>K9m 2015, TOM 147, bmii. 3, rap. 599 604
© 2015
SCALARON PRODUCTION IN CONTRACTING ASTRO PHYSICAL OBJECTS
D. Gorbunova'h* A. Tokarevaa c
" Institute for Nuclear Research, Russian Academy of Sciences 117312. Moscow, Russia
b Moscow Institute of Physics and Technology
141700, Dolgoprudny, Moscow Region, Russia
€Faculty of Physics, Moscow State University 119991, Moscow, Russia
Received October 24, 2014
We study the creation of high-energy SM particles in the Starobinsky model of dark energy (a variant of F{Rj-gravity) inside the regions contracting due to the Jeans instability. In this modification of gravity, the additional degree of freedom — a scalaron — behaves as a particle with the mass depending on matter density. Therefore, when the mass changes, light scalarons could be created at a nonadiabatic stage. Later, the scalaron mass grows and can reach large values, even the value 101J GeV, favored by early time inflation. Heavy scalarons decay contributing to the cosmic ray flux. We analytically calculate the number density of created particles for the exponential (Jeans) contraction and find it negligibly small for the phenomenologically viable and cosmologically interesting range of model parameters. We expect similar results for a generic model of F{Rj-gravity mimicking the cosmological constant.
Contribution for the JETP special issue in honor of V. A. Rubakov's 60th birthday
DOI: 10.7868/S0044451015030209
1. INTRODUCTION
Numerous observational data require a new component in the right-hand side of the Einstein equations, which is called dark energy and which causes the accelerated expansion of the Universe. The simplest and still viable candidate for dark energy is obviously a cosmological constant. But its unnaturally small value engenders investigation of other ways to explain the observational data.
F(R)-gravity provides a framework for constructing models of dark energy with the time dependent equation of state p/p = w = w(t)\ moreover, at some stage we have w < —1 (see fl] for a review). Such models of modified gravity may also explain the inflationary stage of the early Universe, providing a unified mechanism to describe both stages of accelerated expansion.
The choice of the function F(R) is still phenomeno-
E-mail: gorby'fl'ms2.inr.ac.ru
logical to a large extent. It must be self-consistent from the theoretical standpoint, explain the cosmological data, and pass all Solar System and astrophysi-cal tests. The natural question is: How to distinguish F(R)-gravity from other models of dark energy? The most straightforward way is to improve the sensitivity of the overall cosmological analysis to the dark energy equation of state. However, besides serious systematic uncertainties, there are physically motivated degeneracies in the cosmological observables with respect to physical parameters. In particular, specific effects of F(R)-gravity at small spatial scales can be canceled by massive (sterile) neutrinos, whose dynamics works against modified gravity [2].
An attractive idea of probing F(R)-gravity [5] is associated with possible production of high-energy particles (scalarons) in space regions where the matter density changes. It was claimed in Ref. [5] that the growing curvature oscillations that decay into high-energy particles may lead to a significant impact on the flux of ultra-high energy cosmic rays. This result is rather
unexpected, because high-frequency oscillations (i.e., heavy particles) are produced by a slow process (the structure formation) very inefficiently. Moreover, cos-niological evolution naturally gives zero initial amplitude for such oscillations, which can be associated only-wit h the scalar mode (heavy scalaron).
In this paper, we study the processes of quantum particle production in F(R)-gravity using the method of Bogoliubov transformations. We consider F(R)-gravity being equivalent to normal gravity with an additional scalar field (scalaron) having a complicated potential in the Einstein frame. The form of the potential depends on the surrounding density, providing the scalaron to be a chameleon field. Scalaron is very light at densities close to the present energy density of the Universe and heavy at larger densities. It is known that the particle with a time-dependent mass may be created in quantum theory if the typical process rate is close to the IIlclSS value. A scalaron may be born being light and then its mass may grow while the object contracts. When the scalaron becomes heavy, it decays into high-energy Standard Model (SM) particles. We consider the same F(i?)-model as in [3], where such processes can be investigated analytically, and obtain that in contrast to [3], the number density of created particles is unfortunately too small to be observed in all realistic contracting regions (astorphysical objects) in the Universe.
This paper is organized as follows. In Sec. 2, we describe the construction of the function F(R) that is appropriate for both the inflation and dark energy. In Sec. 3, we introduce the Einstein frame approach to the F(R)-gravity in which the additional scalar field has a mass depending on the background energy density. In Sec. 4, we calculate the number density of produced particles an object in contracting due to the Jeans instability and discuss the particle production rate in different contracting regions of the Universe (astorphysical objects). We conclude in Sec. 5.
2. DESCRIPTION OF THE MODEL
The present-day acceleration of the expansion of the Universe can be described in terms of F(R)-gravity by the action [4]1)
3=^1 t^^F(R),
(1)
with F(0) = 0 reflecting the disappearance of the cos-mological constant in the Minkowski flat space limit.
Any viable F(R) function must obey the classical and quantum stability conditions: F'(R) > 0 and F"(R) > 0. It was introduced to mimic the ACDM model in the late-time Universe, and hence in the limit of small curvatures, F(R) « R — 2A with the dark energy density p\ = AM p. Moreover, the second derivative of F(R) must also be bounded from above, F"(R) < const (see [5] for the details) to avoid early time singularities at R. —¥ oc. The last requirement is easily satisfied for any F(R) after adding an i?2-term. This term with a specially selected coefficient may also provide the Starobinsky inflation in the early-Universe [6].
An example of a function appropriate for the dark energy- proposed by Starobinsky- is [4]
F(R) = R + XR0 1 + -52
R2 R
1
(2)
In the regime R. R0, we have F(R) « i? — XRq, providing a cosmological constant. The parameter R0 fixes a scale that corresponds to the dark energy- density- p\ (this is valid for A > 1):
Rn = —
PA
A M j,
(3)
To avoid the early time singularity, we hereafter use the function F(R) with the i?2-term added:
F(R) — R + XRq [(1 +
\ V Rn
1
R2 6 M2
(4)
As discussed above, the last term in (4) is also appropriate for the usual Starobinsky- inflation [6] in the early Universe if we choose M = 3 • 1013 GeV; an impact of the second term in (4) is negligible for the corresponding large values of curvatures.
There is a problem (described in [5]) with the subsequent stage of scalaron oscillations. At this stage, zero and even negative values of R. may be obtained. Then it is easy to see that later the Universe unavoidably arrives at F"(R) < 0, leading to quantum instability of the theory. However it is possible to construct a function F(R) that provides a similar late-time cosmology but does not suffer from instabilities at the post-inflationary epoch. For example (where we neglect the presence of the i?2-term),
F"(R) x
1
(R/Ro)2n+2
(5)
11 The metric signature is (--b ++)•
leads to the results similar to those following from (4) at large curvatures. In what follows, we use function (4)
for simplicity, being interested only in the late time evolution of the Universe. But we need to ensure that the evolution of R does not put its value to the region of forbidden curvatures.
Starobinsky model (4) has two free parameters, A and n (R0 is fixed by (3), while M is fixed to explain the early time inflation). A stringent restriction on the value of n follows from local gravity constraints on the chameleon gravity [7]. It gives n > 1. The parameter A is bounded only from the stability condition for the de Sitter minimum (see [8] for a review). This bound mildly varies with n as
A > f(n) and f(n) « n/(2n - 4/3). (6)
3. EINSTEIN FRAME PICTURE: THE SCALARON AS A CHAMELEON
F(R)-gravity can be considered in the Einstein frame, where it describes the usual Einstein gravity with an additional scalar field (scalaron) coupled to the matter fields as a chameleon field [9]. The scalaron potential is
where
V(<j>) =
\f 2
J'p (RF'(R)^F(R)).
2 F'(R)
where R. = R(<i)) solves the equation
F'(R) = exp ■
(7)
(8)
Through the gravity interaction, the scalaron couples to all the matter fields effectively described in the cosmological context as the ideal fluid with an energy-density p and a pressure p. This coupling modifies the scalaron potential [9]:
VcffW = vw + .
0)
The minimum <f>min of V(:/f can be obtained substituting the solution /('„,„, of the equation
2F(R) - RF'(R) =
P — 3p
TrfT
(10)
in (8), with <f>min written in terms of I!,„,„.
For (4), we can approximate the solution as (the greater A is, the better the accuracy)
Rrnîr,
(4-
4* min —
M'j,'
■log (F'(Rmin)),
(ID
T=(p- 3p)/pA ■ The effective scalaron mass at this minimum is
mhf ~
1
3 F"(Rn
1
Rmin-fi1 {Rri F'iRmin
3 F"(i?.„
(12)
Hence, for model (4), we obtain the scalaron IIlclSS that depends on the surrounding energy density and pressure:
"'< // -
M2m2 M2
m,
where
1
nr =
12n(2n+ 1) \2
\ \ 2"
A \ £ A_
Ml
(4 •
>2)7+2
(13)
(14)
In a particular range of densities in, jy strongly depends on r. Obviously (see Eqs. (14) and (13)), m is small there, in < M, leading to
/MMP\1/in+1) /2\'l/in+1)
4<r<(^f) U) x
x (12>.(2>. + i))i/("-l-i> = (L4 . io
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