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Braneworld Quintessential In?ation and Sum of Exponentials Potentials Carl L.Gardner gardner@https://www.doczj.com/doc/5411511995.html, Department of Mathematics and Statistics Arizona State University Tempe AZ 85287-1804Abstract Quintessential in?ation—in which a single scalar ?eld plays the role of the in?aton and quintessence—from a sum of exponentials potential V =A e 5?+e √
density,allowing[4]slow-roll in?ation to occur for potentials that would be too steep to support in?ation in the standard Friedmann-Robertson-Walker cosmology.In this model,quintessential in?ation—in which a single scalar ?eld plays the role of the in?aton 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?elds or combinations of the dilaton and moduli?elds.
In?ation ends in this model as the quadratic term in energy density in H2decays to roughly the same order of magnitude as the standard linear term(technically,when the slow-roll parameter?=1).Reheating can be accomplished[5]via gravitational particle production at the end of in?ation. The universe subsequently[5]undergoes a transition from the era dominated by the scalar?eld potential energy to an era of“kination”dominated by the scalar?eld 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 inΛCDM.
We will assume a?at universe after in?ation.In the QCDM model, the total energy densityρ=ρm+ρr+ρφ=ρc,whereρc is the critical energy density for a?at universe andρm,ρr,andρφare the energy densities in(nonrelativistic)matter,radiation,and the quintessential in?ation scalar ?eldφ,respectively.Ratios of energy densities to the critical energy density will be denoted by?m=ρm/ρc,?r=ρr/ρc,and?φ=ρφ/ρc,while ratios of present energy densitiesρm0,ρr0,andρφ0to the present critical energy densityρc0will be denoted by?m0,?r0,and?φ0,respectively.
Using WMAP3[6]central values,we will set?φ0=0.74,?r0=8.04×10?5,?m0=1??φ0??r0≈0.26,andρ1/4c0=2.52×10?3eV,with the present time t0=13.7Gyr after the big bang.
For the sum of exponentials potential,we will take a monotonically in-creasing function of?(?will evolve from?i?1at the end of in?ation to |?0|～1today)
V(?)=A eλ?+Beμ? (1)
where A and B are positive constants,λ>μ>0,?=φ/M P,the(reduced)
√
Planck mass M P=2.44×1018GeV,and withλ=5,μ=
potential V(?)=Aeλ?considered in Ref.[5]and the cosh potential V(?)= 2A cosh(λ?)withλ=5.For all four potentials,V～Aeλ?for??1(which will be the case during in?ation and gravitational particle production).The constant A～ρc0for quintessential in?ation(see Table1).
Note that the BBN(z～109–1011),cosmic microwave background(CMB) (z～103–105),and large-scale structure(LSS)(z～10–104)bounds?φ<～0.1 are satis?ed in all the simulations below(Figs.3,8,and12)and that the transition from the era of scalar?eld dominance to the radiation era occurs around z≈1015/λ.
Quintessential in?ation in the standard cosmology from a sum of expo-nentials potential was considered in Ref.[7]forλ=20andμ=0.5or ?20.Ref.[8]extended the analysis to the braneworld case where H2has a quadratic term in energy density,forλ≈20andμ≈?λ.Braneworld quintessential in?ation is also discussed in Ref.[9],forλ=4andμ=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 wφ.We ?nd that the caseλ=4andμ=0.1violates even the looser CMB bound ?φ≤0.2(see Fig.1),and that the recent average of the scalar?eld equation of state parameter
analyses of the evolution of the scalar?eld and its equation of state,the fractional energy densities in the scalar?eld,radiation,and matter,and the acceleration parameter in braneworld quintessential in?ation for the sum of exponentials and cosh potentials.
2Cosmological Equations
The homogeneous scalar?eld—since it is con?ned to the brane—still obeys the Klein-Gordon equation
¨φ+3H˙φ=?dV
a 2=ρ2σ
+Λ4+E
2
˙φ2+V(φ),(4)
σis the four-dimensional brane tension,Λ4is the four-dimensional cosmolog-ical constant,and E is a constant embodying the e?ects of bulk gravitons on the brane.We will set the four-dimensional cosmological constant to zero. The“dark radiation”term E/a4can be ignored here since it will rapidly go to zero during in?ation.Thus for our purposes the modi?ed Friedmann equation takes the form
H2=ρ
2σ .(5)
In the low-energy limitρ?σ,the Friedmann equation reduces to its stan-dard form H2=ρ/(3M2P).
The conservation of energy equation for matter,radiation,and the scalar ?eld is
˙ρ+3H(ρ+P)=0(6)
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where P is the pressure.Except near particle-antiparticle thresholds,P m= 0and P r=ρr/3.Equation(6)gives the evolution ofρm andρr,and the Klein-Gordon equation(2)for the weakly coupled scalar?eld,with
Pφ=
1
a =?
1
σ
(2ρ+3P) (8)
follows from Eqs.(5)and(6).While in?ation occurs in the low-energy(stan-dard cosmology)limit when w≡P/ρ
We will use the logarithmic time variableτ=ln(a/a0)=?ln(1+z).Note that for de Sitter spaceτ=HΛt,where H2Λ=ρΛ/(3M2P),and that HΛt is a natural time variable for the era ofΛ-matter domination(see e.g.Ref.[10]). For0≤z≤z BBN～1010,?23.03≤τ≤0,and for0≤z≤z P l～2.8×1031,?72.41≤τ≤0.
In Eq.(4),we will make the simple approximations
ρr=ρr0e?4τ,ρm=ρm0e?3τ.(9) For the spatially homogeneous scalar?eldφ,the equation of state pa-rameter wφ=wφ(z)=Pφ/ρφ.The recent average of wφis de?ned as
τ τ0wφdτ.(10) We will take the upper limit of integrationτto correspond to z=1.75.The SNe Ia observations[11]bound the recent average
w0≥?1,and measure the transition redshift z t=0.46±0.13from deceleration to acceleration(it is probably premature at this point to say more than that z t≈1).
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?rst-order equations inτplus a scaled version of H:
?H?′=ψ(11)
5
?H(ψ′+ψ)=?3?V
(12)
?
?H2=?ρ 1+?ρ
ψ2+?V+?ρm+?ρr(14)
6
whereψ≡e2τ˙?/H0,?H=e2τH/H0,?V=e4τV/ρc0,?V?=e4τV?/ρc0,?ρ= e4τρ/ρc0,?ρm=e4τρm/ρc0=?m0eτ,?ρr=e4τρr/ρc0=?r0,?σ=e4τσ/ρc0,and where a prime denotes di?erentiation with respect toτ:?′=d?/dτ,etc.
This scaling results in a set of equations that is numerically more robust, especially before the time of BBN.
2? .
Figure2illustrates that while?H spans only ten orders of magnitude between z i～1023and the present,H/H0spans more than?fty orders of magnitude.
3Evolution of the Braneworld Universe
In this section,we will closely follow the analyses of Refs.[5]and[4].
6
3.1Slow-Roll Braneworld In?ation
The in?ationary slow-roll parameter ?is given by [4]
?≡?˙H
2 V φ
(1+V/(2σ))2.(15)
(The slow-roll parameter
η=M 2P V φφ1+V/(2σ)≈?(16)
for the potentials considered here.)In?ation occurs for ?<1.The slow-roll parameter can be approximated during in?ation by
?≈2λ2σ
M 2P φN φe V 2σ dφ≈1
75π2M 6P V 3
2σ 3≈1
M 4P λ2σ3(20)
gives [5]
V 1/4e ≈1.11×1015GeV /λ,σ1/4≈9.37×1014GeV /λ3/2.(21)
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3.2Gravitational Reheating and Initial Conditions At the end of in?ation,gravitational particle production [13]–[15]results in a radiation energy density [5]
ρr
V e ≈2.8×10?4g p V 3
e λ
4≈(1+z i )4ρr 0(23)corresponding [5]to a reheating temperature
T e ～ g p
λ(24)
where g ?(T )is the e?ective number of massless degrees of freedom at temper-ature T .The universe undergoes [5]a transition from the era dominated by the scalar ?eld potential energy to an era of kination (ρφ～1/a 6)dominated by the scalar ?eld kinetic energy as the ρ2/(2σ)term in H 2becomes small compared with ρ.For this era of kination to occur,the potential V must be su?ciently steep.Since during kination ρr /ρφ～a 2,the universe eventually makes a transition [5]to the standard radiation dominated era,around a temperature T ≈103GeV /λ,well before big-bang nucleosynthesis at T ≈1MeV.For the sum of exponentials or cosh potential,the universe then evolves 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 initial conditions at the end of in?ation will be set in the high-energy,slow-roll limit.The initial value for φi is speci?ed by Eq.(18).The initial value for ˙φ
i follows from the high-energy,slow-roll limits H 2≈
ρ26M 2P σ(25)and
¨φ+3H ˙φ≈ 2V
σ˙φ
≈?V φ≈?λV
or[4]
1
1
6≈
w0
2? 5.747.4?1.60?0.78
e5?+e? 2.247.6?1.18?0.89
Table1:Simulation results for the cosh and sum of exponentials potentials.
4.1Exponential Potential
The exponential potential V(?)=Aeλ?[15]–[18]can be derived from M-
theory[19]or from N=2,4D gauged supergravity[20].
For the exponential potential withλ2>3,the cosmological equations have a global attractor with?φ=3/λ2during the matter dominated era (during which wφ=0);and withλ2>4,the cosmological equations have a global attractor with?φ=4/λ2during the radiation dominated era(during which wφ=1/3).Forλ2<3,the cosmological equations have a late time
attractor with?φ=1and wφ=λ2/3?1.
√
Forλ=
2,the universe enters a period of eternal accelera-
√
tion with an event horizon.Forλ>
The ΛCDM cosmology is approached for λ≤1/
√
3.For λ=√w 0is much too high;for a viable present-day QCDM cosmology,λ≤√
2
˙φ2～1/a 6and thus ?′is a (negative)constant since both ?H and ψare proportional to e ?τin Eq.(11).
Figure 3:?for V =2A cosh(5?)(solid)vs.ΛCDM (dotted).
Between z ≈2.5×1014(as the universe enters its radiation dominated
stage)and z ≈10(when m 2φ≈λ2V/M 2P becomes of order H 2≈ρm /(3M 2P )),
?“sits and waits”at a small negative value.
10
Figure4:?for V=2A cosh(5?).
Figure5:Oscillating behavior of?near z=0for V=2A cosh(5?).
11
Figure 6:w φfor V =2A cosh(5?).
With the cosh potential,?is very slowly oscillating and decaying about ?=0at the minimum of the potential for z <
～1(we have neglected any present-day particle production by the oscillating scalar ?eld),with angular
frequency ω≈√t ≈2H ?1
Λ/3(see Fig.5).
Note that w φ→?1for t >t 0(Fig.6).The acceleration parameter ?q in Fig.7closely mimics the ΛCDM curve.
4.3Sum of Exponentials Potential
In the sum of exponentials potential (1),a shift ?→?+χsimply results in a rede?nition of the constants A and B :
V (?)→V (?+χ)=A
e λ?+B e μ? ,A =e λχA,B =e (μ?λ)χB.(29)In a realistic particle theory A and B would be set from ?rst principles;here we choose B =1.It will turn out then that A ～ρc 0for quintessential in?ation.
To produce primordial in?ation in the braneworld scenario with the sum of exponentials potential and the subsequent transition to the standard radia-tion dominated universe satisfying the constraints on ?φ,λ>
～
4.7(this agrees
with the estimate λ>～4.5given in Ref.[5]);for present-day quintessence,μ≤√2or 1(these values appear
naturally in low-energy limits of M/string theory):
V (?)=A e 5?+e √
Figure8:?for V=A e5?+e√
Figure10:wφfor V=A e5?+e√
Figure 12:?for V =A (e 5?+e ?)(solid)vs.ΛCDM (dotted).
Figure 13:?for V =A (e 5?+e ?).
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Figure14:wφfor V=A(e5?+e?).
Note the di?erences in the acceleration parameter?q in Figs.11and15
√
for the two casesμ=
w0≤?0.78 (see Table1),satisfying the observational bound
2)eternal acceleration and an event horizon.
In both the cosh and sum of exponentials potentials considered here, the low z behavior of wφprovides a clear observable signal distinguishing quintessence from a cosmological constant.
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