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40
UDC 530.1; 539.1
COSMOGRAPHIC TRANSITION REDSHIFT IN F(R) GRAVITY
S. Capozziello1"2'3, O. Luongo1"2'4
1 Dipartimento di F'isica, University di Napoii "Federico II Via Cinthia, 1-80126, Napoii, Italy. 2ístituto Nazionale di Física Nucieare (INFN), Sez. di Napoii, Via Cinthia, 1-80126, Napoii, Italy. 3Gran Sasso Science Institute (GSSÍ), Viale F. Crispi, 1-67100, L'Aquila, Italy. 4Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, DF 04510, México, Mexico.
E-mail: capozzie@na.infn.it, luongo@na.infn.it
We propose a strategy to infer the transition redshift zja, which characterizes the passage through the universe decelerated to accelerated phases, in the framework f (R) gravities. To this end, we numerically reconstruct f (z), i. e. the corresponding f (R) function re-expressed in terms of the redshift z and we show how to match f (z) with cosmography. In particular, we relate f(z) and its derivatives to the cosmographic coefficients, i. e. Ho,qo and jo and demonstrate that its corresponding evolution may be framed by means of an effective logarithmic dark energy term Qx> slightly departing from the case of a pure cosmological constant. Afterwards, we show that our model predicts viable transition redshift constraints, which agree with ACDM. To do so, we compute the corresponding zja in terms of cosmographic outcomes and find that zja < 1.
f(z)
To do so, we get numerical constraints employing Monte Carlo fits with the Union 2.1 supernova survey and with the Hubble measurement data set.
f(R)
1 Introduction
The ACDM model predicts the existence of a cosmological constant A, entering the Einstein equations and responsible for the dynamical properties of the universe today. Indeed, the model arguably represents the simplest cosmological paradigm to allow the universe to speed up at late times. Despite its
A
A
energy densities are so comparable today. Nevertheless, cosmological constraints indicate a severe difference
A
quantum field theory's value. It follows that the A
case of more general paradigm, based on the existence of some sort of (dynamical) cosmic fluid, driving the universe today, dubbed dark energy [1]. Dark energy seems to dominate over all cosmological species enabling the universe to accelerate. Dark energy acts as a dynamical source for speeding up the universe, whereas dark matter is responsible for structure formation. Recently, the need of understanding at which time the decelerating-accelerating transition phase occurs has leads cosmologists to directly measure the corresponding transition redshift zda [2,3]. There exists a wide consensus, based on robust observational supports, indicating zda around the unity [2,4]. Even though the ACDM value of zda reproduces the same result of recent observations, we cannot conclude that dark energy is a constant at all stages of universe's evolution. However, several classes of models depart
from the case of zda ~ ^indicating t hat zda represents a new cosmic number which any cosmological model should reproduce to be in accordance with cosmic
A
gravity is considered a realistic alternative to dark energy. In particular, the challenge of finding out
f(R)
describe the gravity as a whole, passes through the reconstruction of the correct cosmological model. In other words, determining viable numerical outcomes of zda f(R)
the late-time bounds in a model independent way [5]. To support this idea, one can consider cosmography, which represents a technique for matching cosmic data with observable quantities without imposing a particular cosmological model. In general, a Taylor expansions evaluated at present-time is adopted in cosmography. Constraining a class of coefficients gives rise to the cosmographic series [6]. Hence, the use of cosmography may suggest possible departures from the A
how and whether dark energy evolves in time [7]. In this paper, we show how to get the Hubble rate
f(R)
models. Further, we bound numerical intervals for
zda
particular classes of f (R) models allow to get zda in the correct observable ranges. Afterwards, we propose an effective evolving dark energy term which extends the A
to the dark energy density. This model predicts that the corresponding acceleration parameter changes sign
around z < 1. Finally, we describe the corresponding f (z) = f (R(z)) function in terms of the cosmographic coefficients and propose a possible expression for it. The paper is structured as follows: in Sec. II we describe the cosmographic approach in the context of f (R) gravity and compute the corresponding transition redshift. We also show a comparison with current data and demonstrate that the dark energy corrections are compatible with observations. We also report numerical outcomes. Finally, Sec. Ill is devoted to conclusions and perspectives.
Best fits performed by considering Monte Carlo techniques based on the Metropolis algorithm evaluated with la error bars for each parameter. We perform a fit by using supernovae without imposing any priors, i.e. adopting a free experimental test. We also use H(z) data without imposing any assumptions. The experimental tests have been carried out by using the luminosity
H
The transition redshifts here reported have been found using the definition of zda for the model, whereas error bars through standard logarithmic propagation. All is adimensional except H0 which is reported in units of km s-1 Mpc-1.
Taylor series around the present epoch to:
fit supernovae H(z)
p value 0.692 0.960
Ho 68.070+3 330 -2.200 68 4 9 0+3 390 68.490-2.500
qo — 0.542+0 072 -0.083 -2.941-00.0433
jo 0.577+0 448 -0.353 -°.955-0..228
a214-0:0i?
a 2.301-0:!69 1.966-0..239
ß 0.760-0:471 0.285-
Zda 0.860—0:272 0.632-00.11410
*(t) = E
dtm
(t - to)m t-t0=o m!
(1)
Keeping in mind Eq. (1), we are able to evaluate f (z) ^d derivatives at z = 0 which corresponds to t = t0. To relate f(z) and derivatives to the cosmographic expansion of a(t), it is necessary to assume the constraint on R [81
R
— = (1 + z) H — 2H
(2)
with Hz = dz- Simple algebra leads to R0 = 6Ho (Hzo — 2Ho) and Rzo = ßH^ — Ho(3Hzo — H2zo)-From Eq. (1), we define the cosmographic coefficients as
h (t) = 1 §
a dt
q(t) = —
j (t)
1 d2a 1H2 dt2
1 d3a
— i + H
aH3 dt3
— (3q + 2) +
1 d2 H
H3 "dt2"'
(3)
(4)
(5)
2 Cosmographic reconstruction of transition
f(R)
f(R)
gravity, can be achieved by means of cosmography. The
f( R)
transition redshift zda showing its compatibility with present-time constraints. In particular, cosmography
f(R)
derivatives by passing through the use of an auxiliary-function f (z). This means to define f (z) = f (R(z)) and its first derivative through df = ^d^ (dff) -To this end, one needs the scale factor expansion into a
H
parameter q, the variation of acceleration, i.e. the jerk parameter j. Constraining the above quantities by means of cosmological data permits one to fix the initial settings on the curvature dark energy
f(R)
scalar curvature is somehow fixed by geometrical observations, cosmography becomes a pure model-independent treatment to fix cosmic bounds and
f(R)
models are really viable to describe current universe expansion. Furthermore, it is easy to show that the measurements of the cosmographic parameters provide the disadvantage that all coefficients H, q and j cannot be measured alone. Assuming, in fact, the luminosity distance as function of the cosmographic series, i.e. Dl = DL(H0, q0, j0), and comparing cosmic data with it, we notice that it is impossible to decouple the terms H—1(1 — q0) and H— 1(3q^ + q0 — j0 — 1), respectively the second and third orders of the Taylor expansion of DL [9]. Thus, to better limit H0, q0 j0
degeneracy problem between coefficients, and also
H0
by considering the first order of DL Taylor expansion
Dl - Ho
(6)
in the observational range z < 0.3. Here, we adopt the Union 2.1 supernova compilation [10] and we compare our results with those predicted by the Planck mission
dm
a
1
z
H0
q0 j0
Hz0
H" = 1 + qo,
2
Ho
jo - qo
Finally, we have
qo jo
fo
2H2
6H2
+ 2,
+ qo + 2.
(7)
(8)
(9) (10)
These expressions are needful for setting priors on f(z)
zda
f(R)
considerations in mind, we can integrate the modified Friedmann equations, directly by using cosmographic measurements as a sort of setting conditions. In so doing, we infer a parameterized cosmological model, which differs from the ACDM model and shows an effective logarithmic dark energy term of the form
Oj = log (a + fiz).
(H)
The whole Hubble rate becomes H (z) =
H0V/Qm(1 + z)3 + QX, with a and P constants to be fixed by present-time bounds. In fact, to get H = H0, terms ^md P are
a = exp(1 — Om), fi G [0.01, 0.1],
(12) (13)
which are consistent with cosmographic bounds. The corresponding cosmographic f (z) function becomes
f (z) « fo + fi a-71 + f a-2 + a,
(14)
with f0 - —10 f1 ~ 7 f2 ~ —3.7 a = 1 and a2 = 2 mid a = (1 + z)-1. This f (z) predicts the above modified Hubble rate and provides a deceleration-acceleration transition redshift of the form [51
qf (r) = —1 +
+
(1 + z) [3Om(1 + z)2 + fi(a + fiz)-1]
2 [Om (1 + z)3 + ln(a + fiz)]
.
By means of Eqs. (12) and (15), the cosmographic reconstruction favors values of the Hubble constant in agreement with other previous estimations [12-
H0
bounds on f0 Mid fz0 suggest that q changes sign at zda < 1, in agreement with previous measurements [2], showing meanwhile that our model provides zda to be A
our cosmological model, providing an evolving dark energy, as presented in Eq. (11) estimates viable Qm mid P, with zda G [0.57,0.97], providing slight
A
our model seems to pass experimental tests, performed using Monte Carlo analyses and the union 2.1 and Hubble rate measurement data sets [2,10,15]. All our numerical outcomes have been listed in Table. Future improvements will better distinguish any significant A
domains. With our results, in fact, it seems that dark energy slightly evolves also at small redshift, although a definitive conclusion remains object of future developments.
3 Concluding remarks
The passage between the deceleration to
acceleration phases happens as the acceleration q
zda f(R)
gravities are able to reproduce suitable bounds on zda f(R)
functions by means of cosmography. Cosmography-represents a strategy to fix constraints on the principal observable quantities by means of Taylor expansions around present-time. Cosmic data may be directly-matched with the cosmographic coefficients and so the cosmographic series permits to understand which models are really favored than others. In the case f( R)
f(z) as f(z) = f(R(z)) and its first derivative as
ddff) = d^Zr (ddf) • Thus, we showed how to relate them to the cosmographic parameters, i.e. H0, q0
j0
f(R)
initial settings the numerical values inferred from cosmography, we found that dark energy seems to evolve as a curvature fluid proportional to a logarithm, at small redshift. This fact departures from the cosmological standard model, in which dark energy is a constant at all stages of universe's evolution. However, albeit our model showed an evolving dark energy term,
zda
time observations. We also found the corresponding f (z) « f + f a-ff1 + f + a. To find out f (z), the degeneracy between cosmographic coefficients are H0
expansion of the luminosity distance in the range z < 0.3. All numerical results are in agreement with present-time observations and it seems possible that dark energy may be framed in terms of a curvature dark energy fluid. To show that paradigm is capable of well reproducing cosmic bounds, we employed experimental tests, making use of the most recent Union 2.1 supernova survey and of the Hubble measurement data
H2zo
f
o
z
set. Future developments will assum different redshift with cosmological data at different epochs. Moreover,
domains to better fix the cosmographic coefficients, another significative aspect will be to directly relate
Thus, the corresponding effective dark energy term the cosmographic coefficients to the transition redshift
would change correspondingly, providing a different itself, opening the possibility to use zda as a direct
transition redshift. This technique will be useful to cosmographic coefficient, understand whether f (R) functions are compatible
References
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Received 08.11.2014
С. Капоциелло, О. Луонго
КОСМОГРАФИЧЕСКОЕ КРАСНОЕ СМЕЩЕНИЕ НА ПУТИ ЛУЧА СВЕТА В f (R)
ГРАВИТАЦИИ
We propose a strategy to infer the transition redshift zja, which characterizes the passage through the universe decelerated to accelerated phases, in the framework f (R) gravities. To this end, we numerically reconstruct f (z), i. e. the corresponding f (R) function re-expressed in terms of the redshift z and we show how to match f (z) with cosmography. In particular, we relate f (z) and its derivatives to the cosmographic со efficients, i. e. Ho,qo and j о and demonstrate that its corresponding evolution may be framed by means of an effective logarithmic dark energy term Пх, slightly departing from the case of a pure cosmological constant. Afterwards, we show that our model predicts viable transition redshift constraints, which agree with ACDM. To do so, we compute the corresponding zja in terms of cosmographic outcomes and find that zja < 1.
f(z)
To do so, we get numerical constraints employing Monte Carlo fits with the Union 2.1 supernova survey and with the Hubble measurement data set.
f(R)
Капоциелло С., доктор.
Университет Неаполя "Федерико II".
Via Cinthia, 1-80126, Napoli, Италия.
Национальный институт ядерной физики, Неаполь.
Via Cinthia, 1-80126, Napoli, Италия.
Gran Sasso Science Institute (GSSI). Víale F. Crispí, 1-67100, L'Aquila, Италия. E-mail: capozzie@na.infn.it
Луонго О., доктор.
Университет Неаполя "Федерико II".
Via Cinthia, 1-80126, Napoli, Италия.
Национальный институт ядерной физики, Неаполь.
Via Cinthia, 1-80126, Napoli, Италия.
Национальный автономный университет Мехико.
DF 04510, México, Мексика. E-mail: luongo@na.infn.it
