aa r X i v : . [ a s t r o - ph ] O c t A Cosmic Model Parameterizing the late Universe
Xin-He Meng , , ∗ Meng Su , , † and Zheng Wang Department of physics, Nankai University, Tianjin 300071, P. R. China Astronomy Department, Harvard University, 60 Garden Street, Cambridge, MA 02138, USA Department of Astronomy, School of Physics, Peking University, Beijing 100871, P. R. China BK21 Division of Advanced Research and Education in physics, Hanyang University,Seoul 133-791, Korea and Department of physics, Hanyang University, Seoul 133-791, Korea and National Laboratory for Information Science and Technology, Tsinghua University, Beijing, 100084, P. R. China (Dated: October 29, 2018)A simple speed-up cosmology model is proposed to account for the dark energy puzzle. We condense contri-butions from dark energy and curvature term into one e ff ective parameter in order to reduce parameter degen-eracies and to find any deviation from flat concordance Λ CDM model, by considering that the discriminationbetween dynamical and non-dynamical sources of cosmic acceleration as the best starting point for analyzingdark energy data sets both at present and in future. We also combine recent Type Ia Supernova (SNIa), CosmicMicrowave Background (CMB) and Baryon Oscillation (BAO) to constrain model parameter space. Degenera-cies between model parameters are discussed by using both degeneracy diagram and data analysis includinghigh redshift information from Gamma Ray Bursts (GRBs) sample. The analysis results show that our modelis consistent with cosmological observations. We try to distinct the curvature e ff ects from the specially scalingdark energy component as parameterized. We study the linear growth of large scale structure, and finally showthe e ff ective dark energy equation of state in our model and how the matter component coincidences with thedark energy numerically. PACS numbers: 04.80.Cc,04.40Dg
I. INTRODUCTION
It is now well-established that the expansion of our uni-verse is currently in an accelerating phase, supported by themost direct and robust evidence from the redshift - apparentmagnitude measurements of the ”cosmic lighthouse” type Iasupernova [1], and indirect others such as the observations ofCosmic Microwave Background (CMB) by the WMAP satel-lite [2, 3, 4, 5, 6, 7, 8, 9, 10], and large-scale galaxy surveysby 2dF and SDSS [11, 12, 13, 15]. Under the assumption thatgeneral relativity is valid on cosmological scale, the combinedanalysis of di ff erent observation data sets indicates a spatially-flat universe with about 70% of the total energy content of theuniverse today as so called dark energy with e ff ectively neg-ative pressure responsible for the accelerating expansion (seeRef. [16] for reviews on this topic). Among multitudinouscandidates of dark energy models, the ”simplest” and theoret-ically attractive one might be the so called vacuum energy, i.e. ρ Λ = Λ / π G , where Λ is the cosmological constant, whichhas been long considered as a leading candidate and worksquite well on explaining observations through out the historyof our universe at di ff erent scales. But the origin or mecha-nisms responsible for the cosmic accelerating expansion arenot very clear. On the other hand, some authors suggest thatmaybe there does not exist such mysterious dark component,but instead the observed cosmic acceleration is a signal ofour first real lack of understanding of gravitational physics[17] on cosmic scale. An example is the braneworld theorywith the extra dimensions compactified or non-compactified ∗ Electronic address: [email protected] † Electronic address: [email protected] [18, 19, 20, 21]. Consequently, finding the di ff erent cosmo-logical implications to distinguish modified gravity modelsand dark energy scenario from observations is essentially fun-damental to physically understanding of our universe[23].Along with the matter (mainly cold dark matter) componentand possible curvature term, the mysterious dark energy domi-nates the fate of our universe (we do not consider the radiationcomponent contribution as it is supposed to be very tiny for thecurrent universe evolution, at least for the present discussioninterests). Ironically so far we do not know much to either ofthem, even full of puzzling to some extends. So any progressor reasonable understanding to each of them is undoubtedlyvaluable. Specifically, the quest to distinguish between darkenergy and modified gravity scenario and further to di ff erenti-ate cosmological constant and dynamical dark energy modelsfrom observations has become the focus of cosmology studysince it holds the key to new fundamental physics.Although we have built up a successful parametrization todescribe the properties and evolution of our universe, and inprinciple distinct dark energy models live at di ff erent sub-space of fully descriptive multi-dimension parameter space,due to serious degeneracies among di ff erent parameters, wecannot get tight enough constraints from observations byglobal fitting various observational data sets. One way to ex-tract useful information from observation data and get hintsfor fundamental physics from cosmology study is to reducethe dimension of parameter space (thus reduce the parameterdegeneracies) with particular purpose in mind without appar-ently biased input to model parametrization. Since we havenot found any evidence of inconsistency of standard Λ CDMmodel, including more parameters which describe detailedproperties of each component if the ”cosmic pie” will com-plicate the situation to constrain model parameters.In order to find deviation of dark energy equation of stateparameter w from − with prior assumption w = −
1. Onthe other hand, typically one looks for evidence of dynam-ical dark energy in the absence of spatial curvature to getbetter constraints (for an exception, see [24]). It has beenconcluded in [25] that the non-curvature assumption can in-duce critically large errors in reconstructing the dark energyequation of state even if the true cosmic curvature is on sub-percent level. These claims motivate us proposing a parame-terized dark component term to mimic the e ff ective contribu-tions from either dark energy or curvature term plus the darkenergy (It is also possible that the parameterized term we pos-tulate may be from a fundamental theory or reasonably mod-ified gravity model we are seeking), besides the conventionalmatter term.In the first step, it is reasonable to introduce only one pa-rameter which stands for any kind of deviation from standardcosmology model. In some limit case, it should be reducedto the simple four dimensional (4D) Λ CDM cosmology. Theconstraint on this parameter from observations should provideinsightful hints to further explore fundamental physics.In the next section, we propose a simple cosmicparametrization for the current universe, a parameterizedmodel for the later evolution of our universe. In section 3 wegive various cosmic probes to this model, with comparison tothe DGP model Universe[18] and the concordance model witha cosmological constant, i.e, the Λ CDM model, with the hopeto locate new features to this new model. Then in section 4,we discuss the new degeneracies between the parameters weintroduced and dark matter content. The possible constraintsfrom high redshift observations are also discussed. The lastsection devotes discussions and conclusions for the generalframework studies to this present model.
II. A PARAMETERIZED LATE UNIVERSE
Firstly we summarize some basics in Standard cosmologyModel (ScM) as a preparation for our present work. TheScM starts with a solution of Einstein’s equation in the 4DFriedmann-Robertson-Walker (FRW) spacetime with cosmoassumed full of perfect fluid in large cosmic scale, mainly de-scribed by the Friedmann equation for the Hubble parameterevolution (expansion rate of the Universe) as H = (˙ a / a ) = ρ/ , (1)where the global scale factor a(t) describes the cosmic evolu-tion history and the isotropic density ρ satisfies the fluid con-tinuum (conserved) equation. A complete expression for theHubble expansion rate, which extends the FRW solutions toinclude all cosmic components so far we know is given by(and we take conventions hereafter, that is we work in naturalunits where c = π G = a = H = Ma − − ka − + Ra − + Λ = H [ Ω m a − − Ω k a − + Ω R a − + Ω Λ ] , (2)where the subscript 0 indicating today’s value, curvature frac-tion Ω k = k / H , similarly to matter component fraction Ω m ,cosmological constant (Dark Energy) contribution Ω Λ and theradiation part Ω R = R / H can be negligible today when com-pared to the mainly dark components.In the year 2000, G. Dvali, G. Gabadadze and M. Porratiproposed a new model that can mimic the 4D Newton poten-tial (with same scaling) in short ranges while it makes a 5Dgravity model (the corresponding potential scaling di ff erentlyfrom the conventional 4D Newton potential) in long ranges. Itturns out interesting to compare this ScM to the DGP model,an extended ScM that in the simplest flat geometry case anadditional term H c contributes a cosmic scale related e ff ectthat deviates the common framework at large distances andwe hope to know when it functions. The Friedmann equationfor Hubble expansion in the DGP model (we take the self-accelerating branch solution with the plus sign in front of theroot term) reads as H − k / a ( t ) + ( H − k / a ( t )) / / r c = ρ/ H = (˙ a / a ) .In the DGP model, gravity is trapped on a four-dimensional(4D) brane world at short distances, but is able to propagateinto a higher-dimensional space at large distances. For theconvenient comparasions we take its flat geometrical form H = (˙ a / a ) = ρ/ + H c (4)where the e ff ective term H c = H / r c that we treat as a param-eter to be fitted in this work and the cross-over length scaledefined by Planck mass over a 5D scale r c = M Pl / M .Conventionally, the redshift is defined by z = / a − a − = + z . Compared with the expression of H for the power-law Λ CDM model, we have known that fromthis 4D cosmological model with cosmological constant, thecosmological observation data analysis can be nicely acco-modated / explained with curvature contribution near zero, sonamed as the concordant model. While the global data fittingsuccessful we are still left with the curiosity that whether thecosmic curvature term is really zero or it can be e ff ectively de-scribed by the accumulated e ff ects from the comsic un-knowndark components[22]?The reduction to the Λ CDM model can be also realized ina more economic form as parameterized below H ( z ) = H [( Ω m ) a − + (1 − Ω m ) a B − ] . (5)where B is a parameter to be determined by data fittings andobviously B = Λ CDM model, we callthe term including B parameter e ff ective dark energy (EDE)in the following. We can compare it with the Λ CDM modelin the flat geometry where the Hubble parameter ˜ H ( z ) is˜ H ( z ) = H [ Ω m (1 + z ) + − Ω m ] . (6) TABLE I: Physics meanings in the Friedmann evolution Eq.Functions or constants Physical meanings Terms in H ˙ a ”Expansion velocity” M Matter (dust) Ω m (1 + z ) k Curvature Ω k (1 + z ) R Radiation Ω R (1 + z ) Λ Cosmological constant Ω Λ Thus, the general case with all possible components we un-derstand so far reads as H ( z ) = ˜ H ( z ) − k (1 + z ) + R (1 + z ) . (7)Of course we can encode relevant physics in the parameter B , but in the 4D Universe with cosmological constant, eacharbitrary parameter and term separately possesses concretephysics meanings compared with the 5D DGP model. So weemploy various cosmological tests to see what physics the pa-rameter B may stand for, the e ff ective e ff ects from both cur-vature and dark energy, or curvature term only with B = ff ectsfrom the dark energy component?Among dark energy candidate models, among which themodified gravity or decaying cosmology term models can ef-fectively describe possibly dark matter interacting with darkenergy[26, 27], to which we also expect this parameterizedmodel can help. The detailed discussion on this topic is be-yond the scope of this paper.We also note that there is a long standing issue on breakingdegeneracies between curvature and dynamical dark energymodel parameters. For example, CMB lensing informationcan e ff ectively help to break such degeneracy[28, 29, 30, 31,32] with only already planned ground-based CMB polariza-tion power spectrum measurements. But the results dependon two strong assumptions, one is that the ground-based CMBsurvey will be able to remove foregrounds and systematics ata level su ffi cient to enable few percent level measurements ofthe lensing B-mode polarization power, another one is that theneutrino masses are fixed by oscillation measurements and atheoretical assumption about the neutrino mass hierarchy[32].So even with ideal future measurements on CMB lensing, ournew parametrization still has its advantages. III. OBSERVATIONAL CONSTRAINTS
In this section, we study the cosmological constraintson our model parameter spaces. There are several meth-ods which have been used or proposed to constrain cos-mological parameters in the literature, e.g. Type Ia super-nove, CMB, linear power spectrum and higher order statis-tics of large scale structure[33], Lyman-alpha forest[34],Alcock-Paczy´nski (AP) e ff ect[35], weak / strong gravitationallensing[36], Gamma ray bursts / ultra-compact radio sourcesas standard candles / rulers[37, 38, 39, 40], X-ray cluster baryon fraction versus redshift test[41], Hubble parametermeasurements on di ff erent redshift[42], cluster counting[43]and so on. In principle, in order to get self-consistent pa-rameter constraints, one should do a global fitting on wholecosmological parameter space with properly chosen observa-tional data sets. However, global fitting is time / CPU consum-ing and it is hard to analyze the degeneracies on parameterspaces.In this paper, in the first setup to look at our model param-eter space and to analyze the parameter degeneracies clearly,we use recent SNe Ia gold sample [44] and SNLS data[45],and combine with information from WMAP three year dataand SDSS analysis results in our explorations. The SNLSsample consists of 44 nearby (0.015 < z < < z < M − m = d L + . (8)Here d L is the luminosity distance in units of Mpc which iswritten as d L = + z √| Ω k | S p | Ω k | Z z dz ′ H ( z ′ ) / H ! (9)where S is defined as S ( x ) = sin( x ) for a closed universe, S ( x ) = sinh( x ) for an open universe and simply S ( x ) = x withnon-curvature universe.To further break the parameter degeneracies, it is useful tostudy the combined constraints with other cosmological ob-servations, we make use of the CMB shift parameter whichincludes the whole shift information of CMB angular powerspectrum. It is defined as R = Ω m √| Ω k | S p | Ω k | Z z l dz ′ H ( z ′ ) / H ! (10)where z l = R = . ± . ff erent dark energy models. Since weonly consider the shift parameter which is determined only bythe background evolution for the constraint from CMB, we donot need to include the e ff ect of the fluctuation of dark en-ergy. In this paper, by using shift parameter, we can confineourselves to considering the e ff ects of the modification of thebackground evolution alone.And we also use the information from observation ofbaryon oscillation acoustic peak which has been detected fromthe SDSS luminous red galaxy sample[13]. The quantity weuse to constrain the cosmological parameters in this paper is B m FIG. 1: The black, grey, and light grey region shows the 1, 2, and 3 σ confidence level contours of Ω m − B parameter space respectivelyon using the SNIa Gold data from Riess et al. defined as A = √ Ω m ( H ( z ) / H ) / " z √| Ω k | S p | Ω k | Z z dz ′ H ( z ′ ) / H ! / (11)where z = .
35 and A is measured as A = . ± . χ mini-mization method. It is well known that parameter estimatesdepend sensitively on the assumed priors on other parameters.In our study, we choose the allowed range of the Hubble con-stant H = ± − Mpc − resulting from the HubbleSpace Telescope Key Project with a uniform prior[47], andmarginalize over H to get two-dimensional constraints forour parameter space.Fig.1 shows confidence-level contours on Ω m − B parameterspace using the SNIa Gold sample. The black, grey, and lightgrey region shows the 1, 2, and 3 σ confidence level contoursof Ω m − B parameter space respectively with the minimum χ = .
42 occurring at Ω m = .
46, and B = .
62. We notethat the best fit point is far from standard concordance cosmol-ogy, but the 2 σ confident contour is consistent with standardconcordance cosmology. The parameter degeneracy proper-ties between model parameters determine the configurationsof constraint contours. We will analyze the degeneracy prop-erties on Ω m − B parameter space in Fig.7. We note that theconstraint results depend sensitively on the prior assumptionsthat one adopts. A strong prior can result in an overestimateon the power of a cosmological probe or make a incorrect con-straints on key parameters, bias our judgement on model se-lection, thus improperly ruling out models. Especially, it is also noted that factitious priors on H can result in stronglybiased constraints.In Fig.2, we show the constraint results from combiningCMB shift parameter with BAO from large scale structure ofgalaxies on di ff erent cosmological models. The black, grey,and light grey regions show the 1, 2, and 3 σ confidence levelcontours of EDE, Λ CDM and DGP model parameter spaces,respectively, with the minimum χ occurring at Ω m = B = Ω M - Ω Λ parameter space in Λ CDM model (middle sub-figure) and Ω M - Ω r c (bottom sub-figure) parameter space in DGP model, cannot constrain ourmodel parameter space tightly. The reason is that in Ω m − B parameter space, CMB shift parameter and BAO factor showsimilar degeneracy properties and thus cannot break the ’ba-nana’ shape of constraint contours. Fortunately, we find thatthe constraint contours on Ω m − B parameter space by usingSNIa are almost perpendicular to contours from CMB + BAOconstraints as two sets of ’mirror bananas’(see figure 3 forcombined results). That means that in our model luminosity-distance measurements from SNIa contributes considerably tothe cosmological constraints comparing with Λ CDM modeland DGP model due to di ff erent degeneracy properties shownon each parameter space.In Fig.3, we plot the results of the combined analysis ofRiess SNIa data + BAO + CMB. Again, the black, grey, andlight grey region shows the 1, 2, and 3 σ confidence level con-tours on Ω m − B parameter space respectively with the min-imum χ = .
09 occurring at Ω M = B = / BAO + CMB can get tight constrains onboth the matter content Ω M and parameter B without sig-nificant degeneracy direction. On the other hand, the twosets of constraints from SNIa / CMB + BAO are largely consis-tent with each other, indicating the feasibility of our Ω m − B parametrization as a successful way to parameterize our lateruniverse. Considering the best fit values of Ω M and Ω Λ ,however, there exist some di ff erences between the constraintresults from SNIa and CMB + BAO. Such discrepancies havealso appeared in data analysis on other cosmology models.These might imply the existence of some systematics for cos-mological observations we have used here and / or potential in-consistencies which deserve further investigations.Fig.4 shows the confidence contours of the combinedanalysis on EDE model Ω m − B parameter space combin-ing CMB and BAO constraints with SNLS data instead ofRiess gold data. The minimum χ locates at Ω M = B = + BAO are less restrictive than combining Riessgold data, but more consistent with standard flat concordancemodel. We note that the di ff erence between two best fit param-eter values is due to di ff erence between Riess gold data andSNLS data. SNLS data gives the minimum χ = .
97 oc-curring at Ω M = B = B m m ClosedOpen mr c FIG. 2: The black, grey, and light grey regions in top, middle andbottom figures show the 1, 2, and 3 σ confidence level contours of Ω m − B , Λ CDM and DGP model parameter spaces, respectively, oncombining CMB shift parameter from WMAP three years data andBAO from SDSS B m FIG. 3: The black, grey, and light grey region shows the 1, 2, and 3 σ confidence level contours of Ω m − B parameter space respectivelyon combining the SNIa Gold data, CMB shift parameter, and BAO. B m FIG. 4: The black, grey, and light grey region shows the 1, 2, and 3 σ confidence level contours of Ω m − B parameter space respectivelyon combining the SNIa SNLS data, CMB shift parameter, and BAO. tance information obtained from di ff erent methods, it is es-pecially helpful to regard the structure formation process asa basis to test our model by using e.g. gravitational lensing,galaxy cluster abundance, galaxy clustering / dynamics and theCMB ISW e ff ect. Further tests are needed to discriminateour model from cosmology models, such as DGP model. TheCMB anisotropies and matter power spectrum provide in prin-ciple suitable discriminatory tests. These tests require a de-tailed understanding of the evolution of density perturbationsin our model. Fig.5 shows the linear growth factor G(a) of Λ CDM model (red curve), DGP model (green curve) and ourmodel (black curve). The growth factor G(a) is defined bysolving the following di ff erential equation[48] dGd ln a + + d ln H d ln a ! G + G + + d ln H d ln a − Ω m ( a ) = , (12)where G = d ln( δ/ a ) / d ln a , H = ˙ a / a is the Hubble parameter.The growth history for a flat universe can be solved as G ( a ) = − + [ a H ( a )] − Z a da ′ a ′ a ′ H ( a ′ ) × " − Ω w ( a ′ ) − G ( a ′ ) . (13)For growth during the matter-dominated era, G will besmall. A reasonable approximation throughout the growthhistory even as dark energy comes to dominate has also beenshown in [48] G ( a ) = − Ω w ( a ) − a − / Z a da ′ a ′ a ′ / Ω w ( a ′ ) . (14)For any particular model of H ( a ), or Ω m ( a ) or Ω w ( a ), we canthen evaluate the growth history.The values of model parameters we chose to plot G(a) inFig.5 correspond to the combined analysis results includingCMB, BAO, and Riess gold SNIa data. We can find that ourbest fit model mimic Λ CDM linear structure formation quitewell both in the early universe and in the late universe. Just forcomparison, we also show the linear growth factor for DGPmodel. The non-linear structure formation in our model isdefinitely worth to study but it is beyond the scope of thispresent paper.In Fig.6, we plot the relative weight of EDE componentand dark matter component with respect to total energy con-tents in our universe versus redshift with best fitting parametervalue from combined analysis of SNIa Gold data, CMB shiftparameter, and BAO. We can see that the DM-EDE equalitytime happened at z ∼ Λ CDM model.
IV. PARAMETER DEGENERACY ANALYSIS
In this section, we discuss the new degeneracies on Ω m − B parameter space, where new introduced parameter B describeseither dark energy or curvature term plus the dark energy. Inthis section, we use the first year of SNLS data in our anal-ysis instead of Riess gold sample, since SNLS data set hasa relatively narrow redshift range with z . G ( a ) FIG. 5: The black, red, and green curve shows the linear growthfactor of our EDE model, Λ CDM and DGP model respectively withbest fitting parameter value from combined analysis of SNIa Golddata, CMB shift parameter, and BAO. F r a c t i on FIG. 6: The red and black curve shows the relative weight of EDEcomponent and dark matter component versus redshift respectivelywith best fitting parameter value from combined analysis of SNIaGold data, CMB shift parameter, and BAO. The DM-EDE equalitytime happened at z ∼ estimated and corrected redshifts to investigate the cosmolog-ical constraints. The redshift of the sample extends to z = > Ω m − B parameter space, but not combing toother cosmological observations to constrain parameter space.In our model, the parameter B represents the deviation fromstandard flat Λ CDM concordance model. we can easily findthat B = Λ CDM model with cosmologicalconstant as dark energy and with flat geometry of our universe,whereas B > ff ective positive curvature geometryof our universe and / or e ff ective dark energy equation of state w < −
1, namely phantom like dark energy, and B < ff ective negative curvature geometry of our universeand / or e ff ective dark energy equation of state w > −
1, namelyquintessence like dark energy. It is well known that there ex-ists significant degeneracies among Ω k , Ω m and dark energyequation of state w parameters. In the first step to explore evi-dences beyond standard cosmology model, it might be helpfuland reasonable to introduce only one parameter which col-lapses both curvature e ff ect and dark energy e ff ect into thissingle parameter, and maybe includes other unknown featuresof new physics beyond flat Λ CDM concordance model, sim-plify the degeneracy relations, thus make the signal of devia-tion from flat Λ CDM model easily spotting out.It is known that the CMB data alone cannot constrain wellthe dynamics of dark energy. Additional information fromlarge scale structure of galaxies helps to tight on dark energyconstraints mostly because they provide a tight limit on Ω m ,which in turn helps to constrain the properties of dark energydue to breaking the degeneracy between Ω m and the equationof state of dark energy in cosmological observable quantities.Here, we concentrate our study on parameter degeneracies inluminosity / angular diameter distance since it can be clearlyand easily understood and it can also give rise to the mostdirect constraints on dark energy models. With flat universeassumption, the luminosity distance can be written as d L = c (1 + z ) Z z dz ′ H [ Ω m (1 + z ′ ) + (1 − Ω m )(1 + z ′ ) − B ] . (15)The degeneracies between B , Ω m and H are clearly seen inthis integral.In order to see the degeneracy between the parameters B and Ω m , in Fig.7 we present the degeneracies in lumi-nosity distance on the Ω m − B parameter plane at di ff er-ent redshifts. The di ff erent color bands describe the pa-rameter spaces of Ω m , B where given the variation of d L isin between ±
1% for z = . . + CMB + BAO before. One can find that, thedegeneracy between Ω m and B varies with the redshift, whichin turn implies that combining the information of d L at di ff er-ent redshifts can indeed helps broken such a degeneracy. Thisis the ideal case for showing the degeneracy between the pa-rameters Ω m and B for di ff erent redshifts, because we fix thenuisance parameter H instead of marginalizing it as we didin fitting procedure. Figure 8 is the results coming from thedata fitting of GRBs (including high redshift information up to z ∼
6) and SNLS with information from much lower redshiftrange. We can find that the rotation of degeneracy directionfrom low redshift to high redshift showing in the plot can beexplained by the degeneracy analysis on Fig. 7. The trend ofdegeneracy rotation in Ω m − B parameter plane is the same forFig.7 and 8. In order to constrain the cosmological parameters Ω m and B well from only distance measurements, one needsdistance determinations for a wide range of redshifts. Or in-stead, one can break the parameter degeneracies by other cos-mological observations with di ff erent degeneracy properties FIG. 7: The di ff erent color region simble ±
1% variation aroundlines of constant d L at redshift 0.1 (black), 0.5(red),1(green),2(blue),3(cyan),10(magenta),1100(yellow), taking fiducial modelwith best fitting combined analysis parameter value from Goldsampe + CMB + BAO. This plot delineates the degeneracy between theparameters Ω m and B at di ff erent redshift z. shown in Fig.7. For current SNIa data, their redshift range islimited with the highest observed redshift ∼ . ∼ .
3. Due to the di ff erentdegeneracies at di ff erent redshift range, the complementarityof GRBs to SNIa is highly expected with assumption of wellcontrolled systematics of using GRBs as standard candles.We note that gravitational radiation opens another windowby providing high redshift information to constrain our model.Observations of the gravitational waves emitted from the coa-lescence of supermassive black holes with independent deter-mination of redshift through an electromagnetic counterpartcan be used as standard sirens to provide an excellent probe ofthe expansion history of the Universe, especially by high red-shift information, thus which can be used to constrain the darkenergy properties[50]. The degeneracy properties of modelparameters are the same as by using standard candle, standardruler or standard siren, as discussed in this paper. Potentially,several well measured standard sirens will be enough to giveus tight constraints on dark energy parameters.In Fig. 9, we plot the e ff ective dark energy equation of state w e ff with di ff erent choices of parameter B. The e ff ective darkenergy equation of state is determined purely by the Hubbleparamter H ( z ) , and there is a general formula that can relate H ( z ) and w e ff [51] as w DE , e ff ( z ) ≡ − + d ln( δ H / H ) d ln(1 + z ) , (16) B m FIG. 8: 1, 2, and 3 σ confidence level contours on Ω m − B parameterspace using the GRB sample and SNIa SNLS data. The red-dashedlines are the results from the GRB sample and the blue-dotted linesshow the constraints resulting from the SNIa SNLS data. w FIG. 9: E ff ective dark energy equation of state w(a). The di ff er-ent color curves correspond to di ff erent value of B, namely B = . B = . B = . B = . B = . V. CONCLUSION
We have presented a cosmic model parameterizing the lateuniverse which collapses curvature and dark energy e ff ectsinto one parameter B that may indicate any deviation fromstandard flat Λ CDM model and we find that we can not con- clude that the cosmic curvature term is constantly zero, in-stead it may contribute rich phenomenological e ff ects. In or-der to show the advantages of our parametrization, we studythe degeneracy properties between B and Ω m , emphasizingthe contribution from high redshift distance information fromGRB or gravitational waves experiments on-going and up-coming. It is well-known that deducing the number of freeparameter without significant physics lost is quite importantto constrain cosmology models and to find new physics be-hind.In this paper we also investigated the DGP cosmologicalmodel in the simplest flat geometry case with the extra di-mension contribution as an e ff ective ”cosmological constant”,compared with our parameterized model and the reduction tothe power-law Λ CDM model for the 4D real Universe. Wefind that the DGP model even in the simplest case is still aninteresting candidate for the current cosmic speed-up expan-sion mechanism at long distances, while we know that in theshort ranges the model behaves as 4D conventional gravity.We will exploit the non-compact extra dimension to see itspossible existence signatures via cosmic e ff ects in the generalDGP model later as a promising model, while we do not in-tent to discuss the quantum aspects of this model as a basictheory[54].As a generalization of the Λ CDM model with naive cos-mological constant as dark energy candidate we has parame-terized a curvature like term with new phenomenological fea-tures via numerical fittings and show the term explicit physicsmeanings when we perform the parameter B reduction di-rectly to zero or 2. It may be interesting also to study thegeneral properties of the parameterized term as the matter-energy contents in our Universe continuous equation to seewhat kind of ”matter” it may describe e ff ectively, withoutspecifying the form of the parameter. Besides, the phantomcase can be realized too, for example, the equation of state pa-rameter w = p /ρ < − B > B < w = p /ρ > − Acknowledgements
X.-H.M. is supported partly by NSFC under No. 10675062and in part by the 2nd stage Brain Korea 21 Program. M.S.acknowledges valuable discussions with Zuhui Fan, XueleiChen, Pengjie Zhang, Hongsheng Zhao, Hong Li, TongjieZhang and technical supports from Bo Liu. Besides, bothauthors XHM and MS would like to thank AS-ICTP for thehospitality where part of this work has been completed. [1] S. Perlmutter el al. Nature 404 (2000) 955; Astroph. J. 517(1999) 565; A. Riess et al. Astroph. J. 116 (1998) 1009; astro-ph / [2] D. N. Spergel et al. , Astrophys. J. Suppl. , 377 (2007).[3] L. Page et al. , Astrophys. J. Suppl. , 335 (2007).[4] G. Hinshaw et al. , Astrophys. J. Suppl. , 288 (2007).[5] N. Jarosik et al. , Astrophys. J. Suppl. , 263 (2007).[6] C. J. MacTavish et al. , Astrophys. J. , 799 (2006).[7] A. C. S. Readhead et al. , Astrophys. J. , 498 (2004).[8] C. Dickinson et al. , Mon. Not. Roy. Astron. Soc. , 732(2004).[9] C. l. Kuo et al. , Astrophys. J. , 32 (2004).[10] Planck Collaboration, arXiv:astro-ph / et al. ,astro-ph / et al. , Mon. Not. Roy. Astron. Soc. (2005) 505.[12] B. Roukema, et al., Astron. Astrophys. 382 (2002) 397[13] D.J. Eisenstein et al., Astrophys. J. 633, 560 (2005);[14] W. J. Percival, S. Cole, D. J. Eisenstein, R. C. Nichol, J. A.Peacock, A. C. Pope, and A. S. Szalay, arXiv: 0705.3323; W. J.Percival et. al., 2007, ApJ, 657, 51[15] M. Tegmark et al. , Phys. Rev. D , 123507 (2006).[16] Linder, Dark energy resource letter, 0705.4102[astro-ph]and refereneces therein; S. Weinberg, Rev. Mod. Phys. ,1 (1989); P.J.E. Peebles and B. Ratra, astro-ph / / / / / ff ayet, Phys. Lett. B 502 (2001)199[19] L. Randall, R. Sundrum, 1999, Phys. Rev. Lett. 83 (1999) 3370,hep-th / / / / / / / ,123002 (2006).[29] W. Hu, Phys. Rev. D , 023003 (2002).[30] A. Lewis and A. Challinor, Phys. Rept. , 1 (2006).[31] M. Zaldarriaga and U. Seljak, Phys. Rev. D , 023003 (1998).[32] W. Hu, D. Huterer, and K. M. Smith, Astrophys.J. 650 (2006)L13[33] M. Takada, & B. Jain, 2004, MNRAS, 348, 897; E. Sefusatti, M. Crocce, S. Pueblas, R. Scoccimarro, Phys. Rev. D 74, (2006)023522[34] U. Seljak, A. Slosar, & P. McDonald, JCAP 0610 (2006) 014[35] C. Alcock, & B. Paczy´nski, 1979, Nature, 281, 358; L. Sun, M.Su and Z. H. Fan, Chin. J. Astron. Astrophys. Vol. 6 (2006),No. 2, 155[36] D. Munshi, P. Valageas, L. Van Waerbeke, A. Heavens,astro-ph / / / / et al. , Astrophys. J. , 98 (2007)[45] P. Astier, J. Guy, N. Regnault et al., 2006, A&A, 447, 31[46] Y. Wang & P. Mukherjee, 2006, ApJ, in press, astro-ph / et al. , Astrophys. J. , 47 (2001)[48] E. V. Linder, & R. N. Cahn, arXiv:astro-ph / / ffff