Table Of ContentQuintom scalar field : Varying dark energy equation of state
obtained from recent SNe Ia, BAO and OHD data
Debabrata Adak†1, Abhijit Bandyopadhyay‡2 and Debasish Majumdar†3
3
1
0
Astroparticle Physics and Cosmology Division,
2 †
Saha Institute of Nuclear Physics,
n
a 1/AF Bidhannagar, Kolkata 700064, India
J
0 Department of Physics
1 ‡
Ramakrishna Mission Vivekananda University,
]
O Belur Math, Howrah 711202, India
C
. Abstract
h
p
From the analysis of Supernova Ia data alongwith Observational Hubble Data (OHD) and
-
o BaryonAcousticOscillation(BAO)data,weattempttofindoutthenatureofascalarpotential
r
t that may be responsible for the Dark Energy of the universe. We demonstrate that in order to
s
a explain the varying dark energy equation of state (ωX(z)) as obtained in a model independent
[
way from the analyses of observational data , we need to invoke a quintom scalar field having
2 a “quintessence” part for ωX(z) > 1 and a “phantom” part for ωX(z) < 1. We consider a
v − −
3 Gaussian type potential for these scalar fields and compare the dark energy equation of state
3 derived from such potential with the one computed from the data analysis.
5
1
.
3
0
1
1
:
v
i
X
r
a
1Email: [email protected]
2Email: [email protected]
3Email: [email protected]
1 Introduction
The recent acceleration of the expansion rate of the universe that has been inferred from the
observationandluminositydistancemeasurementsofdifferentSupernovaIa(SNeIa)[1]atdifferent
redshifts (z) is thought to be caused by the existence of a mysterious energy called dark energy.
Different other observational evidence (such as WMAP [2], Cosmic Microwave Background (CMB)
[3]) etc. suggest that this dark energy accounts for almost 73% of the content of the universe.
Dark energy is a hypothetical energy component with a negative pressure responsible for causing
the universe to experience this accelerated expansion. Although there are efforts to understand its
nature and origin, dark energy still remains an enigma.
An attempt to explain dark energy involvs considering a slowly varying potential V(φ) of a
scalar field φ. Hence for such considerations the knowledge of φ and V(φ) are necessary in order
to get an insight of dark energy and its equation of state. In FRW cosmology, it can be shown
that in cosmological constant scenario , the dark energy equation of state remains constant at the
value -1 all along the evolution history of the universe. However if dark energy varies with time
or equivalently with redshift z, a possible explanation may be given by a scalar field (quintessence
field) for dark energy equation of state ω(z) 1. But one needs to invoke a phantom scalar field
≥ −
σ in case ω(z) falls below 1.
−
In the present work, we have considered five different SNe Ia data sets alongwith Observational
Hubble Data (OHD) [4] and Baryon Acoustic Oscillation Data (BAO) [5]. The data sets are made
available from Riess et al (2007)[6], Wood-Vasey et al (2007)[7], Davis et al[8], Kowalski et al
(2008)[9], Kessler et al (2009) [10] and Amanullah et al (2010) (also known as Union2 data)[11].
The maximum span of redshift z for these compilations for distance modulus µ(z) (related to the
luminosity distance d (z)) areintherange0.001 z 1.76. Aparametric formford (z)is chosen
L L
≤ ≤
whose parameters are obtained from a χ2 minimization of various observational data mentioned
above. The dark energy equation of state can be derived analytically using an analytic form of
d (z),suchasthepresentparametricform. Withthefittedvaluesoftheparametersthedarkenergy
L
equation of state ω (z) can therefore be calculated. On the otherhand, since d (z) is related to
X L
the Hubble parameter H(z) by an analytical expression one can use the same parametric form for
d (z) (with parameters from χ2 analyses) and evaluate both the quintessence scalar field φ and the
L
phantom field σ and their variations with the redshift z (see later). With chosen form of potentials
for both φ and σ, one can then compute the dark energy equation of state (ω(z)) from the φ and
σ values obtained from the analysis. In this work, we have taken the form of this potential as
V = V(φ)+V(σ), where V(φ) and V(σ) are chosen to be V(φ) = V exp[λ (1+(φ(z) φ )2/M2)]
0 1 − 0 pl
andV(σ) = V exp[λ (1+(σ(z) σ )2/M2)], whereφ (σ )isthevalueofφ(σ)atpresentepochand
0 2 − 0 pl 0 0
M is the Planck mass. We refer the dark energy equation of state computed from the scalar fields
pl
with this potential, as ω(z). The ω(z) thus obtained is then compared with the ω (z) computed
X
directly from the analysis of µ (z) data with the assumption that the present universe is spatially
L
2
flat and contains only matter and dark energy.
The paper is organised as follows. In Sect. 2 we discuss the dark energy equation of state as
derived from FRW cosmology. In Sect. 3 we propose a quintom scalar formalism of dark energy.
The calculational procedures are given in Sect. 4. Finally in Sect. 5 we furnish our calculational
results and discussions.
2 Brief overview of model independent reconstruction of dark en-
ergy equation of state
For a homogeneous and isotropic universe the Friedmann-Robertson-Walker metric is given by
dr2
ds2 = dt2+a2(t) +r2(dθ2+sin2θdφ2) . (1)
− "1 Kr2 #
−
where a(t) is the scale factor and K is the curvature parameter. Considering the universe to be a
perfect fluid characterized by energy density ρ and pressure p, the Einstein’s equation alongwith
Eq. (1) leads to two independent equations (for spatially flat universe),
8πG
H2(t) = ρ(t) (2)
3
H˙(t) = 4πG(p+ρ) (3)
−
where H(t) = 1 d(a(t)) is the Hubble parameter that denotes the expansion rate of universe.
a(t)dt
From Eq. (2) and Eq. (3) one gets
a¨(t) 4πG
= (ρ+3p). (4)
a(t) − 3
Assuming the universe contains only matter and dark energy, Eq. (2) leads to the result
H2(z) z dz
= Ω0 (1+z)3+Ω0 exp 3(1+ω (z)) , (5)
H2 m X X 1+z
0 (cid:18)Z0 (cid:19)
where the symbol Ω represents energy density or matter density normalised to the critical density
ρ of the universe and thus Ω0 (Ω0 ) represents the matter (dark energy) density parameter at
c m X
present epoch. As mentioned earlier in this section that universe contains only matter and dark
energy in the redshift range considered here, we have Ω (z) + Ω (z) = 1. From Eq. (5) the
m X
equation of state of dark energy ω (z) can be derived as,
X
2(1+z)dH(z) Ω0 (1+z)3H2
ω (z) = 1+ 3 H(z) dz − m H2(z) 0 (6)
X − 1 Ω0 (1+z)3H2
− m H2(z) 0
The SNe Ia observational data are tabulated as a distance modulus µ(z). This is related to
luminosity distance d (z) by
L
µ(z) = 5log (d /Mpc)+25 (7)
10 L
3
for a redshift value z at which the observation was made. The observed distance modulus µ (z)
obs
is related to the apparent magnitude m (z) of a SNe Ia and absolute magnitude M through
obs
the equation µ (z) = m (z) M. Also for spatially flat universe Hubble parameter and the
obs obs
−
luminosity distance are related through
d d (z) −1
L
H(z) = c . (8)
dz (1+z)
(cid:20) (cid:18) (cid:19)(cid:21)
As stated in Sect. 1 a parametric form of d (z) is considered and the parameters are obtained
L
from a χ2 fit of the observational data. The dark energy equation of state ω (z) can then be
X
reconstructed using the Eqns (1 - 5) and Eq. (6). We also mention here that a combined χ2 fit is
made with SNe Ia data, OHD and BAO data.
3 Dark energy equation of state in a scalar field theory framework
Theaccelerationequation(Eq. (4))suggeststhattheuniversecontainingaperfectfluidcanundergo
accelerated expansion only if the pressureof the fluidis negative. Asuitably chosen real scalar field
can produce negative pressure in the FRW spacetime and may also lead to time varying equation
of state of the perfect fluid depending on the potential of the field. Therefore such a real scalar
field can be a possible candidate for time varying dark energy. The scalar fields naturally arise in
particle physics theories including string theory. There exists in literature a wide variety of scalar
field dark energy models. In this work we consider a quintom scalar field model where the two
scalar fields namely φ(z) and σ(z) are responsible for dark energy and the variation of its equation
of state with redshiftz. Each of the fields has specified form of potential and the two kinds of fields
are not coupled to each other. We like to show that this simple model can explain the varying dark
energy equation of state ω (z) obtained by using the combined χ2 analyses results of SNe data,
X
OHD data, BAO data in Eq. (6).
3.1 Dark energy and quintessence scalar field
The action S for Quintessence scalar field is given by (with V(φ), the potential of the field φ),
1
S = S + d4x√ g gµν(∂ φ)∂ φ)+V(φ) , (9)
m µ ν
− 2
Z (cid:20) (cid:21)
where S is the matter action. The equation of motion of the spatially homogeneous quintessence
m
field is given by
dV(φ)
φ¨(t)+3H(t)φ˙(t)+ = 0 . (10)
dφ
The Eq. (10) will lead to
1 d
φ˙2(t)= (H2) , (11)
−8πGH(t)dt φ
4
where H2 = 8πG 1φ˙2(t)+V(φ) and H2 = 8πG 1φ˙2(t)+V(φ)+ρ . The Energy momentum
φ 3 2 3 2 m
tensor for quintes(cid:16)sence scalar fiel(cid:17)d can be written(cid:16)as (cid:17)
1
Tµ = gµα∂ φ∂ φ δµ gαβ∂ φ∂ φ+V(φ) (12)
ν α ν − ν 2 α β
(cid:18) (cid:19)
From Eq. (12) we have,
1
ρ = T0 = φ˙2+V(φ) (13)
φ − 0 2
1
p = Ti = φ˙2 V(φ) (14)
φ i 2 −
p 1φ˙2 V(φ)
ω (t) = φ = 2 − (15)
φ ρφ 1φ˙2+V(φ)
2
where ρ and p are respectively the pressure and density due to the scalar field φ. It is evident
φ φ
from Eq. (15) that ω can vary between -1 and +1.
φ
3.2 Dark energy and phantom scalar field
The action for Phantom scalar field σ(t) is given by (with V(σ), the potential of the field σ),
1
S = S + d4x√ g gµν(∂ σ)∂ σ)+V(σ) . (16)
m µ ν
− −2
Z (cid:20) (cid:21)
whereS is the matter action. Theequation of motion of thespatially homogeneous phantom field
m
can be written as
dV(σ)
σ¨(t)+3H(t)σ˙(t) = 0 . (17)
− dσ
The above equation leads to
1 d
σ˙2(t) = (H2) , (18)
8πGH(t)dt σ
where H2 = 8πG 1σ˙2(t)+V(σ) and H2 = 8πG 1σ˙2(t)+V(σ)+ρ . The Energy momentum
σ 3 2 3 2 m
tensor for phantom(cid:16) scalar field is(cid:17)given by (cid:16) (cid:17)
1
Tµ = gµα∂ σ∂ σ δµ gαβ∂ σ∂ σ+V(σ) . (19)
ν − α ν − ν 2 α β
(cid:18) (cid:19)
From Eq. (19) we have,
1
ρ = T0 = σ˙2+V(σ) (20)
σ − 0 −2
1
p = Ti = σ˙2 V(σ) (21)
σ i −2 −
p 1σ˙2 V(σ)
ω (t) = σ = −2 − (22)
σ ρ 1σ˙2+V(σ)
σ −2
The above equation shows that ω can vary between -1 and - .
σ
∞
5
3.3 Quintom scalar field motivated dark energy equation of state
We consider here a model [12] which contains a negative kinetic scalar field σ and a normal scalar
field φ with a general potential, described by
R 1 1
S = S + d4x√ g gµν∂ σ∂ σ+ gµν∂ φ∂ φ+V(σ,φ) , (23)
m µ ν µ ν
− 16πG − 2 2
Z (cid:20) (cid:21)
where S = d4x and represents the Lagrangian density of matter fields. Assuming the
m m m
L L
fields to be homogeneous, in a spatially flat FRW cosmological model the equation of motion of the
R
fields and the matter density are given by
dV(φ)
φ¨(t)+3H(t)φ˙(t)+ = 0 (24)
dφ
dV(σ)
σ¨(t)+3H(t)σ˙(t) = 0 (25)
− dσ
ρ˙ +3H(t)(ρ +p ) = 0 , (26)
γ γ γ
where the ρ is the density of fluid (matter) with a barotropic equation of state p = (γ 1)ρ , γ
γ γ γ
−
being a constant with 0 γ 2. For radiation, γ = 4/3 and for dust γ = 1. The corresponding
≤ ≤
effective pressure and energy densities will be
1 1
p = σ˙2+ φ˙2 V(σ,φ) (27)
−2 2 −
1 1
ρ = σ˙2+ φ˙2+V(σ,φ) (28)
−2 2
In this formalism the Friedmann equation becomes
8πG 1 1
H2 = σ˙2+V(σ)+ φ˙2+V(φ)+ρ
γ
3 −2 2
(cid:18) (cid:19)
8πG
= H2+H2+ ρ (29)
σ φ 3 γ
and the equation of state parameter ω(z) as obtained from the present quintom scalar field formal-
ism will then be written as
σ˙2+φ˙2 2V(σ,φ)
ω = − − . (30)
σ˙2+φ˙2+2V(σ,φ)
−
If the two scalar fields (σ and φ) are not directly coupled to each other then V(σ,φ) can be written
as V(σ,φ) = V(σ)+V(φ). With this assumption one can readily see that for φ˙ σ˙, Eq. (30) gives
≥
to ω 1 implying the scenario for quintessence scalar field and for φ˙ <σ˙, ω < 1 signifying the
≥ − −
phantom scalar field model.
4 Calculational Procedure
The purpose of this work is to show the viability of the present formalism of quintom scalar field
model in explaining the dark energy and the variation of its equation of state with z. In order
6
to obtain the dark energy equation of state from observational data, a parametric form for d (z)
L
[13, 14] is first considered whose parameters are fixed by the χ2 minimization of the combined data
(SNe Ia, OHD, BAO). The matter density at the present epoch Ω0 is also made a parameter in
m
this χ2 analyses and its value is obtained from the same χ2 minimization. The form of d (z) can
L
now be used to calculate Hubble parameter H(z) using Eq. (8). Eq. (6) now readily gives the dark
energy equation of state ω (z).
X
The parametric form of d (z) should respect the conditions that d (z) = 0 at z = 0 and
L L
d (z) z for large z. SNe Ia data are available for z < 1.76 from various supernova observations.
L
∝ ∼
A parametrised form of d (z) can be written as [13, 14]
L
c z(1+az)
d (z) = (31)
L
H 1+bz
0 (cid:20) (cid:21)
The parameters a, b and Ω0 are obtained by minimising a suitably defined χ2. In the present case
m
the χ2 is defined as [14],
χ2 = χ2 +χ2 +χ2 (32)
tot SN BAO OHD
whereχ2 correspondstothe SNeIadata, χ2 definestheχ2 forBaryon acoustic oscillation data
SN BAO
and χ2 represents the χ2 for Observational Hubble Data (OHD).
OHD
χ2 [15] is defined as
SN
N (µ (z) µ (a,b,z))2
χ2 (a,b,M′)= obs − th .
SN σ2
i=1 i
X
In the above, the theoretical distance modulus µ (a,b,z) can be obtain from Eq. (7). The above
th
equation can be written in the form [15]
B2
χ2 (a,b,M′ = B/C)= A . (33)
SN − C
where
(5log (D (a,b,z )) m (z ))2
A = 10 L i − obs i , (34)
σ2
zi i
X
(5log (D (a,b,z )) m (z ))
B = 10 L i − obs i , (35)
σ2
zi i
X
1
C = . (36)
σ2
zi i
X
In the above D (z) = H d (z)/c and σ are the errors.
L 0 L i
For the case of BAO data χ2 is defined as
[A(Ω0 ,a,b) 0.469]2
χ2 (Ω0 ,a,b) = m − (37)
BAO m 0.0172
7
where
Ω0 1 z1 dz 2/3
A(Ω0 ,a,b) = m (38)
m [H(z )/H ]1/3 z H(z)/H
p1 0 (cid:20) 1 Z0 0(cid:21)
The experimental value of A = 0.469 0.017 [5] in the above is obtained from the Sloan Digital
±
Sky Survey (SDSS) [5] data at z = 0.35.
For OHD data, χ2 is defined as
15 H (a,b;z ) H (z ) 2
χ2 = th i − obs i . (39)
OHD σ
i=1(cid:20) i (cid:21)
X
In the above, σ denote the errors and H (a,b;z) is obtained as (using Eqs. 8 and 31)
i th
[(1+az)(1+bz)]2
H (a,b;z) = H . (40)
th 0(1+2az+(ab+a b)z2)
−
The parameters a,b (and Ω0 ) are obtained by minimizing the χ2 defined above with SNe Ia, OHD
m
and BAO data together. As described earlier in Sect. 1 we consider five data sets of SNe Ia. Thus
we have five combined data sets of (SNe Ia + OHD + BAO) and from the χ2 minimization five
sets of parameters (a, b, Ω0 ) are obtained [14]. Needless to mention that for all the five combined
m
data sets OHD and BAO data remain the same.
4.1 Variation of ω(z) with scalar fields
The dark energy equation of state obtained from the present quintom scalar field model is denoted
as ω(z). The variation of ω(z) with scalar fields can be obtained using Eq. (30) with a chosen form
for the scalar potentials. In the present work the form of the potentials for the fields φ and σ are
taken as
V(φ) = exp[λ (1+(φ/M )2)]
1 pl
and
V(σ) = exp[λ (1+(σ/M )2)]
2 pl
where M (= 1/8πG) is the Planck mass and λ and λ are two parameters.
pl 1 2
From Eq. (11), we obtain for quintessence scalar field
z dz d
(φ(z) φ )/M = (1+z) (H2) , (41)
− 0 pl −Z0 (1+z)H(z)s dz φ
and from Eq. (18) we get for phantom field as
z dz d
(σ(z) σ )/M = (1+z) (H2) , (42)
− 0 pl −Z0 (1+z)H(z)s− dz σ
8
where φ and σ are values of the fields at present epoch. With the parameters a and b obtained
0 0
from the χ2 fit, the Hubble parameter H(z) appearing in Eqs. (41, 42) can be computed from Eq.
(40).
AsthequantitiesH andH cannotdirectlybeextracted fromdata, itisdifficulttoobtainφ(z)
φ σ
and σ(z) form Eqs. (41, 42). It is also not possible to choose the boundary conditions of both the
fields from observations. In order to circumvent this problem we propose the following formalism.
For a varying dark energy density ω(z) with redshift z, when ω(z) > 1, the quintessence scalar
−
field φ(z) >> σ(z) and in the case when ω(z) < 1, the phantom scalar field σ(z) >> φ(z). In
−
the former case therefore (ω(z) > 1), σ(z) can be taken to be zero for all practical purposes and
−
similarly for the latter case (ω(z) < 1), φ(z) can be taken to be zero. Thus, defining z as the
c
−
value of redshift z at which ω(z) = 1, we can write, using this formalism, that in the redshift
−
limit when z > z , σ(z) = 0 and when z < z (z 0), φ(z) = 0. At z = z , however, φ(z) = σ(z) in
c c c
≥
order that the continuity of the variation of ω(z) with z is not affected.
Under this formalism, for z > z , the quintessence scalar field can be derived from Eq. (11) as,
c
z dz d
(φ(z) φ )/M = (1+z) (H2) , (43)
− c pl −Zzc (1+z)H(z)s dz φ
where H2 = H2+ 8πGρ (neglecting H2 in Eq. (29)). Also, for the case z < z ,z 0, we get from
φ 3 γ σ c ≥
Eq. (18) the phantom field as
z dz d
(σ(z) σ )/M = (1+z) (H2) , (44)
− 0 pl −Z0 (1+z)H(z)s− dz σ
where H2 = H2 + 8πGρ (neglecting H2 in Eq. (29)). In the present work we are considering
σ 3 γ φ
the redshift limit 0 < z < 1.76 the present observational reach of SNe Ia. In this epoch the
−
universe is supposed to be dominated by dark matter and dark energy and therefore ρ in Eq.
γ
(29) denotes the matter density ρ . Hence the matter density can be written in terms of redshift
m
z as 8πGρ = H2Ω0 (1 + z)3, where H and Ω0 are the Hubble parameter and matter density
3 m 0 m 0 m
parameter in the present epoch respectively.
Now as mentioned earlier in the beginning of this section, ω (z) can be calculated from Eq.
σ,φ
(30) with potentials exp[λ (1 + ((φ φ )/M )2)] and exp[λ (1 + ((σ σ )/M )2)] for different
1 c pl 2 0 pl
− −
choices of λ and λ .
1 2
5 Results and Discussions
Using the formalism described in Sect. 4, the χ2 fit was performed [14] for each of the five sets of
SNe Ia data alongwith BAO and OHD data and the best fit values of the parameters a, b and Ω0
m
obtained for all the five sets [14]. The best fit parameters for each set (alongwith BAO and OHD)
are given in Table 1. From the obtained best fit values of the parameters the equation of state
9
MLCS+BAO+OHD SALT2+BAO+OHD
0.6 -0.1
-0.2
0.4
-0.3
0.2
-0.4
0 -0.5
ω(z)) ω(z)) -0.6
(z) (X -0.2 (z) (X -0.7
ω ω
-0.4 -0.8
-0.9
-0.6
-1
-0.8
-1.1
-1 -1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
z z
HST+SNLS+ESSENCE+BAO+OHD UNION+BAO+OHD
-0.1 0.6
-0.2 0.4
-0.3
0.2
-0.4
0
-0.5
-0.2
ω(z)) -0.6 ω(z))
(z) (X -0.7 (z) (X -0.4
ω ω
-0.6
-0.8
-0.8
-0.9
-1
-1
-1.1 -1.2
-1.2 -1.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
z z
UNION2+BAO+OHD
-0.9
-1
-1.1
-1.2
-1.3
ω(z)) -1.4
(z) (X -1.5
ω
-1.6
-1.7
-1.8
-1.9 10
-2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
z
Figure 1: Comparison of ω(z) vs z and ω (z) vs z for the 5 data sets considered (see text). ω(z)
