Table Of ContentBraneworld Quintessential Inflation and Sum
of Exponentials Potentials
7
0
0 Carl L. Gardner
2
[email protected]
n
a Department of Mathematics and Statistics
J
4 Arizona State University
Tempe AZ 85287-1804
1
v
6
3
0 Abstract
1
0
Quintessential inflation—in which a single scalar field plays the
7
0 role of the inflaton and quintessence—from a sum of exponentials
/ potential V = A e5ϕ+e√2ϕ or V = A e5ϕ +eϕ or a cosh poten-
h
p tial V = 2Acosh((cid:16)5ϕ) is consi(cid:17)dered in the(cid:0)context o(cid:1)f five-dimensional
-
p gravitationwithstandardmodelparticlesconfinedtoour3-brane. Re-
e heating is accomplished via gravitational particle production and the
h
: universe undergoes a transition from primordial inflation to radiation
v
domination well before big bang nucleosynthesis. The transition to
i
X an accelerating universe due to quintessence occurs near z 1, as in
≈
r ΛCDM.
a
Braneworldquintessentialinflationcanoccurforpotentialswithor
without a minimum, and with or without eternal acceleration and an
event horizon. The low z behavior of the equation of state parameter
w provides a clear observable signal distinguishing quintessence from
φ
a cosmological constant.
1 Introduction
In five-dimensional gravitation with standard model particles confined to
our 3-brane, the Friedmann equation is modified [1]–[3] at high energies: the
square H2 of the Hubble parameter acquires a term quadratic in the energy
density, allowing [4] slow-roll inflation to occur for potentials that would be
too steep to support inflation in the standard Friedmann-Robertson-Walker
cosmology. In this model, quintessential inflation—in which a single scalar
field plays the role of the inflaton and quintessence—can occur for a sum
of exponentials or cosh potential, a type of potential that arises naturally
in M/string theory for combinations of moduli fields or combinations of the
dilaton and moduli fields.
Inflation ends in this model as the quadratic term in energy density in
H2 decays to roughly the same order of magnitude as the standard linear
term (technically, when the slow-roll parameter ǫ = 1). Reheating can be
accomplished [5] via gravitational particle production at the end of inflation.
The universe subsequently [5] undergoes a transition from the era dominated
by the scalar field potential energy to an era of “kination” dominated by
the scalar field kinetic energy, and then to the standard radiation dominated
era well before big-bang nucleosynthesis (BBN). With a sum of exponen-
tials or cosh potential, the universe evolves at this point according to the
quintessence/cold dark matter (QCDM) model. The transition to an accel-
erating universe due to quintessence occurs late in the matter dominated era
near z 1, as in ΛCDM.
≈
We will assume a flat universe after inflation. In the QCDM model,
the total energy density ρ = ρ + ρ + ρ = ρ , where ρ is the critical
m r φ c c
energy density for a flat universe and ρ , ρ , and ρ are the energy densities
m r φ
in (nonrelativistic) matter, radiation, and the quintessential inflation scalar
field φ, respectively. Ratios of energy densities to the critical energy density
will be denoted by Ω = ρ /ρ , Ω = ρ /ρ , and Ω = ρ /ρ , while ratios
m m c r r c φ φ c
of present energy densities ρ , ρ , and ρ to the present critical energy
m0 r0 φ0
density ρ will be denoted by Ω , Ω , and Ω , respectively.
c0 m0 r0 φ0
Using WMAP3 [6] central values, we will set Ω = 0.74, Ω = 8.04
φ0 r0
×
10 5, Ω = 1 Ω Ω 0.26, and ρ1/4 = 2.52 10 3 eV, with the
− m0 − φ0 − r0 ≈ c0 × −
present time t = 13.7 Gyr after the big bang.
0
For the sum of exponentials potential, we will take a monotonically in-
creasing function of ϕ (ϕ will evolve from ϕ 1 at the end of inflation to
i
≫
ϕ 1 today)
0
| | ∼
V(ϕ) = A eλϕ +Beµϕ (1)
(cid:16) (cid:17)
where A and B are positive constants, λ > µ > 0, ϕ = φ/M , the (reduced)
P
Planck mass M = 2.44 1018 GeV, and with λ = 5, µ = √2 or 1, and
P
×
B = 1 for the simulations. We will also discuss the simple exponential
2
potential V(ϕ) = Aeλϕ considered in Ref. [5] and the cosh potential V(ϕ) =
2Acosh(λϕ) with λ = 5. For all four potentials, V Aeλϕ for ϕ 1 (which
∼ ≫
will be the case during inflation and gravitational particle production). The
constant A ρ for quintessential inflation (see Table 1).
c0
∼
NotethattheBBN(z 109–1011), cosmic microwave background (CMB)
∼
(z 103–105), and large-scale structure (LSS) (z 10–104) bounds Ω < 0.1
φ
∼ ∼ ∼
are satisfied in all the simulations below (Figs. 3, 8, and 12) and that the
transition from the era of scalar field dominance to the radiation era occurs
around z 1015/λ.
≈
Quintessential inflation in the standard cosmology from a sum of expo-
nentials potential was considered in Ref. [7] for λ = 20 and µ = 0.5 or
20. Ref. [8] extended the analysis to the braneworld case where H2 has
−
a quadratic term in energy density, for λ 20 and µ λ. Braneworld
≈ ≈ −
quintessential inflation is also discussed in Ref. [9], for λ = 4 and µ = 0.1.
Only one simulation for the sum of exponentials potential braneworld sce-
nario is presented, in Ref. [9] for the equation of state parameter w . We
φ
find that the case λ = 4 and µ = 0.1 violates even the looser CMB bound
Ω 0.2 (see Fig. 1), and that the recent average of the scalar field equation
φ
≤
of state parameter w = 0.68 is too large.
0
−
Figure 1: Ω for V = A(e4ϕ +e0.1ϕ) (solid) vs. ΛCDM (dotted). The light
yellow rectangles are the bounds Ω 0.1 from LSS, CMB, and BBN.
φ
≤
The current investigation furthermore presents detailed simulations and
3
analyses of the evolution of the scalar field and its equation of state, the
fractional energy densities in the scalar field, radiation, and matter, and the
acceleration parameter in braneworld quintessential inflation for the sum of
exponentials and cosh potentials.
2 Cosmological Equations
The homogeneous scalar field—since it is confined to the brane—still obeys
the Klein-Gordon equation
dV
¨ ˙
φ+3Hφ = V . (2)
φ
−dφ ≡ −
The Hubble parameter H is related to the scale factor a and the energy
densities in matter, radiation, and the scalar field through the braneworld
modified [4] Friedmann equation
a˙ 2 ρ ρ
H2 = = 1+ +Λ + E (3)
a 3M2 2σ 4 a4
(cid:18) (cid:19) P (cid:18) (cid:19)
where the energy density
1
ρ = ρ +ρ +ρ , ρ = φ˙2 +V(φ), (4)
φ m r φ
2
σ is the four-dimensional brane tension, Λ is the four-dimensional cosmolog-
4
ical constant, and is a constant embodying the effects of bulk gravitons on
E
the brane. We will set the four-dimensional cosmological constant to zero.
The “dark radiation” term /a4 can be ignored here since it will rapidly
E
go to zero during inflation. Thus for our purposes the modified Friedmann
equation takes the form
ρ ρ
H2 = 1+ . (5)
3M2 2σ
P (cid:18) (cid:19)
In the low-energy limit ρ σ, the Friedmann equation reduces to its stan-
≪
dard form H2 = ρ/(3M2).
P
The conservation of energy equation for matter, radiation, and the scalar
field is
ρ˙ +3H(ρ+P) = 0 (6)
4
where P is the pressure. Except near particle-antiparticle thresholds, P =
m
0 and P = ρ /3. Equation (6) gives the evolution of ρ and ρ , and the
r r m r
Klein-Gordon equation (2) for the weakly coupled scalar field, with
1
P = φ˙2 V(φ). (7)
φ
2 −
The acceleration equation
a¨ 1 ρ
= ρ+3P + (2ρ+3P) (8)
a −6M2 σ
P (cid:18) (cid:19)
follows from Eqs. (5) and (6). While inflation occurs in the low-energy (stan-
dard cosmology) limit when w P/ρ < 1/3, for inflation to occur in the
≡ −
high-energy limit w < 2/3.
−
Wewillusethelogarithmictimevariableτ = ln(a/a ) = ln(1+z). Note
0
−
that for de Sitter space τ = H t, where H2 = ρ /(3M2), and that H t is a
Λ Λ Λ P Λ
natural time variable for the era of Λ-matter domination (see e.g. Ref. [10]).
For 0 z z 1010, 23.03 τ 0, and for 0 z z 2.8 1031,
BBN Pl
≤ ≤ ∼ − ≤ ≤ ≤ ≤ ∼ ×
72.41 τ 0.
− ≤ ≤
In Eq. (4), we will make the simple approximations
ρ = ρ e 4τ, ρ = ρ e 3τ. (9)
r r0 − m m0 −
For the spatially homogeneous scalar field φ, the equation of state pa-
rameter w = w (z) = P /ρ . The recent average of w is defined as
φ φ φ φ φ
1 τ
w = w dτ. (10)
0 φ
τ
Z0
We will take the upper limit of integration τ to correspond to z = 1.75. The
SNe Ia observations [11] bound the recent average w < 0.76 (95% CL)
0
−
assuming w 1, and measure the transition redshift z = 0.46 0.13 from
0 t
≥ − ±
deceleration to acceleration (it is probably premature at this point to say
more than that z 1).
t
≈
For numerical simulations, the cosmological equations should be put into
a scaled, dimensionless form. Equations (2) and (5) can be cast [12] in the
form of a system of two first-order equations in τ plus a scaled version of H:
H˜ϕ = ψ (11)
′
5
H˜(ψ +ψ) = 3V˜ (12)
′ ϕ
−
ρ˜
H˜2 = ρ˜ 1+ (13)
2σ˜
(cid:18) (cid:19)
1
ρ˜= ψ2 +V˜ +ρ˜ +ρ˜ (14)
m r
6
where ψ e2τϕ˙/H , H˜ = e2τH/H , V˜ = e4τV/ρ , V˜ = e4τV /ρ , ρ˜ =
0 0 c0 ϕ ϕ c0
≡
e4τρ/ρ , ρ˜ = e4τρ /ρ = Ω eτ, ρ˜ = e4τρ /ρ = Ω , σ˜ = e4τσ/ρ , and
c0 m m c0 m0 r r c0 r0 c0
where a prime denotes differentiation with respect to τ: ϕ = dϕ/dτ, etc.
′
This scaling results in a set of equations that is numerically more robust,
especially before the time of BBN.
Figure 2: Log of H˜ (blue, bottom) and H/H (cyan, top) vs. log (1+z)
10 0 10
for V = A e5ϕ +e√2ϕ .
(cid:16) (cid:17)
Figure 2 illustrates that while H˜ spans only ten orders of magnitude
between z 1023 and the present, H/H spans more than fifty orders of
i 0
∼
magnitude.
3 Evolution of the Braneworld Universe
In this section, we will closely follow the analyses of Refs. [5] and [4].
6
3.1 Slow-Roll Braneworld Inflation
The inflationary slow-roll parameter ǫ is given by [4]
H˙ M2 V 2 1+V/σ
ǫ P φ . (15)
≡ −H2 ≈ 2 V (1+V/(2σ))2
(cid:18) (cid:19)
(The slow-roll parameter
V 1
η = M2 φφ ǫ (16)
P V 1+V/(2σ) ≈
for the potentials considered here.) Inflation occurs for ǫ < 1. The slow-roll
parameter can be approximated during inflation by
2λ2σ
ǫ (17)
≈ V
for the sum of exponentials potential, the simple exponential potential, and
the cosh potential, since V σ and ϕ 1 for our models during inflation.
≫ ≫
Inflation ends when ǫ = 1, implying [5] that the potential V at the end of
e
inflation is
V 2λ2σ. (18)
e
≈
To evaluate V and σ, the COBE-measured amplitude of primordial density
e
perturbations is matched against the theoretical value atN = 50 e-folds from
the end of inflation, where
te 1 φN V V 1
N = Hdt 1+ dφ (V V ) (19)
≈ M2 V 2σ ≈ 2λ2σ N − e
ZtN P Zφe φ (cid:18) (cid:19)
yielding V (N+1)V . Now matching against the amplitude [4] of density
N e
≈
fluctuations
2 1 V3 V 3 1 V4
2 10 5 = A2 1+ (20)
× − S ≈ 75π2M6 V2 2σ ≈ 600π2M4λ2σ3
(cid:16) (cid:17) P φ (cid:18) (cid:19) P
gives [5]
V1/4 1.11 1015 GeV/λ, σ1/4 9.37 1014 GeV/λ3/2. (21)
e ≈ × ≈ ×
7
3.2 Gravitational Reheating and Initial Conditions
At the end of inflation, gravitational particle production [13]–[15] results in
a radiation energy density [5]
ρ H4 V3
r 0.01g e 2.8 10 4g e 4.86 10 17g (22)
ρ ≈ p V ≈ × − pM4σ2 ≈ × − p
φ e P
where g 10–100 is the number of particle species that are gravitationally
p
≈
produced. This relation fixes the “initial conditions” redshift z 3.89
i
≈ ×
1023g1/4/λ atwhich gravitationalparticle productionoccurs, fromthescaling
p
1011 GeV 4
ρ 4.86 10 17g ρ g (1+z )4ρ (23)
r − p φ p i r0
≈ × ≈ λ ! ≈
corresponding [5] to a reheating temperature
g 1/4 1011 GeV
p
T (24)
e
∼ g ! λ
∗
where g (T) is the effective number of massless degrees of freedom at temper-
∗
ature T. The universe undergoes [5] a transition from the era dominated by
the scalar field potential energy to an era of kination (ρ 1/a6) dominated
φ
∼
by the scalar field kinetic energy as the ρ2/(2σ) term in H2 becomes small
compared with ρ. For this era of kination to occur, the potential V must be
sufficiently steep. Since during kination ρ /ρ a2, the universe eventually
r φ
∼
makes a transition [5] to the standard radiation dominated era, around a
temperature T 103 GeV/λ, well before big-bang nucleosynthesis at T 1
≈ ≈
MeV.Forthesum ofexponentials or coshpotential, theuniverse thenevolves
according to the QCDM model. The transition to an accelerating universe
due to quintessence occurs late in the matter dominated era near z 1, as
≈
in ΛCDM.
The initialconditions at the end of inflation will be set in the high-energy,
slow-roll limit. The initial value for φ is specified by Eq. (18). The initial
i
˙
value for φ follows from the high-energy, slow-roll limits
i
ρ2 V2
H2 (25)
≈ 6M2σ ≈ 6M2σ
P P
and
3 V V
¨ ˙ ˙
φ+3Hφ φ V λ (26)
φ
≈ s2M √σ ≈ − ≈ − M
P P
8
or [4]
1 V 1
φ˙2 e λ2σ. (27)
2 i ≈ 6 ≈ 3
4 Simulations
Forthecomputationsbelow, wewilluseEqs. (11)–(14)withinitialconditions
specified at z by ϕ and ϕ˙ ψ (Eqs. (18) and (27)). We set λ = 5 and g
i i i i p
∝
= 100. The constant A in the potentials is adjusted so that Ω = 0.74. This
φ0
involves the usual single fine tuning. The final time t is set by z = 0.999
f f
−
corresponding to t 100 Gyr.
f
∼
First we briefly summarize the properties of the exponential potential,
and then turn our attention to the quintessential inflation potentials.
V(ϕ)/A A/ρ ϕ ϕ w
c0 i 0 0
2cosh(5ϕ) 0.4 48.0 0.03 0.87
− −
e5ϕ +e√2ϕ 5.7 47.4 1.60 0.78
− −
e5ϕ +eϕ 2.2 47.6 1.18 0.89
− −
Table 1: Simulation results for the cosh and sum of exponentials potentials.
4.1 Exponential Potential
The exponential potential V(ϕ) = Aeλϕ [15]–[18] can be derived from M-
theory [19] or from N = 2, 4D gauged supergravity [20].
For the exponential potential with λ2 > 3, the cosmological equations
have a global attractor with Ω = 3/λ2 during the matter dominated era
φ
(during which w = 0); and with λ2 > 4, the cosmological equations have a
φ
global attractor with Ω = 4/λ2 during the radiation dominated era (during
φ
which w = 1/3). For λ2 < 3, the cosmological equations have a late time
φ
attractor with Ω = 1 and w = λ2/3 1.
φ φ
−
For λ = √2 and ρ = 0, a¨ 0 asymptotically; if ρ > 0, the universe
m m
→
eventually enters a future epoch of deceleration. In either case, there is no
event horizon. For λ < √2, the universe enters a period of eternal accelera-
tion with an event horizon. For λ > √2, the universe eventually decelerates
and there is no event horizon.
9
The ΛCDM cosmology is approached for λ 1/√3. Significant acceler-
≤
ation occurs only for λ < √3. For λ = √3, w is much too high; for a viable
0
∼
present-day QCDM cosmology, λ √2 [12] in the exponential potential.
≤
4.2 Cosh Potential
In the cosh potential model
V = 2Acosh(5ϕ) ρ cosh(5ϕ), (28)
Λ
≈
dark energy derives from the value of the potential near its minimum. This
is the simplest way to “correct” the exponential potential to incorporate
quintessence.
In the simulations for the cosh potential (Figs. 3–7), initially ϕ 50 and
≈
then evolves to ϕ = 0.03 at t . The linear decay of ϕ(τ) in Fig. 4 during
0 0
the kination era occur−s because ρ 1φ˙2 1/a6 and thus ϕ is a (negative)
φ ≈ 2 ∼ ′
constant since both H˜ and ψ are proportional to e τ in Eq. (11).
−
Figure 3: Ω for V = 2Acosh(5ϕ) (solid) vs. ΛCDM (dotted).
Between z 2.5 1014 (as the universe enters its radiation dominated
≈ ×
stage) and z 10 (when m2 λ2V/M2 becomes of order H2 ρ /(3M2)),
≈ φ ≈ P ≈ m P
ϕ “sits and waits” at a small negative value.
10
