Dark Energy as an Inverse Problem
Cristina España i Bonet1 and Pilar Ruiz-Lapuente2
Departament d'Astronomia i Meteorologia, Universitat de Barcelona,
Martí i Franquès 1, E-08028 Barcelona, Catalunya, Spain
1
2 [email protected]
[email protected]
Abstract
Inverse Problem
The aim of an inverse problem is to determine the values of a set of parameters appearing in
a theoretical expression from a set of observables. So studying dark energy using this
approach is an inverse problem. For the linear discrete case the solution of an inverse
problem can be solved via the least squares method, but for nonlinear and/or continuos
cases the method must be generalized. The version used here is a Bayesian approach to
this generalization [1].
We consider a flat universe with only two dominant constituents (at present): cold matter and
dark energy. Therefore we characterize the cosmological model by the density of matter, ΩM,
and by the parameter w(z) of the equation of state of the dark energy.
In order to improve the information on dark energy it is not only important
to have a large number of data of a good quality, but also to know where are
these data more profitable and then explode all the statistical methods to
extract the information. We apply here the Inverse Problem Theory to
determine the parameters appearing in the equation of state (EoS) and the
functional form itself. Using this method it is also determined which would
be the best distribution of high redshift data to study the equation of state
of dark energy, i.e., with which distribution it is obtained a best quality
of the inversion. Supernovae magnitudes are used alone and together with
other sources such as radio galaxies and compact radio sources.
Data
The main data used in this work are SNe Ia at high redshift [2] although we also consider
other sources at even higher z such as radio galaxies (RG) [3] and compact radio sources
(CRS) [4]. In order to join all these data it is useful to define the dimensionless coordinate
distance y as [5]:
yi ≡
mi − M
5
zi
10
dz '
,
=∫
3
c(1 + zi ) 0 Ω (1 + z ') + Ω ( z ' )
M
X
σy =
i
(
Continuous case
Just as in the discrete case there is an iterative equation to obtain the EoS.
The difference now is that it can be calculated at every desired redshift:
)
ln 10
yi σ mi + σ M .
5
Using the Inverse Problem Theory to
determine the discrete parameters
appearing in an EoS of the form
i =1
0
Where Cw is the covariance function, g is the kernel of the derivatives and W
is the same vector as before.
z
1+ z
w(z) = w0 + wa
we have obtained an iterative value for
ΩM, w0 and wa [6]:
∂yith
Wi
∑
∂Ω M
i =1
This way we don’t have to make any
hypothesis about the specific form of
the EoS, but the price to pay is the
introduction of a priori information.
N
2
ΩM
w0[ k +1]
th
∂
y
= w00 + σ w20 ∑ Wi i
∂w0
i =1
wa[ k +1]
∂yith
= w + σ ∑ Wi
∂wa
i =1
N
N
0
a
zi
w[ k +1](z) = w0 ( z ) + ∑ Wi ∫ C w ( z , z ' ) g w ( z , z ' )dz '
Discrete Case
Ω M [ k +1] = Ω M 0 + σ
N
2
wa
The table shows the results with the
different sources used and combinations of them. Next to each parameter
there is the mean index I, a very useful
parameter defined to see how the data
restrict the model, being the ideal case
that with I =1. Data of SNe Ia alone are
the best ones to determine the EoS,
but the results can be improved up to a
50% with the inclusion of these other
sources at higher redshift. Current data
slightly favour an EoS near the one of
a cosmological constant but allowing a
positive evolution.
The left panel shows the results with
data coming from de gold set of [2].
Top figure represents the evolution of
the EoS with the 1σ intervals when it
is assumed an a priori value of
w(z)=-1±1 and the density of matter is
fixed to ΩM=0.3. So, a cosmological
constant is compatible with current
SNe Ia data.
The lower panels are used to see the
reliability of the result. The resolving
kernel K(z,z’) informs about how well
determined is the redshift z’ . Low
redshifts are better determined, as
there is a larger number of data, and
it is reflected with a sharper K(z,z’).
The mean index I(z) has the same
interpretation as in the discrete case,
and such low values indicate that the
results can only be trust when the a
priori is totally justified.
Estimating the best distribution of data
Nowadays several experiments are being designed in order to detect new sets of
SNe Ia at high redshift. It is then important to know where should be these data
concentrated to determine with a minimum error the parameters appearing in the
EoS.
We need then the best distribution, i.e., the distribution which gives the best result.
In order to quantify this quality we will ask for two characteristics in the result: it
must have a small uncertainty (precision) and the best value must be near the
“true” one (accuracy). As we are going to simulate gaussian distributions of data,
the “true” value will be the “seed” of the simulation. So, we define the quality factor
as




1
1



.
,
Qw0 = log
Qwa = log
seed
seed
 σ w w0 − w0 
 σ w wa − wa 
 0

 a

Conclusions
The results are shown in the figure on the right. Zones with dark blue represent
distributions with the highest quality of the inversion, whereas the lightest zone are
those with a bad quality, as shown in the colour scale next to the figures.
It has been applied the method of resolution
of inverse problems to the dark energy
equation of state.
The column shows the results for a fiducial model of CC (w0=-1, wa=0) inverted
using these “seed” values as a priori. In general, we observe that all the
distributions determine much better w0 than wa. Furthermore, distributions centred
only at high redshift give very poor results even for wa. When we join together the
qualities in both parameters we see more clearly that there is another poor section
at low redshift with a small width. So, we see the necessity to extend the number of
data at high redshift. This has been already done within the GOODS and HST
Treasury Program [7] for example, and must be continued and extended in order to
be in the best condition to study dark energy. Alternatives to a CC are considered
in the top row where it is shown a SUperGRAvity model (w0=-0.8, wa=0.6) and a
similar one with (w0=-0.8, wa=-0.3).
Nowadays SNe Ia data and this method
determine w0 with a relative error of ~30%
and wa with one of ~100%, depending the
exact value on the number of studied
parameters.
References
[1] A. Tarantola and B. Valette, Rev. Geophys. & Space Phys.,
20(2), 219 (1982).
[2] A.G. Riess et al., Astrophys. J., 607, 665 (2004).
[3] R.A. Daly and S.G. Djorgovski, Astrophys. J., 612 (2004).
[4] L.I. Gurvits et al., Astron. Astrophys., 342, 378 (1999).
[5] R.A. Daly and S.G. Djorgovski, Astrophys. J., 597, 9 (2003).
[6] C. España-Bonet and P. Ruiz-Lapuente, in preparation.
Adding other sources at high z such as RG
and CRS reduces these errors in almost a
50%. As with SNe Ia alone, the high mean
index indicates a very good inversion and a
high reliability on the results.
The inverse method can also be applied to
determine the function w(z) itself. A pure cosmological constant can not be discarded from
these results. In this case the low mean index
demands using a very well motivated priors.
A gaussian distribution of SNe Ia centred at
z0~0.7 with a width of σz~1 would give us the
best determination of w0 and wa.
[7] http://www.stsci.edu/science/goods
Photo courtesy of Nik Szymanek and Ian King for the Isaac Newton Group of Telescopes