Cosmological Constraints on the Undulant Universe

Abstract

We use the redshift Hubble parameter $H(z)$ data derived from relative galaxy
ages, distant type Ia supernovae (SNe Ia), the Baryonic Acoustic Oscillation
(BAO) peak, and the Cosmic Microwave Background (CMB) shift parameter data, to
constrain cosmological parameters in the Undulant Universe. We marginalize the
likelihood functions over $h$ by integrating the probability density $P\propto
e^{-\chi^2/2}$. By using the Markov Chain Monte Carlo (MCMC) technique, we
obtain the best fitting results and give the confidence regions on the
$b-\Omega_{\rm m0}$ plane. Then we compare their constraints. Our results show
that the $H(z)$ data play a similar role with the SNe Ia data in cosmological
study. By presenting the independent and joint constraints, we find that the
BAO and CMB data play very important roles in breaking the degeneracy compared
with the $H(z)$ and SNe Ia data alone. Combined with the BAO or CMB data, one
can improve the constraints remarkably. The SNe Ia data sets constrain
$\Omega_{\rm m0}$ much tighter than the $H(z)$ data sets, but the $H(z)$ data
sets constrain $b$ much tighter than the SNe Ia data sets. All these results
show that the Undulant Universe approaches the $\Lambda \rm$CDM model. We
expect more $H(z)$ data to constrain cosmological parameters in future.

Full text

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arXiv:1006.3353v1 [astro-ph.CO] 17 Jun 2010
Cosmological Constraints on the Undulant Universe
Tian Lan1,2, Yan Gong2,3, Hao-Yi Wan4and Tong-Jie Zhang1,5
1Department of Astronomy, Beijing Normal University, Beijing, 100875, China;
2National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China;
3Graduate School of Chinese Academy of Sciences, Beijing 100049, China;
4Business Office, Beijing Planetarium, No. 138 Xizhimenwai Street, Beijing 100044, China;
5Center for High Energy Physics, Peking University, Beijing 100871, P. R. China
tjzhang@bnu.edu.cn
Abstract We use the redshift Hubble parameter H(z) data derived from relative galaxy
ages, distant type Ia supernovae(SNe Ia), the Baryonic Acoustic Oscillation (BAO) peak,
and the Cosmic Microwave Background (CMB) shift parameter data, to constrain cos-
mological parameters in the Undulant Universe. We marginalize the likelihood functions
over h by integrating the probability density P ∝ e−χ2/2. By using the Markov Chain
Monte Carlo (MCMC) technique, we obtain the best fitting results and give the confi-
dence regions on the b−Ωm0plane. Then we compare their constraints. Our results show
that the H(z) data play a similar role with the SNe Ia data in cosmological study. By pre-
senting the independent and joint constraints, we find that the BAO and CMB data play
very important roles in breaking the degeneracycomparedwith the H(z) and SNe Ia data
alone. Combined with the BAO or CMB data, one can improve the constraints remark-
ably. The SNe Ia data sets constrain Ωm0much tighter than the H(z) data sets, but the
H(z) data sets constrain b much tighter than the SNe Ia data sets. All these results show
that the Undulant Universe approaches the ΛCDM model. We expect more H(z) data to
constrain cosmological parameters in future.
Key words: Cosmology – Observational cosmology – Dark matter
1 INTRODUCTION
Many cosmological observations (Bennett et al. (2003); Spergel et al. (2007); Eisenstein et al. (2005);
Kowalski et al. (2008)), such as distant SNe Ia (Riess et al. (1998); Perlmutter et al. (1999)), the
subsequent CMB measurement by Wilkinson Microwave Anisotropy Probe (WMAP) (Spergel et al.
(2003)) and the large scale structure survey by Sloan Digital Sky Survey (SDSS) (Tegmark et al.
(2004a); Tegmark et al. (2004b)) etc. support that our Universe is undergoing an accelerated expan-
sion, and these data have mapped the universe successfully by constraining cosmological parameters
(Perlmutter et al. (1999); Tegmark et al. (2000); Zhu (2004); Zhan et al. (2006); Lineweaver (1998);
Efstathiou (1999); Zhan et al. (2005); Maartens et al. (2006)). Meanwhile, many different observa-
tions, such as the redshift-dependent quantities, are used usually. The luminosity distance to a par-
ticular class of objects, e.g. SNe Ia and Gamma-Ray Bursts (GRBs) (Nesseris & Perivolaropoulos
(2004)), the size of the BAO peak detected in the large-scale correlation function of luminous
red galaxies from SDSS (Eisenstein et al. (2005)) and the CMB data obtained from the three-year

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2Tian Lan, Yan Gong, Hao-Yi Wan and Tong-Jie Zhang
WMAP estimate (Wang & Mukherjee (2006); Spergel et al. (2007)) are redshift-dependent. The BAO
and CMB data have been combined to constrain the cosmological parameters widely. Recently,
the Hubble expansion rate at different redshifts can be obtained from the measurement of relative
galaxy ages directly. As a function of redshift z, the H(z) data have been used to test cosmologi-
cal models (Gong et al. (2008); Yi & Zhang (2007); Samushia & Ratra (2006); Wei & Zhang (2007);
Zhang et al. (2007); Wei & Zhang (2007); Zhang & Wu (2007); Dantas et al. (2007); Zhang & Zhu
(2007); Wu & Yu (2007); Wei & Zhang (2007)). Using different data to constrain parameters can pro-
vide the consistency checks between each other, and the combination of different data can also make
the constraints tighter (Li et al. (2008); Zhao et al. (2007); Xia et al. (2006)). So far, few work has been
done in comparison. We’d like to study the comparison of the SNe Ia, H(z), BAO and CMB data in
this letter. Although lots of cosmological models, e.g. the Quintessence (Caldwell et al. (1998)), the
brane world (Deffayet et al. (2002)), the Chaplygin Gas (Alcaniz et al. (2003)) and the holographicdark
energy models (Ke & Li (2005)), are extensively explored to explain the acceleration of the universe.
ΛCDM model, which is a standard and popular model, has been studied by Lin et al. (Lin et al. (2008)).
In fact, there are some problems in the standard model, and this model is not consistent with the real
universe. Scientists proposed a Undulant Universe, which is characterized by alternating periods of ac-
celeration and deceleration(Gabriela (2005)). This model can removethe fine tuningproblem (Vilenkin
(2001); Jaume et al. (1999); Bludman (2000); Garriga & Vilenkin (2001); Stewart (2000)), and solve
the coincidence scandal between the observed vacuum energy and the current matter density, with some
details in (Peebles & Ratra (2003)). The equation of state (EOS) of the Undulant Universe is
ω(a) = −cos(blna),
(1)
where the dimensionless parameter b controls the frequency of the accelerating epochs. Due to most
inflation models predicting Ωk < 10−5(Tegmark & Rees (1998)), we assume spatial flatness, so the
Hubble parameter is
H2(a) = H2
0(Ωm0a−3+ ΩΛ0a−3exp[3
bsin(blna)]),
(2)
where Ωm0, ΩΛ0 are the density parameter of matter and dark energy, respectively. Furthermore,
S. Nesseris and L. Perivolaropoulos have enhanced the fact that an oscillating expansion rate
ansatz has provided the best fit to the data among many ansatz by using recent supernova
data(Nesseris & Perivolaropoulos (2004)).
In this letter, we concentrate on the comparison of different data sets on the Undulant Universe. In
§II, we simply introduce these data sets that we use. Then the constraints are shown in §III. Finally, we
show some important conclusions and discussions in §IV.
2 OBSERVATIONAL DATA
2.1 SNe Ia data
There has been some calibrated SNe Ia data with high confidence in (Kowalski et al. (2008)). This
data set contains 307 data and the redshifts of these data span from about 0.01 to 1.75. These data give
luminositydistancesdL(zi) andthe redshiftsziofthecorrespondingSNe Ia.Ina Friedmann-Robertson-
Walker (FRW) cosmology, considering a flat universe, the luminosity distance is,
dL(z) =c(1 + z)
H0
F(z),
?z
(3)
The function F(z) is defined as F(z)
ΩM0a−3+ ΩΛ0a−3exp[3
=
0dz/E(z), with E(z)=H(z)/H0
=
?
bsin(blna)]. H0is the Hubble constant. The distance modulus is
µ(z) = m − M = 5log
dL
10pc= 42.39+ 5log1 + z
h
F(z),
(4)

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Cosmological Constraints on the Undulant Universe3
with h = H0/100 km s−1Mpc−1, where m and M are the apparent and absolute magnitudes,
respectively.
2.2 The BAO Data
As the acoustic oscillations in the relativistic plasma of the early universewill also be imprintedonto the
late-timepowerspectrumofthenon-relativisticmatter(Eisenstein & Hu(1998)),theacousticsignatures
in the large-scale clustering of galaxies yield additional tests for cosmology (Spergel et al. (2003)).
Eisenstein et al. (Eisenstein et al. (2005)) choosea spectroscopicsampleof 46748luminousred galaxies
from SDSS to find the peaks successfully. These galaxies cover 3816 square degrees and have been
observed at redshifts up to z = 0.47. The peaks are described by the model-independent A-parameter.
A-parameter is independent of Hubble constant H0,
√Ωm
z1
A =
[
z1
E(z1)F2(z1))]1/3,
(5)
wherez1= 0.35is theredshiftatwhichtheacousticscalehasbeenmeasured.Eisensteinetal.suggested
that the measured value of the A-parameter is A = 0.469± 0.017 (Eisenstein et al. (2005)).
2.3 The CMB Data
The expansionof the Universe has transformedthe black body radiation, which is left over from the Big
Bang, into the nearly isotropic 2.73 K cosmic microwave background (CMB) (Bernardis et al. (2000)).
The whole shift of the CMB angular power spectrum is determined by the CMB shift parameter R. The
shift parameter perhaps the most model-independent parameter, is also independent of H0. It can be
derived from CMB data (Bond et al. (1997); Odman et al. (2003)),
R =
?
ΩmF(zr),
(6)
where zr = 1089 is the redshift of recombination. Using the Markov Chain Monte Carlo (MCMC)
chains from the analysis of the three-year results of WMAP (Spergel et al. (2007)), Wang & Mukherjee
compute the CMB shift parameter to get R = 1.70 ± 0.03 and demonstrate that its measured value is
mostly independent of assumptions about dark energy (Wang & Mukherjee (2006)).
2.4 the H(z) data from relative galaxy ages
The Hubble parameter H(z) data can be derived from the derivative of redshift z with respect to the
cosmic time t, i.e. dz/dt (Jimenez & Loeb (2002)),
H(z) = −
1
1 + z
dz
dt.
(7)
Therefore,an applicationofthe differentialagemethodto oldelliptical galaxiesin thelocal universecan
determine the value of the current Hubble constant. This is a direct measurement for H(z) through a
determination of dz/dt. Jimenez et al. (Jimenez et al. (2003)) have applied the method to a z ≤ 0.2
sample. Paying careful attention to uncertainties in the distance, systematics, and model uncertain-
ties, they have selected Monte-Carlo techniques to demonstrate the feasibility by evaluating the er-
rors. They also use many other different methods and yield consistent value for the Hubble constant.
Hence, the H(z) data are reliable. With the availability of new galaxy surveys, it becomes possible
to determine H(z) at z > 0. By using the differential ages of passively evolving galaxies determined
from the Gemini Deep Deep Survey (GDDS) (Abraham et al. (2004)) and archival data (Treu et al.
(2001); Treu et al. (2002); Nolan et al. (2003a); Nolan et al. (2003b)), Simon et al. derive a set of H(z)
data, which is listed in Table.1 (Simon et al. (2005), see also Samushia & Ratra (2006); Jimenez et al.
(2003)). The estimation method in detail can be found in the work (Simon et al. (2005)). As z has a

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4 Tian Lan, Yan Gong, Hao-Yi Wan and Tong-Jie Zhang
Table 1 the H(z) data in units of km s−1Mpc−1(Simon et al. (2005)) (see
Samushia & Ratra (2006); Jimenez et al. (2003) also)
z(redshift)
H(z)
1σ error
0.09
69
12.0
0.17
83
8.3
0.27
70
14.0
0.40
87
17.4
0.88
117
23.4
1.30
168
13.4
1.43
177
14.2
1.53
140
14.0
1.75
202
40.4
Fig.1 The contour maps of b − Ωm0for different data sets. The left panel is H(z) only, and
the right panel is SNe Ia only. Confidenceregions are at 68.3%,95.4% and 99.7% levels from
inner to outer respectively, for a flat Undulant Universe without a prior of h.
relatively wide range, 0.1 < z < 1.8, these data are expected to provide a full-scale description for the
dynamical evolution of our universe. The application of the H(z) data to cosmology can be referred
to (Simon et al. (2005); Yi & Zhang (2007); Samushia & Ratra (2006); Wei & Zhang (2007); Lin et al.
(2008); Gong et al. (2008)) and so on.
3 CONSTRAINTS ON THE UNDULANT UNIVERSE
Considering that the contents of the universe are mainly dark matter and dark energy, we just show
b − Ωm0plane contour maps, where b traces dark energy, and Ωm0traces dark matter respectively. We
estimate the best fit to the set of parameters by using individual χ2statistics, with
χ2
|data=
N
?
i=1
[di
t(z) − di
o(z)]2
σ2
i
,
(8)
whereN is thenumberofdatain eachset, anddi
di
In order to strengthen the constraints, we use MCMC technique to modulate the process (Gong et al.
(2007)).
It is worth noting that the value of h may strongly affect the constraint process, because the proba-
bility distribution function (PDF) of h for each data is different. We can see the PDF for H(z) and SNe
t(z) is thevalueat zigivenbyabovetheoreticalanalysis.
o(z) is individual observational value in each data set at zi, and σiis the error due to uncertainties.

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Cosmological Constraints on the Undulant Universe5
Fig.2 The contour maps of b − Ωm0for different sets. The left panel is H(z)+BAO, and the
right panel is SNe Ia+BAO. Confidence regions are at 68.3%, 95.4% and 99.7% levels from
inner to outer respectively, for a flat Undulant Universe without a prior of h.
Ia data in Fig.5. For H(z) data, we find the PDF of h covers a large range, while it is small for SNe Ia
date. Because the distribution of h constrainedby SNe Ia is very narrow,if we fix h far from the best-fit,
then the constraints will be totally wrong.Forexample, we test the effect on fixing a ”wrong” h by using
SNe Ia+BAO+CMB. As can be seen in Fig.6, with a ”wrong” h prior, the constraints of the left panel
are very different from that of the right panel. This is because if we assume a ”wrong” prior for h, the
χ2would become verylarge (χ2∼ 426 for h = 0.65 while χ2∼ 312 for h = 0.7), and then we will get
a false minimum χ2to lead to a wrong result.However, if we marginalize the likelihood functions over
h by integrating the probability density P ∝ e−χ2/2, we can remove the effect of the distribution of h,
and illustrate the reliable distribution of the other parameters. Therefore, we marginalize the likelihood
functions over h to obtain the best fitting results. Then we get the confidence regions and the b − Ωm0
plane.
As it is seen from Eq.(1), ω(a) is an even function. Correspondingly, the values of b are along the
axis of Ωm0symmetrical distribution. Thus, the constraints on b just in the range from 0 to +∞ are
enoughto demonstratethe comparison.Fig.1 shows the confidenceregionsdeterminedby the H(z) and
SNe Ia data alone respectively. We find that the constraints on Ωm0from the H(z) data are weaker than
that from the SNe Ia data. Although we just have 9 data in H(z) and 307 data in SNe Ia, there is still
some consistency between them in the constraints. The joint constraints are shown in Fig.2, Fig.3, and
Fig.4. We find that the constraints are weak if we use the H(z) or SNe Ia data alone, however, after
inclusion of the other observational data (the BAO or CMB data), the constraints are improved remark-
ably. Especially, the CMB data improve the constraints more remarkably. For breaking the degeneracy,
the BAO and CMB data play very important roles in the constraint compared with the H(z) and SNe
Ia data alone. In the left contour map of Fig.1, and the two contour maps of Fig.2, Fig.3, and Fig.4,
all contours at the 1σ confidence for b are open-ended. It shows that b takes possible value around 0.
We have not demonstrated the details of the constraints, because MCMC technique cannot determine
exact values reliably. These contours correspond to 1σ,2σ, and 3σ confidence from inside to outside.
Table.2 summarizes 68.3% confidence intervals. From the table, we find that the constraints on Ωm0