Intrinsic scatter of the luminosity relation, redshift distribution of the standard candles, and the constraining capability
aa r X i v : . [ a s t r o - ph . C O ] S e p Intrinsic scatter of the luminosity relation, redshift distribution of the standardcandles, and the constraining capability
Shi Qi ∗ Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, ChinaKey Laboratory of Dark Matter and Space Astronomy,Chinese Academy of Sciences, Nanjing 210008, ChinaJoint Center for Particle, Nuclear Physics and Cosmology,Nanjing University—Purple Mountain Observatory, Nanjing 210093, China andKavli Institute for Theoretical Physics China, Chinese Academy of Sciences, Beijing 100190, China
Standard candles are one of the most important tools to study the universe. In this paper,the constraints of standards candles on the cosmological parameters are estimated for differentcases. The dependence of the constraints on the intrinsic scatter of the luminosity relation andthe redshift distribution of the standard candles is specifically investigated. The results, especiallyfor the constraints on the components of the universe, clearly show that constraints from standardcandles at different redshifts have different degeneracy orientations, thus standard candles with awide redshift distribution can self break the degeneracy and improve the constraints significantly.As a result of this, even with the current level of tightness of known luminosity relations, gamma-ray bursts (GRBs) can give comparable tightness of constraint with type Ia supernovae (SNe Ia)on the components of the universe as long as the redshifts of the GRBs are diversifying enough.However, for a substantial constraint on the dark energy EOS, tighter luminosity relations for GRBsare needed, since the constraints on the dark energy from standard candles at high redshifts arevery weak and are thus less helpful in the degeneracy breaking.
Standard candles are one of the most important toolsto study the universe. Type Ia supernovae (SNe Ia) arecurrently the maturest standard candles on cosmologicalscales, studies on which lead to the discovery of cosmicacceleration [1, 2]. Gamma-ray bursts (GRBs) have alsoattracted much attention as standard candles (see e.g. [3]and references therein). GRBs cover much wider redshiftrange than SNe Ia, but, on the other hand, have larger in-trinsic scatters in their luminosity relations, which makesthem less ideal as standard candles than SNe Ia. In thispaper, the constraints of standards candles on the cos-mological parameters are estimated for different cases.The dependence of the constraints on the intrinsic scat-ter of the luminosity relation and the redshift distributionof the standard candles is specifically investigated. Theinvestigation is done by keeping in mind the current de-velopment of the GRBs as standard candles (see e.g. [3]).In [4], a general procedure for estimating constraints ofstandard candles on cosmological parameters using mockdata was discussed and, as a result, analytical formulaefor the marginal likelihood of the cosmological parame-ters were derived. Consider a general luminosity relationof the form y = c + X i c i x i + ε, (1)where x i s are some luminosity indicators which can be di-rectly measured from observation, ε is a random variableaccounting for the intrinsic scatter σ int , of the relation,and y has the form of y = log (cid:0) πd L F (cid:1) , (2) ∗ [email protected] where d L is the luminosity distance and F may be anyphysical quantity that can be directly measured from ob-servation. Define l ( z, θ, θ ) = 2 log d L ( z, θ ) d L ( z, θ ) (3)and use l i as the abbreviation for l ( z i , θ, θ ). Ignoring themeasurement uncertainties, we have the marginal likeli-hood of the cosmological parameters θ L ( θ ) ∝ (cid:0) σ , + σ l (cid:1) − N − p , (4)where p is the number of the calibration parameterswhich include the coefficients c and the intrinsic scatter σ int . See [4] for more details.From Eqs. (3) and (4), we can see that, to estimate theconstraining capability of a sample of standard candleson the cosmological parameters (without considering themeasurement uncertainties), we only need to input theinformation1. about the luminosity relation: its intrinsic scatterand the number of luminosity indicators involved;2. about the sample: the number of the standard can-dles and their redshifts.No further detailed information is needed. This muchsimplifies the procedure and makes things transparentand clear. For example, one can immediately tell fromEq. (4) that the smaller the intrinsic scatter σ int , isand/or the larger the sample size N is, the more sen-sitive the marginal likelihood L ( θ ) is to the variation ofthe cosmological parameters θ , which means that the con-straint is tighter. The Hubble constant only contributesto l ( z, θ, θ ) a constant that is same for all the standardcandles and has no effect on σ l , so L ( θ ) is independentof the Hubble constant and we cannot directly constrainit in this way.Here, utilizing Eqs. (3) and (4), the constraining ca-pability of standard candles on cosmological parameterswas estimated using mock data. The flat ΛCDM withΩ m = 0 . p = 3 wasassumed. The constraints on (Ω m , Ω Λ ) for the ΛCDMmodel and on (Ω m , w ) for the flat w CDM model werestudied. The dependence of the constraints on the in-trinsic scatter of the luminosity relation and the redshiftdistribution of the standard candles was specifically in-vestigated. For the intrinsic scatter of the luminosityrelation, σ int , = 0 .
4, 0 .
3, 0 .
2, and 0 . . , , , , . , σ int , is, the tighter the con-straints are. For the dependence of the constraints onthe redshift distribution of the standard candles, we cansee, especially from Fig. 1, that the constraints from stan-dard candles of different redshifts show different degener-acy orientations, thus a combination of standard candlesfrom a wide redshift range can self break the degener-acy and improve the constraints significantly. This is be-cause that L ( θ ) depends on l ( z, θ, θ ) through its variancealong the redshift. Standard candles with a narrow red-shift distribution can only reflect the local redshift vari-ance of l ( z, θ, θ ), and the redshift variance of l ( z, θ, θ )has different features at different redshifts, which leadsto different degeneracy orientations of L ( θ ). In contrast,standard candles with a wide redshift distribution can re-flect the global redshift variance of l ( z, θ, θ ), the derived L ( θ ) is more sensitive to the variation of the cosmologicalparameters θ , which means tighter constraints. Such afeature of standard candles was also shown in [5, 6] byusing stripes of constant d L at different redshifts in thecosmological parameter space.The self degeneracy breaking feature of a wide red-shift distribution of standard candles means that redshiftdistribution of standard candles can play a similar role as the intrinsic scatter of the luminosity relation in de-termining the tightness of the constraints. For a givenintrinsic scatter, standard candles of same sample sizewith wider redshift distributions give tighter constraints.A loose luminosity relation combined with a wide red-shift distribution of standard candles can have compa-rable tightness of constraint with a tight luminosity re-lation combined with a narrow redshift distribution ofstandard candles. For example, in Fig. 1, the contourplots of the top right ( σ int , = 0 . . , σ int , = 0 . . , w mainly come from standard can-dles at redshifts less than, say, about 2. The universe ismatter dominated at high redshifts, where dark energydoes not contribute much in determining the evolution ofthe universe, so the constraints on the dark energy fromstandard candles at high redshifts are very weak and arethus less helpful in the degeneracy breaking.As mentioned earlier, the intrinsic scatter of the lumi-nosity relation and the redshift distribution of the stan-dard candles are chosen by keeping in mind the currentdevelopment of GRBs as standard candles. From the re-sults, we can conclude that, even with the current levelof tightness of known luminosity relations (see e.g. [3]),GRBs can give comparable tightness of constraint withSNe Ia on the components of the universe as long as theredshifts of the GRBs are diversifying enough. However,for a substantial constraint on the dark energy EOS, weneed tighter luminosity relations for GRBs. ACKNOWLEDGMENTS
This work was supported by the National Natu-ral Science Foundation of China (Grant No. 10973039,11203079, and 11373068), Key Laboratory of Dark Mat-ter and Space Astronomy (Grant No. DMS2011KT001),and the Project of Knowledge Innovation Program(PKIP) of Chinese Academy of Sciences (GrantNo. KJCX2.YW.W10). [1] A. G. Riess et al. (Supernova Search Team), Astron. J. , 1009 (1998), astro-ph/9805201.[2] S. Perlmutter et al. (Supernova Cosmology Project), As-trophys. J. , 565 (1999), astro-ph/9812133.[3] S. Qi and T. Lu, Astrophys.J. , 99 (2012),arXiv:1111.6249 [astro-ph.CO]. [4] S. Qi, (2015), arXiv:1509.07064 [astro-ph.CO].[5] G. Ghisellini, G. Ghirlanda, C. Firmani, D. Laz-zati, and V. Avila-Reese, Nuovo Cim.
C28 , 639 (2005),arXiv:astro-ph/0504306 [astro-ph].[6] C. Firmani, V. Avila-Reese, G. Ghisellini, andG. Ghirlanda, Rev.Mex.Astron.Astrofis. , 203 (2007),arXiv:astro-ph/0605267 [astro-ph]. Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m Ω Λ Ω m FIG. 1. 68 .
3% and 95 .
4% confidence regions in the (Ω m , Ω Λ ) plane for the ΛCDM model. The flat ΛCDM with Ω m = 0 . σ int , = 0 .
4, 0 .
3, 0 .
2, 0 .
1. The columnsrepresent different redshift distributions of standard candles. From left to right, 500 standard candles uniformly distributing inthe redshift range [0 . , , , , . ,
7] were used. The luminosity relation was assumed to have only one luminosityindicator involved. The upper left gray region in the figures represent the parameter space for which the universe does notexperience a big bang in the past. w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 w Ω m -3-2.5-2-1.5-1-0.50 0 0.2 0.4 0.6 0.8 1 FIG. 2. 68 .
3% and 95 .
4% confidence regions in the (Ω m , w ) plane for the flat w CDM model. The flat ΛCDM with Ω m = 0 . σ int , = 0 .
4, 0 .
3, 0 .
2, 0 .
1. The columnsrepresent different redshift distributions of standard candles. From left to right, 500 standard candles uniformly distributing inthe redshift range [0 . , , , , . ,,