Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mrao.cam.ac.uk/ppeuc/astronomy/papers/lukash.ps Äàòà èçìåíåíèÿ: Wed Jul 23 14:53:34 1997 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 10:18:24 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: inflationary universe

Gravitational wave contribution to CMBR anisotropy
V.N. Lukash 1 and E.V. Mikheeva 2
Astro Space Center of P.N.Lebedev Physical Institute 3 , Profsoyuznaya 84/32, Moscow 117810, Russia
We argue that the contribution of cosmic gravitational waves into the large­scale CMBR
anisotropy can exceed 50% for a broad class of the inflationary models with effective \Lambda­term
producing the standard local spectrum of primordial cosmological perturbations.
1 Introduction
It is the Scalar and Tensor modes of Gaussian metric perturbations ffi g ik that are standardly produced
in any cosmological inflation with one fundamental scalar field '. The contributions of both modes
due to the Sachs--Wolfe effect in the CMB temperature anisotropy convolved with the COBE DMR
antenna, are commonly presented in terms of S and T parts:
D `
\DeltaT
T
' 2 E
10 0
= S +T: (1)
Available observational data cannot yet discriminate the ratio T/S, thus, the latter currently remains
a pure theoretical tool in the analysis.
2 Discussion
A general myth created by the Chaotic Inflation, that ``T/S is negligible for the Harrison--Zel'dovich
spectrum'', proved to be an artifact of the model, namely, the '­potential­choice there (the power­
law inflation has demonstrated the consistency with the myth displaying that large T=S – 1 can be
obtained only while rejecting from the HZ­slope of density perturbations, n S Ÿ 0:8). The reason is that
the CI­models generate cosmological­scale­perturbations only for ' AE 1 (as the process of inflation
is to be intrinsicly stopped at ' ¸ 1) so, the T=S ¸ ' \Gamma2 Ü 1 remains always there. At the same
time, smooth '­potentials produce ``red'' HZ spectra of scalar perturbations, which is also a general
property of CI. Although both properties, small T/S and n S ¸ 1, originate from a single condition
-- the ' AE 1 inflation, -- the latter is not a general case for inflationary process at all. More of that,
the relationship between T/S and n S is not confirmed (in particular, it is violated when the slow­roll
approximation is broken). Instead, the connection of T/S to n T seems a fundamental one (see below).
Obviously, it is clear that T/S becomes large if the cosmological perturbations are generated at
' ¸ 1. In this case the inflation must continue to smaller '(! 1). This possibility is realised for a
rather general class of the fundamental inflations with one scalar field and the effective \Lambda­term (Lukash
& Mikheeva (1996)):
V = V 0 + 1
2 m 2 ' 2 + 1
4 –' 4 : (2)
The vacuum density V 0 (¸ 10 14 GeV) should be metastable (otherwise inflation will proceed infinitely)
and decay at some ' ¸ ' ? ! 1. The mechanism of the V 0 decay may be arbitrary and is not fixed in
this simple model (as an example of \Lambda­decay see hybrid inflation, Linde (1994)).
Regarding that the inflation proceeds up to small ' ! 1, eq(2) can be relevantly understood as the
decomposition of V (') near the local­minimum­point ' = 0.
Another important parameter of the model (besides ' ? ) is ' cr where V (' cr ) = 2V 0 . For
' ? !'!' cr ; ' cr ? ¸ 1:3; (3)
1 mailto:lukash@dpc.asc.rssi.ru
2 mailto:mikheeva@dpc.asc.rssi.ru
3 http://www.asc.rssi.ru/
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Figure 1: The spectra of scalar (q k ) and tensor (h k ) metric perturbations in the model (2) with – = 0
in arbitrary normalisation. In the ``blue'' asymptotic n S = 1:5.
the process of the inflation is dominated by the V 0 ­term. Such a stage, being impossible in the chaotic
inflation, brings about the generation of ``blue'' spectra of S­perturbations. Taking into account that
for ' ? ' cr the spectrum is ``red'', we come to very generic properties of the S and T spectra in the
models with \Lambda­term.
ffl The power­spectra of scalar perturbations have a broad minimum near k ¸ k cr , corresponding
to '¸' cr .
ffl Nearby the minimum of S­perturbations the amplitudes of both modes, S and T, are close to
each other, so the expected T/S get its maximum when measured at those scales (k ¸ k cr ¸
1000 h \Gamma1 Mpc for COBE).
ffl In the region where T=S ¸ 1
-- the effective spectrum of temperature fluctuations is close to HZ­one;
-- the local slope of S­mode is scale­free: n S ' 1;
-- the deviation of T­mode from the HZ­spectrum gets its maximum.
ffl The approximation T=S ' \Gamma6n T works well at any scale, where n T should be understood as the
effective (local) spectral index of the T­mode where T/S is measured.
Figure 1 presents S and T perturbation spectra (dashed and solid lines, respectively) and a ratio
between them (dotted line). The gauge­invariant metric perturbations generated in the inflation, are
determined in the synchronous reference system comoving to the '­field in large scales (– AE ct):
ds 2 = dt 2 \Gamma a 2 [(1 +2q)ffi fffi +2G fffi ] dx ff dx fi ; (4)
where q = q(x) and G fffi =G fffi (x) are random Gaussian functions of spatial coordinates, G ff
ff =G fi
ff;fi =
0,
hq 2 i =
Z 1
0
dk
k
q 2
k ; hG fffi G fffi i =
Z 1
0
dk
k
h 2
k : (5)
For HZ­spectra q k and h k would be k­independent (nS = 1 and n T = 0, respectively). The normalisa­
tion of both spectra is arbitrary and can be calculated exactly after defining k cr .
For the case – = 0, m 6= 0, T/S was calculated by the authors for the following model parameters:
1:008 ! nS ! 2:8, 10 \Gamma5 k hor ! k cr ! 10 7 k hor (Figure 2). Figure 3 shows the same two­parametric
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Figure 2: T/S ratio for the model (2) with – = 0.
Figure 3: The slices for T/S.
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function T/S as a set of the slices of constant T/S values. We see that the probability to find T=S ?
¸ 1
in the model plane is roughly 50%.
As the ``blue'' S­spectra appearing naturally in the \Lambda­dominated inflation models have important
implications for LSS formation theories, below we write down explicitly (k AE k cr ).
For – = 0:
q k ¸ k (nS \Gamma1)=2 ; nS = 4 \Gamma 3
`
1 \Gamma 4m 2
3V 0
' 1=2
: (6)
For m= 0:
q k ' 0:3
p
–ln 3=2 (k=k cr ): (7)
For\Omega 0 = 1 the dimensionless power spectrum of density perturbations is as follows:
\Delta k = 0:4 (k=H 0 ) 2
q k ' 3:6 \Theta 10 6 (k=h) 2 q k : (8)
Acknowledgements
The work was partly supported by Russian Foundation for Fundamental Research (96­16689­a), and
German Scientific Foundation (96­02­00086G). We are grateful to the PPEUC Organising Committee
for financial support.
References
Lukash V.N. & Mikheeva E.V., 1996, Gravitation & Cosmology, 2, 247.
Linde A.D., 1994, Phys.Rev.D, 49, 748.
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