The Astrophysical Journal, 605:599–606, 2004 April 20 # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.
CONSTRAINTS ON A CARDASSIAN MODEL FROM TYPE Ia SUPERNOVA DATA, REVISITED WxODZIMIERZ GODxOWSKI AND MAREK SZYDxOWSKI Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Krako´w, Poland;
[email protected],
[email protected]
and Adam Krawiec Institute of Public Affairs, Jagiellonian University, Rynek Gyo´wny 8, 31-042 Krako´w, Poland;
[email protected] Received 2003 September 26; accepted 2004 January 9
ABSTRACT We discuss some observational constraints, resulting from Type Ia supernova (SN Ia) observations, imposed on the behavior of the original flat Cardassian model and on its extension with the curvature term included. We test the models using the Perlmutter SN Ia data, as well as the new Knop and Tonry samples. We estimate the Cardassian model parameters using the best-fit procedure and the likelihood method. In the fitting procedure we use density variables for matter, Cardassian fluid, and curvature and include the errors in redshift measurement. For the Perlmutter sample in the nonflat Cardassian model we obtain a high-density or normal (m;0 0:3) universe, while for the flat Cardassian model we have a high-density universe. For sample A in the high-density universe, we find negative values of estimates of n, which can be interpreted as a phantom fluid effect. For the likelihood method we find that a nearly flat universe is preferred. We show that if we assume that matter density is 0.3, then n 0 in the flat Cardassian model, which corresponds to the Perlmutter model with a cosmological constant. Tests with the Knop and Tonry SN Ia samples show no significant differences in results. Subject headings: cosmology: theory — distance scale — supernovae: general
1. INTRODUCTION
was also examined by Frith (2003), who used the Tonry SN Ia data. However, such a choice of an interval for this parameter is not physically justified, and both negative and positive values are admissible.
Freese & Lewis (2002) have recently proposed an alternative to the cosmological-constant model explaining the currently accelerating universe. In this model the standard Friedmann-Robertson-Walker (FRW) equation is modified by the presence of an additional term n , namely, H ¼ þ Bn ; 3 2
2. BASIC EQUATION We add a curvature term to the original Cardassian model. Then equation (1) assumes the following form:
ð1Þ
H2 ¼
where H a˙ =a is the Hubble parameter, a is the scale factor, is the energy density of matter and radiation, and B is a positive constant. For simplicity it is assumed that the density parameter for radiation matter r;0 ¼ 0. This proposal seems to be attractive because the expansion of the universe is accelerated automatically because of the presence of the additional term (if we put B ¼ 0, then the standard FRW equation is recovered). Because the Cardassian model offers an alternative to the cosmological-constant model, the agreement of this model with available observations of Type Ia supernovae (SNe Ia) was immediately verified (Zhu & Fujimoto 2003; Sen & Sen 2003). Some interesting results on observational constraints in the generalized Cardassian model were also obtained from the statistical analysis of gravitationally lensed quasars (Dev, Alcaniz, & Jain 2003). In our analysis we use three SN Ia data sets compiled by Perlmutter et al. (1999), Knop et al. (2003), and Tonry et al. (2003) to test both the original Cardassian model and its extension with a curvature term. Our results are inconsistent with the prediction of Zhu & Fujimoto (2003) of a low-density universe. Noteworthy results from the joint analysis of SN Ia data and cosmic microwave background radiation (CMBR) were obtained by Sen & Sen (2003), who also confirmed the prediction of a low-density universe, but their analysis was restricted to the case of n > 0. This case
k þ Bn þ 2 : 3 a
ð2Þ
It is useful to rewrite equation (2) in the dimensionless variables i;0 , i ¼ ðm; Card; kÞ, 3 3n 2 H2 a a a ¼ m;0 þ Card;0 þ k;0 ; ð3Þ a0 a0 a0 H02 where the energy density for dust is assumed ( p ¼ 0) (and hence / a3 from a conservation equation), m;0 is the matter density parameter, and Card;0 ¼ 3Bn =3H 02 is the density parameter for the fictitious noninteracting fluid that mimics term n and yields additional density in the model. The subscript zero denotes a present value of model parameters. Then ˙ ¼ 3Hð þ pÞ:
ð4Þ
For a ¼ a0 (the present value of scale factor) we obtain the following constraint: m;0 þ Card;0 þ k;0 ¼ 1:
ð5Þ
For the fictitious Cardassian fluid we obtain p ¼ ðn 1Þ if / a3 . In this interpretation the negative value of n is 599
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equivalent to the existence of phantom matter with supernegative pressure. Let us note that in the analogous equation describing FRW dynamics with dust and an additional fictitious fluid with the equation of state p ¼ ðn 1Þ presented in Sen & Sen (2003), only the n > 0 case was considered.
It is well known that various cosmic distance measures, for example the luminosity distance, depend sensitively both on the spatial geometry (curvature) and dynamics. Therefore, luminosity depends on the present densities of the different components of matter content and its equation of state. For this reason, the magnitude-redshift relation for distant objects is proposed as a potential test of cosmological models and plays an important role in determining cosmological parameters. Let us consider an observer located at r ¼ 0 at the moment t ¼ t0 who receives light emitted at t ¼ t1 from a source of absolute luminosity L located at the radial distance r1 . It is known that the cosmological redshift z of the source is related to t1 and t0 by the relation 1 þ z ¼ aðt0 Þ=aðt1 Þ. If the apparent luminosity l of the source measured by the observer is defined by L ; 4dL2
where 8 > : sinh x with K ¼ k;0
k;0 < 0; k;0 ¼ 0; k;0 > 0;
ð12Þ
and the density parameter for hypothetical curvature fluid is
3. MAGNITUDE-REDSHIFT RELATION
l¼
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ð6Þ
k;0 ¼
k : a˙ 20
Thus, for a given M, m;0 , Card;0 , and k;0 , equations (8) and (11) give the predicted value of mðzÞ at a given z. The goodness of fit is characterized by the parameter 2 ¼
X j00;i t0;i j i
20;i þ 2z;i
;
ð13Þ
where 00;i is the measured value, t0;i is the value calculated 2 is the measurement error, in the model described above, 0;i and 2z;i is the dispersion in the distance modulus due to peculiar velocities of galaxies. We assume that supernovae measurements come with uncorrelated Gaussian errors and that in this case the likelihood function L can be determined from the 2 statistic, L / exp ð2 =2Þ (Perlmutter et al. 1999; Riess et al. 1998).
then the luminosity distance dL of the source is dL ¼ ð1 þ zÞa0 r1 :
ð7Þ
For historical reasons, the observed and absolute luminosities are defined in terms of K-corrected observed magnitudes and absolute magnitudes m and M, respectively [l ¼ 102m=5 ð2:52 ; 105 Þ ergs cm2 s2, L ¼ 102M =5 ð3:02 ; 1035 Þ ergs s2; Weinberg 1972]. When written in terms of m and M, equation (6) yields mðz; M; m;0 ; Card;0 ; k;0 Þ ¼ M þ 5 log10 ½DL ðz; m;0 ; Card;0 ; k;0 Þ;
ð8Þ
M ¼ M 5 log10 H0 þ 25
ð9Þ
where
and DL ðz; m;0 ; Card;0 ; k;0 Þ H0 dL ðz; m;0 ; Card;0 ; k;0 ; H0 Þ ð10Þ is the dimensionless luminosity distance, while dL is in megaparsecs. The standard analysis yields the following relationship for the dimensionless luminosity distance: 1þz DL z; m;0 ; Card;0 ; k;0 ¼ pffiffiffiffi K ( Z z pffiffiffiffi 2 ; K 1 m;0 Card;0 ð1 þ z 0 Þ 0 0 3
0 3n
þ m;0 ð1 þ z Þ þ Card;0 ð1 þ z Þ
)
1=2 dz
0
;
ð11Þ
4. STATISTICAL ANALYSIS WITH THE PERLMUTTER SAMPLE We decided to test our model using the Perlmutter samples of supernovae (Perlmutter et al. 1999). We estimate the value of M (eq. [9]) separately for the full sample of 60 supernovae (sample A) and for the sample of 54 supernovae (sample C; in this sample we exclude two supernovae as outliers and two as likely reddened ones from the sample of 42 high-redshift supernovae, and two as outliers from the sample of 18 lowredshift supernovae). We obtain a value of M ¼ 3:39 for sample A and M ¼ 3:43 for sample C. To test the model we calculate the best fit with minimum 2 , as well as estimating the model parameters using the likelihood method (Riess et al. 1998). For both statistical methods we take the parameter n in the interval [3.33, 2], m;0 in the interval [0, 1], and k;0 in the interval [1, 1], while an interval for Card;0 is obtained from equation (5). First, we estimate best-fit parameters for the nonflat Cardassian model and obtain k;0 ¼ 1:00, m;0 ¼ 0:83, n ¼ 0:67, and Card;0 ¼ 1:17 for sample A and k;0 ¼ 1:00, m;0 ¼ 0:60, n ¼ 0:50, and Card;0 ¼ 1:40 for sample C. This model is characterized with strong negative curvature, which is similar to the Perlmutter results. Let us note that such a high value of k;0 is in contradiction with CMB observations, which prefer a flat universe. When we assume that k;0 ¼ 0, we obtain a best-fit flat Cardassian model with m;0 ¼ 0:52, n ¼ 0:93, and Card;0 ¼ 0:48 for sample A and m;0 ¼ 0:46, n ¼ 0:60, and Card;0 ¼ 0:54 for sample C. The detailed values are in Tables 1 and 2. For sample C we draw the best-fit nonflat (upper middle curve) and flat (lower middle curve) Cardassian models, as well as the best-fit Perlmutter model with m;0 ¼ 0:28, ;0 ¼ 0:72 (upper curve), against the flat Einstein–de Sitter model (zero line) on the Hubble diagram (Fig. 1). One can observe that the difference between the best-fit Cardassian model and
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TABLE 1 Cardassian Model, Perlmutter Samples A and C Card
k;0
m;0
M
2
Method
3.39 3.39 3.39 3.39
94.8 ... 95.7 ...
BF L BF L
3.43 3.43 3.43 3.43
52.8 ... 53.2 ...
BF L BF L
n Sample A
1.17 0.37 1.65 0.27
1.00 0.12 0.95 0.45
0.67 0.13 0.37 0.33
0.83 0.50 0.30 0.30
Sample C 1.40 0.33 1.70 0.29
1.00 0.17 1.00 0.41
0.60 0.29 0.30 0.30
0.50 0.10 0.36 0.30
Notes.—Results of the statistical analysis of the Cardassian model obtained for Perlmutter samples A and C both from the best fit with minimum 2 (denoted with BF) and from the likelihood method (denoted with L). The same analysis was repeated with fixed m;0 ¼ 0:3.
the Einstein–de Sitter model with ;0 ¼ 0 assumes the largest value for z 0:9 and significantly decreases for higher redshifts. While the best-fit flat Cardassian model and Perlmutter one have increasing differences from the flat Einstein–de Sitter model for higher redshifts, the differences between the best-fit flat Cardassian model and Perlmutter model increase for higher redshifts. This gives us a possibility of discriminating between the Perlmutter model and Cardassian models if z ’ 1 supernovae data are available. This is very important because for the present data the Cardassian models are only marginally better than the Perlmutter model. However, knowing the best-fit values alone does not have much scientific relevance if the confidence levels for the
BF L
parameter intervals are not presented too. Therefore, we carry out model parameter estimation using the minimization procedure, based on the likelihood method. We obtain parameter values for samples A and C at the 68.3% confidence level (Table 3). It should be noted that the values obtained with each method are different but that the minimization procedure seems to be more adequate for analyzing our problem. The density distributions f (k;0 ) and f (n) in the Cardassian model are presented on Figures 2 and 3, respectively. These figures show that the preferred model of the universe is the nearly flat one, which is in agreement with CMBR data. For sample C the most probable value of m;0 is 0.29, which is in agreement with the present CMBR and extragalactic data (Peebles & Ratra 2003; Lahav 2002). Therefore, a detailed analysis of the flat case of the Cardassian model is carried out. In Figure 4 we present confidence levels on the plane ðm;0 ; nÞ minimized over M for the flat Cardassian model. Figure 4 shows the preferred values of parameters m;0 and n. We find that the expected value of m;0 increases when n decreases. A similar result has already been obtained by Sen & Sen (2003) for n > 0. It should be pointed out that these authors argued that m;0 could be less than 0.3 from the confidence level obtained from the joint analysis of supernovae data and the positions of the CMBR peaks. In the Cardassian model the structure formation would be significantly different from, e.g., the classical model with a cosmological constant. Consequently, in order to calculate the consequence for the CMB, simple angular rescaling does not suffice. The detailed results of our analysis for the flat model are summarized in Table 2. The best-fit procedure suggests that n
BF L
TABLE 3 Nonflat Cardassian Model Parameter Values Obtained from Minimization Procedure Carried Out on Perlmutter Samples
TABLE 2 Cardassian Flat Model, Perlmutter Samples A and C Card;0
m;0
M
2
3.39 3.39 3.44 3.41 3.39 3.39 3.39 3.39
95.3 ... 95.0 ... 96.1 ... 96.1 ...
BF L M, M, BF L M, M,
95.3 ... 95.0 ... 53.3 ... 53.3 ...
BF L M, M, BF L M, M,
n
Method
Sample A 0.48 0.40 0.46 0.46 0.70 0.70 0.70 0.70
0.52 0.60 0.54 0.54 0.30 0.30 0.30 0.30
0.93 0.53 1.43 1.20 0.03 0.03 0.03 0.03
BF L
BF L
Sample C 0.54 0.42 0.52 0.48 0.7 0.7 0.7 0.7
0.46 0.58 0.48 0.52 0.30 0.30 0.30 0.30
0.60 0.10 0.96 0.17 0.03 0.03 0.03 0.03
3.43 3.43 3.46 3.45 3.43 3.43 3.43 3.43
Fig. 1.—Residuals (in magnitudes) between the Einstein–de Sitter model and four cases: the Einstein–de Sitter model itself (zero line), the Perlmutter flat model (upper curve), the best-fit flat Cardassian model (upper middle curve; k;0 ¼ 0, m;0 ¼ 0:46, n ¼ 0:60, and Card;0 ¼ 0:54), and the best-fit (nonflat) Cardassian model (lower middle curve; k;0 ¼ 1:00, m;0 ¼ 0:60, n ¼ 0:50, and Card;0 ¼ 1:40).
Notes.—Results of the statistical analysis of the Cardassian flat model for Perlmutter samples A and C obtained both from the best fit with minimum 2 (denoted with BF) and from the likelihood method (denoted with L). The case in which we marginalize over M is denoted with M. The same analysis was repeated with fixed m;0 ¼ 0:3.
Sample
k;0
Card;0
m;0
n
A..................... C.....................
0:12þ0:54 0:50 0:17þ0:44 0:66
þ0:38 0:370:20 þ0:43 0:330:17
0:50þ0:28 0:32 0:29þ0:28 0:28
0:13þ0:56 1:37 0:10þ0:36 1:46
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Fig. 2.—Density distribution for k;0 in the Cardassian model (sample C). We obtain k;0 ¼ 0:17þ0:46 0:66 at the confidence level of 68:3% (inner dashed lines). Both positive and negative values of k;0 are formally possible.
should be negative and that consequently m;0 is greater than 0.3. While we find that the parameter n is negative for the minimum value of the 2 statistic, the confidence level allows positive values of n. For the maximum likelihood method þ0:86 on sample A, we obtain m;0 ¼ 0:60þ0:06 0:15 and n ¼ 0:531:27 at the confidence level of 68:3% for M ¼ 3:39, while we þ0:53 obtain m;0 ¼ 0:54þ0:10 0:11 and n ¼ 1:201:36 at the same confidence level when we marginalize over M. In turn, for sample C we obtain m;0 ¼ 0:58þ0:06 0:13 and n ¼ at the confidence level of 68:3% for M ¼ 3:43 0:10þ0:30 1:57 (Figs. 5 and 6), while we obtain m;0 ¼ 0:52þ0:10 0:13 and n ¼
Fig. 3.—Density distribution for n in the Cardassian model (sample C). We obtain n ¼ 0:10þ0:36 1:46 at the confidence level of 68:3% (inner dashed lines). Both positive and negative values of n are formally possible.
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Fig. 4.—Confidence levels on the plane (m;0 ; n) minimized over M for the flat model (k;0 ¼ 0), and with Card;0 ¼ 1 m;0 . The figure shows of the preferred value of m;0 and n.
0:17þ0:20 2:04 at the same confidence level when we marginalize over M. For the flat model with m;0 ¼ 0:3 we obtain for sample A n ¼ 0:03 with ðnÞ ¼ 0:13 for M ¼ 3:39 and with ðnÞ ¼ 0:33 when we marginalize over M. In turn, for sample C we obtain n ¼ 0:03 with ðnÞ ¼ 0:12 for M ¼ 3:43 (Fig. 7) and ðnÞ ¼ 0:23 when we marginalize over M. 5. STATISTICAL ANALYSIS WITH THE KNOP SAMPLE Because the Perlmutter sample was completed 4 years ago, it would be interesting to use newer supernovae observations. Recently, Knop et al. (2003) have reexamined the Perlmutter sample with host-galaxy extinction correctly applied. They
Fig. 5.—Density distribution for m;0 in the Cardassian flat model. We obtain m;0 ¼ 0:58þ0:06 0:13 at the confidence level of 68:3% (inner dashed lines). In addition, the 95:4% confidence level is marked (outer dashed lines).
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CONSTRAINTS ON CARDASSIAN MODEL
Fig. 6.—Density distribution for n in the Cardassian flat model. We obtain n ¼ 0:10þ0:30 1:57 at the confidence level of 68:3% (inner dashed lines). In addition, the 95:4% confidence level is marked (outer dashed lines). Both positive and negative values of n are formally possible.
chose from the Perlmutter sample those supernovae that were more securely spectrally identified as Type Ia and that have reasonable color measurements. They also included 11 new high-redshift supernovae and a well-known sample with lowredshift supernovae. We have also decided to test the Cardassian model using this new sample of supernovae. They distinguished a few subsets of supernovae from this sample. We consider two of them. The first is a subset of 58 supernovae with extinction correction (Knop subsample 6, hereafter K6), and the second one is a subset of 54 supernovae with low extinction
(Knop subsample 3, hereafter K3). Sample C and K3 are similarly constructed because both contain only low-extinction supernovae. It should be pointed out that in contrast to Perlmutter et al. (1999), who included errors in measurement of redshift z, Knop et al. (2003) took only uncertainties in the redshift due to peculiar velocities. This is the reason that for security we separately repeated our analysis including errors in measurement of redshift z (subsamples K6z and K3z). The errors were taken from Perlmutter et al. (1999). In cases where errors were not available, we assumed (z) ¼ 0:001. As one can see above, this change has only marginal influence on the results. At first we estimate the value of M (eq. [9]) separately for both samples K6 and K3. We obtain a value of M ¼ 3:53 for sample K6 and M ¼ 3:48 for sample K3. This is in agreement with Knop et al. (2003). First, we estimate best-fit parameters for the nonflat Cardassian model and obtain k;0 ¼ 0:24, m;0 ¼ 0:65, n ¼ 1:10, and Card;0 ¼ 0:59 for sample K6 and k;0 ¼ 0:36, m;0 ¼ 0:49, n ¼ 0:20, and Card;0 ¼ 0:87 for sample K3. When we assume that k;0 ¼ 0 (flat universe), we obtain a best-fit flat Cardassian model with m;0 ¼ 0:53, n ¼ 1:50, and Card;0 ¼ 0:47 for sample K6 and m;0 ¼ 0:41, n ¼ 0:63, and Card;0 ¼ 0:59 for sample K3. The detailed values are in Tables 4 and 5. We also carry out model parameter estimation using the minimization procedure, based on the likelihood method. We obtain parameter values for samples K6 and K3 at the confidence level of 68:3% (Table 6). It should be noted that the TABLE 4 Cardassian Model, Knop Samples Card
k;0
m;0
n
M
2
Method
3.53 3.53 3.53 3.53
54.8 ... 54.8 ...
BF L BF L
3.53 3.53 3.53 3.53
43.6 ... 43.7 ...
BF L BF L
3.48 3.48 3.48 3.48
60.3 ... 60.4 ...
BF L BF L
3.48 3.48 3.48 3.48
52.0 ... 52.1 ...
BF L BF L
Sample K6 0.59 0.39 0.31 0.32
0.24 0.02 0.39 0.38
0.63 0.45 0.30 0.30
1.10 0.43 2.43 0.27 Sample K6z
0.48 0.43 0.29 0.33
0.08 0.15 0.41 0.37
0.60 0.52 0.30 0.30
1.77 0.50 3.17 0.27 Sample K3
0.87 0.34 0.45 0.35
0.36 0.29 0.25 0.35
0.49 0.27 0.30 0.30
0.20 0.03 0.97 0.27 Sample K3z
0.80 0.34 0.47 0.35 Fig. 7.—Density distribution for n in the Cardassian flat model with m;0 ¼ 0:3. We obtain n ¼ 0:03 with (n) ¼ 0:12. Both positive and negative values of n are formally possible.
0.27 0.30 0.23 0.35
0.47 0.25 0.30 0.30
0.27 0.00 0.83 0.27
Notes.—Results of the statistical analysis of the Cardassian model obtained both for the Knop samples from the best fit with minimum 2 (denoted with BF) and from the likelihood method (denoted with L). The same analysis was repeated with fixed m;0 ¼ 0:3.
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GODxOWSKI, SZYDxOWSKI, & KRAWIEC TABLE 5 Cardassian Flat Model, Knop Samples
Card;0
m;0
M
n
2
Method
Sample K6 0.47 0.44 0.46 0.48 0.70 0.70 0.70 0.70
0.53 0.56 0.54 0.52 0.30 0.30 0.30 0.30
1.50 1.13 3.33 3.33 0.17 0.17 0.10 0.13
3.53 3.53 3.60 3.55 3.53 3.53 3.52 3.53
54.6 ... 53.5 ... 55.7 ... 55.6 ...
BF L M, M, BF L M, M,
43.6 ... 42.7 ... 44.9 ... 44.8 ...
BF L M, M, BF L M, M,
60.3 ... 60.3 ... 60.4 ... 60.4 ...
BF L M, M, BF L M, M,
52.0 ... 52.0 ... 52.2 ... 52.2 ...
BF L M, M, BF L M, M,
BF L
BF L
Sample K6z 0.45 0.43 0.46 0.48 0.70 0.70 0.70 0.70
0.55 0.57 0.54 0.52 0.30 0.30 0.30 0.30
1.87 1.57 3.33 3.33 0.17 0.17 0.07 0.10
3.53 3.53 3.60 3.54 3.53 3.53 3.51 3.51
BF L
BF L
Sample K3 0.59 0.50 0.58 0.52 0.70 0.70 0.70 0.70
0.41 0.50 0.42 0.48 0.30 0.30 0.30 0.30
0.63 0.37 0.77 0.40 0.20 0.20 0.13 0.17
3.48 3.48 3.49 3.49 3.48 3.48 3.47 3.48
BF L
BF L
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þ0:77 while we obtain m;0 ¼ 0:48þ0:08 0:13 and n ¼ 0:401:24 at the same confidence level when we marginalize over M. For the flat model with m;0 ¼ 0:3 we obtain for sample K6 n ¼ 0:17 with ðnÞ ¼ 0:16 for M ¼ 3:39 and n ¼ 0:13 with ðnÞ ¼ 0:23 when we marginalize over M. In turn, for sample K3 we obtain n ¼ 0:20 with ðnÞ ¼ 0:13 for M ¼ 3:48 and n ¼ 0:20 with ðnÞ ¼ 0:17 when we marginalize over M.
6. STATISTICAL ANALYSIS WITH THE TONRY SAMPLE Another sample was presented by Tonry et al. (2003), who collected a large number of supernovae published by different authors and added eight new high-redshift SNe Ia. This sample of 230 SNe Ia was recalibrated with a consistent zero point. Whenever possible, the extinction estimates and distance fitting were recomputed. However, none of the methods applied to all supernovae (for details, see Table 8 of Tonry et al. 2003). Despite this problem, the analysis of the Cardassian model using this sample of supernovae could be interesting. We decide to analyze four subsamples. First, the full Tonry sample of 230 SNe Ia (hereafter sample Ta) is considered. The sample of 197 SNe Ia (hereafter sample Tb) consists of low-extinction supernovae only (median V-band extinction AV < 0:5). Because the Tonry sample has a lot of outliers especially at low redshift, the sample of 195 SNe Ia is such that all low-redshift (z < 0:01) supernovae are excluded (hereafter sample Tc). In the sample of 172 SNe Ia, all supernovae with low redshift and high extinction are omitted (hereafter sample Td). Tonry et al. (2003) presented redshift and luminosity distance observations for their sample of supernovae. Therefore, equations (8) and (9) should be modified (Williams et al. 2003):
Sample K3z 0.60 0.49 0.58 0.52 0.70 0.70 0.70 0.70
0.40 0.51 0.42 0.48 0.30 0.30 0.30 0.30
0.57 0.30 0.83 0.40 0.17 0.17 0.17 0.17
3.48 3.48 3.50 3.50 3.48 3.48 3.48 3.48
m M ¼ 5 log10 ðDL ÞTonry 5 log10 65 þ 25;
ð14Þ
M ¼ 5 log10 H0 þ 25:
ð15Þ
BF L
BF L
Notes.—Results of the statistical analysis of the Cardassian flat model for the Knop samples obtained both from the best fit with minimum 2 (denoted with BF) and from the likelihood method (denoted with L). The case in which we marginalize over M is denoted with M. The same analysis was repeated with fixed m;0 ¼ 0:3.
values obtained in both methods are different but that the differences are much smaller than in the case of the original Perlmutter sample. The detailed results of our analysis for the flat model are summarized in Table 5. The best-fit procedure suggests that n should be negative and that consequently m;0 is greater than 0.3. While we find that the parameter n is negative for the minimum value of the 2 statistic, the confidence level allows positive values of n. For the maximum likelihood method þ1:07 on sample K6 we obtain m;0 ¼ 0:44þ0:14 0:08 and n ¼ 1:131:13 at the confidence level of 68:3% for M ¼ 3:53, while we þ2:00 obtain m;0 ¼ 0:48þ0:09 0:09 and n ¼ 3:330:00 at the same confidence level when we marginalize over M. In turn, for sample K3 we obtain m;0 ¼ 0:50þ0:07 0:15 and n ¼ 0:37þ0:63 1:00 at the 68:3% confidence level for M ¼ 3:48,
For H0 ¼ 65 km s1 Mpc1 , we obtain M ¼ 15:935. First, we estimate best-fit parameters for the nonflat Cardassian model and obtain k;0 ¼ 1, m;0 ¼ 0:48, n ¼ 0:27, and Card;0 ¼ 1:52 for sample Ta. When we assume that k;0 ¼ 0 (flat universe), for the sample Ta we obtain a best-fit flat Cardassian model with m;0 ¼ 0:46, n ¼ 1:50, and Card;0 ¼ 0:54. For samples Tb, Tc, and Td, the results differ only marginally. The detailed values are presented in Tables 7 and 8. As in previous sections, we also carry out model parameter estimation using the minimization procedure based on the likelihood method. We present the parameter values obtained for the Tonry samples at the 68:3% confidence level in Table 9.
TABLE 6 Model Parameter Values Obtained from Minimization Procedure Carried Out on Knop Samples Sample
k;0
Card;0
m;0
n
K6................... K6z................. K3................... K3z.................
þ0:49 0:020:54 þ0:56 0:150:45 þ0:38 0:290:66 þ0:38 0:300:66
0:39þ0:34 0:18 0:43þ0:28 0:21 0:34þ0:44 0:13 0:34þ0:45 0:14
0:45þ0:30 0:31 0:52þ0:30 0:32 0:27þ0:13 0:23 0:25þ0:23 0:23
þ0:73 0:431:50 þ0:63 0:501:63 þ0:33 0:031:37 þ0:40 0:001:40
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TABLE 7 Cardassian Model, Tonry Samples Card
k;0
m;0
n
M
2
Method
15.935 15.935 15.935 15.935
252.1 ... 252.5 ...
BF L BF L
15.935 15.935 15.935 15.935
185.3 ... 185.7 ...
BF L BF L
procedure we use the more natural variable Card;0 (or m;0 ) instead of their parameter zeq [zeq : ð=3Þ ¼ Bn ; ¼ 0 a3 ; 1 þ z ¼ a1 ]. The two parameters m;0 and zeq are related by the formula
Sample Ta 1.52 0.54 1.68 0.28
1.00 0.64 0.98 0.42
0.48 0.49 0.30 0.30
0.27 0.10 0.37 0.33 Sample Tb
1.50 0.64 1.70 0.31
1.00 0.78 1.00 0.39
0.50 0.49 0.30 0.30
0.23 0.07 0.37 0.33
h i1 m;0 ¼ 1 þ ð1 þ zeq Þ3ð1nÞ :
1.00 0.64 0.98 0.42
0.48 0.49 0.53 0.49
0.23 0.10 0.37 0.33
m;0 ¼ 1 Card;0 k;0 ;
1.00 0.78 1.00 0.39
0.50 0.49 0.30 0.30
0.23 0.17 0.37 0.33
ð17Þ
TABLE 8 Cardassian Flat Model, Tonry Samples 15.935 15.935 15.935 15.935
201.0 ... 201.3 ...
BF L BF L
Card;0
BF L BF L
0.54 0.49 0.48 0.47 0.70 0.70 0.70 0.70
Sample Td 1.50 0.63 1.70 0.31
ð16Þ
In Figure 8 relation (16) for different n is presented. We can observe that taking a constant step in zeq leads to varying steps in m;0 . In our method we have
Sample Tc 1.47 0.54 1.68 0.28
605
CONSTRAINTS ON CARDASSIAN MODEL
15.935 15.935 15.935 15.935
164.9 ... 165.4 ...
Notes.—Results of the statistical analysis of the Cardassian model obtained for the Tonry samples both from the best fit with minimum 2 (denoted with BF) and from the likelihood method (denoted with L). The same analysis was repeated with fixed m;0 ¼ 0:3.
The detailed results of our analysis for the flat model, based on the likelihood method, are summarized in Table 8. For þ0:80 sample Ta we obtain m;0 ¼ 0:51þ0:08 0:12 and n ¼ 0:400:63 at the 68:3% confidence level for M ¼ 15:935, while we obtain þ0:83 m;0 ¼ 0:53þ0:04 0:06 and n ¼ 1:871:00 at the same confidence level when we marginalize over M. In turn, for sample Td we obtain m;0 ¼ 0:49þ0:08 0:12 and n ¼ 0:43þ0:50 0:63 at the 68:3% confidence level for M ¼ 15:935, þ0:77 while we obtain m;0 ¼ 0:51þ0:05 0:06 and n ¼ 1:201:06 at the same confidence level when we marginalize over M. For the flat model with m;0 ¼ 0:3 we obtain for subsample Ta n ¼ 0:03 with (n) ¼ 0:10 for M ¼ 15:935 and n ¼ 0:10 with (n) ¼ 0:15 when we marginalize over M. For other subsamples the results are very similar. All these results suggest that n is negative and that consequently m;0 is greater than 0.3. Generally, we find that the results obtained with the Tonry sample are similar to those obtained with the Perlmutter sample, apart from the nonflat case, where values of the n-parameter are positive (both positive and negative values are in the Perlmutter samples). 7. DISCUSSION In the present paper we have discussed the problem of universe acceleration in the Cardassian model. Our results are different from those obtained by Zhu & Fujimoto (2003), who suggested a universe with very low matter density. Our results indicate a high or normal (m;0 0:3) matter density. This difference may come from not including the errors in redshifts and using a variable different from theirs. In our fitting
m;0
M
n
2
Method
Sample Ta 0.46 0.51 0.52 0.53 0.30 0.30 0.30 0.30
0.50 0.40 2.00 1.87 0.03 0.03 0.10 0.10
15.935 15.935 15.855 15.855 15.935 15.935 15.895 15.895
253.6 ... 247.7 ... 254.7 ... 252.1 ...
BF L M, M, BF L M, M,
187.1 ... 183.4 ... 188.1 ... 186.5 ...
BF L M, M, BF L M, M,
202.5 ... 198.7 ... 203.6 ... 202.2 ...
BF L M, M, BF L M, M,
166.7 ... 164.1 ... 167.7 ... 166.8 ...
BF L M, M, BF L M, M,
BF L
BF L
Sample Tb 0.56 0.51 0.51 0.49 0.70 0.70 0.70 0.70
0.44 0.49 0.49 0.51 0.30 0.30 0.30 0.30
0.50 0.40 1.47 1.40 0.03 0.03 0.13 0.17
15.935 15.935 15.875 15.875 15.935 15.935 15.905 15.905
BF L
BF L
Sample Tc 0.54 0.49 0.49 0.47 0.70 0.70 0.70 0.70
0.46 0.51 0.51 0.53 0.30 0.30 0.30 0.30
0.50 0.40 1.53 1.47 0.03 0.03 0.07 0.07
15.935 15.935 15.875 15.875 15.935 15.935 15.905 15.905
BF L
BF L
Sample Td 0.56 0.51 0.52 0.49 0.70 0.70 0.70 0.70
0.44 0.49 0.48 0.51 0.30 0.30 0.30 0.30
0.50 0.43 1.23 1.20 0.03 0.03 0.13 0.13
15.935 15.935 15.885 15.885 15.935 15.935 15.905 15.905
BF L
BF L
Notes.—Results of the statistical analysis of the Cardassian flat model for the Tonry samples obtained both from the best fit with minimum 2 (denoted with BF) and from the likelihood method (denoted with L). The case in which we marginalize over M is denoted with M. The same analysis was repeated with fixed m;0 ¼ 0:3.
606
GODxOWSKI, SZYDxOWSKI, & KRAWIEC TABLE 9 Model Parameter Values Obtained from Minimization Procedure Carried Out on Tonry Sample
Sample
k;0
Card;0
m;0
n
Ta.................... Tb ................... Tc.................... Td ...................
0:64þ0:72 0:61 0:78þ0:76 0:55 0:64þ0:72 0:61 0:78þ0:77 0:55
þ0:52 0:540:27 þ0:50 0:640:32 þ0:52 0:540:27 þ0:51 0:630:31
þ0:21 0:490:23 þ0:21 0:490:23 þ0:21 0:490:22 þ0:21 0:490:23
þ0:36 0:100:70 þ0:36 0:070:67 þ0:33 0:100:70 þ0:28 0:170:87
Note.—To obtain errors of the parameter k;0 we enlarge the estimation interval of k;0 to [2, 2].
and we take a constant step of 0.01 in m;0 . For the flat Cardassian model both Zhu & Fujimoto (2003) and we obtained negative values of n with the best-fit method. Assuming the curvature term k;0 in the Cardassian model, we obtained its statistical estimation as close to zero. In this natural way we got that a nearly flat universe is preferred. Moreover, in this case for the Perlmutter sample C, the preferred value of m;0 is 0.29, which is in agreement with CMBR and extragalactic data, while for sample A, we got that a universe with higher matter content is preferred. Let us note that Riess et al. (1998) and Perlmutter et al. (1999) obtained a high negative value of k;0 as the best fit, although a zero value of k;0 is statistically admissible. To find the curvature, they also used the data from CMBR and extragalactic astronomy. But in the Cardassian model, in a natural way, a nearly flat universe is favored. When a flat universe was assumed, in the analysis of the Cardassian universe with sample C, we got low but positive values of n and a matter density higher than 0.3. For all other samples we obtained negative values of n (this could indicate phantom fluid with supernegative pressure). In turn, when m;0 is fixed to be 0.3, then n 0 is favored. In general, the results obtained using the Perlmutter and Knop samples are not significantly different. The main
Dev, A., Alcaniz, J. S., & Jain, D. 2003, preprint (astro-ph/0305068) Freese, K., & Lewis, M. 2002, Phys. Lett. B, 540, 1 Frith, W. J. 2003, preprint (astro-ph/0311211) Knop, R. A., et al. 2003, ApJ, 598, 102 Lahav, O. 2002, preprint (astro-ph/0208297) Peebles, P. J. E., & Ratra, B. 2003, Rev. Mod. Phys., 75, 559 Perlmutter, S., et al. 1999, ApJ, 517, 565
Fig. 8.—Relation between m;0 and zeq for different values of n
advantage of using the Knop sample is the lower errors of estimated parameters in the model. The Tonry sample also gave results similar to the Perlmutter sample. We note that the Tonry sample does not improve the estimation errors compared with the Perlmutter sample. From the statistical analysis, we found that the Perlmutter and Cardassian models are indistinguishable. Hence, using the philosophy of Occam’s razor, the Cardassian model should be rejected on the basis of current SN Ia data. However, the Cardassian model has its own advantages and fits the present observational data well. Therefore, we should not reject this model until we have more precise data.
M. S. was supported by KBN grant 1 P03D 003 26. The page charges were covered by Administracja Budynko´w ASMG.
REFERENCES Riess, A. G., et al. 1998, AJ, 116, 1009 Sen, S., & Sen, A. A. 2003, ApJ, 588, 1 Tonry, J. L., et al. 2003, ApJ, 594, 1 Weinberg, S. 1972, Gravitation and Cosmology (New York: Wiley) Williams, B. F., et al. 2003, AJ, 126, 2608 Zhu, Z.-H., & Fujimoto, M.-K. 2003, ApJ, 585, 52
