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38
SarniM.,1 Dadhich N.2 UNIFYING BRANE WORLD INFLATION WITH QUINTESSENCE
11ntroduction
Universe seems to exhibit an interesting symmetry with regard to accelerated expansion. It has gone under inflation at early epochs and is believed to be accelerating at present.
The inflationary paradigm was orinaly introduced to address the initial value problems of the standard hot big bang model. Only later it became clear that the scenario could provide important clues for the origin of structure in the universe. The recent measurement of the Wilkinson Microwave Anisotropy Probe (WMAP) [2,3] in the Cosmic Microwave Background (CMB) made it clear that (i) the current state of the universe is very close to a critical density and that (ii) primordial density perturbations that seeded large-scale structure in the universe are nearly scale-invariant and Gaussian, which are consistent with the inflationary paradigm. Inflation is often implemented with a single or multiple scalar-field models [4] (also see the excellent review on inflation by Shinji Tsujikawa [5]). In most of these models, the scalar field undergoes a slow-roll period allowing an accelerated expansion of the universe. After drawing the required amount of inflation, the inflaton enters the regime of quasi-periodic oscillation where it quickly oscillates and decays into particles leading to (p)reheating.
As for the current accelerating of universe, it is supported by observations of high redshift type la supernovae treated as standardized candles and, more indirectly, by observations of the cosmic microwave background and galaxy clustering. Within the framework of general relativity, cosmic acceleration should be sourced by an energy-momentum tensor which has a large negative pressure {dark energy) [6]. Therefore, the standard model should, in order to comply with the logical consistency and observation, be sandwiched between inflation at early epochs and quintessence at late times. It is natural to ask whether one can build a model with scalar fields to join the two ends without disturbing the thermal history of universe. Attempts have been made to unify both these concepts using models with a single scalar field [7]. In these models, the scalar field exhibits the properties of tracker field. As a result it goes into hiding after the commencement of radiation domination; it emerges from the shadow only at late times to account for the observed accelerated expansion of universe. These models belong to the category of non oscillating
1 Email: sami@iucaa.ernet.in
2 Email: nkd@iucaa.ernet.in
models in which the standard reheating mechanism does not work. In this case, one can employ an alternative mechanism of reheating via quantum-mechanical particle production in time varying gravitational field at the end of inflation [8]. However, then the inflaton energy density should red-shift faster than that of the produced particles so that radiation domination could commence. And this requires a steep field potential, which of course, cannot support inflation in the standard FRW cosmology. This is precisely where the brane [9,10] assisted inflation comes to the rescue.
The presence of the quadratic density term (high energy corrections) in the Friedman equation on the brane changes the expansion dynamics at early epochs [11] (see Ref. [12] for details on the dynamics of brane worlds). Consequently, the field experiences greater damping and rolls down its potential slower than it would during the conventional inflation.Thus, inflation in the brane world scenario can successfully occur for very steep potentials [13,14], The, model of quintessential inflation based upon reheating via gravitational particle production is faced with
itifH/*n|i!*»C ecqo/^^tprl Wjjil ^Wfacciyp 'prrtrlfliM'irt?} nf
» • ' ' >l'' ' * Is . «.. ' \
!,.. • <!>. Ii. , , • k • • <i . 1 . , 1\
' ' „ * O ■«> V.'j . <i i ., • .-\jl ' . < i >
1 j' i J ni I » .. > •. i'll I • 1 . _ii ¡1} • ,
11'tlv.u >1) * - -1 Itijlt .¡i, . in.-.;" fl'-l
Kinetic. regime during wnich the amplitude ol primordial gravity waves enhances and violates the nucleosynthesis constraint [15] (see also [16]). Hence, it is necessary to look for alternative mechanisms more efficient than the gravitational particle production to address the problem.
A proposal of reheating with Born-lnfeld matter was made in Ref. [17] (see also Ref. [18,19] on the related theme). It was shown that reheating is quite efficient and the model does not require any additional fine tuning of parameters [17]. However, the model works under several assumptions which are not easy to justify.
The problems associated with reheating mechanisms discussed above can be circumvented if one invokes an alternative method of reheating, namely "instant preheating" proposed by Felder, Kofman and Linde [20] (see also Ref. [21] on the related theme. For other approaches to reheating in quintessential inflation
see [22]). This mechanism, is quite efficient arid robust, and is well suited to non-oscillating models. It describes a new method of realizing quintessential inflation on the brane in which inflation is followed by "instant preheating". The larger reheating temperature in this model results in a smaller amplitude of relic gravity waves which is consistent with the nucleosynthesis bounds [23]. However, the recent measurement of CMB anisotropics by WMA.P places fairly strong constraints on inflationary models [24,25]. It seems that the steep brane world inflation is on the the verge of being ruled out by the observations [26]. Steep inflation in a Gauss-Bonnet braneworld may appear to be in better agreement with observations than inflation in a RS scenario [27].
2 Quintessential inflation
Quintessential inflation aims to describe a scenario in which both inflation and dark energy (quintessence)
ar. d' «-riivd by die *ame scalar field. The unification o; v ,at.cv;/is in a single scalar field model imposes cc.nu- ,« .inN >\hich were spelled oat in the ir •-•••> •• 'firv concepts can be put together c< • • • \ • i ■ with brane ••. i • '>'■■■ u . det
us below list the building blocks of such a model
H2=-
1
3 Mi
1+-
2 A.
+ -
where £ is an integration constant which transmits bulk graviton influence onto the brane and Xb is the three dimensional brane tension which provides a relationship between the four and five-dimensional Planck masses and also relates the four-dimensional cosmological constant A4 to its five-dimensional counterpart.
The four dimensional cosmological constant A4 can be made to vanish by appropriately tuning the brane tension. The "dark radiation" £/a* is expected to rapidly disappear once inflation has commenced so that
we effectively get [11,13]
H2=-
1
3 Ml
1+-
2K
(2)
1
where p = p? = -^<i>2 if one is dealing with a
universe dominated by a single minimally coupled scalar field. The equation of motion of a scalar field propagating on the brane is
<j> + 3H<j> + V'(<j>) = 0. (3)
From (2) and (3) we find that the presence of the additional term p2/ll} increases the damping experienced by the scalar field as it rolls down its potential. This effect is reflected in the siow-roli parameters which have the fonts [13,14] 1+V7L,
6 =
'!1 = TW where
(1 +V/2.L,)*' '
,(1 + 1 //2 V1.
(4)
; i • s, iii • I-, >,;: ¡1
(1) Potentials which become shallow at late time (such as inverse power law potentials)
(2) Potentials reducing to particular power law type at late times.
2.1 Steep Brane World inflation
In what follows we shall work with the steep exponential potential which exhibits the aforementioned features necessary for the description of inflationary as well as post inflationary regimes. The brane world inflation with steep potentials becomes possible due to high energy corrections in the Friedmann equation. The exit from inflation also takes place naturally when the high energy corrections become unimportant.
In the 4+1 dimensional brane scenario inspired by the Randall-Sundrum [9] model, the standard 0-0 Friedman equation is modified to [11]
f \ ^ n
(5)
eF№---2* 1 ~Y J ' ~ f I y"
are slow roll parameters in the absence of brane corrections. The influence of the brane term becomes important when V Ikb »1 and in this case we get
e - (1/ / A,r1, il = (V'K)'!• (6)
Clearly slow-roll (e,T|«l) is easier to achieve when V/Xb »1 and on this basis one can expect inflation to occur even for relatively steep potentials, such the exponential and the inverse power-law which we discuss below.
2.2 Exponential Potentials
The exponential potential V(<j>) = V0e°*'Uf
(7)
with <b > 0
(1)
i> y
with <j)<0 (equivalently V(<t>) = V0e' has traditionally played an important role within the inflationary framework since, in the absence of matter, it gives rise to power law inflation a f, c- 2/ or
provided a < \/2 . For a > -s/2 the potential becomes too steep to sustain inflation and for larger values a> the field enters a kinetic regime during which field energy density <= of6. Thus within the standard
general relativistic framework, steep potentials are not capable of sustaining inflation. However extra-dimensional effects lead to interesting new possibilities for the inflationary scenario. The increased damping of the scalar field when V /Xb »1 leads to a decrease in the value of the slow-roll parameters e = tj = 2a2kb / V, so that slow-roll (£,t]<k1) leading to inflation now becomes possible even for large values of a. The steep exponential potentials satisfies the post inflationary requirements mentioned earlier. Intact, the cosmological dynamics with steep exponential potential in presence of background (radiation/matter) admits scaling solution as the attractor of the system. The attractor is characterized by the tracking behavior of the field energy density p^.
During the "tracking regime", the ratio of p? to the background energy density ps is held fixed P* 3(1 + Wg)
■<0.2
or
(8)
Po+P«
where wB is the equation of state parameter for background (w„= 0,1/3 for matter and radiation respectively) and the inequality (8) reflects the nucleosynthesis constraint which requires aï!:5. It is therefore clear that the field energy density in the post inflationary regime would keep tracking the background being subdominant such that it does not interfere with the thermal history of the universe.
Within the framework of the braneworld scenario, the field equations (2) and (3) can be solved exactly in the slow-roll limit when p/Xb » 1. In this case â(t) 1
(9)
ait) JmX
which, when, substituted in 3//<j> = -V'(4>) (10) leads to
# = -a/2V3 (11)
Hie expression for number of inflationary e-foldings is
easy to establish
N = \og^= I H(t')dt'
(12)
2lba2
(p —em)
2 k„a2
1 - expi -
3 Ml
a2 (f-/,.))
(13)
where vi = V^e®4''. From Eq, (13) we find that the expansion factor passes through an inflection point marking the end of inflation and leading to
(14)
=21^ (15)
The COBE normalized value for the amplitude of scalar density perturbations allows to estimate vad and the brane tension L
V =
"ma
3xi0"7 m
h.
a
1.3x10~7 a6
x4
iV+1,
My
N + l
(16)
We work here under the assumption that scalar density perturbations are responsible for most of the COBE signal. We shall, however, come back to the important question about the tensor perturbations later in our discussion.
The scenario of quintessential inflation we are discussing here be'npgs *o tV rbi« of non-fwHiHery
models where the .v »>.-, 1 ! .»' •< does not work. . . - 't -
production to do
process which lea 1 i • • ..
species quantum i ■
when the space tiiii • < .
Unlike the cosrve. >• • • - ' • .
process does not i ... • « u 1
fields. The radiation ueiuuy creawsu via m is
mechanism at the end of inflation is given by
Pr-O.Olx^/C (17)
where g ~ 100 is the number of different particle species created from vacuum. Using the relation (17) and the expressions of and V^ obtained above, it can easily be shown that
P.
•2x10"
N +1 51
>p
(18)
Pr /end
This leads to a prolonged "kinetic regime" during which scalar matter lias the "stiff equation of state.
Using Eqs. (12) & (13) one can demonstrate that inflation proceeds at an exponential rate during early epochs which plays an important role for the generation of relic gravity waves during inflation.
3 Late Time Evolution
As discussed ; above, the ' scalar field' with exponential potential (7) leads to a viable evolution at early times. We should, however, ensure that the scalar field becomes quintessence at late times which demands a particular behavior of the .scalar field
potential as discussed above. Indeed, any scalar field potential which interpolates between an exponential at early epochs and the power law type potential at late times could lead to a viable cosmological evolution. The cosine hyperbolic potential provides one such example [29]
V (f) = V0 [cosh(ouj) /MP)~ if, p> 0 (19)
which has asymptotic forms
à(J>/M, »1, #>0
(20)
\d4/Mp |«Kl
(21)
where a = pâ . As the cosine hyperbolic potential (19) -*- fiihits power law type of behavior near the origin, i"; Id oscillations build up in the system at late times. T'. - i particular choice of power law, the average > i> ».ion of state parameter may turn negative [29,30]
L___
42
£zl
p+i
(22)'
de
Logi0(a/â#reJ)
Figure 1: The post-inflationary energy density in the scalar field (solid line) radiation {dashed line) and cold dark matter (dotted line) is shown as a function of the scale factor for the model described by (19) with
V0 --5x10'
1-46 GeV4, a = 5 =5 and p = 0.2. The enormously
large value of the scalar field kinetic energy (relative to the potential)
ensures that the scalar field density overshoots the background radiation value, after which p, remains approximately constant for a
substantially long period of time. At late times the scalar field briefly
tracks the background matter density before becoming dominant and driving the current accelerated expansion of the universe. From Sahni, Sami and Souradeep [15]
-tve> • • '•>'. >• sUte •
. . i,0 i i i I . ; . « iii
| • i> !>' i i ..'I >c. 1if;i , i , i ■ •" im a", it fici-'Y'; 1 az 2. sVc .'h*hI , uit. cje 1 "> if; cjry I. ■ i c vahir of the seal tr field kiiVH< energy „i he • (Mouiicc-inent oi the radidne iegt ae, the scalar fidd il.i-Iiy owrshooti the radiation energy density. Alter J <\ ff v,-»M»r of Dj, sphidi,". and renrnim vhtivrly ,i.ich;lng"d toi ^ consic».-i,«b1e lengtn uf time dnr ng > ivch the vcalai hold equat;on of sit,a is n, »-1.
Tracking commences late into the matter dominated epoch and the universe accelerates today during rapid oscillations of the scalar field. This model provides an interesting example of "quintessential inflation". However as we shall discuss next, the long duration of the kinetic regime in this model results in a large gravity wave background which comes into conflict with nucleosynthesis constraints.
3,1 Relic Gravity Waves and Nucleosynthesis
Constraint
The tensor perturbations or gravity waves get quantum mechanically generated during inflation and leave imprints on the micro-wave background.
where t= jdt/a(t) is the conformai time coordinate
and k = 2m/X is the comoving wavenumber. Since brane driven inflation is near-exponential we can write a = x0/x | x |<) t01, in this case normalized positive frequency solutions of (23) corresponding to the adiabatic vacuum in the "in state" are given by [15]
€(Jc>T) =
JtT„
1/2 f \3/2 T
V^oy
H™{kT)F(Hb/ÇL) (24)
where p. = AiltM1 and Hin=-Ux0 is the inflationary Hubble parameter. The term
F(x) =
4i
-x2log<j —+ Jl + -~
(25)
is responsible for the increased gravity wave amplitude in braneworld inflation [28]. The "out state" is described by a linear superposition of positive and negative frequency solutions to (23). For power law expansion
a = (tltaY S(z/xa)m-\
we have
Kinetic
Radiative
Log10{a/a.„d)
Figure 2: The dimensionless density parameter Û is plotted as a
function of the scale factor for the model in figure 1. Late time oscillations of the scalar field ensure that the mean equation of state turns negative (w^) = -2/3 , giving rise to the current epoch of
cosmic acceleration with a(t) « t2 and present day values Q0ll) = 0.7, Q0ffl = 0.3. From Sahni, Sami and Souradeep [15]
^(¿.x^aO^ + PC'i**) (26)
The energy density of relic gravity waves is given by [15]
P,=<r0"
eV
jdkk'lßf.
(27)
Computing the Bogolyubov coefficient (3, one can show that the spectral energy density of gravity waves produced during slow-roll inflation is [15,23]
H
^Pr I 3rt
\2
7
\ ei
(29)
commencement of the kinetic regime Tlin is related to the temperature at the end of inflation as
T = T
'km 1,md
■'ml
V "»«
~TaiE(a)
(30)
a*
where F,(a)= c + —r , c = 0.142, ¿ = -1.057 and
Tmi =(?7T.
The equality between scalar field matter and radiation takes place at the temperature
F2m
T ~T
eq end
(pJPX
(31)
with F,(a) =
e +
f
e = 0.0265. / =0.176. The
a' '
fitting formulas (30) and (31) are obtained by numerical integration of equations of motion.
Using equations (30), (31) and (29) we obtain the ratio of scalar field energy density to radiation euer density at the end of inflation
P,
^ end
3n 64
1
Kh^w(P;(a)/F2(a)Y
(32)
p ,(*)«* (28)
where w is the equation of state parameter which characterizes the post-inflationary epoch. In braneworld model under consideration, w = l during the kinetic regime, consequently the gravity wave background generated during this epoch will have a blue spectrum pg (k)°c k .
We imagine that radiation through some mechanism was generated at the end of inflation with radiation density pr. Then the ratio of energy in gravity waves to pr at the commencement of radiative regime is given by [15]
where hcw is the dimensionless amplitude of gravity waves (from COBE normalization, -1.7xl0~10, for J\f ~ 70). We should mention that the commencement of the kinetic regime is not instaneous and the brane effects petrsist for some time after inflation has ended. The temperature at the
r '•■!,.ii •« i.'2) is an import,1 -t r.-u » • t • • 'i '' < \ of scalar fie' » -ir • , • > • .
1 _ " ity at the end t ; ■ icsis constraii1
. .sty in gravit • quality (pg/p
potential (a i> 5), we have )7. (33)
>■ ' ..- ieci earlier, this ratio is of the order of ID'6 >.' c . » vitational particle production and exceeds the nucleo-synthesis constraint by nine orders of magnitudes. An interesting proposal which can circumvent this difficulty has recently been suggested by Liddle and Lopez [22]. The authors have employed a new method of reheating via curvaton to address the problems associated with gravitational particle production mechanism. The curvaton model as shown in Ref. [22] can in principal resolve the difficulties related to excessive amplitude of short-scale gravitational waves. Although this model is interesting, it operates through a very complex network of constraints dictated by the fine tuning of parameters of the model.
In the following section, we shall examine an alternative mechanism based upon Born-Infeld reheating.
4 Born-lnfeid Brane Worlds
The D-branes are fundamental objects in string theory. The end points of the open string to which the gauge fields are attached are constrained to lie on the branes. As the string theory contains gravity, the D-
branes are the dynamical objects. The effective D-
brane action is given by the Born-Infeld action
Sa = (34)
where F is the elecromagnetic field tensor (Non-Abelian gauge fields could also be included in the action) and Xb is the brane tension. The Born-Infeld action, in general, also includes Fermi fields and scalars which have been dropped here for simplicity. In the brane world scenario a la Randall-Sundrum one adopts the Nambu-Goto action instead of the Born-Infeld action. Shiromizu et al have suggested that in the true spirit of the string theory, the total action in the brane world cosmology be composed of the bulk and D-brane actionsfl 8]
^ — ShtM + S „
(35)
where Sbuli is the five dimensional Einstein-Hilbert
action with the negative cosmol.ogi.cal constant. The stress tensor -appearing on the right hand side (RH5) of the Einstein equations on the brane will now be sourced by the Born-Infeld action. The modified Friedman equation on a spatially flat FEW brane acquires the form
H
with
whei
Pm
2L,
(36)
(37)
6. The tension X,. is tuned so that the
net ( constant on the brane vanishes. We
have "dark radiation" term, in the equation
(36) ii '^piJly disappear once inflation sets in. Spatial averaging is assumed while computing pB1 and Pm from the stress-tensor corresponding to action (34). The. scaling of energy density of the Born-Infeld matter, as usual, can be established from the conservation equation Born-Infeld matter, as usual, can be established from the conservation equation pB1+3H(pm+PBl) = 0 (38)
where
_2
(39)
P =£__!!
w 3 6XI:/
Interestingly, the pressure due to the Born-Infeld matter becomes negative in the high energy regime allowing the accelerated expansion at early dines without the introduction of a scalar field. As shown in [18], the energy density pBI scales as radiation when 6 « 6Xb. For e > 6Xb, the Born-Infeld matter energy density starts scaling slowly (logarithmically) with the scale factor to mimic the cosmological constant like behavior. The point is that the Born-Infeld matter is
subdominant during the inflationary stage. It comes to play the important role after the end of inflation when it behaves like radiation and hence serves as an alternative to reheating mechanism.
The brane world cosmology based upon the Born-Infeld action looks promising as it is perfectly tuned with the D -brane ideology. But since the Born-Infeld action is composed of the non-linear elecromagnetic field, the D-brane cosmology proposed in Ref. [18] can not accommodate density perturbations at least in its present formulation. One could include a scalar field in the Born-Infeld action, say, a tachyon condensate to correct the situation. However, such a scenario faces the difficulties associated with reheating [32,31 j and formation of acoustics/kinks [33]. We shall therefore not follow this track. We shall assume that the scalar field driving the inflation (quintessence) on the brane is described by the usual four dimensional action for the scalar fields. We should remark here that the problems faced by rolling tachyon models are beautifully circumvented in the scenario based 'upon massive Born-Infeld scalar field on the .£)., brane of KKLT vacua [34].
The total action that we are trying to motivate here is given by
5=s(,wt;s8/+s4d_hr (40)
where
S,,^ + (41)
field f which
(42)
The scalar field propagating on the brane modifies the Friedman equation to
H2=-
3 M
1+-
2X,
(43)
b J
where p1((( is given by
P«=P»+Pbi (44)
As mentioned earlier, in the scenario based upon reheating via quantum mechanical particle production during inflation, the radiation density is very small, typically one part in 1016 and the ratio of the field energy density to that of radiation has no free parameter to tune. This leads to long kinetic regime which results in an unacceptably large gravity background. The Born-Infeld matter which behaves like radiation (at the end of inflation) has no such problem and can be used for reheating without conflicting with the nucleosynthesis constraint. Indeed, at the end of inflation pBI can be chosen such that
pj/ <k 6Xb. Such an initial condition for pB1 is consistent with the nucleo-synthesis constraint [17]. In
that case the Born-Infeld matter energy density would scale like radiation at the end of inflation. At this epoch the scale factor will be initialized at aend = 1. The
energy density pBI would continue scaling as 1/a4
below a - armi. The scaling would slow down as pBI
reaches 6'kb which is much smaller than Vmd for
generic steep potentials, say for a > 5 . Hence pBi
remains subdominant to scalar field energy density p0
for the entire inflationary evolution. The Born-Infeld matter comes to play the important role only at the end of inflation which is in a sense similar to curvaton. But unlike curvaton, it does not contain any new parameter. The numerical results for a specific choice of parameters is shown in Fig, 3. In contrast to the "quintessential inflation' based upon the gravitational particle production mechanism where the scalar field spends long time in the kinetic regime and makes deep undershoot followed by long locking period with very brief tracking, the scalar field in the present scenario tracks the background for a very long time (see figure 3). This pattern of evolution is consistent with the thermal history of the universe. We note that "quintessential inflation' can also be implemented by inverse power law potentials. Unfortunately, one has to make several assumptions to make the scenario working:
(i) The tension of the D3 brane appearing in the Born-Infeld action is treated as constant and is identified with the brane tension in the Randall-Sundrum scenario.
(ii) The fluctuations in the Born-Infeld matter are neglected.
(iii)The series expansion of the Born-Infeld action is truncated beyond a certain order.
In what follows, we shall examine the instant reheating mechanism discovered by Felder, Kofman and Linde and show that their mechanism is superior to other reheating mechanism mentioned above.
0
•20
-40
I
.'100
.-120
,140
0 5 10 IS 20 25
Log,„(a/a,)
Figure 3: The post-inflationary evolution of the scalar field energy density (solid line), radiation (dashed line) and cold dark matter (dotted line) is shown as a function of the scale factor for the quintessential
inflation model described by (19) with V01/4 = 1(T%>Mp, a = 50
and p = 0,1 ( a = pâ-5 ), After brane effects have ended, the
field energy density p^ enters the kinetic regime and soon drops
below the radiation density, After a brief interval during which '
(»it,) - -1. the scalar field begins to track first radiation and then
matter, At very late times (present epoch) the scalar field plays the role of quintessence and makes the universe accelerate, The evolution of the energy density is shown from the end of inflation until the present epoch. From Sami, Dadhich and Shiromizu [17].
5 Braneworld Inflation Followed by Instant Preheating
Braneworld Inflation induced by the steep exponential potential (7) ends when <j> = §frul, see (14), Without loss of generality, we can make the inflation end at the origin by translating the field
V(^)^Vm = V0ea''n1", (45)
where Vj, = V0e+a4>*"' IM and <{>' = (|)-<t»raii. In order to achieve reheating after inflation has ended we assume that the inflaton <j> interacts with another scalar field % which has a Yukawa-type interaction with a Fermi field \ji. The interaction Lagrangian is
- .... (46)
To avoid confusion, • '
remembering that <|><C • • »,• . . •
should be noticed that tl effective mass being det value of the coupling constant g ( m = g j è | ).
The production of % particles commences as soon as m begins changing non-adiabatically [20] |»J>mi or (47)
Figure 4; The post inflationary evolution of Y = .109x | <j) 1 / M2p , /M2p is shown as a function of the scale factor for the model described by Eq (19) with V0 = 10"30 Mp, & = 50 'and /> = 0.1,
The solid curve corresponds to M2 whereas the dashed and dotted curves correspond to gf21M2 for g =10^ (extreme left)
and g = 1CT9 (extreme right) respectively. The violation of the adiabaticity condition (necessary for particle production to take place) arises in the region bounded by the solid and dashed curves. The time duration of particle production is seen to be smaller for larger values of the coupling g . For the range of g allowed by the nucleosynthesis constraint, particle production is almost instantaneous. From Sami and Sahni [23}
The production of % particles commences as soon as m begins changing non-adiabatically [20]
or lil>g#2. (47)
The condition for particle production (47) is satisfied when
I4>N<UI=
li
№
(48)
from (16) we find that <j)i>w «tff for g » UT9. The production time for % particles can be estimated to be 1
Ai Jit
lèl igv
Ml and
(49)
The uncertainty relation provides an estimate for the momentum of % particles created non-adiabatically:
••(At^y' ~gl,zvjfj. Proceeding as in [20] we
occupation number of % panicles
"piva T| . i-0\v •'■If
The
p'dj
X -partieie number
(54)
M „ T ip Y
\Pr J end UJ
Comparing (54) with (33) we find that, in order for relic gravity waves to respect the nucleosynthesis constraint, we should have g > 4 x 1CT3. (The energy density created by instant preheating (Pr= {g! 1Û)2 can clearly be much larger than the energy density produced by quantum particle production, for which (pr/p,) — 1 CT16 g J. The
constraint g > 4x1 (T3, implies that the particle production time-scale (49) is much smaller than the Hubble time since
1
At,mxfimd
à300a , a»1.
(55)
Thus the effects of expansion can safely be neglected during the very short time interval in which "instant preheating" takes place. We also find, from equation (48), that |<j»lim(i \/Mp <10^3 implying that particle
production takes place in a very narrow band around 4> = 0 . Figure 4 demonstrates the violation of
i ;.>!> i.-t'.t t -, ,,r i . , -,
>fl
*
IS I >
Quanta of the x-i;,-iu ¿re created during the time interval Atpmd that the field f spends in the vicinity of 0 = 0. Thereafter the mass of the % -particle begins to grow since m% = g | <j}(t) [, and the energy density of particles of the % -field created in this manner is given by
/ \ 3 7 ft S
accounts
P i^'Vt
a
8K3
-«I
(52)
where the
(amd / af term accounts for the cosmological dilution of the energy density with time. As shown above, the process of % particle-production takes place immediately after inflation has ended, provided g > 10~'}. In what follows we will show that the % -field can rapidly decay into fermions. It is easy to show that if the quanta of the x-i\t\A were converted (thermalized) into radiation instantaneously, the radiation energy density would become
Pr=Pr
(£p3;2 8n3
prod
•low«,.
(53)
From equation (53) follows the important result
the bacK-reaction ot % particles in the evolution equation is negligible during the time scale -(here ~ Hll characterizes the epoch the kinetic regime commences. HUn = //t(lrf (0.085-0.688/or) [15]).
We now turn to the matter of reheating which occurs through the decay of y particles to fermions, as a consequence of the interaction term in the Lagrangian (46). The decay rate of % particles is given by
Tf¥ = h2mx / 8it, where ml = g (<j) |. Clearly the decay rate is faster for larger values of|0|. For r^ > Hkin, the decay process will be completed within the time that back-reaction effects (of % particles) remain small. Using the expression for HkiK this requirement translates into
h^>^LVJlF{ a). (56)
Smp
For reheating to be completed by <j>/Af < I, we find from equations (33) and (56) that h>WAg~m (g >4xl0~3) fora = 5. This along with the constraint imposed by the back reaction defines the
allowed region in the parameter space (g, h). We observe that there is a wide region in the parameter space for which
(i) reheating is rapid and (ii) the relic gravity background in non-oscillatory braneworld models of quintessential inflation is consistent with nucleosynthesis constraints. However, this is not the complete story. One should further subject the model to the recent WMAP observations. The measurement of CMB anisotropics places fairly strong constraints on inflationary models [24,251. It appears that the tensor perturbations are not adequately suppressed in the models of steep brane world inflation and as a result these models are on the verge of being ruled out. As indicated by Lidsey and Nunes, inflation in a Gauss-Bonnet braneworld could appear to be in better agreement with observations than inflation in a RS II scenario [27], In tie following section, we briefly discuss the prospects of brane world inflation with the Gauss-Bonnet correction term in the bulk.
8 Gauss-Bonnet Brane Worlds
Though we are trying to motivate the GB term in ■i :«-ii . ¡w'l: .i;j{-hc. ic > m mind, the
• .' !• (.,••. .! i- ■ >..!>" "'-\v\ its own right
at the very ng physical . , It is the
t , i • .■1 .'h . 'itation. Zero
speed which 'his can only 1. • i .. Since space
■ i •"■••> >■■ icorporation
• . » M..- pa 1 v- \» rnciuiui/,, \u>',;tationaI field
thus cuiws spovuiiiic. in other wuiua it can truly be described by curvature of spacetirne and it thus becomes a property of spacetirne - no longer an external field [35].
From the physical standpoint the new feature that GR has to incorporate is that gravitational field itself has energy and hence like any other energy it must also link to gravity. That is, field has gravitational charge and hence it is self interacting. Field energy density will go as square of first derivative of the metric and it must be included in the Einstein field equation. It is indeed included for the Riemann curvature involves the second derivative and square of the first derivative. However in the specific case of field of an isolated body, we obtain Mr potential, the same as in the Newtonian case. Where has the square of V<1> (<D denotes the gravitational potential) gone? It turns out that its contribution has gone into curving the space, grr component of the metric being different from 1.
The main point is that gravitational field equation should follow from the curvature of spacetirne and they
should be second order quasilinear differential equations (quasilinear means the highest order of derivative must occur linearly so that the equation admits a unique solution). Riemann curvature through the Bianchi identities leads to the Einstein equation with the A term. We should emphasize here that A enters here as naturally as the stress energy tensor. It is indeed a true new constant of the Einsteinian gravity [36]. It is a pertinent question to ask, is this the most general second order quasilinear equation one can obtain from curvature of spacetirne? The answer is No. There exists a remarkable combination of square of Riemann tensor and its contractions, which when added to the action gives a second order quasilinear equation involving second and fourth power of the first derivative. This is what is the famous Gauss-Bonnet (GB) term. Thus GB term too appears naturally and should have some non-trivial meaning.
However GB term is topological in D<5 and hence has no dynamics. It attains dynamics in I) > 4. Note that gravity does not have its full dynamics in D< 4 and hence the minimum number of dimensions . 1 ¡'. t d 1 i , , • • i i ¡«1',, !•,.:<!.' i
first derivative of the metric, Self interaction should
yields the quasilinear second order equation with high-
('■»».' i : ! i1 i in
Lovelock Lagrangian represent higher order loop corrections to the Einstein gravity.
They do however make non-trivial contribution classically only for D> 4 dimensions. This is rather important. If. GB term had made a non-trivial contribution in 4-dimensions, it would have conflicted with the 1 / r character of the potential because of the presence of (¥cf>)4 terms in the equation. The square terms (to account for contribution of gravitational field, energy) were taken care of by the space curvature ( grr in the metric) and now nothing more is left to accommodate the fourth (and higher) power term. However we can not tamper with the inverse square law (i.e. 1 / r potential) which is independently required by the Gauss law of conservation of flux. That can not be defied at any cost. Thus it is not for nothing that the GB and its Lovelock generalization term makes no contribution for D = 4. It further carries an important message that gravitational field cannot be kept confined to 4-dimensions, It is indeed a higher dimensional interaction where the higher order iterations attain meaning and dynamics. This is the
most profound message the GB term indicates. This is yet another independent and new motivation for higher dimensional gravity [36].
Self interaction is always to be evaluated iteratively. For gravity iteration is on the curvature of spaeetime. It is then not surprising that GB term arises naturally from the one loop correction to classical gravity. String theory should however encompass whatever is obtained by iterative the iterative process. GB term is therefore strongly motivated by string theoretic considerations as well. Further GB is topological in 4-D but in quantum considerations it defines new vacuum state. It is quantum mechanically non-trivial. In higher dimensions, it attains dynamics even at classical level. In the simplest case in higher dimension it should have a classical analogue of 4-D quantum case. That is what indeed happens. Space of constant curvature or equivalent!}' conformally flat Einstein space solves the equation with GB term with redefined vacuum. This is a general result for all D > 4. It is interesting to see quantum in lower dimension becoming classical in higher dimension.
In the context of the brane bulk system we should therefore include GB term in the bulk and see its
density. In order to regain general relativity at low energies, the effective 41) Newton constant is defined
by [39]
1
«¿a-
__KP"b
Ml 6(l-4aA1/9)'
(60)
■P ■ —»'5'
When a = 0, we recover the RS expression. We can
fine-tune the brane tension to achieve zero cosmological constant on the brane [39]:
kX = ~4A5 +1 a
1- 1+-OA,
3/2
(61)
The modified Friedman equation (58), together with
Eq. (59), shows that there is a characteristic GB energy scale Mm [40] such that,
P=>h!
16a
6?w
p\
(62)
(63)
(64)
It should be noted that Hubble law acauires an unusual -r r, ,V>< i.b-i'u '!< i m,, r
for
live
t jdsX\f~g {% - 2AS +am{7l2
,jy ABCD i
'4BCD
Jd\x,ph(£m ~Xb),
-mMnAB
+
(57)
71, R refer to the Ricci scalars in the bulk metric gAB and the induced metric on the brane hAB; am has dimensions of (length)1 and is the Gauss-Bonnet coupling, while Xb is the brane tension and A, (< 0) is the bulk cosmological constant. The constant k, contains thsM5, the 5D fundamental energy scale
(Kj = M5"3). The modified Friedman equation on the (spatially flat) brane may be written as [38,39,27] (see alsoRef. [4J])
H2 =
_1_ 4a
(l-4afi2)cosh 2(1-ap.2)3
-1
a
2%
3 ,
/2
sinhx,
(58)
(59)
where % is a dimensionless measure of the energy-
7 Summary
In this paper we have reviewed the recent work on unification of inflation with quintessence in the frame work of brane worlds. These models belong to the class of non-oscillatory models in which the underlying alternative reheating mechanism plays a crucial role. The popular reheating alternative via quantum mechanical production of particle during inflation leads to an unacceptable relic gravity wave background which violates the nucleo-synthesis constraint at the commencement of radiative regime. We have mentioned other alternatives to conventional (p)reheating and have shown that "instant preheating" discovered by Felder, Linde and Kofman is superior and best suited to brane world models of quintessential inflation. The recent measurement of CMB anisotropics by WMAP, appears to heavily constraint these models. The Gauss-Bonnet correction in the bulk could come to rescue the brane world inflation and, in our opinion, this is an important point which requires further investigation.
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González-Díaz. P.F. Of '<YON AND SI-Í-QI / ¡'Jim PHANTOM COSMOLOGIES
Colina de los Chopos, Instituía ae iviatemancas y Ksiea Fundamental, uortsejo Superior de Investigaciones Científicas,
Serrano 121, 28006 Madrid, SPAIN
11ntroduction < ' ■ . i > ;1
....... . terms which anmar to character»/« nhanrom wwsrpv
• ,.« ■ a,- 1 , .
i i. i.. ', a. ,,
• ■■I v-iC ' 1 ' i at -r< 1 , ii i
1 • „■ Ul i r, | »;. I , . \ III
Hi'" ¡hi pHi;e;-.l,, - ,lJ(/"< i.uladt .>' . n en:<.:_, density incieasiug with tune, naively unphysical superluminal speed of sound, violation of the dominant energy condition which would eventually allow existence of inflating wormholes and ringholes [4], and ultimately emergence of a doomsday singularity in the finite future which is known as the big rip [5].
The regime for phantom energy takes place for state, equation parameters (o = plp<~) and has been shown to occur in all current dark-energy models. However, whereas the big rip singularity is allowed to happen in quintessence [5] and k-essence [6] models, it is no longer present in models based on generalized Chaplygin-gas equations of state [7,3] having the form F = ~,4/p", with A and n being constants. No discussion has been so far made nevertheless on the occurrence of phantom energy and big rip in the other major contender model for dark energy: the tachyon matter scenario of Padmanabhan et al. [9] or its sub-quantum generalization [10]. Abrarno and Finelli have in fact used [11] a Bom-Infeld Lagrangian with a power-law potential and recovered a nice dark-energy
! " • d i < •• i •' ,. 1 . j - ■. 1 u\,
may occur. Under „.e .ism. .» -i .•< .. - ."Ui.v ...t parameter for the equate oi state, e deriv. in till» report a rather general solution for the scale factor of a universe dominated by tachyon matter, and show that, quite similarly to as it happens in current quintessence models, that solution predicts the occurrence of a big-rip singularity. In the case that a sub-quantum potential is added to the theory, we also obtain the same result, tho i-h in this case the scale factor differs from the t vp esMon obtained for current, quintessence and 'classical" tachyon fields by terms which generally depend on the sub-quantum potential, and the regime, of phantom energy is restricted by the smallness of the field kinetic term which may even make such a regime to vanish.
The paper can be outlined as follows. In Sec. 2 we discuss a rather general solution for the scale factor which satisfies all requirements and constraints imposed by tachyon theory. The phantom regime for such a solution is then investigated in Sec. 3, where it is seen that it shares all funny properties of quintessential phantom energy. The sub-quantum generalization of the tachyon theory is also considered
