The cosmology resulting from two coupled scalar fields was studied, one which is either a new inflation or chaotic type inflation, and the other which has an exponentially decaying potential. Such a potential may appear in the conformally transformed frame of generalized Einstein theories like the Jordan-Brans-Dicke theory. The constraints necessary for successful inflation are examined. Conventional GUT models such as SU(5) were found to be compatible with new inflation, while restrictions on the self-coupling constant are significantly loosened for chaotic inflation

Berkin, Andrew L.

Maeda, Kei-Ichi

Yokoyama, Junichi

English

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Fermi National Accelerator Laboratory
WU-AP/07/90
UTAP-107/90
FERMILAB-PUB-90/52-A
Soft Inflation - C/_
ANDREW L. BERKIN 1.(a), KEI-ICHI )¢_AEDA ],3,(b) and JUN'ICHI YOKOYAMA 2'3'(c)
1 Department of Physics, Waseda University, Shinjuku-ku, Tokyo 169, Japan
2 Department of Physics, Faculty of Sciences, The University of Tokyo, Tokyo 113, Japan
and
3 NASA/Fermilab Astrophysics Center
Fermi National Accelerator Laboratory, Batavia, IL 60510
Abstract --_5 :_-
We study the cosmology resulting from two coupled scalar fields, one which is either
a new inflation or chaotic type inflaton and the other which has an exponentially
decaying potential. Such a potential may appear in the conformally transformed frame
of generalized Einstein theories like the Jordan-Brans-Dicke theory. The constraints
necessary for successful inflation are examined. We find conventional GUT models
such as SU(5) are compatible with new inflation, while restrictions on the self-coupling
constant are significantly loosened for chaotic inflation.
(NASA-CR-1865_3)
Univ.) 12 p
SgFT INFLATION (Waseda N90-22500
CSCL 03B
April, 1990
Oncl as
G3/90 0279351
(a) BITNET: berkin@jpnwas00
(b) BITNET: maeda@jpnwas00
(c) DECNET: tkyvax::yokoyama
Operated by Universities Research Association Inc. under contract with the United States Department of Energy
https://ntrs.nasa.gov/search.jsp?R=19900013190 2020-03-19T23:12:29+00:00Z

The inflationary cosmology[l] purports to solve many longstanding cosmological prob-
lems, and has been the subject of much investigation during the previous decade[2]. However,
there is no fully satisfactory model for the source of inflation yet. In this letter, we consider
a matter content described by two coupled scalar fields, one of which has a decaying poten-
tial and the other which serves as the inflaton driving the expansion. In this theory, the
constraints placed upon previous models are considerably loosened.
Inflation with two scalar fields has been considered before, with effects such as consequent[3]
and double[4] inflation arising. The extended inflationary scenario[5] uses the Jordan-Brans-
Dicke (JBD) scalar[6] to achieve expansion which is slower than exponential. Applied to the
original old inflationary scenario, a successful completion of the inflationary phase may be
possible. The chaotic inflation model in the JBD theory has also been investigated[7].
The potential which will be discussed here arises in the following two cases: (i) Some
supergravity or superstring models produce an exponential potential coupled to other scalar
fields [8]. (ii) In the generalized Einstein theories, which are conformally equivalent to stan-
dard Einstein gravity with a scalar field, the potential of a standard scalar field is modified
to obtain such coupling in the conformal frame[9]. The generalized Einstein theories include
the JBD theory, induced gravity[10], any theory with non-minimal coupling, the curvature
squared theory and effective four-dimensional theories arising from a higher-dimensional uni-
fied theory. In such cases we should also discuss what happens in the original physical frame.
Decaying potentials have recently been considered by Peebles and Ratra[ll] to account for
the dark matter in the universe.
The decaying exponential potential in our model produces the same effect as the JBD
scalar in the extended inflation scenario, to reduce the rate of expansion. As the inflaton rolls
down a flat plateau, the other scalar field evolves on the exponential potential, resulting in
power-law inflation[12,13]. Although we find weaker inflation than in the conventional new
and chaotic models, we obtain a longer period of inflation due to this coupling and the much
smaller potential energy at the end of inflation. Naively speaking, since the energy scale at
the period of horizon crossing is related to density perturbations, we expect much weaker
constraints on the parameters of inflation. When the inflaton is of new inflation type, the fine
tuning of initial conditions is lessened and density perturbations are suppressed. For a chaotic
type field, the restrictions placed upon the coupling parameter by density perturbations are
reduced considerably. We will investigate whether the Coleman-Weinberg potential in the
standard GUT model yields successful new inflation and what restrictions on the coupling
A are necessary for successful chaotic inflation.
We use as our action
where t¢2 = 8rG, V(¢) is the inflaton potential and g is a dimensionless coupling constant,
which must be smaller than _ to guarantee power-law inflation. Although presently known
supergravity models yield/3 > v_, we may realize power-law inflation with the help of vis-
cosity due to particle production[13]. Furthermore, it is possible in the generalized Einstein
theories to have small enough/3, e. 9. /3 = 2(,_ + 3/2) -1/2 for the JBD theory, with the JBD
parameter w > 1/2 [9].
Working in the spatially flat Friedmann-Robertson-Walker metrlc With cosmic scale factor
a(t), we find from the field equations:
4- 3H¢-/3_¢e-a"%'(¢) = 0, (2)
,_ + 3H,], + V'(¢)e -z'_ = O, (3)
"if" _2 + ¢2 + V(¢)e-_, , (4)
where H = h/a is the Hubble parameter, an overdot denotes time derivation, and a prime
denotes differentiation with respect to the argument of the function.
Inflation occurs as ¢ slowly rolls down the potential V(¢) with its amplitude further
decreasing as e-a*_. During this period, _ and ¢2/2 can be neglected in (3) and (4),
respectively. If/32 << 6, this ansatz is true until _b rolls down into the field-oscillation region.
We then find a unique attractor of the power-law inflationary solution [13]:
2 (5)
a = ao(t/to) 2/_2, (6)
f(_,) --- f(_'o)- (1 - 32/6)ln(a/ao), (7)
where
exp[3_¢c] = 12(1 -/32/6) ' f(¢) _ d_b , (8)
and the subscript 0 denotes the value at to when the universe enters the inflationary phase.
While the initial value of ¢ may be arbitrary, the spacetime evolves into the above power-law
inflationary phase in a few expansion times, since the above solution is the unique attractor.
Thus, any value of ¢ approaches ¢0 - ¢c + 2(3x) -1 In(to x) at to.
We consider the three constraints imposed by the horizon problem, density perturba-
tions and the reheating temperature, both with a new inflation type Coleman-Weinberg
potential[14], and a chaotic type potential. The former may be approximated for small ¢
v(e) = yo- (9)
where V0 is the GUT scale and A ,-_ 1/2 for the SU(5) model, although A depends weakly on
V0 as well. For the latter we consider
v(¢) = (lo)
n
with n an even integer and A" arbitrary for now. A main problem of ordinary chaotic inflation
is the fine-tuned small value of the coupling parameter imposed by density perturbations[l],
in particular _4_10 -12 for the quartic'case.
When the scalar field ¢ evolves into the oscillation phase, inflation will end. Hence the
condition l¢/3H¢ls _ 1, or equivalently I(ha V)"]t ", 3_2(1 - 3_/6) -', fixes the value of ¢I,
wherea subscript f denotes the value at the end of inflation. Then we find
f(¢) = ,¢2V012A¢2 and
f(¢) = ,¢2_2/2n and
_¢f = [_'y0/_(1 - Z_/6)]'/2
_¢¢f= [n(1- fl2/6)/3]1/2 for
The horizon problem will be solved if[12]
for new inflation (11)
chaotic inflation (12)
a-z_ 1-_2/2 >103° TRu''_ax (13)
a 01 "" WlpL
where mPL is the Planck mass and TRH, max is the maximal possible reheating temperature,
given by H_ = tc27r2g.T_H,m_x/90, with g.(T) ~ 100 the effective number of particle species.
Another constraint comes from density perturbations. In particular, results from the
cosmic microwave background radiation imply that _Sp/p < I0 -4, where _p is the perturbation
in the density p. Generalizing the result of Lucchin and Matarrese[12], the amplitude of
density fluctuations of an arbitrary scale at the epoch when that scale re-enters the Hubble
horizon may be calculated as follows:
{ v__p~ H_max{l_l,ld'l} ~ '_2H° _q h (1 -/_2/6)-1 (ah/ao) -zU2 for ld'lh> I¢1_p ,_2 +_,'_ h _go,8-' (ah/ao)-_'2/_ for I¢lh < I'_1,,
(14)
where a value with subscript h is evaluated at the time the perturbation originally left the
horizon, and of/h - ln(a//ah) is -,_ 50 - 70 for scales currently entering the horizon. Because
of the factor (ah/ao) -_2/_ in (14), we expect much smaller density perturbations than those
in the conventional model, loosening the parameter constraints.
If we account for the observed baryon asymmetry in terms of the standard baryogenesis
mechanism[15] through heavy Higgs bosons, the reheating temperature should be larger than
101° GeV[16]. For efficient reheating, we find
TRH,max TRH,conv e -_'_14 ( al _-_14= -- (15)
\a0 /
where TRu,¢om, -- (30V(¢l)/Tr2g.) 1/4 is the maximum reheating temperature possible in the
4
corresponding conventional model. We expect a lower reheating temperature than in the
conventional model, although this depends on the detail of each reheating mechanism.
The three constraints, (13),(14) and (15), restrict the initial values ¢o and ¢o, the coupling
parameter/3, and parameters in the potential V(¢). We next examine the permitted values
for both new and chaotic inflationary potentials.
(i) New inflation type
From the above constraints, we obtain three conditions for the initial value ¢o:
¢0 < ¢cr,H---= O.16Ho A-'/2ez'_'°/2 [1 + 0.0171n (l/ol/4/lO'SGeV)] -'/2
(horizon problem)
4'0 > _be,.,l:ttt =- O.19Ho/_-l/2_e _'_4'°/2 [1 + 0.0941n (V_/4/IO'SGeV)] -1/2 (reheating)
¢0 < ¢cr,D _ O.15HoA-l/_(al/h/70) -1/2ez'¢°/2
×{1-0 [1+0 0
(density perturbations)
Note that [¢[h > [_'[h unless 3_lO-l°A-l/_(a]/h/70)-3/2(Vol/4/lOlsGeV) 2. Imposing
min{_bc_,g, ¢c_,D} > _c_,RH, we find that/3 < 0.79(af/_/70) -1/2. If the condition A<0.02e z'¢°
is satisfied, the natural initial condition of ¢o_Ho is allowed. These values are around the
natural ones of unity, without need for fine-tuning. The constraints on A and Vo are plotted
together in figure 1 for fl = 0.1,_b0 = Ho and ¢0 = lOmpL. There is a significant region in
A - Vo space where all the constraints are satisfied. The SU(5) model with V_/4 = 101SGeV
and A ,,_ I/2 is inside the allowed region. With this scenario, the slower expansion rate
during inflation allows for the suppression of density fluctuations, which had been the main
drawback of the regular new inflationary model.
5
(ii) Chaotic inflation type
Assuming the initial energy scale is the Planck one, we obtain the constraints:
¢0 > ¢c_,n _ 2.3rnpL na/_ (horizon problem)
¢0 < ¢c_,nH = 2.6mpLnl/2_ -1 (reheating)
1.5mPLnl/2_8-1[1 + 2.5f12(o,I/h/70)] I/2 for _ < (n/2af/h) '/2¢0 > ¢c_,n = 1.4rnpLnl/2fl-l[1 + 2.9f12(of/h/70)] 1/2 for fl > (n/2o_f/h) 1/2
(density perturbations)
where/3 < correspondsto I¢lh< I /'lh,respectively. Just as in the new inflation
model, we find fl < 0.9(Of h 70) -a/2. Most significantly, much lower values of if'0 than in
the standard case are now possible, leading to considerably larger and hence less fine-tuned
values of the coupling constant A,,. For example, A4 < 5.8 x 10-3f14e _"_° for the quartic case
and A2 < O.15rn2pL_32eO"¢° for the massive scalar field, so we find a natural coupling constant
A,, is possible, A different mechanism to obtain a similar constraint on _4 has been discussed
by use of a non-minimal coupling term with _ < -2 x 104 [17].
If our model is derived from the generalized Einstein theories via a conformal transfor-
mation; we must transform back t o th e original system. We present the JBD model as one
example. The JBD scalar field _JBO is related to our ¢ via the conformal transformation
= eB_¢/2 oJBDa.¢;
g_v - al_.v --
x¢ = (w + 3/2) '/2 In(@JBDG) + _¢p,_,,t (16)
with/3 = 2(w + 3/2)-a/219]. Since the JBD parameter w is constrained by observations to
be larger than 500[18],/3 is smaller than 0.09 in this model. The Cosmic time tjBD and the
scale factor aJBD are given by dtJBD = e-a_¢/4dt and ajBD = e-a_'/4a. Hence we obtain the
formula[19]:
(,,fl,,obBo "-"
JBD 2 p
Vo/V:)'/' (17)
(18)
6
TRHJBD
,m_x _ e_'_¢/4TnH,m_ "" TnH,conv (19)
Therefore, in the original system, the constraints from the horizon problem and density
perturbations do not change much, while the reheating temperature is the same as in the
conventional model, where we have assumed instantaneous reheating. Hence the constraint
from successful reheating is loosened in the JBD theory (see Fig.l).
We have presented a scenario in which an inflaton potential coupled to an exponentially
decaying potential leads to a less rapidly expanding inflationary stage. Hence, the problem
of excessive density perturbations in the new inflationary model is removed and fine-tuning
in the chaotic model is lessened. In addition, the less rapid expansion of the present model
may allow successful old inflation, just as with extended inflation.
Finally, we mention that if the inflaton potential V is nonzero at the termination of in-
flation, then a decaying cosmological constant will exist. Though primordial nucleosynthesis
sets a stringent bound on the vacuum energy at that epoch [20], it may still be possible that
the relic vacuum energy contributes significantly enough to the present total energy density
to close the universe. Work on this possibility is in progress.
Acknowledgements
KM would like to thank J. Hwang at Univ. of Texas for useful discussions, especially
regarding density perturbations in the conformal frame. ALB acknowledges a postdoctoral
fellowship from the Japanese Society for the Promotion of Science, administered in con-
junction with the National Science Foundation. KM and JY acknowledge Rocky Kolb and
Theoretical Astrophysics Group for their hospitality at Fermilab. Thanks are also due to the
Japan-US cooperative science program No. MPCR-185(JSPS), which supported their visit
to Fermilab, where this work was completed. This work was supported in part by NASA
grant # NAGW-1340 and in part by the Grant-in-Aid for Scientific Research Fund of the
Ministry of Education, Science and Culture No. 01795079.
References
[1] A.H. Guth, Phys. Rev.D23 (1981) 347;
K. Sato, Mon. Not. Roy. Astron. Soc.195 (1981) 467;
A. Albrecht and P.J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220;
A.D. Linde, Phys. Lett. 108B (1982) 389; Phys. Left 129B (1983) 177.
[2]R. Brandenberger, Rev. Mod. Phys. 57 (1985) 1;
A. Linde, Rep. Prog. Phys. 47 (1984) 925;
M.S. Turner, in the Proceedings of the Cargese School on Fundamental Physics and
Cosmology, ed. by J. Audouze and J. Tran Thanh Van (Gif-sur-Yvette, 1985)
[3] L.A. Kofman, A.D. Linde and A.A. Starobinsky, Phys. Lett. B157 (1985) 36.
[4] A.A. Starobinsky, JETP Lett. 42 (1985) 152;
J. Silk and M. Turner, Phys. Rev. D35 (1986) 419.
[5] D. La and P.J. Steinhardt, Phys. Rev. Lett. 62 (1989) 376;
D. La, P.J. Steinhardt and E.W. Bertschinger, Phys. Lett. B231 (1989) 231.
[6] C. Brans and R.H. Dicke, Phys. Rev. 24 (1961) 925.
[7] A. Linde, CERN preprint CERN-TH.5619/89 (1989).
[8] E. Witten, Phys. Lett. 155B (1985) 151;
J.-P. Derendinger, L. Ibanez and H. Nilles, Nucl. Phys. B267 (1986) 365.
[9] G. Magnano, M. Ferraris and M. Francaviglia, Gen. Rel. Grav. 19 (1987) 465;
A. Jakubiec and J. Kijowski, Phys. Rev. D37 (1988) 1406;
K. Maeda, Phys. Rev. D39 (1989) 3i59;
J.D. Barrow and K. Maeda, Waseda University preprint WU-AP/04/89 (1989).
[10] A. Zee, Phys. Rev. Lett. 42 (1979) 417.
[11] P.J.E. Peebles and B. Ratra, Astrophys. J. 325 L17;
B. Ratra and P.J.E. Peebles, Phys. Rev. D37 (1988) 3406.
[12] L.F. Abbott and M.B. Wise, Nucl. Phys. B244 (1984) 541;
F. Lucchin and S. Matarrese, Phys. Rev. D32 (1985) 1316.
[13] J.J. Halliwell, Phys. Lett. 185B (1987) 341;
J. Yokoyama and K. Maeda, Phys. Lett. 207B (1988) 31.
[14] S. Coleman and E.J. Weinberg, Phys. Rev. D7 (1973) 1888.
[15] M. Yoshimura, Phys. Rev. Lett. 41 (1978) 281;
E.W. Kolb and M.S. Turner, Ann. Rev. Nucl. Part. Sci. 33 (1983) 645.
[16] J. Yokoyarna, H. Kodama, K. Sato and N. Sato, Intern. J. Mod. Phys. A 2 (1987) 1809.
[17] B.L. Spokoiny, Phys. Lett. 129B (1984) 39;
R. Fakir and W.G. Unruh, Phys. Rev. D41 (1990) 1783;
D.S. Salopek, J.R. Bond and J.M. Bardeen, Phys. Rev. D40 (1989) 1953.
8
[18] R.D. Reasenberget al, Astrophys. J. 234 (1979) L219.
[19] J. Hwang, Univ. of Texas preprint (1990).
[20] K. Sato, N. Terasawa, and J. Yokoyama, in the Proceedings of IX th Moriond Astro-
physics Meeting on The Quest for the Fundamental Constants on Cosmology, ed. by J.
Audouze (1990, to be published)
Fig.i
Figure Caption
The constraints from the horizonproblem, reheatingtemperature (TRH> 10l°
GeV) and density perturbations (_Sp/p< 10-4) for /3 = 0.1, _0 = H0 and
_o = lOmpL. The shaded region is permitted. The curves RH and D are the
constraints from the reheating temperature and density perturbations, respec-
tively. The point + corresponds to the SU(5) model with Vo_/4 = 10 is GeV and
X -,_ 1/2. The dotted line RHjBD shows the reheating constraint for the JBD
theory with a; -,_ 500.
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