Dark radiation from a unified dark fluid model

We present a unified dark fluid model to describe the possible evolutionary behavior of $\Delta N_\mathrm{eff}$ in dark radiation. This model can be viewed as an interacting model for the dark sectors, in which dark matter interacts with dark radiation. We show that the evolution of $\Delta N_\mathrm{eff}$ can be nicely explained without some drawbacks, such as the blowup of $\Delta N_\mathrm{eff}$ and the non-vanishing interaction at the late time.


I. INTRODUCTION
The ΛCDM model has successfully explained many important cosmological observations such as the acceleration of the universe and the radial velocity distribution of the galaxies as well as the cosmic microwave background (CMB) fluctuations [1,2]. Besides the motivation of the theoretical completeness, from the viewpoint of the observational data there still leaves some room for the existence of physics beyond ΛCDM. Recently, the analysis of the pure CMB data shows that the effective number of relativistic degrees of freedom is N eff = 3.36 +0.68 −0.64 (95% CL) [2], which accommodates the standard model (SM) prediction of N SM eff = 3.046 [3] within 1σ range, while the combined analysis with the measurement of H 0 gives N eff = 3.62 +0.50 −0.48 (95% CL) [2], which is larger than the SM value at around 2σ level. The extra degree of freedom is usually referred to as dark radiation (DR). It is worth noticing that the extra radiating component can be extracted by the probe of the primordial deuterium and helium abundances at the big bang nucleosynthesis (BBN) epoch [4]. For instance, it has been recently shown that N eff = 3.71 +0.47 −0.45 and 3.50 ± 0.20 in Refs. [5] and [6], respectively.
Many models have been used to describe ∆N eff ≡ N eff − N SM eff . Among them, imposing a new relativistic degree of freedom beyond the SM is a straightforward way [7], but such a scenario can only explain the case in which DR is in an equal amount at the BBN and CMB epoches, namely, N BBN eff = N CMB eff . Note that there may be a tension between BBN and CMB for N eff as the current data seems to indicate that N CMB eff < N BBN eff . In order to understand such a decrease (or increase) of N eff at CMB, various subtle models have been proposed in which some interactions between DR and dark matter (DM) are assumed. For example, if heavy DM particles 1 can decay into relativistic states, the increase in DR could be interpreted; see, e.g., Refs. [9][10][11][12] for model dependent and independent analyses.
Since there is no evidence that the dark sectors are independent to each other, an interaction between DM and DR is quite possible. Models related to this possibility have been widely discussed in the literature [13][14][15][16]. However, it should be pointed out that there are still some drawbacks in these models: some of them blow up ∆N eff in the late time, which is also equivalent to the existence of non-vanishing interaction between DM and DR in the 1 For the ultra light DM candidates, some interesting properties were discussed in Ref. [8].
present. In this paper, we propose a unified dark fluid model describing both DM and DR, which can nicely yield the decrease (or increase) in ∆N eff without the above drawbacks. This paper is organized as follows. In Sec. II, we introduce the unified dark fluid model.
In Sec. III, we discuss the extra effective relativistic degree of freedom ∆N eff . Conclusions are given in Sec. IV.

II. A UNIFIED DARK FLUID
We start with a dark fluid, in which the energy density is expressed as where a is the scale factor of the universe, α is a small real number, and A and B are positive, which can be determined by the initial condition at some specific time. Note that this dark fluid can be viewed as a mixture of DM and DR. In fact, it is a special case of the new generalized Chaplygin gas (NGCG) model with the equation of state (EOS) w = 1/3 proposed in Ref. [17]. We remark that this model is also inspired by the generalized Chaplygin gas (GCG) scenario which unifies DM and dark energy (in the case of the cosmological constant) in a single fluid [18]. For α = 0 in Eq. (1), the energy density reduces to the sum of matter and radiation forms. For α being a small real number, the fluid can exhibit the behavior of both matter and radiation.
From the continuity equation,ρ + 3H(ρ + P ) = 0, the pressure of the dark fluid P dark can be derived, and then the EOS parameter of the dark fluid can be obtained to be For a small a, we have w dark ≃ 1/3, which is the same as the radiation fluid, while for a large a, the fluid behaves like matter with w dark ≃ 0. Similar to GCG, this fluid can be naturally decomposed into two interacting components with constant EOS parameters, w = 1/3 and w = 0, respectively. As a result, this unified dark fluid model can also be regarded as an interacting dark-sector model in which DM interacts with DR.
Subsequently, we can write ρ dark = ρ dm + ρ dr and P dark = P dm + P dr . By using P dm = 0 and P dr = (1/3)ρ dr , we derive where K ≡ Aa −4(1+α) + Ba −3(1+α) . Evidently, A and B can be naturally determined by the initial condition of the two components, e.g., the DR and DM densities at the present time.
The energy transfer from DM to DR in unit volume and in unit time can be derived as where the sign of α fixes the direction of the energy flow. A positive α makes the energy flow from DR to DM, whereas the negative one reverses the direction. By this definition, the energy continuity equations for DM and DR are given byρ dm + 3Hρ dm = −Q anḋ ρ dr + 4Hρ dr = +Q, respectively. Note that if ρ dr ≫ (≪)ρ dm , the energy transfer Q can be reduced to Q = −αHρ dm(dr) . This kind of the interaction simultaneously involves the two important forms Q = −αHρ dm and −αHρ dr , studied extensively in the interacting dark energy models [19]. These forms, similar to those obtained from the GCG fluid, are crucial features of the GCG-like model [18]. We remark that once Q is proportional to the Hubble expansion rate H, there is a factor of T 2 in the radiation dominated epoch. We will discuss the effect of the interactions on the time evolution of ∆N eff in the next section.
It should be pointed out that although our model is inspired by the GCG and NGCG models, there are some significant differences between the DM-DR interacting model and the DM-dark energy interacting model, in particular when the cosmological perturbations are considered. For example, for the GCG model, when it is considered as a unified model the perturbation calculations force it to be extremely close to the ΛCDM model (α < 10 −6 ) [20,21], whereas a much wider range of α is allowed, i.e., α may be of the order O(10 −1 ) [22], when it is treated as a model of vacuum energy interacting with DM. The case of the NGCG model is discussed in Ref. [23]. The primary cause is that dark energy is a non-adiabatic fluid so that how to treat its pressure perturbation is obscure to some extent. 2 Nevertheless, for the model considered in this paper, since both DM and DR are adiabatic fluids, our model can be treated as a model of unified dark fluid as well as a model 2 In the case of dark energy, since w is negative, its adiabatic sound speed c a would be imaginary due to c 2 a = dp de /dρ de = w (for the example of constant w), leading to instability in the theory. In order to fix this problem, it is necessary to assume that dark energy is a non-adiabatic fluid and impose a physical sound speed c 2 s > 0 by hand. Usually, c s is set to be the light speed as if the dark energy fluid is realized by a scalar field, which is what is done in the CAMB and CMBFAST codes. But such a treatment would also lead to some instabilities, in particular for the w = −1-crossing models and some specific interacting dark energy models. For more detailed discussions, see Ref. [24]. of DM interacting with DR. As a result, we expect that the constraints on our model from "geometry measurements" and "structure's growth measurements" will be consistent owing to the fact that both DM and DR are adiabatic fluids with well defined sound speeds and well treated pressure perturbations.

III. N eff IN INTERACTING MODELS
From the definition of N eff , the extra relativistic energy density exceeding the ΛCDM model is given by where (T ν /T γ ) = (4/11) 1/3 after the photon was heated at the e + e − annihilation epoch.
where ρ dark has been re-parameterized by the value ρ 0 dark at the present time and a dimensionless parameter r, taken around 10 −5 , which is of the same order as the radiation fractional density now.
All curves with r fixed approach the same ∆N eff at a = 1, which are sensitive to the initial condition ρ 0 dark for a 10 −2 . For α = 0.1, ∆N eff is a decreasing function, while for α = −0.1 or α = −0.3, it behaves as an increasing one. Notice that α = 0 gives the constant value of ∆N eff due to the vanishing of interacting term Q. Different choices of r will lead to different results. From the figure, it is clear that α > 0 with the energy flowing from DM to DR is favored. In Fig. 1b, we illustrate the correlations between the two parameters r and α  Fig. 1a, which implies the existence of extra energy density apart from that given by ΛCDM. In addition, the EOS parameter w eff versus a is given in Fig. 2b. In the figure, we also show the result (dashed curve) for ΛCDM. It is worth noting that the evolution of the Hubble parameter H could also provide some effect on the anisotropic CMB power spectrum. The increasing of H at the CMB epoch (log 10 a ≃ −3) for any values of α would not only suppress the damping tail to equivalently solve the anomaly of DR, but also shift the acoustic peak slightly toward a smaller angular scale (larger ℓ), while the value of the first acoustic peak could be lifted up. Typically, by taking r ≃ 5 × 10 −6 and α = 0.15, the first peak could rise about the same amount as that in the scenario of adding an additional massless sterile neutrino into ΛCDM. For a larger r, the amplitude of the power spectrum increases rapidly.
To illustrate our results, we now compare our model with other two interacting models, Models A and B, in which the energy transfers are Q A = α 1 Hρ dm and Q B = ρ dm /τ dm , respectively, with α 1 and τ dm being the free parameters. Model A is a simple interacting scenario between DM and DR, which is studied in Ref. [13], while Model B is examined in Refs. [9,10], in which the interacting term ρ dm /τ dm can be directly interpreted as the energy transferring into the DR component from the decaying of heavy particles with the life time τ dm , around the BBN epoch. Unlike our model and Model A, the energy density ρ dm for a heavy particle in Model B is unlikely to be linked with DM due to the short life time τ dm of only a few orders of seconds [10]. In Fig. 4a, we present ∆N eff as a function of the scale factor a in different models. For Model A (B), we will use α 1 = 0.03 and 0.01 (τ dm = 2000 s and 500 s) as input parameters. ρ 0 dm in Model A is identical to the DM density in the present, while for Model B we will fix the comoving energy density (ρ dm /s) = (2 × 10 −3 )MeV at BBN, with s being the entropy density at that time. In both models ρ dr × a 4 at very early time is taken to be zero as the other initial condition. We see that in Model A, ∆N eff coincides with the observation at the CMB era, but it blows up in the late time. In Model B, ∆N eff only increases at very early time and behaves as a constant after a 10 −8 . The average rate of the change in ∆N eff from BBN to CMB for our model is faster than Model B, but gently than Model A.
Moreover, both increasing and decreasing behaviors of N eff can be described in our model, which could be a potential target for probing this model in the future observations. In addition, the dimensionless relative energy transfer q ≡ |Q|/(ρ t H) with ρ t being the sum of energy densities of DM and DR is plotted in Fig. 4b for each case. With the same parameter values in Fig. 4a, q in Model A always behaves as a constant due to the crucial feature (ρ dr /ρ dm ) ≃ α [13], whereas in the late time the nonzero value of q indicates that the interaction between DM and DR is still rather strong even at present. In Model B, the region of the nonzero q centralizes at the beginning of BBN with the order of magnitude around the peak as large as order unity. In our model, |Q|/H is proportional to ρ dm and ρ dr in very early and late times, respectively, so that a nonzero value of q can only be confined in some range of time. Obviously, the behaviors of ∆N eff and q in our model are more reasonable than Model A.

IV. CONCLUSIONS
We have proposed a unified dark fluid model to understand the possible evolutionary behavior of ∆N eff in DR. Inspired by the GCG model, the dark fluid can be viewed as a scheme for the unification of DM and DR. Such a fluid behaves like radiation and matter in the radiation and matter dominated epochs, respectively. Interestingly, this model can also be regarded as an interacting model in the dark sectors as DM interacts with DR with the form explicitly obtained. Moreover, we have evaluated the evolution of ∆N eff in DR, which is favored by the current observational data for α > 0. Comparisons with the other two interacting models, Q = α 1 ρ dm H and ρ dm /τ dm , have been also given. We have shown that our predicted values of ∆N eff and q in the unified dark fluid model are more reasonable than Model A. In particular, in our model there are no drawbacks, such as the blowup of ∆N eff and the non-vanishing interaction at the late time. Clearly, more accurate analyses on N eff and its evolution in the future could help to identify if our model is a viable scenario.