely small when the extra driving term behaves as if it is
constant. The present model may imply that the cosmological
constant is related to a small deviation from the Bekenstein?
Hawking entropy. In addition, interestingly, the order of the
constantlike term is naturally consistent with the order of the
cosmological constant measured by observations, because
the specific condition constrains the value of the
constantlike term. The present model may provide new insights into
the cosmological constant problem.
This paper focuses on the derivation of the extra driving
term and the specific condition for the constantlike term.
Accordingly, general solutions for the present model that
includes a logarithmic driving term have been separately
studied in Appendix A. To solve the cosmological
equations, the present model is assumed to be a particular case
of (t )CDM models. From the obtained solution, the
background evolutions of the universe in the present model are
found to agree well with those in the standard CDM model
when both 2 h /H? 2 and 2 h are small. For lower redshift,
the present model gradually deviates from the CDM model,
due to the logarithmic term. Therefore, the present model can
be distinguished from the standard CDM model.
Acknowledgements The author wishes to thank H. Iguchi and V. G.
Czinner for very valuable comments.
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Appendix A: General solutions for the present model
and background evolutions of the late universe
So far, a constant term under a specific condition in the
present model has been focused on, without tuning h given
by Eq. (54). However, in general, the present model has a
logarithmic driving term. Accordingly, in this appendix,
general solutions for the present model are examined,
assuming that it is a particular case of (t )CDM models. Using
the solution, background evolutions of the late universe are
briefly observed. In the following, p is set to be zero for
nonrelativistic matter. In addition, h is considered to be
a free constant parameter (i.e., a kind of density parameter
for effective dark energy), as for in the standard CDM
model.
1. General solutions for the present model
In this subsection, the general solution for the present model
is examined. To this end, the present model is assumed to be a
particular case of (t )CDM models, as shown in Eqs. (44)?
(46). From Eq. (47), the extra driving term f (H ) is written
as
Substituting H = H? H0 and a = a? a0 into Eq. (A5), and
arranging the resultant equation, we have
In addition, a parameter N is defined by
da?
N ? ln a? , and therefore d N = a .
?
Using Eq. (A11), Eq. (A10) can be written as
This solution is given by
= ?
h is constant, Eq. (A12) can be integrated as
f (H ) = H 2 1 ?
When f (H ) = ?K /2, the solution is given by Eq. (53). The
solution method for the present model is partially based on
Refs. [82?85]. (When f (H ) is a power series of H and H? ;
see, e.g., the recent work of G?mezValent et al. [45?47] and
Basilakos et al. [80,81].)
Using a? /a = H? + H 2, coupling Eq. (44) with Eq. (45)
?2, and setting p = 0, we obtain
From Eq. (A3), we have (d H/da)a given by
Substituting Eq. (A1) into Eq. (A4) gives
a =
1 ?
H 2 1 ? ln[1 +?K(?/KH/2H2)] ?
H 2 ?
where the following equation, given by Eq. (55), has also
been used:
Note that, from Eq. (54), h is defined by
and the normalized Hubble parameter H? is defined as
Similarly, the normalized scale factor a? is defined as
? 13 li 1 + 2H? 2h = ?N + C = ? ln a? + C, (A14)
where C is an integral constant and ?li? represents the
logarithmic integral [138,139] defined as
li(x ) ?
From Eqs. (A8) and (A9), the present values of H? and a? are
1. Substituting H? = 1 and a? = 1 into Eq. (A14), the integral
constant C can be written as
C = ? 31 li(1 + 2 h ). (A16)
Substituting Eq. (A16) into Eq. (A14), and solving the
resultant equation with respect to a? , we obtain
a? = exp li 1 + 2H 2h ? li(1 + 2 h ) . (A17)
?
This equation is the general solution for the present model.
The relationship between a? and H? for each h can be
calculated from Eq. (A17). Equations (A17) and (53) approach
the same approximate equation, respectively, when both
2 h /H? 2 1 and 2 h 1 and when both /H? 2 1
and 1. The two conditions, 2 h 1 and 1,
are related to the integral constants.
2. Background evolutions of the late universe
In this subsection, the background evolutions of the late
universe in the present model are briefly examined. For this
purpose, h is considered to be a free parameter. Note that,
when Eq. (A17) for h = 0 is calculated numerically, h is
set to be 10?4.
]
dL 0.0
)
c
/
H00.5
(
[
10 1.0
g
o
l
Observed data points
Fig. 2 Dependence of the luminosity distance dL on redshift z. The
solid lines represent dL for the present model. The open diamonds with
error bars are supernova data points taken from Refs. [3,4]. For the
supernova data points, H0 is set to 67.8 km/s/Mpc from the Planck
2015 results [5]
To observe the properties of the present model, the
luminosity distance is examined. The luminosity distance [140]
is generally given by
dL = (1 + z)
where the integrating variable y, the function F (y), and the
redshift z are given by
y = a? ?1,
F (y) = H? ,
z ? a? ?1 ? 1.
The relationship between a? and H? for each h is obtained
from Eq. (A17). Using this relationship, the luminosity
distance is calculated from Eqs. (A18) and (A19).
For typical results, the luminosity distances for h = 0,
0.4, and 0.8 are plotted in Fig. 2. The luminosity distance for
the present model tends to be more consistent with supernova
data points with increasing h . This result indicates that the
present model can describe an accelerated expansion of the
late universe. However, the influence of the increase in h
on dL is likely to be smaller than the influence of the increase
in in the CDM model on dL (the result for the CDM
model is not shown). To compare the two models, a temporal
deceleration parameter is examined.
The temporal deceleration parameter q is defined by
q ? ?
where positive q represents deceleration and negative q
represents acceleration. It should be noted that the q examined
here is not the nonextensive parameter discussed in previous
sections. Substituting Eq. (A3) into a? /a = H? + H 2, dividing
the resultant equation by ? H 2, and applying Eq. (A20), we
have
Fig. 3 Dependence of the temporal deceleration parameter q on
redshift z. The solid lines represent q for the present model, while the
symbols represent q for the CDM model. Note that positive q
represents deceleration and negative q represents acceleration
In addition, substituting Eq. (A1) into Eq. (A21), and
applying Eq. (A6), we obtain
The temporal deceleration parameter q for the present model
can be calculated from Eq. (A22). Note that the relationship
between a? and H? is obtained from Eq. (A17).
For the CDM model, substituting f ( H ) = /3 into
Eq. (A21), and using = /(3 H02) and H? = H / H0, we
have
where H? 2 is given by H? 2 = (1 ? )a? ?3 + . In a flat
universe, the density parameter m for matter is given by
m = 1 ? , neglecting the density parameter for the
radiation.
Figure 3 shows the dependence of the temporal
deceleration parameter q on the redshift z. As expected, q for
h = 0 and 0.2 is consistent with that for = 0 and
0.2, respectively. However, q for the present model
gradually deviates from that for the CDM model, with increasing
h , especially for low redshift. In fact, even for h = 0.2,
q gradually deviates from that for = 0.2 with
decreasing z, although the two results agree well for high redshift.
0 0.2 0.4 0.6 0.8
Fig. 4 Dependence of the parameter 2 h /H? 2 on redshift z. Note that
2 h /H? 2 is equivalent to X ? ?K /H 2 shown in Fig. 1
As shown in Fig. 4, 2 h / H? 2 increases with decreasing z,
because H? decreases with decreasing z. That is, the deviation
from the CDM model for low redshift (shown in Fig. 3) is
related to the increase in 2 h / H? 2. In addition, for h = 0.6
and 0.8, q varies from positive to negative with decreasing z,
as shown in Fig. 3. Therefore, the present model can describe
a decelerated and accelerated expansion of the late universe.
In this appendix, the general solution for the present model
has been examined, assuming that the present model is a
particular case of (t )CDM models. In addition, background
evolutions of the late universe in the present model have
been discussed. Consequently, the present model is found
to agree well with the standard CDM model, when both
2 h / H? 2 and 2 h are small. For lower redshift, the present
model gradually deviates from the CDM model, due to the
properties of the logarithmic driving term. Accordingly, the
present model examined here can be distinguished from the
CDM model.
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