Cosmological Constraints from Hubble parameter H(z) and SN Ia observations
Hui Lin, Cheng Hao, Xiao Wang, Qiang Yuan, Ze-Long Yi, Tong-Jie Zhang, Bao-Quan Wang
aa r X i v : . [ a s t r o - ph ] D ec Observational H(z) Data as a complementarity to other cosmological probes
Hui Lin , Cheng Hao , Xiao Wang , Qiang Yuan , Ze-Long Yi , Tong-Jie Zhang , , ∗ and Bao-Quan Wang Department of Statistics, School of Mathematical Sciences,Beijing Normal University, Beijing, 100875, P.R.China Department of Applied Mathematics, School of Mathematical Sciences,Beijing Normal University, Beijing, 100875, P.R.China Key Laboratory of Particle Astrophysics, Institute of High Energy Physics,Chinese Academy of Sciences, P.O.Box 918-3, Beijing 100049, P.R.China Department of Astronomy, Beijing Normal University, Beijing, 100875, P.R.China Kavli Institute for theoretical Physics China, Institute of Theoretical Physics,Chinese Academy of Sciences(KITPC/ITP-CAS),P.O.Box 2735,Beijing 100875,P.R. China and Department of Physics, Dezhou University, Dezhou, 253023, P. R. China
In this paper, we use a set of observational H ( z ) data (OHD) to constrain the ΛCDM cosmology.This data set can be derived from the differential ages of the passively evolving galaxies. Meanwhile,the A -parameter, which describes the Baryonic Acoustic Oscillation (BAO) peak, and the newlymeasured value of the Cosmic Microwave Background (CMB) shift parameter R are used to presentcombinational constraints on the same cosmology. The combinational constraints favor an acceler-ating flat universe while the flat ΛCDM cosmology is also analyzed in the same way. We obtaina result compatible with that by many other independent cosmological observations. We find thatthe observational H ( z ) data set is a complementarity to other cosmological probes. PACS numbers: 98.80.Cq; 98.80.Es; 04.50.+h; 95.36.+x
I. INTRODUCTION
Several observations, such as the type Ia Supernovae (SNe Ia) [1, 2], Wilkinson Microwave Anisotropy Probe(WMAP) [3] and Sloan Digital Sky Survey (SDSS) [4, 5] support an accelerating expanding universe. Many cosmo-logical models such as the Quintessence [6], the Braneworld cosmology[7], the Chaplygin Gas [8] and the holographicdark energy models [9, 10] are extensively explored to explain the acceleration of the universe. One of the mostcommon candidates is composed of cold dark matter with the equation of state ω = p/ρ = 0 and a cosmologicalconstant Λ with ω = −
1. It is usually called the ΛCDM cosmology [11], which is characterized by H ( z ) = H [Ω m (1 + z ) + Ω Λ + Ω k (1 + z ) ] , (1)where Ω m , Ω Λ and Ω k are proportion of the matter density, the cosmological constant and the curvature termrespectively, and H = 100 h km s − Mpc − is the current value of the Hubble parameter. We can get the relationΩ k = 1 − Ω m − Ω Λ from Eq.(1) by setting z = 0, and Ω k >
0, Ω k = 0 and Ω k < H ( z ) data (OHD for simplicityhereafter), which are related to the differential ages of the oldest galaxies, has been used to test cosmological models[21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33].In this paper, we first examine the non-flat ΛCDM cosmology using the observational H ( z ) data, which can bederived from the derivative of redshift z with respect to the cosmic time t (i.e., d z/ d t ) [34]. Meanwhile, we do ∗ Electronic address: [email protected] the combinational analyses using observations of BAO and CMB. We obtain a compatible result with many otherindependent cosmological probes. We find that the observational H ( z ) data are effective for cosmological constraints.We organize this paper as follows: In Sec.2, we briefly overview the observational H ( z ) data, the BAO data and theCMB data. In Sec.3, we present the constraints on the ΛCDM model. In Sec.4, the conclusions and more discussionsare given. II. OHD, BAO AND CMB AS COMBINATIONAL TESTSA. The Observational H ( z ) Data
The Hubble parameter H ( z ) depends on the differential ages of the universe in this form H ( z ) = −
11 + z d z d t , (2)which provides a direct measurement for H ( z ) through a determination of d z/ d t . In the work of Jimenez et al. [35],they demonstrated the feasibility of the method by applying it to a z ∼ H and it was showed that its value was in good agreement with other independent methods. Withthe availability of new galaxy surveys, it becomes possible to determine H ( z ) at z > H ( z ) data.The data set was derived from the absolute ages of 32 passively evolving galaxies with high-quality spectroscopy.Synthetic stellar population models were used to constrain the ages of the oldest stars in the galaxy (after marginalisingover the metallicity and star formation history), similar to the work of Jimenez et al. [35]. In order to estimate thedifferential ages, these galaxies were further divided into three subsamples. The first subsample was composed of 10field early-type galaxies, after discarding galaxies for which the spectral fit indicated an extended star formation. Theages of this sample were derived using the SPEED models [42]. The second subsample was composed of 20 old passivegalaxies from GDDS. They calculated the absolute ages using the SPEED models too, obtaining harmonious resultswith the GDDS collaboration which estimated the ages using different models. The third subsample consisted of twored radio galaxies 53W091 and 53W069 [40, 43, 44]. After grouping together all the galaxies within ∆ z = 0 .
03 andexcluding unperfect ones, they computed differential ages only for those bins within 0 . ≤ ∆ z ≤ .
15. The interval∆ z = 0 .
03 is set small in order to avoid incorporating galaxies that have already evolved in age, but large enoughfor the sparse sample to have more than one galaxy in most of the bins. The lower limit is imposed so that the ageevolution between the two bins is larger than the error in the age determination. This provides a robust determinationof dz/dt . The differential ages are less sensitive to systematics errors than absolute ages [42]. Then a set of differentialages dz/dt , equivalently H ( z ), was obtained. As z has a relatively wide range, 0 . < z < .
8, these data are expectedto provide a more full-scale description of the dynamical evolution of our universe. But what a pity, the data amountis not sufficient enough and the corresponding errors are quite large [22, 23].These observational H ( z ) data have been used to constrain the dark energy potential and its redshift dependence bySimon et al. [41]. Using this data set, one can constrain various cosmological models too. Yi & Zhang [21] first usedthem to analyze the holography-inspired dark energy models in which the parameter c plays a key role. The caseswith c = 0 . , . , . c free are discussed in detail. The results are consistent with some other independentcosmological tests. Samushia & Ratra [22] used the data set to constrain the parameters of ΛCDM, XCDM and φ CDMmodels. Wei & Zhang [23] compared a series of other cosmological models with interaction between dark matter anddark energy. And they find that the best models have an oscillating feature for both the Hubble parameter and theequation of state.
B. The BAO Data
The acoustic peaks in the CMB anisotropy power spectrum has been found efficient to constrain cosmologicalparameters [3]. As the acoustic oscillations in the relativistic plasma of the early universe will also be imprinted on tothe late-time power spectrum of the non-relativistic matter [45], the acoustic signatures in the large-scale clusteringof galaxies yield additional tests for cosmology.Using a large spectroscopic sample of 46748 luminous red galaxies covering 3816 square degrees out to z = 0 . A -parameterwhich is independent on H , A = √ Ω m z [ z E ( z ) 1 | Ω k | sinn ( p | Ω k |F ( z ))] / , (3)where E ( z ) = H ( z ) /H , z = 0 .
35 is the redshift at which the acoustic scale has been measured, the function sinn(x)is defined as sinn(x) ≡ sinh(x) if Ω k > k = 0;sin(x) if Ω k < , (4)and the function F ( z ) is defined as F ( z ) ≡ Z z dzE ( z ) . (5)Eisenstein et al. [18] suggested the measured value of the A -parameter as A = 0 . ± . C. The CMB Data
The shift parameter R is perhaps the most model-independent parameter which can be derived from CMB dataand it does not depend on H . It is defined as [46, 47] R = √ Ω m p | Ω k | sinn[ p | Ω k |F ( z r )] , (6)where z r = 1089 is the redshift of recombination. From the three-year result of WMAP [20], Wang & Mukherjee [19]estimated the CMB shift parameter R and showed that its measured value is mostly independent on assumptionsabout dark energy. The observational result is suggested as R = 1 . ± .
03 [19].BAO and CMB have been widely used to do combinational constraints on the cosmological parameters. In thework of Guo et al. [48], BAO was used to constrain the Dvali-Gabadadze-Porrati (DGP) braneworld cosmology anda closed universe is strongly favored. Wu & Yu [49] combined BAO with CMB to determine parameters of a darkenergy model with ω = ω / [1 + b ln(1 + z )] . They suggested that a varying dark energy model and a crossing with ω = − ω is very likely less than -1. Pires et al. [50] combined BAO and thelookback time data to make a joint statistic analysis for the DGP braneworld cosmology and they suggested a closeduniverse. Wang & Mukherjee [19] used the R parameter, combing several other cosmological probes including BAO,to derive model-independent constraints on the dark energy density and the Hubble parameter. III. CONSTRAINTS ON THE Λ CDM COSMOLOGY
In this paper, we combine the observational H ( z ) data with BAO and CMB to make a constraint on the ΛCDMcosmology. The best-fit parameters can be determined through the χ minimization method. For the non-flat ΛCDMcosmology, we assume a prior of h = 0 . ± .
08 suggested by the Hubble Space Telescope (HST) Key Project [51]. Thuswe have only two free parameters, i.e., Ω m and Ω Λ . We get the fitting results Ω m = 0 . ± .
34 and Ω Λ = 0 . ± . χ -value per degree of freedom χ /d.o.f = 8 . /
7. The best-fit results suggest Ω k = 0 .
29. We plot theconfidence regions in the Ω m − Ω Λ plane in Fig.1. It seems that this constraint is quite weak and requires deeperdiscussions such as combinational analysis with other cosmological probes.In order to further explore the role of the observational H ( z ) data, we study the combinational constraints fromthe other cosmological probes. If we combine OHD and BAO, we get Ω m = 0 . ± .
02 and Ω Λ = 0 . ± .
09, with χ /d.o.f = 9 . /
8. The best-fit results correspond to a universe with Ω k = − .
04. We also plot the confidenceregions in the Ω m − Ω Λ plane in Fig.1. It is clear that the constraint from OHD+BAO is much more restrict thanusing only OHD. We also find that Ω m is constrained more effectively than Ω Λ .By combing OHD and CMB, we get Ω m = 0 . ± .
04 and Ω Λ = 0 . ± .
03, with χ /d.o.f = 9 . /
8. The best-fitvalues correspond to a universe with Ω k = − .
02. The fitting results are not too far from the result for combiningOHD+BAO, but they suggest a universe closer to being flat. We plot the confidence regions in the Ω m − Ω Λ planein Fig.1 too, and the parameters are constrained more strictly than that using OHD and OHD+BAO, especially Ω Λ . OHD+BAO A c c e l e r a t i n g D e c e l e r a t i n g O pen C l o s ed OHD+CMBOHD+BAO+CMB Ω m Ω Λ FIG. 1: Confidence regions in the Ω m − Ω Λ plane for a non-flat ΛCDM universe, for different observational data sets as labeledin the figure. The shaded regions from inner to outer stand for confidence levels at 68.3%, 95.4% and 99.7% respectively.For a comparison, confidence regions for considering only the observational H ( z ) data are plotted with solid lines. And thesolid lines from inner to outer correspond to confidence levels at 68.3%, 95.4% and 99.7% respectively. The dash-dot straightline represents a flat universe which satisfies Ω k = 0, i.e., Ω m + Ω Λ = 1. The dashed straight line is the critical line for anaccelerating and a decelerating universe.TABLE I: Fitting results for a non-flat universeTest Ω m Ω Λ Ω k χ /d.o.f OHD 0 . ± .
34 0 . ± .
57 0.29 8.94/7OHD+BAO 0 . ± .
02 0 . ± .
09 -0.04 9.01/8OHD+CMB 0 . ± .
04 0 . ± .
03 -0.02 9.06/8OHD+BAO+CMB 0 . ± .
02 0 . ± .
02 -0.01 9.74/9
And the tendency of the confidence regions keeps along with the dash-dot straight line which corresponds to a flatuniverse. Note Fig.4 in the work of Su et al. [52] for more comparisons with results from GRBs.If we combine OHD, BAO and CMB, we get Ω m = 0 . ± .
02 and Ω Λ = 0 . ± .
02, with χ /d.o.f = 9 . /
9. Thiscase corresponds to a universe with Ω k ≃ − .
01, which is a flat universe at a confidence level of 68.3%. Confidenceregions in the Ω m − Ω Λ plane are plotted in Fig.1. Both Ω m and Ω Λ are constrained more strictly. Clearly, all pointswithin the confidence region at 99.7% confidence level are near the dash-dot straight line which stands for a flatuniverse. And our confidence regions are very similar to those from BAO+CMB+GRBs [52]. All the fitting resultsdiscussed above are listed in Table I.For a comparison, we present a constraint on the flat cosmology. Thus only one free parameter needs to be fittedif the prior of h is taken too. All the fitting results are listed in Table II. All the tests suggest Ω m ≃ .
28 which isconsistent with WMAP which suggested Ω m = 0 . ± .
04 [3]. In addition, we discuss the flat ΛCDM cosmology if h is set free. All the fitting results are listed in Table III. The information on Ω m is consistent with that of WMAP [3]and the value of h is roughly concordant with the prior we have taken in the above discussions. We plot the confidenceregions in the Ω m − h plane in Fig.2. IV. IHE ROLES OF OHD AND SNE IA DATA IN COSMOLOGICAL CONSTRAINTS
The purpose of this section is to examine the role of OHD and SNe Ia data in the constraints on cosmologicalparameters. Thus we just consider the ΛCDM cosmological model with a curvature term. The likelihood for thecosmological parameters can be determined from a χ ( h, Ω M , Ω Λ ) statistic. We marginalize the likelihood functions TABLE II: Fitting results for a flat universe with a prior of h Test Ω m Ω Λ χ /d.o.f OHD 0 . ± .
04 0 .
70 9.04/8OHD+BAO 0 . ± .
02 0 .
72 9.47/9OHD+CMB 0 . ± .
03 0 .
73 10.97/9OHD+BAO+CMB 0 . ± .
02 0 .
73 10.97/10TABLE III: Fitting results for a flat universe, setting h freeTest h Ω m Ω Λ χ /d.o.f OHD 0 . ± .
07 0 . ± .
09 0 .
69 9.02/7OHD+BAO 0 . ± .
03 0 . ± .
02 0 .
72 9.23/8OHD+CMB 0 . ± .
04 0 . ± .
04 0 .
75 10.03/8OHD+BAO+CMB 0 . ± .
03 0 . ± .
02 0 .
73 10.39/9 over h by integrating the probability density P ∝ e − χ / to obtain the fitting results and the confidence regions inthe Ω M -Ω Λ plane.We further do the joint constraint using several data sets including BAO and CMB. In order to avoid double-countingthe constraints from CMB observation, we do not employ the prior of h . In Fig.3, we show the combination results ofOHD+BAO+CMB (solid lines) and SNe Ia+BAO+CMB (dotted lines) respectively. For the combined constraint, weget Ω M = 0 . Λ = 0 .
74 for OHD+BAO+CMB and Ω M = 0 .
28, Ω Λ = 0 .
74 for SNe Ia+BAO+CMB respectively.The best-fit results for OHD+BAO+CMB are almost identical to those for SNe Ia+BAO+CMB combination, andare consistent with that from WMAP five year’s results [53]. From Fig.3, it is shown that there are slight differencesbetween the confidence regions at 68.3%, 95.4% and 99.7% levels for OHD+BAO+CMB and SNe Ia+BAO+CMB.The one-dimensional probability distribution functions (PDF) p for selections of parameters Ω M and Ω Λ for thecombination data analysis are shown in Fig.4. It is also shown that there are only slight discrepancies betweenOHD+BAO+CMB and SNe Ia+BAO+CMB combination data on both PDF of Ω M and Ω Λ . Therefore, we find thatthe OHD play an almost same role as SNe Ia for the joint constraints. In the recent work by Carvalho et al.[54], the Ω m h OHD+BAOOHD+CMBOHD+BAO+CMB A cc e l e r a t i ng D e c e l e r a t i ng FIG. 2: Confidence regions in the Ω m − h plane for a flat ΛCDM universe. The most outer regions correspond to OHD, frominner to outer for confidence levels at 68.3%, 95.4% and 99.7% respectively. Regions for different observational data sets arelabeled in the figure, from inner to outer for confidence levels at 68.3%, 95.4% and 99.7% respectively. The dashed straightline is the critical line for an accelerating and a decelerating universe. Ω M Ω Λ Flat universeOHD+BAO+CMBBest fitting valueSNe Ia+BAO+CMBBest fitting value
FIG. 3: Confidence regions at 68.3%, 95.4% and 99.7% levels from inner to outer respectively in the Ω M − Ω Λ plane for a non-flatΛCDM universe. The dotted lines correspond to the results using SNe Ia+BAO+CMB, while solid lines for OHD+BAO+CMB.The crosses in the center of confidence regions indicate the best-fit values respectively. The dash straight line represents a flatuniverse with Ω k = 0. very similar conclusions are also drawn in the test of f ( R ) cosmology with the same data as our work. V. CONCLUSIONS AND DISCUSSIONS
Recent observations have provided many robust tools to analyze the dynamical behavior of the universe. Most ofthem are based on distance measurements, such as SNe Ia. It is also important to use other different probes to setbounds on the cosmological parameters. In this paper, we have followed this direction and used the observational H ( z ) data from the differential ages of the passively evolving galaxies to constrain the ΛCDM cosmology, combiningBAO and CMB. For the non-flat case, the value of χ /d.o.f from OHD+BAO+CMB is the smallest while thatfrom OHD is the largest. In other words, OHD fails to provide a more restrict constraint on this model. It is mainlydue to its lackness in quality and big observational errors which can be seen in Fig.1 of Yi and Zhang [21]. However,most of the combinational results suggest a universe with small absolute values of Ω k , i.e., close to being flat, whichcan be easily found in Fig.1. This is consistent with most observations that support a flat universe [3].For a flat universe with a prior, the value of χ /d.o.f from OHD+BAO is the smallest while that from OHD+CMBis the largest. If we set h free, the value of χ /d.o.f from OHD+CMB is the smallest while that from OHD+BAO isthe largest. From Fig.2, we can see clearly that most of the confidence regions are overlapped, in agreement with eachother very well. Meanwhile, most of the fitting results suggest an accelerating expanding universe at 99.7% confidencelevel.From the above comparison and previous works [21, 22], we find that our results from the observational H ( z ) dataare believable and the observational H ( z ) data can be seen as a complementarity to other cosmological probes. VI. ACKNOWLEDGMENTS
We are very grateful to the anonymous referee for his valuable comments that greatly improve this paper. Ze-LongYi would like to thank Yuan Qiang and Jie Zhou for their valuable suggestions. This work was supported by theNational Science Foundation of China (Grants No.10473002, 10533010), 2009CB24901 and the Scientific Research Ω M p OHD+BAO+CMBSNe Ia+BAO+CMB Ω Λ p OHD+BAO+CMBSNe Ia+BAO+CMB
FIG. 4: The one-dimensional probability distribution function (PDF) p for selections of parameters Ω M ( Top panel ) andΩ Λ ( Bottom panel ) for the combination analysis OHD+BAO+CMB (solid) and SNe Ia+BAO+CMB (dotted) respectively.
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