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OCR for page 68
Chaotic Inflationary Universe and the Anisotropy
of the Large-Scale Structure
G.V. CHIBISOV AND YU.V. SHTANOV
P.N. Lebedev Physical Institute
INTRODUCTION
Inflationary universe models began their history from the seminal paper
by Guth (1981), and since then they have won great recognition among
physicists. The last years were marked by a considerable progress towards
understanding the actual picture of inflation. It was realized that the
inflationary universe is in fact chaotic (see Goncharov 1987 for a review),
that globally it is strongly inhomogeneous, and that the inflation in the
universe as a whole is eternal In such a picture the region available to
modern observations is just a tiny part of the universe, in which inflation
finished about 101° years ago.
In spite of the great popularity of the chaotic inflationary universe
models, it is usually taken for granted that their specific features (such
as strong global inhomogeneity of the universe) can hardly lead to any
observable consequences. The argument is that all we see is jUSt a tiny
part of the universe, a region about 1028 cm, and the typical scales of
considerable inhomogeneities are much greater than this size. In contrast to
this opinion we want to show that such observable consequences really can
Ernst. In spite of the tremendous spatial size of the inhomogeneities under
consideration they are well inside the actual horizon of the modern observer.
Hence their observable manifestation does not contradict causality. The
problem Is to discover such a manifestation.
The phenomenon we are going to discuss is closely connected with the
origin of structure (galaxies, clusters, etc.) in the observable region. As it is
now well Clown (for a review see Brandenberger 1985), primordial density
fluctuations relevant to structure formation could originate from vacuum
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HIGH-ENERGY ASTROPHYSICS
69
fluctuations at the inflationary stage during the permanent stretching of
fluctuations scales. In most of the works on this topic the vacuum Ductua-
tions evolution was considered on a spatially homogeneous background. At
the same time, as already emphasized above, the inflationary universe is not
homogeneous even on the classical level. Indeed, permanently produced
fluctuations on scales bigger than the Hubble scale (i.e., with wave numbers
k < aH) can be treated as a classical background inhomogeneity of the
inflationary universe.
Our main idea then is to consider the vacuum fluctuations evolution
on the inhomogeneous background. An essential change is the change of
the mode functions which determine the vacuum state. We will see that this
leads to a distortion of the resulting primordial fluctuations spectrum as
compared with that in a homogeneous background model The primordial
fluctuations spectrum becomes anisotropic. And this in its turn results in the
anisotropy of the observable large-scale structure. The phenomenon can
be understood in terms of vacuum polarization. The classical background
inhomogeneity polarizes the vacuum state, and this results in the distortion
of the primordial fluctuations spectrum.
One can see that our phenomenon is very similar to the well-known
phenomenon discovered by Sachs and Wolfe (1967~. In this phenomenon
the an~sotropy of the microwave background radiation is due to large-scale
metric inhomogeneity. In our phenomenon the large-scale metric inhomo-
geneity in a similar way influences vacuum fluctuations at the inflationary
stage, resulting in the structure anisotropy. The main difference between
these effects is that in our case the inhomogeneity on scales much bigger
than the size of the observable region are significant. Indeed, vacuum
fluctuations spreading with a speed close to the speed of light pass the
distance to the true horizon, which at the present tune exceeds by many
orders of magnitude the size of the observable region.
Using the above-mentioned analogy with the Sachs and Wolfe phe-
nomenon it is not difficult to make simple estimates for the value expected
of the large-scale structure anisotropy. On the inflationary stage the small-
scale vacuum fluctuations spread almost with the speed of light. Then they
pass a Hubble distance in a Hubble time. As it can be shown, metric
fluctuations on this distance are of the order
h2' . 2/M4
(1)
where y is the scalar field, the dot denotes its time denvative, and Mp is the
P~nck mass (we are using the units in which ~ = c = 1~. The expression
(1) also gives the order of the squared anisotropy produced in a Hubble
time. 1b obtain an estimate for the squared anisotropy produced during
the whole evolution we should integrate (1)
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70
2 irk ~2
where ~ is He effective anisotropy degree and
AMERICAN AND SOVIET PERSPECTIVES
(2)
(3)
is a timelike parameter the expansion index. The upper integration limit
in (23 corresponds to the moment of the Hubble radius crossing (when
k = aH) and the lower one-to some initial moment If the scalar field
evolution is the so-called slow rolling-down
T = lNa
_ i dV(~)
3H dy '
then using also the approximate expression valid at the inflationary stage
H2 8~ V( ~
we obtain from (2)
(4)
(5)
Vo-Ok
3Mp '
(6,
where VO is the initial value of the scalar field potential energy, and Vk
is the value at the moment of He Hubble radius crossing. Our simple
estimate shows that the anistropy can be an appreciable value as typical
values of VO are (Linde 1985)
Vo ~ Mp,
and Vk ~ VO.
FLUCTUATIONS ON THE: INHOMOGENEOUS BACKGROUND
(7)
1b obtain more exact expressions we must consider the evolution of the
mode functions bake Hat derange He scalar field smaD-s~le fluctuation,
on the nonhomogeneous background metric We will take into account only
the fluctuations of the scalar type. Then we would have for the classical
background metric (see Chibisov and Mukhanov 1983; Mukhanov 1985,
1988)
ds2 = aid) t1 + 2~C)dy2-(1 - 2¢C)dX2]
(8)
where He is the so-called relativistic potential It describes the metric
fluctuations on scales bigger than the Hubble scale
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Representative terms from entire chapter:
correlation function
HIGH-ENER~ ~TROP~SICS
71
Be = | (2fi)3(q>(- ah + Ark al ),
(9)
where A) are unperturbed mode functions, and ak, a+ are the usual
annihilation~reation operators. In (8) and (9) ~ is the conformal time. For
A) we can use the approximate expression
A) ~ g2'r97~ k3/2 exp(-iky + it.
We must solve the equation for Ark
Ok + m (I) Ark = 0,
(10)
(11)
where O is me d'Alambertian on the background with the metric (8), and
mom) = d2V(So)/d~2. We find the solution for t
72
AMERICAN AND SOVIET PERSPECTIVES
link the mode Ark to the mode ski ~ is easily done by meam of the
linear perturbation theory equation
Then we have
(^ + 42.22~2) ~ 42 2 ~ d (tY
Ok = ~(- )(1-iCk(9' X)),
(16)
(17)
where Ck is a complex valued function, linear me and similar to Sk in (12~.
Developing Ck in powers of 1/k similar to (13)
Ci=k(Co+kCi+. . .),
and counting powers of 1/k we obtain from (16)
Co _ So, C:-So ~ i(n~V)So + Disc
(18)
(19)
The evolution of Kim ~ after the Hubble radio crms~g is deter-
mined by the linear perturbation theory (see Mukhanov 1988~. After this
time the spatial shape of irk ~ "frozen" and only in amplitude changes
with time. Thus at the moment of the end of the inflationary stage (~7'~ we
have
Okay], X)-Ok Off ~ ~1-~Ck-~0k' X).
Note that A) is the unperturbed solution.
PRIMORDIAL FLUCTUATIONS SPECTRUM
(20,
The expression (20) allows us to calculate the primordial fluctuations
spectrum. 1b do this we calculate the correlation function (~(~)~),
where (A--q>(pf,x). Developing Ck(:i~ in powers of ~ and then pro
ceeding to the integration over new wave numbers k-VC0 ~=0 we will
come to the standard shape of the correlation function
(~(X)~(~)) = / (2 )3 ~ Hi- ) ~2 (1 + ~k)e3~, (21)
with
dln(k3 1 (I) 12
irk-- ~dlak
-
(22)
HIGH-ENERGY ASTROPHYSICS
ark
* J do (Vn)[~c(?7, x + n(rlk - ?7)) - ~C(rt' X - n(Rk - al))] I x = 0
770
73
It can be shown that the anisotropy which appeared in (21) and which
is described by the value ZJk is quadrupole at least at 84%, that is
Vk = no~n,~3 [~e,~ + {a;' M]'
(23)
where Ao<' is a traceless matrix with very weak (logarit}unic) dependence
on k. A value which will characterize the degree of the anisotropy will be
{,2 = tr(A2). (24)
~ obtain a characteristic value for the anistropy we must average (24) with
respect to the random background inhomogeneities ~c. In fact this is av-
eraging over the vacuum state of which the inhomogeneities me originated.
Using (22) we can obtain an approximate result
irk y2
where
(25)
d In~k3 1~°) {2
.
(26)
The expression (25) differs from the estimate (2) only by a factor. Hence
instead of the estimate (6) we will have
~ Ilk Vo-Vk
3 Me
(27)
It can be shown by direct calculations that for the values of k typical for
the present clustering
~lk/3 ~ 10 - 2
Thus for typical values of VO Even by (7) we obtain
(~ 10~1
that is, the allotropy of the order of 10%.
CHAOTIC INFLATION
(28)
(~29)
Now let us take into account one of the peculiar features of the chaotic
inflationary universe. Namely, that such a universe consists of independent
74
AMERICAN AND SOVIET PERSPECTIVES
domains of a physical size [ph = 0~-~)- Hey are independent ~ a sense
that physical processes inside any such domains do not depend on the
processes inside any other domain (Goncharov et al. 1987~. From this it
follows that the integration in (25) must be taken over a history of the
domain of which our observable region has originated. The size of this
domain at all times during the inidationa~y stage must be taken to be of
the order of the Hubble size. Then the history of such a domain becomes
random. For the random history of the scalar field evolution ANTI a Fo~er-
Planck equation can be obtained (Starobinsky 1986; see also Goncharov et
al. 1987 for a review)
where
({' ~ = ~ ~-APT 2(BP)~],
A ~' B H2
(30)
(31)
P($o, r) is a -dependent scalar field distn~ution in the domain considered.
Now let us denote the solution to (303 with the initial conditions P($o,0) =
b(~-~0) by Z(~,TO,T). Then a formal solution for Z(SP,~0,7) can be written
in terms of path integral over all trajectories that start at ~ and at A:
Z(~, SooT) = /(D~) exp(-St~)
where
is the "action," and
(32)
Sty= (ad-A)2B-l
ODD) = n B 2 (
HIGH-ENERGY ASTROPHYSICS
75
effect is conserved on scales on which nonlinear processes do not yet play
an important role, that is, scales more than 50 Mpc. We can easily cal-
culate the two-point correlation function (by) = (~0~), where Aid) =
[p(~/(p>. The integrals over k-space must be taken with some small-scale
cutoff (Peebles 1980~. This can be done by a simple cut-off function exp
(-1~1) where e is a cut-off scale. Then the isotropic part of the correlation
function is found to be
Q4y4 (y~2 + it
and the anisotropic part is
where
(36)
((x) =-hit ~2' ~(37)
h(X) = e4y4 (y~2 + i)3
(38)
y = x/¢, and C in (36) and (38) is some t-independent constant.
One can use the correlation function calculated to test experimentally
the phenomenon discussed In this paper.
DISCUSSION
In conclusion, it is easy to show that during the greater part of
the inflationary stage ache vacuum fluctuations scales of interest are much
smaller than the Planck scale. One such scale, an old-fashioned field
theory, is likely to be invalid and one is supposed to use, for example, the
superstring theory (Green e' al. 1987~. This last however is still in progress,
and we don't yet know how to handle it when dealing with such small scales.
So we use a field theoretical approach hoping that further investigations
based on a more advanced theory will not change our result dramatically.
In context of the superstring theory, the phenomenon discussed might be
caused by string vacuum polarization due to large-scale inhomogeneity of
the inflationary universe.
The problem of small scales disappears to some extent if one takes
into account that the fluctuation scale is not a gauge invariant variable and
thus can be changed by me change of the reference frame. This can be well
illustrated in the case of flat space-time: a photon can be of any energy, in
particular, of energy exceeding the Planck energy.:
1Ihe authors are grateful to B.L Spokoiny and AA Starobinsly for this remark
76
AMERICAN AND SOVIET PERSPECTIVES
From the observational point of view our result means a possibility of
the large-scale structure anisotropy together with highly isotropic microwave
background radiation. Observational discovery of the phenomenon consid-
ered might serge as confirmation to the chaotic inflation scenario and, in
fact, throw some light upon physics on very small scales, in the context
discussed just above.
NOTE ADDED IN PROOF: The extended version of this work is
published in the Int. J. Mod. Phys. A52625 (1990~.
ACKNOWLEDGEMENTS
The authors egress their gratitude to UP. Gnshchok, N~ Gurench,
AD. Linde, OF. Mukhanov, B.L. Spokoiny, and NA Starobinsly for
valuable discussions.
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Green, M.B., J.H. Schwarz, and E. Written. 1987. Superseding Theory. Cambridge Universitr
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Guth, A.H. 1981. Phyla Rev. D23: 347.
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