Datasets:
alx-ai
/
arxiv_papers

Dataset card Data Studio Files Community
text
stringlengths
0
44.4k
2V0
V2
; (11)
=M2
plV00
V; (12)
whereMplis the reduced Planck mass and ' denotes deriva-
tive with respect to the in
aton eld. The observables are
then expressed in terms of the above slow roll parameters
as:
nS= 16+ 2; (13)
r= 16 (14)
4nt=2; (15)
2
R=VM4
pl
242: (16)
nSis the slope of the scalar primordial power spectrum,
ntis the corresponding tensor one, ris the scalar to tensor
ratio and 2
Ris the amplitude of the initial perturbations.
All these numbers are constrained by current cosmological
observations [10, 11, 12]. We will use their constraints to
explore the naturalness of our in
ationary trajectories. In-
ation takes place when the slow-roll parameters are much
smaller than 1.
We will use the amplitude of initial perturbations and
the number of efoldings to t some of the paramenters
of the toy model in the previous section. Recall that the
potential in the range of interest is:
V=f2(1 +A1( 2+ 2) +B1( 2 2) +:::);(17)
which appears in gure 1. We can compute ;while
rolling in the direction:
= 2 ( (A1B1) )2+::: (18)
= 2 (A1B1) +:::; (19)
since << 1,is naturally small. We can make small
by a slight ne tuning of the di erence A1B1. We will
writelater as a ratio of the in
aton and gravitino masses.
Once the slow roll conditions are satis ed, we can compute
the number of efoldings (see for instance [16, 17]):
N=1
M Zdxp
2 = Z f
id
2p (20)
From (19) we get:
N=1p
2jA1B1j log f
i : (21)
In most models of supersymmetry breaking, the gravitino
mass is given by:
m3=2=f
M; (22)
hence, we can rewrite the parameters and masses in (9)
as:
jA1B1j=1
2m2
m2
3=2;jA1+B1j=1
2m2
m2
3=2;(23)thus:
N=p
2m3=2
m 2 log f
i (24)
The number of efoldings is considered normally to be be-
tween 50100. Finally we will use the amplitude of initial
perturbations to get one extra condition in the parameters
of our potential. Using [11] (16) can be written as:
V
1=4
=f1=2
21=4(jA1B1j )1=2=:027M; (25)
where is taken atN-efoldings before the end of in
ation.
Summarizing, the two cosmological constraints we get on
the parameters of our potential can be written as:
N=p
2m3=2
m 2 log f
i ; (26)
21=4m3=2
m pf
M1=2
= 0:027; (27)
and theparameter can be written as:
=m
m3=22
: (28)
We take iabove the supersymmetry breaking scale
pf=M ==M , and fclose tomsoft=M, therefore we
can easily get values for Nbetween 50100 for moderate
values of, which is expressed here as the square of the