Redshift drift exploration for interacting dark energy

By detecting redshift drift in the spectra of Lyman-$\alpha$ forest of distant quasars, Sandage-Loeb (SL) test directly measures the expansion of the universe, covering the"redshift desert"of $2 \lesssim z \lesssim5$. Thus this method is definitely an important supplement to the other geometric measurements and will play a crucial role in cosmological constraints. In this paper, we quantify the ability of SL test signal by a CODEX-like spectrograph for constraining interacting dark energy. Four typical interacting dark energy models are considered: (i) $Q=\gamma H\rho_c$, (ii) $Q=\gamma H\rho_{de}$, (iii) $Q=\gamma H_0\rho_c$, and (iv) $Q=\gamma H_0\rho_{de}$. The results show that for all the considered interacting dark energy models, relative to the current joint SN+BAO+CMB+$H_0$ observations, the constraints on $\Omega_m$ and $H_0$ would be improved by about 60\% and 30--40\%, while the constraints on $w$ and $\gamma$ would be slightly improved, with a 30-yr observation of SL test. We also explore the impact of SL test on future joint geometric observations. In this analysis, we take the model with $Q=\gamma H\rho_c$ as an example, and simulate future SN and BAO data based on the space-based project WFIRST. We find that in the future geometric constraints, the redshift drift observations would help break the geometric degeneracies in a meaningful way, thus the measurement precisions of $\Omega_m$, $H_0$, $w$, and $\gamma$ could be substantially improved using future probes.


I. INTRODUCTION
Since interactions are ubiquitous in nature, it is rather natural to imagine that dark energy might directly interact with cold dark matter. Actually, that there is no any direct interaction between dark energy and dark matter is an additional, strong assumption. Currently, one of the most important missions in the field of dark energy research is to provide positive/negative evidence for (i.e., certify/falsify) the scenario of interacting dark energy in the light of observational data. Synthetically using the measurements of expansion history and growth of structure to consistently test the scenario is a fairly important way. However, for the scenario of interacting dark energy, it is difficult to finely test models using the measurements of growth of structure due to some complexness, such as the diversity of the construction of covariant 4-vector interaction, the large-scale gravity instability, and the lack of abundant, highly accurate data of growth of structure. Although the progress has been made in this aspect since the parametrized post-Friedmann theoretical framework for interacting dark energy was proposed and applied [1,2], obstacle due to other factors still exists. Under such circumstance, in this work, we only consider to use the geometric measurements to constrain the interacting dark energy models; in particular, we focus on the future redshift drift data.
As a purely geometric measurement, Sandage-Loeb (SL) test will be crucial to probe the "redshift desert" (2 z 5) by directly measuring the expansion of the universe. It was firstly proposed by Sandage [3] to directly measure the variation of redshift of distant sources. Then Loeb [4] found a realistic way of detect- * Electronic address: zhangxin@mail.neu.edu.cn ing redshift drift in the spectra of Lyman-α forest of distant quasars (QSOs). The 39-meter European Extremely Large Telescope (E-ELT) in built was equipped with a high-resolution spectrograph called CODEX (COsmic Dynamics Experiment), which was designed to achieve this goal. Cosmological constraints with SL test have been studied by numerous works [5][6][7][8][9][10][11][12][13], some of which simulated 240 or 150 quasars to be observed. However, as pointed in Ref. [14], only about 30 quasars are bright enough or lying at high enough redshift for the actual observation, based on a Monte Carlo simulation using a telescope with a spectrograph like CODEX. Besides, as far as we know, in the most existing papers, the bestfit ΛCDM model to current data is chosen as the fiducial model, based on which SL test data are simulated. Thus, when these SL test data are further combined with other actual data to constrain dark energy models, tension between the simulated SL data and other actual data may occur. This is inappropriate and may not give convincing conclusion on the impact of future SL test data on cosmological constraints.
To avoid inconsistency in data, in our previous work [15], we suggested that the best-fit model in study to current actual data is chosen as the fiducial model in simulating 30 mock SL test data. To give a typical example, we only focused on the dark energy model with constant w (referred to as wCDM model). In our recent work [16], we extended the discussion to time-evolving dark energy model and explored the impact of SL test data on dark energy constraints in the future geometric measurements.
Though in Ref. [15], a preliminary SL test analysis has been made for two simple interacting dark energy models, a synthetical analysis in depth in quantifying the impact of future redshift drift data on testing different types of interacting dark energy models is still absent. This paper will provide such an analysis. In this paper, we will quantify the constraining power of future SL test data on different interacting dark energy models, and show how the SL test impacts on the parameter estimation.
For interacting dark energy models, the energy balance equations for dark energy and cold dark matter arė where ρ de and ρ c are the background energy densities of dark energy and cold dark matter, respectively. The Hubble parameter H =ȧ/a describes the expansion rate of the universe and the interacting term Q describes the energy transfer rate between dark energy and dark matter densities. We consider four typical interacting dark energy models: (i) Q = γHρ c , with which the model is called the IwCDM1 model, (ii) Q = γHρ de , with which the model is called the IwCDM2 model, (iii) Q = γH 0 ρ c , with which the model is called the IwCDM3 model, and (iv) Q = γH 0 ρ de , with which the model is called the IwCDM4 model. Here γ is the dimensionless coupling parameter, and the equation-of-state parameter w is considered to be a constant in this paper.
In fact, when the CODEX experiment is ready to deliver its redshift drift data, other future geometric measurements data will also be available. Therefore, it seems that a more meaningful issue is to ask what role the SL test will play in parameter estimation for interacting dark energy models using the future geometric measurements. To address this issue, we also simulate the future SN and BAO data based on the long-term space-based project WFIRST (Wide-Field Infrared Survey Telescope). It is quite common that program in built is changed or postponed. Besides, because science and technology develop rapidly, it is hard to well and truly quantify the percentages of the parameter estimation by SL test data. We only take WFIRST as an example. Owing to the fact that it covers the redshift desert (2 z 5), SL test is a valuable supplementary to other geometric measurements. Moreover, because of the different degeneracy orientations of SL test and other geometric measurements, it is very credible that SL test can effectively break the strong degeneracy in other geometric measurements and greatly improve the corresponding cosmological constraint results. It's necessary to point out that this analysis does not mean the necessity of combining the data from WFIRST and CODEX in the future, and that we merely explore the ability of redshift drift to break degeneracy in other geometric measurements and to improve the accuracy of cosmological constraints.

II. METHODOLOGY
Our procedure is as follows. Interacting dark energy models are first constrained by using the current joint SN+BAO+CMB+H 0 data, and then for each case the best-fit model is chosen to be the fiducial model in producing the simulated mock SL test data. The obtained SL test data are thus well consistent with the current data. Therefore, it is rather appropriate to combine the mock SL test data with the current data for further constraining the interacting dark energy models. We perform an MCMC likelihood analysis [17] to obtain O(10 6 ) samples for each model.
For current data, the most typical geometric measurements are chosen, i.e., the observations of SN, BAO, CMB, and H 0 . For the SN data, the SNLS compilation [18] with a sample of 472 SNe is used. For the BAO data, we consider the r s /D V (z) measurements from 6dFGS (z = 0.1) [19], SDSS-DR7 (z = 0.35) [20], SDSS-DR9 (z = 0.57) [21], and WiggleZ (z = 0.44, 0.60, and 0.73) [22] surveys. For the CMB data, we use the Planck distance posterior given by Ref. [23]. As dark energy only affects the CMB through the comoving angular diameter distance to the decoupling epoch (and the late-time ISW effect), the distance information given by the CMB distance posterior is sufficient for the joint geometric constraint on dark energy. We also use the direct measurement result of the Hubble constant in the light of the cosmic distance ladder from the HST, H 0 = 73.8 ± 2.4 km s −1 Mpc −1 [24].
Next, we briefly review how to simulate the SL test data. This method is just to directly measure the redshift variation of quasar Lyman-α absorption lines. The redshift variation is defined as a spectroscopic velocity shift [4], where ∆t o is the time interval of observation, and E(z) = H(z)/H 0 is given by specific dark energy models. According to the Monte Carlo simulations, the uncertainty of ∆v expected by CODEX can be expressed as [14] σ ∆v = 1.35 S/N 2370 where S/N is the signal-to-noise ratio defined per 0.0125 A pixel, N QSO is the number of observed quasars, z QSO represents their redshift, and the last exponent x = −1.7 for 2 < z < 4 and x = −0.9 for z > 4.
To simulate the SL test data, we first constrain the interacting dark energy models by using the current data combination. The obtained best-fit parameters are substituted into Eq. (3) to get the central values of the SL test data. We choose N QSO = 30 mock SL data uniformly distributed among six redshift bins of z QSO ∈ [2, 5] and typically take ∆t o = 30 yr in our analysis. The error bars are computed from Eq. (4) with S/N = 3000.
In order to quantify the power of SL test in future high-precision joint geometric constraints on dark energy, we simulate future SN and BAO data based on the longterm space-based project WFIRST using the method pre-sented in Ref. [25], and take the interacting dark energy model with Q = γHρ c as an example. We simulate 2000 future SNe distributed in 16 bins over the range z = 0.1 to z = 1.7. The observables are apparent magnitudes m i = M +µ(z i ), where M represents the absolute magnitude, and µ(z i ) is the distance modulus. We also include an additional "near sample" of 500 SNe at z ≈ 0.025. For future BAO data, we simulate 10000 mock BAO data uniformly distributed among 10 redshift bins of z ∈ [0.5, 2], with each ∆z i centered on the grid z i . The observables are expansion rate H(z) and comoving angular diameter distance d co For details, we also refer the reader to Ref. [16].

III. RESULTS AND DISCUSSION
First, we constrain the wCDM model and four typical interacting dark energy models from the current SN+BAO+CMB+H 0 data combination, and present the detailed fit results in Table I. From this table, one can clearly see that for wCDM model, w < −1 is preferred at about the 1.8σ level, while w < −1 is preferred at more than 2.2σ level for all the four interacting dark energy models. For the wCDM, IwCDM2, IwCDM3 and IwCDM4 models, Ω c h 2 can be tightly constrained, and a smaller value is preferred. But for the IwCDM1 model, Ω c h 2 cannot be well constrained, and a bigger value is more favored in this case. The coupling γ is tightly constrained in the IwCDM1 model, but its constraint is much weaker in the IwCDM2, IwCDM3 and IwCDM4 models. For IwCDM1 model, γ < 0 is preferred at about 2.1σ level, while γ < 0 is slightly favored at about 1.4σ level for IwCDM2, IwCDM3 and IwCDM4 models.
Then we combine the simulated 30-yr SL test data with current data and show the constraint results in Figs. 1-3. To give an intuitive comparison, the results from current only data are also presented. Figures 1 and 2 show the joint constraints on the wCDM, IwCDM1, IwCDM2, IwCDM3 and IwCDM4 models in the Ω m -H 0 and Ω m -w planes, respectively. Figure 3 shows the joint constraints on the four interacting dark energy models in the Ω m -γ plane. In these three figures, the 68.3% and 95.4% CL posterior distribution contours are shown. The data combinations used are the current only and the current+SL 30-yr combinations, and their constraint results are shown with red and blue contours, respectively. Here we use "current" to denote the current SN+BAO+CMB+H 0 data combination for convenience. One can clearly see that the degeneracies are well broken with the SL test data for all the models.
The 1σ errors of the parameters w, γ, Ω m , and H 0 for the five models with current only and current+SL 30-yr data are given in Table II  for the IwCDM3 model, and by 58.3% and 43.3% for the IwCDM4 model. Therefore, we can see that with a 30yr observation of the SL test the geometric constraints on dark energy would be improved enormously. For all the considered interacting dark energy models, the constraints on Ω m and H 0 would be improved, relative to the current joint observations, by about 60% and 30-40%, with the SL 30-yr observation.
We also discuss the impact of the SL test data on constraining the equation of state w and the coupling γ. The SL 30-yr observation helps improve the constraints on w by 24.4%, 20.5%, 14.6%, 10.9% and 13.1% for the wCDM, IwCDM1, IwCDM2, IwCDM3 and IwCDM4 models, respectively. The SL 30-yr observation helps improve the constraints on γ by 30.5%, 9.0%, 9.7% and 8.8% for the IwCDM1, IwCDM2, IwCDM3 and IwCDM4 models, respectively. Therefore, we find that among the four interacting dark energy models, the IwCDM1 model is the best one in the sense that the constraint results could be improved by the SL test data.
We have discussed the quantification of the impact of future SL test data on constraining interacting dark energy from current SN+BAO+CMB+H 0 observations. The results show that future SL test data can effectively break the degeneracies in the current data for interacting dark energy models and thus will provide fairly important   supplement to the other observations. In the following, we will further explore what role the SL test will play in the future geometric constraints on interacting dark energy. We take the IwCDM1 model (with Q = γHρ c ) as an example for this analysis and compare the result with that of the wCDM model. As mentioned above, we simulate future SN and BAO data based on the WFIRST mission in this analysis. Figure 4 shows the joint constraints on the wCDM model in the Ω m -H 0 and Ω m -w planes. Figure 5 shows the joint constraints on the IwCDM1 model in the Ω m -H 0 , Ω m -w, Ω m -γ, and w-γ planes, respectively. The 68.3% and 95.4% CL posterior distribution contours are shown. The data combinations used are the future only and the future + SL 30-yr combinations, and their constraint results are shown with red and blue contours, respectively. The 1σ errors of the parameters w, γ, Ω m , and H 0 for the wCDM model and the IwCDM1 model for the above data combinations are given in Table III. Note that here we use "future" to denote the data combination of future SN and BAO for convenience. It is shown , we find that in the future geometric constraints, the redshift drift observation could not break the degeneracies for the wCDM model, but could efficiently break the degeneracies for the IwCDM1 model. In the future geometric constraints, for the IwCDM1 model, the SL 30-yr observation would help improve the measurement precisions of Ω m , H 0 , w and γ by more than 75%, 15%, 65%, and 80%, respectively.
The main purpose of this work is to investigate the possible impact of future SL test data on existing geometric measurements. In the future, in case ΛCDM model is excluded, one important work is to constrain the coupling parameter γ, if we wish to find evidence for the existence of interaction between dark sectors. Therefore, it is quite meaningful to investigate the impact of future SL test data on parameter estimation for interacting dark energy models. For all considered interacting wCDM models in this work, it is shown that SL test can effectively break the existing parameter degeneracies and greatly improve the precisions of parameter estimation. The results are consistent with the cases of the ΛCDM, wCDM, and w 0 w a CDM models [15,16]. By considering more models, we can conclude that the improvement of parameter estimation by SL test data is independent of the cosmological models in the background. This shows the importance of including SL test data in future cosmological constraints.

IV. SUMMARY
In this paper, we have analyzed how the redshift drift measurement (i.e. SL test signal) would impact on parameter estimation for the interacting dark energy models. By detecting redshift drift in the spectra of Lyman-α forest of distant quasars, SL test directly measures the expansion rate of the universe in the redshift range of 2 z 5, providing an important supplement to other probes in dark energy constraints. We consider four typical interacting dark energy models: (i) Q = γHρ c , (ii) Q = γHρ de , (iii) Q = γH 0 ρ c , and (iv) Q = γH 0 ρ de .
Following our previous works [15,16], in order to guarantee that the simulated SL test data are consistent with the other geometric measurement data, we used the bestfitting dark energy models constrained by the current combined geometric measurement data as the fiducial models to produce the mock SL test data.
We showed that the SL test data are extremely helpful in breaking the existing parameter degeneracies. Compared to the current SN+BAO+CMB+H 0 constraint results, the 30-yr observation of SL test could improve the constraints for all the considered interacting dark energy models on Ω m and H 0 by about 60% and 30-40%, while the constraints on w and γ can be only slightly improved.
We also quantified the impact of SL test data on interacting dark energy constraints in the future geometric measurements. To do this analysis, we simulated the future SN and BAO data based on the long-term spacebased project WFIRST. We found that the SL test could also play a crucial role in the future joint geometric constraints, especially for the constraints on w and γ. Taking the interacting dark energy model with Q = γHρ c as an example, the 30-yr observation of SL test would help improve the measurement precision of Ω m , H 0 , w and γ by more than 75%, 15%, 65%, and 80%, respectively.