Dark energy, inflation and the cosmic coincidence problem

We show that holographic dark energy could explain why the current dark energy density is so small, if there was an inflation with a sufficient expansion in the early universe. It is also suggested that an inflation with the number of e-folds $N\simeq 65$ may solve the cosmic coincidence problem in this context. Assuming the inflation and the power-law acceleration phase today we obtain approximate formulas for the event horizon size of the universe and dark energy density as functions of time. A simple numerical study exploiting the formula well reproduces the observed evolution of dark energy. This nontrivial match between the theory and the observational data supports both inflation and holographic dark energy models.

The type Ia supernova (SN Ia) observations [1,2] strongly suggest that the current universe is in an accelerating phase, which can be explained by dark energy ( a generalization of the cosmological constant) having pressure p Λ and density ρ Λ such that ω Λ ≡ p Λ /ρ Λ < −1/3. There are various dark energy models rely on exotic materials such as quintessence [3,4], k-essence [5,6], phantom [7], and Chaplygin gas [8,9]. Being one of the most important unsolved puzzles in modern physics, the cosmological constant problem consists of three sub-problems; why the cosmological constant is so small, nonzero, and comparable to the critical density at the present.
In this paper we show that, in the holographic dark energy model, an inflation with a sufficient expansion explain why the current dark energy density is so small. We also suggest that the last problem, the cosmic coincidence problem, could be solved, if there was an inflation with a specific expansion. Note that, in many other dark energy models, it is not easy to explain the current ratio of dark energy density to matter energy density, because usually dark energy density and matter energy density reduce at different rates [10] for a long cosmological time scale.
It is well known [11] that a simple combination of the reduced Planck mass M P = m P / √ 8π and the Hubble parameter H = H 0 ∼ 10 −33 eV , gives a value ρ Λ ≃ M 2 P H 2 0 comparable to the observed dark energy density ∼ 10 −10 eV 4 [2]. This interesting coincidence, on one hand, is of the cosmic coincidence problem and, on the other hand, motivated holographic dark energy models. The holographic dark energy models are based on the holographic principle proposed by 't Hooft and Susskind [12,13,14], claiming that all of the information in a volume can be described by the physics at the boundary of the volume. With the base on the principle, Cohen et al [15] proposed a relation between an UV cutoff (a) and an IR cutoff (L) by considering that the total energy in a region of size L can not be larger than the mass of a black hole of that size. Saturating the bound, one can obtain where d is a constant. Hsu [16] pointed out that for L = H −1 , the holographic dark energy behaves like matter rather than dark energy. Many attempts [17,18,19,20,21,22] have been made to overcome this IR cutoff problem, for example, by using non-minimal coupling to a scalar field [20,21] or an interaction between dark energy and dark matter [22,23,24,25,26]. Li [27,28] suggested that an ansatz for the holographic dark energy density would give a correct accelerating universe, where the future event horizon (R h ) is used instead of the Hubble horizon as the IR cutoff L.
To solve the coincidence problem many attempts have been done [24,29,30,31,31,32,33,34]. An interaction of dark matter [35] with dark energy was introduced in [23,36,37]. In [32] inflation at the GUT scale with the minimal number of e-folds N ≃ 60 was suggested as a solution. In this paper we suggest a solution similar to the later. One motivation to study the cosmic coincidence problem in the context of inflationary cosmology is that if there was no inflation, there could be no 'now' (t 0 = 1.37 × 10 10 years) for the 'why now' question. According to astronomical observations and cosmological theory there are at least two inflationary periods in the history of the universe. As is well known, the first inflation at the early universe with N > 60 is need to solve the problems of the standard big-bang cosmology. This inflation is often assumed to be related to vacuum energy of a scalar field (inflaton). The second inflation (re-inflation) is a period of an accelerated expansion today due to dark energy. (Usually, the first inflation is related to a phase transition of the inflaton and has a different origin from that of the re-inflation due to dark energy. In this paper we assume this case. ) Thus, we assume that in the universe there are the inflaton, holographic dark energy, radiation and matter (mostly, cold dark matter). We also assume that after reheating inflaton energy decays to radiation perfectly. During the first inflation holographic dark energy is diluted exponentially. In this work we suggest that if there is holographic dark energy in the universe, the first inflation with N ≃ 65 leads to onset of the second inflation at the time t a = O(10 9 ) years as observed, and, hence, the inflation solves the cosmic coincidence problem in the context of holographic dark energy.
In this paper we consider the flat (k = 0) Friedmann universe which is favored by observations [38] and described by the metric where R(t) is the scale factor. In the holographic dark energy model a typical length scale of the system with the horizon is given by the future event horizon which is a key quantity. It is a subtle task to obtain an explicit form for R h (t), because R h (t) depends on the whole history of the universe after t. To tackle this problem we divide the history of the universe into two phases; the inflation (phase 1) is followed by phase 2 which are consecutive radiation dominated era (RDE; R(t) ∝ t 1/2 ) and dark energy dominated power-law accelerating era (DDE; R(t) ∝ t n , n > 1), respectively. For simplicity, we ignore the matter dominated era (MDE) as often done in an order of magnitude estimate in cosmology. ( In appendix, we perform a similar calculation with MDE. The main results are similar.) 1) inflation phase ( t i ≤ t < t f ) The inflation starts at t = t i and ends at t f . The scale factor evolves in this phase as follows where R i is the initial scale factor at t = t i and H i = M 2 i /( √ 3M P ) is the Hubble parameter with the energy scale M i of the inflation. Hence, the number of e-folds of expansion N ≡ H i (t f − t i ).
2) power-law expansion phase ( t f ≤ t < ∞) This phase consists of RDE ( t f ≤ t < t a ) followed by DDE ( t a ≤ t < ∞). The universe starts to accelerate at an inflection point t = t a , i.e.,R(t a ) = 0. We assume that the scale factor evolves in this phase as where α ≃ (t f /t a ) 1/2 is a constant. The scale factor R(t) grows as t 1 2 during the RDE and as t n+ 1 2 during the DDE later. R(t) of this form gives a smooth transition from RDE to DDE. Note that R(t) for each era is well-known and can be derived from the Friedmann equation depending on the dominant energy source. The power-law acceleration is a generic feature of DDE if d > 1. (Alternatively, one can divide this phase into RDE and DDE and choose the scale factor as R(t) ∝ (t/t f ) 1/2 and R(t) ∝ (t/t a ) n for RDE and DDE, respectively. This choice gives almost the same results except for a slightly decreasing R h as t → t a . Thus, we can use the specific form in Eq. (6) without loss of generality.) Since observational data favor d ≃ 1 [39,40] and ω Λ close to −1, the power index n = (1 + d)/(2d − 2) is much larger than 1. If we choose d = 1.0513, then n = 20. The inflection point t a is determined by the value at which the second derivative of R(t) vanishes: From R(t) we obtain R h (t) using Eq. (4). During the inflationary phase (phase 1): and, hence, the existence of DDE. This constant should be determined by the initial condition at t i . Thus the distance to the future horizon R h (t) during the phase 1 is To determine the value of C(t f ), we use an initial condition R h (t i ) for R h : Inserting Eq. (10) into Eq. (9) we can rewrite Eq. (9) as where A is a dimensionless constant which depends on the initial condition at t = t i , given by Therefore, if H i R h (t i ) > 1, i.e., A > 0, the event horizon grows exponentially during the inflation. At the same time ρ Λ decreases exponentially. This is also noted in Ref. [41], where the correction to the inflation due to holographic dark energy was investigated. It is a reasonable assumption that dark energy density at t i is comparable to other energy densities, that is, If not, we need either fine tuning or a special mechanism to make initial dark energy density parameter Ω Λ be much smaller than 1 at the extremely early universe, which is implausible. This can be also seen from the following relation [28]; which holographic dark energy model should satisfy all the time. It is O(1) for Ω Λ not too much smaller than 1.
Now consider the phase 2. Using R(t) in Eq. (6), it is straightforward to obtain the following relations, Therefore, during the phase 2 the event horizon at t is at the distance Now, the horizon distance at t = t f is Comparing Eq. (14) with Eq. (17), we have Therefore, we have Inserting this into Eq. (16) we obtain Now we have approximate analytical formulas for R h (t) for the whole history of the universe since the inflationary era. Note that R h (t) is a monotonically increasing function of time. We next consider the behaviors of R h (t) for t f < t ≪ t a . In this case, we have which is proportional to R(t). From Eq. (7) the inflection point t a is given by the initial conditions A and N to be This relation is interesting and informative. The ratio of the two time scale t a and t f is related to the initial condition. For A > 0 and N ≫ 1, this ratio explain why dark energy dominates so lately. Eq. (22) implies that t a and, hence, evolution of the universe is more sensitive to N than to A or n.
From now on all quantities are given in natural units; m P = 1. For t f ≃ 10 7 (GUT scale inflation) and N ≃ 66, a reasonable value for inflation to solve the problems of the big-bang cosmology, this equation gives observed t a ≃ 10 60 . In this way, the holographic dark energy model could solve the cosmic coincidence problem. Interestingly, Eq. (22) gives a lower bound for the energy scale of the inflation. The usual bound N > ∼ 60 for inflation returns t f < ∼ 10 12 and hence M i > ∼ 10 −7 m P ∼ 10 12 GeV . This can rule out low energy scale inflation models. On the other hand, an obvious condition t f > t P = 1 returns N < 75, where t P is the Planck time. To determine the true value of N we need to go beyond the approximation used in this work.
Let us explain more physically how our model could solve the coincidence problem. Using Eq. (11) we obtain R h (t f )/R h (t i ) ≃ e N , which means that the event horizon expands exponentially during the inflation. At the same time the dark energy density ρ Λ = 3d 2 M 2 P /R 2 h rapidly decreases (see Fig. 2); where we used ρ Λ (t i ) ≃ M 2 P H 2 i ≃ M 4 i and A ∼ O(1). This is the dark energy density just after the inflation. After the inflation, dark energy is sub-dominant, i.e., Ω Λ ≪ 1, and behaves like matter with a constant equation of state [32] In this case ρ Λ ∼ R −3(1+ω) ∼ R −2 , while the radiation energy density, decreases more rapidly than the dark energy density. (From Eq. (21) one can also see that R h (t) ∝ R(t) ∝ t 1/2 during the RDE). Here we assume an instant reheating after the inflation for simplicity. Therefore, during the RDE which should be comparable to M 4 a at t a , where M a ∼ 10 −3 eV is the observed energy scale of the universe at the inflection point t a . From the above relation, the required e-folds is which is slightly larger than the minimal N for the inflation to solve the many problems of the standard big bang cosmology. Here we have used t f ∼ M P /M 2 i ∼ M P /(10 16 GeV ) 2 . This result is comparable with heuristic arguments of Li [32,42,43]. Hence, we see again that an inflation with N ≃ 65 could solve the cosmic coincidence problem in a self-consistent manner in the holographic dark energy context.
To be more concrete we perform a numerical study using the analytic formulas to fit parameters for the inflation such as N and M i onto the observed cosmological parameters such as Ω Λ (t 0 ) and ρ Λ (t 0 ). Once we know R h (t) and R(t), it is easy to obtain ρ Λ (t) and ρ r (t) by using Eqs. (25) and (26) for a given N . We choose reasonable values M i = 10 16 GeV , A = 1 and n = 20. We will show later that our results are not so sensitive to the value of A or n as long as n ≫ 1. From Eqs. (21) and (22) one can see that t a and are sensitive to N . For N = 65.7 our theory gives Ω Λ (t 0 ) = 0.73, ω Λ = −0.876, and ρ Λ (t 0 ) ≃ 2.4 × 10 −123 , which are comparable with current observations. Note that this is not a fine tuning of N . Since the expansion during the inflation is a history already happened, N is obviously a fixed value. Thus, N value is, like other cosmological parameters, something which should be predicted by a theory and then be verified by observations. One can assert fine-tuning only when a required parameter value is unnatural. Our model predicts a value N ≃ 65 which satisfies all known observational constraints and is consistent with inflation theory. Thus, the possibility of determining N is not necessary a flaw but a possible merit of our theory. Although we can not rule out N ≫ 60, interestingly, there is an asserted upper bound, N < ∼ 65 from the holographic principle [44,45,46] and from the density perturbation generation [47,48]. If this upper bound is correct, one can say the holographic dark energy can solve the cosmic coincidence problem.
On the other hand, t a = 0.072 t 0 is smaller than the observed value. This discrepancy can be attributed to approximations we used such as an instant reheating after the inflation and ignoring the matter dominated era. If we choose n = 100 instead of n = 20, N = 65.715 gives the same results. Thus, the results are not sensitive to n. We do not need a fine tuning for A too. For example, if we choose A = 10, then we need N = 63.39 to reproduce the observed universe and N = 68.01 for A = 10 −1 . As mentioned above a natural value for this dimensionless quantity A without fine tuning is O(1). Let us recall the inputs and the outputs in our theory. We have assumed that there are inflationary era, RDE, and DDE in the observed evolution of our universe and used the typical forms of R(t) for these phases. With reasonable input values for n, A (our results are not sensitive to these values) and N ≃ 65, we have obtained output values for current density parameter Ω Λ (t 0 ) and equation of state ω Λ (t 0 ) for dark energy, which are comparable with observed values. This could solve the cosmic coincidence problem. Note that assuming DDE without an appropriate inflation does not automatically explain why t a ∼ t 0 . Considering the long time scale involved (O(10 10 years)) and the difference between time dependency of the dark energy density and that of the matter density, it is remarkable that with the parameter N , and the reasonable assumptions, our analysis reproduces the observed universe with the correct order of magnitude as shown in the figures. This indicates that the holographic dark energy models with d ≃ 1 are promising candidates for a correct dark energy model.
In summary, we show that an inflation with a sufficient expansion make the current holographic dark energy density exponentially small. It is also possible that an inflation of N ≃ 65 could solve the cosmic coincidence problem without introducing an interaction with dark matter or modifying gravity. The holographic dark energy models have an intrinsic advantage over non-holographic models in that it does not need fine tuning of parameters or an ad hoc mechanism to cancel the zero-point energy of the vacuum, simply because it has no O(M 4 P ) zero-point vacuum energy from the start. Quantum field theory over-counts the independent physical degrees of freedom inside the volume. Furthermore, as suggested in this paper, the cosmic coincidence problem could be also solved if there was an inflation with N ≃ 65. All these results support not only the inflation theory but also the holographic dark energy models with d ≃ 1.

APPENDIX A: INCLUDING MDE
In this appendix, we investigate the effect of matter dominated era (MDE) on the evolution of holographic dark energy. We assume that there are a period of the inflation followed by the radiation dominated era (RDE), a slow transition from matter dominated era to dark energy dominated era (MDE+DDE) of which scale factors are given by respectively, where α is a constant. We set the transition from the radiation dominated era (RDE) to the matter dominated era (MDE) happens at t = t eq , the equipartition time. That is, ρ r (t eq ) = ρ m (t eq ), where ρ m is matter energy density. The last phase consists of the matter dominated era (MDE) ( t eq ≤ t < t a ) followed by a dark energy dominated era (DDE) ( t a ≤ t < ∞). The inflection point t a is determined by the value in which the second derivative of R(t) vanishes during the third phase: Using the scale factors in Eq. (A1), it is easy to derive the event horizon size R h and dark energy density ρ Λ by following the procedure in the main text. After some straightforward calculation we obtain R h for each phase; and used this α for following calculations. In generic holographic dark energy models, dark matter is independent of dark energy, and we need an parameter describing the nature of dark matter. We choose the observed equipartition time t eq ≃ 10 −7 t 0 (z eq ≃ 3200) for the parameter. Fig. 4 shows the results for N = 61.9. N becomes smaller compared to the case in Fig. 3 , because the matter energy density decreases slowly (ρ m ∝ R −3 ) than the radiation energy density (ρ r ∝ R −4 ). The other parameters are the same as those of Fig. 3. As assumed in the main text, including MDE in our consideration does not significantly change the results. Compared to the case without MDE (Fig. 3), Ω Λ (t) ≡ ρ Λ (t)/(ρ Λ (t) + ρ r (t) + ρ m (t)) curve is more flat and t a ≃ 5 × 10 9 years is later. These results are more consistent with observations, while ρ Λ (t 0 ) ≃ 2.3 × 10 −124 is slightly smaller than the observed value. Since our holographic dark energy density changes about 10 107 times in scale from the inflation to the present, this level of coincidence is interesting, considering the approximations we have used. To check the accuracy of our calculation using the Friedmann equation, we plot the total energy density and 3H 2 (t)M 2 P in Fig. 5. The graph shows the level of accuracy mentioned above. In the case with MDE considered in this appendix, due to the freedom of t eq , there was no guarantee that the inflation with N ∼ 65 could solve the cosmic coincidence problem. However, interestingly, it turns out that even in this case the required N value is similar to that of the case without MDE. This is due to the fact that the observed initial dark matter density is much smaller than that of radiation. Even in the worst case that MDE started just after the reheating of the inflation and there was no RDE, Eq. (26) with R ∼ t 2/3 gives a value N ≃ 2 3 ln Mi Ma ∼ 43.