Observational constraints on a decaying cosmological term

Abstract

We investigate the evolution of a universe with a decaying cosmological term (vacuum energy) that is assumed to be a function of the scale factor. In this model, while the cosmological term increases to the early universe, the radiation energy density is lower than the model with the cosmological "constant". We find that the effects of the decaying cosmological term on the expansion rate at the redshift z<2 is negligible. However, the decrease in the radiation density affects on the thermal history of the universe; e.g. the photon decoupling occurs at higher $z$ compared to the case of the standard \Lambda CDM model. As a consequence, a decaying cosmological term affects on the cosmic microwave background anisotropy. We show the angular power spectrum in D\Lambda CDM model and compare with the Wilkinson Microwave Anisotropy Probe (WMAP) data.

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arXiv:astro-ph/0611258v2 12 Mar 2007
Observational constraints on a decaying
cosmological term
Riou Nakamura∗
Department of Physics, Graduate school of sciences, Kyushu University, Fukuoka 810-8560,
Japan
E-mail: riou@gemini.rc.kyushu-u.ac.jp
Masa-aki Hashimoto
Department of Physics, Faculty of sciences, Kyushu University Fukuoka 810-8560, Japan
E-mail: hashi@gemini.rc.kyushu-u.ac.jp
Kiyotomo Ichiki
Research Center for the Early Universe, Graduate School of Science, The University of Tokyo
Tokyo 113-0033,Japan
E-mail: ichiki@resceu.s.u-tokyo.ac.jp
We investigate the evolution of a universe with a decaying cosmological term (vacuum energy)
that is assumed to be a function of the scale factor. In this model, while the cosmological term
increases to the early universe, the radiation energy density is lower than the model with the cos-
mological"constant". We find thatthe effects ofthe decayingcosmologicaltermonthe expansion
rate at the redshift z < 2 is negligible.
However, the decrease in the radiation density affects on the thermal history of the universe;
e.g. the photon decoupling occurs at higher z compared to the case of the standard ΛCDM
model. As a consequence, a decaying cosmological term affects on the cosmic microwave back-
ground anisotropy. We show the angular power spectrum in DΛCDM model and compare with
the Wilkinson Microwave Anisotropy Probe (WMAP) data.
International Symposium on Nuclear Astrophysics — Nuclei in the Cosmos — IX
June 25-30 2006
CERN, Geneva, Switzerland
∗Speaker.
c ? Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence.
http://pos.sissa.it/

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Observational constraints on a decaying cosmological term
Riou Nakamura
1. Introduction
Recent observations (e.g. type Ia supernovae [1], the cosmic microwave background (CMB)
[2]) indicate that the cosmological term is necessary. If the cosmological term is constant from the
Planck time to the present, there is the cosmological constant problem: the present value of the
cosmological constant is extraordinarily small compared with an inferred vacuum energy during
the Planck time. To solve this problem, it is natural to consider that the cosmological term de-
creases from the large value at the early epoch to the present value. Many functional form of the
cosmological term has been suggested, e.g. the function of the scalar field [3]. On the other hand,
more physically motivated researches of varying vacuum energy (e.g. cosmic quintessence) have
been presented [4].
Cosmological constrains on and results of a decaying vacuum energy density have been inves-
tigated, where the ratio of vacuum to radiation energy was ∼ 4×10−4[5]. We note that vacuum
energy corresponds to a cosmological term in the present paper. The model with a decaying-Λ term
into the radiation has been found to affect the thermal evolution of the universe. Since the radiation
temperature is lower compared with the standard ΛCDM (SΛCDM) model [6], the molecular for-
mation occurs at earlier epoch compared to the case of the SΛCDM [7]. Furthermore the model is
found to be consistent with the CMB temperature observations at z < 4 if appropriate parameters
are adopted [8].
In the present paper, we assume that the Λ term decays into the photon (hereafter we call
this the DΛCDM model) and investigate the CMB temperature fluctuation in the DΛCDM model.
Then, we constrain the parameters of the DΛCDM model.
2. A decaying Λ cosmology
Usingthe Friedmann-Robertson-Walker metric, theEinstein equation andthe energy-momentum
conservation law are written as follows:
?˙ a
a
?2
=8πGN
3
ρ −k
a2+Λ
3,
(2.1)
˙ ρ +
˙Λ
8πGN
= −3˙ a
a(ρ + p),
(2.2)
where a is the cosmic scale factor, k is the curvature and Λ is the cosmological term. The total
energy density ρ and the pressure are written as
ρ = ρm+ργ+ρν,
p =1
3
?ργ+ρν
?,
(2.3)
where the subscripts m,γ and ν are the non-relativistic matter (baryon and cold dark matter), pho-
ton, neutrino, respectively. Here the energy density of matter and neutrinos varies as ρm= ρm0a−3
and ρν0= ρνa−4, where the subscript 0 means the present value. From eqs. (2.2) and (2.3), we get
the evolution of the photon energy density:
dΩγ
da
+4Ωγ
a
= −dΩΛ
da.
(2.4)
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Observational constraints on a decaying cosmological term
Riou Nakamura
101
102
103
Temperature [K]
SΛCDM (m=0)
m=0.5
m=1.0
m=1.2
10-3
10-2
10-1
100
scale factor
0.4
0.6
0.8
1
Ratio
Figure 1: Upper panel: the evolution of the photon temperature Tγ as a function of the scale factor in a
decaying Λ model. Lower panel: the ratio of Tγfor m = 0.5, 1.0, 1.2 relative to m = 0.
with
Ωi≡
ρi
ρcrit
,
ρcrit≡
3H2
8πGN
0
,
ΩΛ≡
Λ
3H2
0
,
where H0is the Hubble constant in unit of km/sec/Mpc.
We assume a functional form of Λ as follow [6, 7, 8, 9]:
ΩΛ= ΩΛ1+ΩΛ2a−m.
(2.5)
Note the present value of ΩΛ: ΩΛ0= ΩΛ1+ΩΛ2.
Following the Stefan-Boltzmann’s law ργ∝T4
γ, the photon temperature evolves as follows [8]:
Tγ=
Tγ0
a
?
1+
mΩΛ2
(4−m)Ωγ0
?a4−m−1??1/4
?1/4
for m ?= 4(2.6)
Tγ=
Tγ0
a
?
1+4ΩΛ2
Ωγ0
lna
for m = 4(2.7)
where Ωγ0= 2.471×10−5h−2(Tγ0/2.725 K)4is the present photon energy density, h is the Hub-
ble constant (H0≡ 100h km/sec/Mpc). If m and/or ΩΛ2are too large, the photon temperature is
negative at some epoch of a < 1. By excluding this kind of solution, we obtain limits on ΩΛ2and
m from eq. (2.6): mΩΛ2/(4−m) < Ωγ0for m < 4. In this parameter region, the first term in eq.
(2.5) dominate the universe at low-z. As the result, the effects of the second term in eq. (2.5) on
the expansion rate is negligible. For ΩΛ2< 0 or m < 0, we find that Tγbecomes negative at a > 1.
Therefore we calculate under the condition of ΩΛ2≥ 0 and m ≥ 0.
Figure 1 illustrates the evolution of the photon temperature in the DΛCDM model. The
adopted cosmological parameters are as follows: h = 0.73, Tγ0= 2.725 K, ΩΛ0= 0.763 and
k = 0 (flat universe). The photon temperature evolves as Tγ∝ a−1and Tγ∝ a−m/4before and
during the Λ dominant epoch, respectively. Changes in Tγaffects the cosmic thermal history sig-
nificantly [7], which should be constrained by the observations such as CMB anisotropy as we shall
show below.
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Observational constraints on a decaying cosmological term
Riou Nakamura
0 200400
multipole l
600 800 1000
0
1000
2000
3000
4000
5000
6000
l (l+1) Cl / 2π [µK2]
SΛCDM (m=0)
m=0.5
m=0.8
m=1.0
WMAP (2006)
Figure 2: The angular power spectrum in a decaying Λ cosmology and WMAP observation data [2]. The
solid line is the result for the SΛCDM model. The dashed, the dot-dashed and the dotted lines are those of
DΛCDM model with (Ω,Λ2,m) = (10−4,0.5), (10−4,1.0) and (10−4,1.2), respectively.
3. CMB constraint
The CMB anisotropy observed by the WMAP constrains the cosmological model with very
high accuracy. In this section, we investigate the consistency of DΛCDM model with WMAP and
give the limit to the model parameters.
We calculate the CMB power spectrum by modifying CMBFAST code [10]. Figure 2 shows
the angular power spectrum in the DΛCDM model. We adopt the following cosmological param-
eters: the baryon density parameter Ωbh2= 0.0223 and the CDM density parameter ΩCDMh2=
0.104. We neglect reionization. If m (and/or ΩΛ2) is small, the amplitude of the power spectrum
decreases. If we take larger values of m, the first and third peaks of the CMB power spectrum in-
creases because of the large baryon density relative to the photon energy density. Furthermore the
CMB power spectrum shifts toward higher-l because the photon last scattering occurs at an earlier
epoch.
To obtain the upper limit of ΩΛ2and m, we calculate the likelihood function given by Ref.
[11]. Figure 3 shows 68.3%, 95.4% and 99.7% confidence limits on the m−ΩΛ2plane from CMB.
We obtain the constraint mΩΛ2/(4−m) < 4.9×10−6at 95 % confidence limit. With the value of
this upper limit, the photon last scattering occurs at the earlier epoch by ∆z∼30 compared with that
in the SΛCDM model. Our constraint is severer than that from the observed radiation temperature
: |m| ≤ 1,|ΩΛ,2| ≤ 10−4[8]. Therefore, the results indicate that a decaying-Λ contribution to the
cosmic thermal evolution should be small.
It should be noted that Ωbh2has been fixed during the calculation for simplicity. If we operate
CMBFAST with Ωbvarying, we can get the more reasonable parameter regions for the DΛCDM
model. Then the primordial abundances of He, D and Li would be different from those predicted
by WMAP.
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Observational constraints on a decaying cosmological term
Riou Nakamura
0
1
234
m
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
ΩΛ2
99.7 % C.L.
95.4 % C.L.
68.3 % C.L.
no Big-Bang
allowed region
Figure 3: Constraint on m−ΩΛ2plane from WMAP. Drawn lines correspond to 1, 2 and 3σ confidence
limit. The labeled no Big-Bang region means that Tγis negative at a < 1.
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