Constraints on a Vacuum Energy from Both SNIa and CMB Temperature Observations

Abstract

We investigate the cosmic thermal evolution with a vacuum energy which decays into photon at the low redshift. We assume that the vacuum energy is a function of the scale factor that increases toward the early universe. We put on the constraints using recent observations of both type Ia supernovae (SNIa) by Union-2 compilation and the cosmic microwave background (CMB) temperature at the range of the redshift 0.01<𝑧<3. From SNIa, we find that the effects of a decaying vacuum energy on the cosmic expansion rate should be very small but could be possible for 𝑧<1.5. On the other hand, we obtain the severe constraints for parameters from the CMB temperature observations. Although the temperature can be still lower than the case of the standard cosmological model, it should only affect the thermal evolution at the early epoch.

1. Introduction

One of the biggest cosmological mysteries is the accelerating cosmic expansion which was discovered by the observations of distant type Ia supernovae (SNIa) starting from more than 10 years ago. For the origin of the accelerating expansion, the following possibilities have been proposed: the modified gravity, such as 𝑓(𝑅) gravity [1, 2], brane-world cosmology [3], inhomogeneous cosmology [4, 5], and existence of unknown energy called as dark energy which is equivalent to a cosmic fluid with a negative pressure.

It is strongly suggested that the dark energy amounts to 70% of the total energy density of the universe from the astronomical observations such as SNIa [6–8], the anisotropy of cosmic microwave background (CMB) [9–11], and the baryon acoustic oscillation [12]. Although there are various theoretical models of dark energy (details are shown in a review [13, 14]), its physical nature is still unknown. Considering above-referenced observations, we can not exclude the constant Λ term which is the most simple model of dark energy.

The interacting dark energy with CMB photon has been discussed, whereas decaying vacuum into photon is related to a primordial light-element abundances [15] and the CMB intensity [16]. From the point of the thermodynamical evolution in the universe, a decaying vacuum energy modifies the temperature-redshift relation [17–19]. This modification affects the cosmic thermal evolution after a hydrogen recombination [20], such as formation of molecules [21, 22] and the first star [21]. In addition to this model, the angular power spectrum of CMB could be also modified [23]. In the previous analysis, thermal history at the higher redshift has been studied. However, as seen in [20, 21], the time variation of temperature at low redshift differs from the model without a decaying vacuum.

In the present study, we update the observational consistency of a vacuum energy (hereafter we denote as Λ) coupled with CMB photon. Hereafter, we call this model DΛCDM (model). To examine the consistency with observations at low redshift, we focus on the cosmic evolution comparing both the type Ia supernovae and the redshift dependence of CMB temperatures. This is because the time dependence of the CMB temperature for a DΛCDM differs from CDM model with a constant Λ term of which hereafter we call SΛCDM (model). In Section 2, formulation of DΛCDM is reviewed. In Section 3, the 𝑚−𝑧 relation is investigated for DΛCDM in a flat universe. Parameters inherent in this model are constrained from the CMB temperature observations in Section 4. Concluding remarks are given in Section 5.

2. Dynamics of the Decaying Λ Model

The Einstein’s field equation is written as follows (e.g., [24]): 𝑅_𝜇𝜈−12𝑔_𝜇𝜈𝑅=8𝜋𝐺𝑇_𝜇𝜈, (1) where 𝐺 is the gravitational constant and 𝑇_𝜇𝜈 is the energy momentum tensor. If we assume the perfect fluid, 𝑇_𝜇𝜈 is written as follows: 𝑇_𝜇𝜈=diag−𝜌,𝑝,𝑝,𝑝. (2) Here 𝜌 and 𝑝 are the energy density and the pressure, respectively. Note that we choose the unit of 𝑐=1.

The equation of motion is obtained with use of the Friedmann-Robertson-Walker metric of the homogeneous and isotropic principle: 𝑑𝑠²=−𝑑𝑡²+𝑎(𝑡)²𝑑𝑟²1−𝑘𝑟²+𝑟²𝑑𝜃²+𝑟²sin²𝜃𝑑𝜙², (3) where 𝑎(𝑡) is the scale factor and 𝑘 is the specific curvature constant. This leads to the Friedmann equations: 𝐻²≡𝑎̇𝑎²=8𝜋𝐺3𝑘𝜌−𝑎². (4)

From the conservation’s law of energy, we can obtain the equation of the energy density: 𝜌̇=−3(1+𝑤)𝑎̇𝑎𝜌, (5) where 𝑤 is the coefficients of the equation of state, 𝑝=𝑤𝜌, and defined by 1𝑤≡3forphotonandneutrino,0forbaryonandcolddarkmatter,−1forvacuumenergy. (6)

We take the component of the energy density as 𝜌=𝜌_𝛾+𝜌_𝜈+𝜌_𝑚+𝜌_Λ, (7) where 𝛾,𝜈,𝑚, and Λ indicate photon, neutrino, nonrelativistic matter (baryon plus cold dark matter), and the vacuum, respectively. Here we neglect the energy contribution of Λ on other components except for photon: the 𝜌_𝜈 and 𝜌_𝑚 evolves as 𝜌_𝜈∝𝑎⁻⁴ and 𝜌_𝑚∝𝑎⁻³, respectively. From (5), the equation of the photon coupled with Λ is written as 𝜌̇_𝛾+4𝐻𝜌_𝛾=−𝜌̇_Λ. (8) Let us define the parameter of the energy density as follows: Ω_𝑖≡𝜌_𝑖𝜌_𝑐𝑟=8𝜋𝐺3𝐻²₀𝜌_𝑖, (9) where 𝜌_𝑐𝑟 is the critical density defined by the present Hubble constant 𝐻₀. We can rewrite (8) as follows (for details also see [22, 23]): 𝑑Ω_𝛾Ω𝑑𝑎+4^𝛾𝑎=−𝑑Ω_Λ.𝑑𝑎 (10)

Models with time-dependent Λ-term have been studied as summarized in [25, 26]. We adopt the energy of the vacuum which varies with the scale factor [20–23]: Ω_Λ(𝑎)=Ω_Λ1+Ω_Λ2𝑎^−𝑚, (11) where Ω_Λ1,Ω_Λ2, and 𝑚 are constants. Note that the present value of Ω_Λ is expressed by Ω_Λ0=Ω_Λ1+Ω_Λ2.

Integrating (10) with (11), we obtain the photon energy density as a function of 𝑎 [22]: Ω_𝛾=Ω_𝛾0+𝛼𝑎^4−𝑚−1𝑎⁻⁴(),Ω𝑚≠4_𝛾0+4Ω_Λ2ln𝑎𝑎⁻⁴(𝑚=4), (12) where Ω_𝛾0=2.471×10⁻⁵ℎ⁻²(𝑇_𝛾0/2.725K)⁴ is the present photon energy density, ℎ is the normalized Hubble constant (𝐻₀=100ℎ km/sec/Mpc), and 𝑇_𝛾0 is the CMB temperature at the present epoch. We define 𝛼 as 𝛼=𝑚Ω_Λ2/(4−𝑚).

Since we assume the flat geometry (𝑘=0) in this work, we can write the condition of Ωs as follows: Ω_𝑚0+Ω_Λ1+Ω_Λ2=1. (13) From now on, we adopt the present cosmological parameters: Hubble constant ℎ=0.738 [27, 28], and the present density parameter of matter Ω_𝑚=0.2735 [11]. The present temperature is 𝑇_𝛾0=2.725K observed by COBE [29]. In our study, we search the parameter regions in the range of Ω_Λ2 and 𝑚∶10⁻⁷<Ω_Λ2<10⁻² and 0<𝑚<4. The range of 𝑚 has already been obtained from the previous analysis [23].

Here we note the range of Ω_Λ2 in the present analysis. Kimura et al. [20] found that the radiation temperature (i.e., the energy density of radiation) in DΛCDM model becomes numerically negative at a point of 𝑎<1 when 𝑚 and/or Ω_Λ2 is too large. To avoid this unreasonable situation, that is, 𝑇_𝛾<0, Nakamura et al. [23] imposed the limit of Ω_Λ2 and 𝑚 as follows: 𝛼<Ω_𝛾0(),Ω𝑚<4_Λ2<Ω_𝛾0/92(𝑚=4). (14) From the condition (14), we obtain the limit of Ω_Λ2 as 4.9×10⁻⁷ for 𝑚=4. We can consider that the effect of Ω_Λ2 is negligible when Ω_Λ2 is smaller than 10⁻⁷.

3. SNIa Constraints

The cosmic distance measures depend sensitively on the spatial curvature and expansion dynamics of the models. Therefore, the magnitude-redshift relation for distant standard candles are proposed to constrain the cosmological parameters.

To calculate the effects of the comic expansion at low-𝑧, we calculate the theoretical distance modules, 𝜇_th=𝑚_𝐵−𝑀=5log𝑑_𝐿+25, (15) where 𝑚_𝐵,𝑀, and 𝑑_𝐿 are the apparent and absolute magnitudes, and the luminosity distance, respectively. Here 𝑑_𝐿 is related to the radial distance 𝑟 in the metric as follows [24]: 𝑑_𝐿=(1+𝑧)𝑟. (16) We can obtain 𝑟 as follows: 𝑟=𝑑𝑡𝑎(𝑡)=1𝐻₀𝑑𝑧𝐸(𝑧), (17) where 𝑧 is the redshift defined by 𝑎=1/(1+𝑧). Now 𝐸(𝑧) is defined as 𝐸(𝑧)²=𝐻(𝑧)𝐻₀²=Ω_𝑚0(1+𝑧)³+Ω_Λ1+Ω_Λ2(1+𝑧)^𝑚. (18) Then, (16) is rewritten as follows: 𝑑_𝐿=(1+𝑧)𝐻₀𝑑𝑧𝐸(𝑧). (19) As a consequence, the theoretical distance modules are calculated by (15) and (19).

Recently, the Supernovae Cosmology Project (SCP) collaboration released their Union2 sample of 557 SNIa data [8]. The Union2 compilation is the largest published and spectroscopically confirmed SNIa sample. The observations are in the redshift range of 0.01<𝑧<2. For the SNIa data set, we evaluate 𝜒² value of the distance modules, 𝜒²=^𝑁_𝑖=1𝜇_th,𝑖−𝜇_obs,𝑖²𝜎²_𝜇obs,𝑖+𝜎²_𝑣,𝑖, (20) where 𝜇_obs,𝑖 and 𝜇_th,𝑖 are the observed and the theoretical values of the distance modules. 𝜎_𝑣 is the dispersion in the redshift due to the peculiar velocity 𝑣 is written as: 𝜎_𝑣=𝑣𝑐𝑑𝜇_th.𝑑𝑧 (21) We adopt 𝑣=300km sec⁻¹ [40]. The total number of the united sample 𝑁 is 557 for the present analysis.

We confirm that Ω_Λ2 has done some contribution to change the 𝜇−𝑧 relation. When Ω_Λ2 increases, the expansion rate in the universe decreases, which is similar to the behavior of the matter dominant universe in the Friedmann model.

Figure 1 indicates the relation between the distance modules and the redshift against the SNIa observations in terms of SΛCDM and DΛCDM with several values of 𝑚 for a fixed value of Ω_Λ2=10⁻². As the value of 𝑚 increases, 𝜇 tends to decrease because of the increasing cosmic expansion rate. We recognize that 𝑚 is not seriously effective to change the 𝜇−𝑧 relation. Even if we chose the larger value of 𝑚>4, the effect is still small.

Figure 1 

Illustration of the magnitude-redshift relation in SΛCDM model (𝑚=0) and DΛCDM with ΩΛ2=10−2 compared with SNIa observations (cross-shaped ones with error bars). Note that the theoretical curve for 𝑚=3 has no difference with that of 𝑚=0.

Figure 2 shows the allowed parameter region due to the 𝜒² fitting of (20) in DΛCDM from SNIa constraints. We obtain the minimum value of 𝜒²=677 (the reduced 𝜒²_𝑟 is defined to be 𝜒²_𝑟=𝜒²/𝑁≃1.215, where 𝑁 is the degree of freedom). The best fit parameter region of Ω_Λ2 for 𝑚<4 is investigated as Ω_Λ2<4.5×10⁻³ at 1𝜎 confidence level (C.L.). Since the parameter region is very loose compared with the previous analysis [22, 23], this result implies that the decaying-Λ has minor effects on the expansion rate at 𝑧<2.

Figure 2 

Constraints on the 𝑚−ΩΛ2 plane from the recent observations of CMB. The lines indicate upper bounds of 1, 2, and 3𝜎 confidence level. The dashed and solid lines are the limits deduced from SNIa and CMB temperature observations. The shaded region shows the excluded region obtained from [23].

4. Temperature Constraints

It has been shown that the decaying-Λ term affects the photon temperature evolution at the early epoch [20]. In this section, we study about the consistency of DΛCDM with recent temperature observations.

The temperature evolution is obtained from (10). Following the Stefan-Boltzmann Law, 𝜌_𝛾∝𝑇⁴_𝛾, the relation between the photon temperature and the cosmological redshift is obtained as follows [22, 23]: 𝑇_𝛾(𝑧)=𝑇_𝛾0()Ω1+𝑧1+^𝛾ΛΩ_𝛾0^1/4,Ω_𝛾Λ≡𝛼(1+𝑧)^𝑚−4−1(𝑚≠4),−4Ω_Λ2ln(1+𝑧)(𝑚=4). (22)

From (22), the temperature-redshift relation at higher-𝑧 approaches that of SΛCDM model, 𝑇_𝛾∝(1+𝑧). At lower-𝑧, the temperature evolution deviates from the proportional relation.

In the meanwhile, recently, the temperatures at higher-𝑧 are observed using the molecules such as the fine structure of excitation levels of the neutral or ionized carbon [30–34] and the rotational excitation of CO [35–37] and Sunyaev-Zel’dovich (S-Z) effect [38, 39]. Furthermore, there are a lot of temperature observations for the lower-𝑧 owing to the S-Z effect as shown in Table 1. Therefore, we expect to constrain more severely the allowable region on the 𝑚−Ω_Λ2 plane from the new temperature observations.

Table 1 

Observational temperatures obtained from molecular excitation levels and S-Z effects.

Figure 3 illustrates the temperature evolution in SΛCDM and DΛCDM model with Ω_Λ2=10⁻². It is shown that the larger value of 𝑚 results in the lower-𝑇_𝛾. When we adopt 𝑚=0.15, the theoretical curve of 𝑇_𝛾 should be inconsistent with observations even for the temperature of the largest uncertainty [32]. We can conclude that the deviation from SΛCDM model might be small for 𝑚<0.1.

Figure 3 

Illustration of the temperature evolutions compared with CMB observations. The left panel is the results at 𝑧>0.6. The right is same, but at 𝑧<0.6.

Using the recent observations of the CMB temperature, whose accuracy has been improved quantitatively, we can put severe constraints on the temperature evolution in DΛCDM model. We calculate 𝜒²-analysis as follows: 𝜒²=^𝑁_𝑖𝑇_𝛾(𝑧)−𝑇_obs,𝑖²𝜎²_𝑖, (23) where 𝑇_𝛾(𝑧) is theoretical temperature calculated by (22) and 𝑇_obs,𝑖 the observational data as shown in Table 1. 𝜎_𝑖 is the uncertainty of the observation. 𝑁=25 is the number of the observational data. Note that the result of Srianand et al. [32] does not have best-fit value. Therefore, we assume the best-fit value is the same as the mean value, 𝑇_Srianand=10.0±4.0K.

In Figure 2, we show the limits in the 𝑚−Ω_Λ2 plane calculated by (23). The shaded region indicates the parameter regions which should be excluded and are from (14). We can obtain the allowed parameter range as follows: Ω_Λ2<6.1×10⁻⁴at1𝜎C.L.,1.7×10⁻³at2𝜎C.L. (24) Although temperature constraints give the upper bound, we can obtain the parameter range which is more severe than that obtained in the previous temperature constraint [22].

Finally, we identify the influence of temperature observations at 𝑧<0.6 by S-Z effect. We also calculate the 𝜒²-fitting (23) using observations at 𝑧>0.6 and 𝑧<0.6, separately. As the results, we obtain the upper bound of 1𝜎(2𝜎) C.L. as follows: Ω_Λ2<2.8×10⁻³4.7×10⁻³ΩbyS−Zat𝑧<0.6,_Λ2<5.5×10⁻⁴1.9×10⁻³byCI,COat𝑧>0.6. (25) These results suggested that the temperature observations at a higher-𝑧 are important to constrain DΛCDM model.

5. Concluding Remarks

We have investigated the consistency of the decaying-Λ model with both the 𝜇−𝑧 relation of SNIa and CMB temperature. We obtain the upper bound of our model parameters from both observations. First, we have constrained our model from SNIa: we can obtain the upper bound of parameters, which region is wider compared with previous constraints [22, 23]. From this result, a decaying-Λ has minor effects on the cosmic expansion at low-𝑧 and its parameters cannot be well constrained.

On the other hand, we can obtain more severe parameter constraints from CMB temperature. We acquire the limit of Ω_Λ2 as Ω_Λ2<1.7×10⁻³ at 2𝜎 C.L. From this result, the temperature can become lower by a few percentages. That should affect the hydrogen recombination. We note that we cannot get the bound of 𝑚. Because change in Ω_Λ2 cancels the effects of the variation of 𝑚.

Our parameter is mainly obtained by the observations at 𝑧>2. It has been suggested that the observations of the temperature at higher redshift is important to constrain the cosmological models with energy flow into photon [23]. The upper limit of Ω_Λ2 obtained from the present constraints is still larger than the previous constraints of Ω_Λ2<1.7×10⁻⁴ at 95.4% C.L. [23]. The analysis using the CMB temperature fluctuation of the recent observation seems to be unable to give the tight constraints. We plan to perform more detailed analysis, for instance using CMB anisotropy of the newest WMAP data [10, 11].

Acknowledgment

This work has been supported in part by a Grant-in-Aid for Scientific Research (18540279, 19104006, and 21540272) of the Ministry of Education, Culture, Sports, Science and Technology of Japan.