Inverse Problem

Continuous case

Abstract

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.

Photo courtesy of Nik Szymanek and Ian King for the Isaac Newton Group of Telescopes

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

Estimating the best distribution of data

Using the Inverse Problem Theory to

determine the discrete parameters

appearing in an EoS of the form

we have obtained an iterative value for

ΩM, w0 and wa [6]:

The table shows the results with the

different sources used and combina-

tions 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.

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:

Where Cw is the covariance function, g is the kernel of the derivatives and W

is the same vector as before.

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.

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.

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.

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]:

Discrete Case

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

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.

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).

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.

[7] http://www.stsci.edu/science/goods

Conclusions

It has been applied the method of resolution

of inverse problems to the dark energy

equation of state.

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.

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 cos-

mological 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.

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1 cespana@am.ub.es 2 pilar@am.ub.es