c ESO 2010°

Astronomy & Astrophysics manuscript no. text˙final
December 23, 2010

Hubble parameter reconstruction from PCA: minimizing the bias
E. E. O. Ishida1,2 and R. S. de Souza1,2
1

IAG, Universidade de São Paulo, Rua do Matão 1226, Cidade Universitária,
CEP 05508-900, São Paulo, SP, Brazil
2
Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8568, Japan
Released Xxxxx XX
ABSTRACT
Aims. A model-independent reconstruction of the cosmic expansion rate is essential to a robust analysis of cosmologicalobservations. Our goal is to demonstrate that current data are able to provide reasonable constraints on the behavior of the
Hubble parameter with redshift, independently of any cosmological model or underlying gravity theory.
Methods. Using type Ia supernova data, we show that it is possible to analytically calculate the Fisher matrix components ina Hubble parameter analysis without assumptions about the energy content of the Universe. We used a principal component
analysis to reconstruct the Hubble parameter as a linear combination of the Fisher matrix eigenvectors (principal components).
To suppress the bias introduced by the high redshift behavior of the components, we considered the value of the Hubble
parameter at high redshift as a free parameter. We first tested our procedure using a mock sample of type Ia supernova
observations, we then applied it to the real data compiled by the Sloan Digital Sky Survey (SDSS) group.
Results. In the mock sample analysis, we demonstrate that it is possible to drastically suppress the bias introduced by the highredshift behavior of the principal components. Applying our procedure to the real data, we show that it allows us to determine
the behavior of the Hubble parameter with reasonable uncertainty, without introducing any ad-hoc parameterizations. Beyond
that, our reconstruction agrees with completely independent measurements of the Hubble parameter obtained from red-envelope
galaxies.

Key words. cosmology: cosmological parameters, methods: statistical

1. Introduction
At the end of the 20th century, observations of type Ia
supernovae (SNIa) revealed that the Universe expansion
is accelerating (Riess et al. 1998; Perlmutter et al. 1999).
Since these publications, several efforts have been made to
explain these observations (Cunha et al. (2009); Frieman
et al. (2008); Linder (2008); Linder & Huterer (2005);
Samsing & Linder (2010); Freaza et al. (2002); Ishida(2005); Ishida et al. (2008) and references therein). In astandard analysis, dark-energy models are characterized
by a small set of parameters. These are placed into the cosmic expansion rate by means of the Friedman equations,in substitution for the conventional cosmological-constantSend offprint requests to: Emille E. O. Ishida e-mail:emille.ishida@ipmu.jp

term. This approach assumes a specific dependence of thedark-energy equation of state (w) on redshift and provides
some insight into the probable values of the parameters
involved. However, the results remain restricted to thatparticular parametrization. An interesting question to attempt to answer is what can be inferred about the cosmic
expansion rate from observations without any reference toa specific model for the energy content of the Universe?
To perform an independent analysis, we used principal component analysis (PCA). In simple terms, PCA
identifies the directions of data points clustering in thephase space defined by the parameters of a given model.
Consequently, it allows a dimensionality reduction withas minimum an information loss as possible (Tegmark
et al. 1997). The importance of a model-independent reconstruction of the cosmic expansion rate has already been

2

Ishida & de Souza: Hubble parameter reconstruction

investigated in the literature (Huterer & Turner 1999,
2000; Tegmark 2002; Wang & Tegmark 2005; Mignone &Bartelmann 2008). In this context, PCA has been used toreconstruct the dark-energy equation of state (Huterer &Starkman 2003; Crittenden et al. 2009; Simpson & Bridle
2006) and the deceleration parameter (Shapiro & Turner
2006) as a function of redshift. The use of PCA was also
proposed in the interpretation of future experiments results by Albrecht et al. (2009). In the face of growing interest in the application of PCA to cosmology, Kitching& Amara (2009) recall that some care must be taken inchoosing the basic expansion functions and the interpretation assigned to the components.
The main goal of this work is to apply PCA to reconstruct directly the Hubble parameter redshift dependence
without any reference to a specific cosmological model. In
this context, the eigenvectors and eigenvalues of the Fisher
matrix form a new basis in which the Hubble parameteris expanded. For the first time, we show that it is possible to derive analytical expressions for the Fisher matrix
if we focus on the Hubble parameter (H(z)) as a sum of
step functions. The reader should realize throughout thiswork that our procedure is mostly driven by the data, although there is a weak dependence of the components on
our starting choices of parameter values. In other words,
the functional form of each eigenvector is not of primary
importance, we are more interested in how they are linearly combined. This approach allows us to avoid many
interpretation problems pointed out by Kitching & Amara(2009). Our only assumption is that the Universe is spatially homogeneous and isotropic and can be described byFriedmann-Robertson-Walker (FRW) metric.
The paper is organized as follows. In section 2, we
briefly review our knowledge of PCA and demonstrate
how it can be applied to a Hubble parameter analysis using type Ia supernova observations. Section 3 shows theresults obtained with a simulated supernova data set, following the standard procedure for dealing with the linear
combination coefficients. We demonstrate that the qualityof our results derived from the simulated data are greatly
improved if we consider the Hubble parameter value in theupper redshift bound as a free parameter. We apply thesame procedure to real type Ia supernova data compiledby the Sloan Digital Sky Survey team (Kessler et al. 2009).
The results are shown in section 4. Finally, in section 5,
we present our conclusions.

2. Principal component analysis
2.1. The Fisher matrix
The procedure used to find the principal components(PCs) begins with the definition of the Fisher information

matrix (F). Owing to its relation to the covariance matrix(F=C−1 ), it can be shown that the PCs and their associated uncertainties are related to the eigenvectors andeigenvalues of the Fisher matrix, respectively.
We consider that our data set is formed by N independent observations, each one characterized by a Gaussian
probability density function, fi (xi , σi ; β ). In our notation,
xi represents the i − th measurement, σi the uncertainty
associated with it, and β is the vector of parameters of
our theoretical model. In other words, we investigate a
specific quantity, x, which can be written as a function ofβ )). In this context, the likelihood
the parameters βi , (x(βQNfunction is given by L = i=1 fi and the Fisher matrix isdefined as¿ 2Àβ)∂ ln L(βFkl ≡ −.
(1)∂βk ∂βlThe brackets in equation (1) represents the expectation
value.
We can write F=DT Λ D, where the rows of the decorrelation matrix (D) are the eigenvectors (eei ) of F, andΛ is a diagonal matrix whose non-zero elements are theeigenvalues (λi ) of F. Choosing D to be orthogonal, withdet(D)=1, ei forms an orthonormal basis of decorrelated
vectors (or modes). After finding the eigenvectors andeigenvalues of F, we rewrite x as a linear combinationof ei . Our ability to determine each coefficient of this lin−1/2
ear expansion (αi ) is given by σαi = λi
. Following thestandard convention, we enumerate ei from the larger tothe smaller associated eigenvalue.
The main goal of PCA is the dimensionality reductionof our initial parameter space. This arises in the number of PCs we use to rewrite x. The most accurately determined modes (smaller σαi ) correspond to directions ofhigh data clustering in the original parameter space. As aconsequence, they represent a larger part of the variancepresent in the original data set. In the same way, the mostpoorly determined modes correspond to a small portionof the variance in the data, describing features that might
not be important in our particular analysis. In this context, we must determine the number of PCs that will be
used in the reconstruction. Our decision must be balanced
between how much information we are willing to discard
and the amount of uncertainty that will not compromise
our results. The constraint on x reconstructed with M
modes (where M 6 NP C and NP C is the total numberof PCs), is given by a simple error propagation of the uncertainties associated with each PC (Huterer & Starkman
2003)
σP2 CS (z) ≡

N
PC
X
i=1

2
[σαi ei (z)] ≈

M
X
i=1

2
[σαi ei (z)] .

(2)

Ishida & de Souza: Hubble parameter reconstruction

From this expression, it is clear that adding one more PC
adds also its associated uncertainty. At this point, we notethat to calculate Fkl we must choose numerical values foreach parameter βi . This corresponds to specifying a basemodel as our starting point. As a consequence, the results provided by PCA are interpreted as deviations fromthis initial model. The uncertainty derived from fitting thedata to this base model should also be added in quadrature to equation (2), to compute the total uncertainty inthe final reconstruction.
The question of how many PCs should be used in the
final reconstruction is far from simple, and there is no
standard quantitative procedure to determine it. In manycases, the decision depends on the particular data set analyzed and our expectation towards them (for a complete
review see Jollife (2002), chapter 6). One practical way offacing the problem is to consider how many components
are inconsistent with zero in a particular reconstruction. In
most cases, the coefficients αi tend to decrease in modulus
for higher i, at the same time as the uncertainties associated with them increases. In this context, we can choose
the final reconstruction as the one whose coefficients are
all inconsistent with zero.
The determination of one final reconstruction is beyond the scope of this work. However, to provide an ideaof how much of the initial variance is included in our plots,
we shall order them following their cumulative percentage
of total variance.
The total variance present in the data is represented
well by the sum of all λi , and a reconstruction with thefirst M PCs encloses a percentage of this value (tM ), givenbyPM
λitM = 100 PNi=1.
(3)PC
j=1 λjAs a consequence, the question of how many PCs turns
into a matter of what percentage of total variance we arewilling to enclose.

2.2. Investigating the Hubble parameter from SNIa
observations
From now on, we consider the distance modulus, µ, provided by type Ia supernova observations as our observed
quantity (xi = µi ). In a very simple approach, if we
consider a flat, homogeneous and isotropic Universe, described by the FRW metric, the distance modulus relates
to cosmology according toµ(z) = 5 log10 [dL (z)] + µ0 ,Z z
du,dL (z) ≡ (1 + z)
H(u)
0

(4)
(5)

3

where µ0 is called intercept, dL (z) is the luminosity distance, and H(z) is the Hubble parameter. We use H0 = 72
km s−1 Mpc−1 as the current value of the Hubble parameter (Komatsu et al. 2009).
To make H(z) as general as possible, we write it as asum of step functions
H(z; β ) =

N
bin
X

βi ci (z),

(6)

i=1

where
½
ci (z) =10

if(i − 1)∆z < z ≤ i∆z,otherwise

βi are constants, β is the vector formed by all βi , Nbin is
the number of redshift bins and ∆z is the width of eachbin. This approach was proposed by Shapiro & Turner(2006) in the context of deceleration parameter analysis.
Although, when it is used for the Hubble parameter, theFisher matrix calculations are simplified and we still get
pretty general results. Given the definition above, βi are
now the parameters of our theory. Physically, they represent the value of the Hubble parameter in each redshiftbin. We can obviously express any functional form using
this prescription, with higher resolution for a larger number of bins.
In this context, we are able to obtain analytical expressions for the luminosity distance


L(z)
X 1
z
−
L(z)∆z , (7)dL (z, β ) = (1 + z) ∆z+ββL(z)+1i=0 i+1
and its derivatives,L(z)X δi+1,k∂dL (z, β )+= −(1 + z) ∆z
2
∂βk
βi+1
i=0
#
δL(z)+1,k (z − L(z)∆z)+,2βL(z)+1(8)

where L(z) corresponds to the integer part of z/∆z. Fromequations (1), (4), and (5), we can calculate the Fisher
matrix components as
"N
X
25
1
Fkl =
×
2
(ln 10) i=1 (σdatai dL (zi ; β ))2¸∂dL (zi , β ) ∂dL (zi , β )
×
.
(9)
∂βk
∂βl
The Hubble parameter may now be reconstructed as
the sum of Hbase and a linear combination of the new

4Ishida & de Souza: Hubble parameter reconstruction

uncorrelated variables represented by the eigenvectors of
the Fisher matrix. Mathematically,

Mock sample
0.5

αiei (z),

(10)

i=1

ei

Hrec (z; α ) = Hbase (z) +

M
X

1.0

where αi are constants and α is the vector formed by all
the αi . Using equation (10) in equations (4) and (5), we
can write the reconstructed distance modulus. The data
set is then used to find values for the parameters αi thatminimize the expression(µi − µrec (zi ; α))2
.
2σi2
i=1

(11)

This minimization procedure will also generate an uncertainty in the value of parameters αi (σαmin), which shouldibe taken into account in the final reconstruction error budget.

3. Application
3.1. Mock sample
To test the expressions and procedures presented before,
we used a simulated type Ia supernova data set. We consider 34 redshift bins of ∆z = 0.05 (0 ≤ z ≤ 1.7), each one
containing 50 supernovae. We tested configurations witha larger number of bins, but the results are consistent for
any configuration with more than ∼ 25 redshift bins. The
uncertainty in the i − th bin was calculated according to
the prescription proposed in Kim et al. (2004)sµ¶2∆z(i − 0.5)
0.152
σi =
+ 0.02
.
(12)
50
1.7
We performed 1000 simulations of a flat Universe containing a cosmological constant and dark matter, with matter
density parameter Ωm = 0.27 as our fiducial model.
Our main goal in using this simulation is to obtainan idea of how the procedure proposed here behaves inan almost ideal scenario. It represents an simplified version of future data, as for the Joint Dark Energy Mission(JDEM)1 , but it is enough to allow us to check the consistency of our procedure.
Using the equations shown in the previous section,
we calculated the Fisher matrix components. We foundthat the modes are weakly sensitive to the choice of initial base model (values for the parameters βi , hereafter
Hbase ). However, if we use a specific cosmological model
to attribute values to the parameters βi (ΛCDM, for example), all the results derived from this initial choice can1

http://jdem.lbl.gov/

1 st PC
2 nd PC

-0.5
-1.0
1.0

0.5

3 rd PC
4 th PC
1.0

1.5

z

0.5
ei

α) =
χ21 (α

N
X

0.0

0.0

5 th PC
6 th PC

-0.5
-1.0

0.5

7 th PC
8 th PC
1.0

1.5

z

Fig. 1. PCs obtained from our mock sample as a functionof redshift. All PCs are shown according to the convention
that ei (z = 0) > 0. Top: First (red-full), second (bluedashed), third (green-dotdashed), and fourth (brown-dotted)
PC. Bottom: Fifth (purple-full), sixth (cyan-dashed), seventh(magenta-dotdashed), and eighth (dark blue-dotted) PC.

only be analyzed in the face of that model. As our goal
is to make a model-independent analysis, the best choice
is to calculate the PCs based on a model where there isno evolution with redshift (βi = cte > 0, ∀i). The PCs
will then denote deviations from a constant behavior, regardless of the value attributed to βi . A constant Hubble
parameter is obviously an extremely unrealistic model, although, it does allow us to have a better idea of whichcharacteristics of our results are extracted from the data
and which are only a consequence of our initial choices.
We do not currently have well constrained information about the evolution of the Hubble parameter with
redshift, but we do have independent measurements of its
value today, H0 (e.g. Komatsu et al. (2009)). Hence, we
present our results in units of H0 and use a base modelin which βi = 1, ∀i. The resulting eigenvectors with larger
eigenvalues are shown in Fig. 1.
The comparison between the PCs obtained from using
Hbase = ΛCDM and Hbase = cte is shown in figure (2).
From this plot, we can see that the difference exist, but
the overall shape of the PCs are not very sensitive to the
choice of Hbase .
To clear illustrate the standard-procedure results of
PCA reconstruction in the specific case studied here, weshow in Fig. 3 reconstructions using one to six PCs with

Ishida & de Souza: Hubble parameter reconstruction
1.0

1 st PC

0.0
-0.5
-1.0
1.0

Βi =H LCDM

0.5

1.0

-1.0
1.0

1.5
3 rd PC

z

0.0

-0.5

-1.0
1.0

-1.0
1.0
1.5
5 th PC

0.5

1.0
z

ei

-0.5

1.5
4 th PC

3.2. Minimizing the bias in the reconstruction

0.5

1.0

1.5
6 th PC

1.0

1.5

z

0.5

0.0

1.0

0.0
-0.5

-1.0

0.5

-1.0
1.5

1.0

0.5

z

z

3.5 t M =2 =73.9 %
2 PC
3.

2.5

2.5
H0

3.5 t M =1 =54.4 %
1 PC
3.
H rec

H0

H rec

Fig. 2. Principal components obtained using a constant basemodel (full-red line) and using ΛCDM as a base model (dottedbrown line).

2.
1.5
1.

1.
0.5

Mock sample

0.5
3.5 t M =3 =83.1 %
3.

3 PC

1.

1.5

z

H0

2.
1.5

1.5

1.

1.5

1.

1.5

2.

1.

1.

0.5

0.5

3.

1.
z

1.5

0.5
3.5 t M =5 =91.2 %

5 PC

1.

1.5

z

0.5
3.5 t M =6 =93.3 %
3.

6 PC

z

2.

H0

2.5
H rec

2.5
H0

0.5
3.5 t M =4 =88.1 %
4 PC
3.
2.5

H rec

H0

H rec

2.5

H rec

2.
1.5

0.5

1.5

2.
1.5

1.

1.

0.5

0.5
0.5

1.
z

1.5

corresponding values of tM . From this plot, it is clear that
our attempt to reconstruct H(z) using a few PCs does notprovide the expected results. We have two main problems
here: the reconstruction merely oscillates around the fiducial model (blue-dashed line) and we can clearly see that
there is a bias dominating the high-redshift behavior. We
address both problems in the next section.

z

0.0

-0.5

0.5

0.5

0.5
ei

0.5
ei

0.0
-0.5

Βi =cte >0ei

2 nd PC

0.5
ei

ei

0.5

1.0

5

0.5
z

Fig. 3. Reconstruction of the Hubble parameter using equations (10) and (11). Hubble parameter reconstructed with one(top-left) to six PCs (bottom-right), in units of H0 . The black(solid) line represents our best-fit reconstruction and the red(dotted) curves shows 2σ confidence levels. The blue (dashed)line corresponds to the the behavior of the Hubble parameterin our fiducial model.

From Fig. 1, we realize that all the first eight PCs go tozero at high-redshift, which means that at these redshifts
our data provides little information 2 . Consequently, nomatter how many PCs we use or which values we attributeto the parameters αi , the reconstructed function will always be biased in the direction of our previously chosenbase model for high z (in this work, “high z” corresponds
to the upper redshift bound of our data set. In our mocksample, zmax = 1.7).
At this point, we must pay attention to the crucialrole played by Hbase in the standard procedure describedso far. Although the PCs depend weakly on our choiceof Hbase , the final reconstruction is extremely sensitive tothe choice. Figure (4) shows how different Hbase lead tocompletely different final reconstructions.
Searching the literature, we found two different approaches to dealing with this problem. We could ignore
the reconstruction in the region of high redshift (Huterer& Starkman 2003; Shapiro & Turner 2006) or add a physically motivated model for Hbase in equation (10), which
would provide us with the value we expect to measurein the upper redshift bound (Tang et al. 2008). We consider that the first alternative does not represent a goodsolution. The problem is not only the bias at high z, butalso the weird behavior present in the reconstruction asa whole. Beyond that, our intention is not only to improve the fit quality, but also to make it independent of
our initial choice of Hbase . The second alternative would
produce results in good agreement with the fiducial model(corresponding to the dotted-brown reconstruction in figure (4)), in a simulated situation. Defining a physicallymotivated Hbase would only, however, introduce anotherbias. As in reality we do not have access to the “true”
value of H(z), this would require us to make a hypothesis
about the energy content and dark energy model, which
we are trying to avoid.
In this context, we believe that it is reasonable to consider the behavior of H(z) at high z as a free parameter.
2This kind of behavior is also present in PCs from the darkenergy equation of state (Huterer & Starkman 2003) and deceleration parameter (Shapiro & Turner 2006).

Ishida & de Souza: Hubble parameter reconstruction

1.
Hbase =0.5
Hbase =1.0
3 PC

Hbase =4.0
Hbase =HL CDM

0.5

1.

0.5

1.

2.5

H0

2.
Hrec

H0

1.5

z

Hbase = cte Î R+
Hbase =HL CDM
HL CDM
3 PC - hz0.5
free
max

1.
0.5

1.

1.5
2.5

z

2.

2.

H0

H0

Hrec

1.

1.

0.

0.5

0.5

6 PC

0.5

1.

2.5

2.

H0

H0

2.

5 PC - hz0.5
free
max

1.

1.5
2.5

z

2.

3.
Hrec

3.

1.5

z

4.

1.

1.

1.

0.

0.

0.5

0.5

1.

1.5

0.5

1.

z

1.5

Fig. 4. Reconstructions using 1 (top-left) to 6 (right-bottom)
PCs for different Hbase , obtained from our mock sample. The
black (full) line represents the fiducial model ΛCDM, the brown(dotted) line corresponds to the final reconstructions in thecase Hbase (z) = ΛCDM , the red (full) line corresponds to thecase Hbase (z) = 0.5, the blue (dashed) line shows the reconstruction considering Hbase = 1.0, and the green (dot-dashed)line is the reconstruction for Hbase (z) = 4.0.

This means adding another parameter (hzmax ) to equation(10), which becomes
Hrec (z) = hzmax + Hbase (z) +NXαiei (z).

(13)

i=1

As a consequence, the new χ2 will be given byχ2 (hzmax , α ) =NX(µi − µrec (zi ; hz
i=1

2σi2

max

, α ))2,(14)

and the uncertainty associated with the determination of
hzmax (σhzmax ) is added in quadrature to the right handside of equation (2), leading to2σrec(z) ≈ σh2 zmax +

0.5

1.
z

z

M h
X
¡
¢2 i
2
(σαi ei (z)) + σαmin
.
i

(15)

i=1

The effect of this choice is shown in figure (5). We cansee that, no matter which Hbase we use, if hzmax is consider to be a free parameter, the reconstruction at high z
is driven by the data. As a consequence, we obtain good
agreement for the reconstruction using a constant as well
as a ΛCDM model for Hbase . Even though the PCs are not

6 PC - hz0.5
free
max

1.

1.5

1.

1.5

z

1.5

1.

0.5

1.5

2.

1.5

H0

1.5

Hrec

1.
z

1.
z

1.5

1.

0.5

4 PC - hz0.5
free
max

2.

1.5

0.

Hrec

H0

Hrec

4 PC

2 PC - hzmax free

1.5

1.

5 PC

H0

0.5

3.

2.

4.

Hrec

1.

0.

4.

3.

2.5

1.5

1.

1.5

z

4.

1 PC - hzmax free

2.

2.

Hrec

0.

2.5

Hrec

3.

2.

H0

3.
Hrec

H0

Hrec

2 PC

4.

H0

HL CDMH0

1 PC

4.

Hrec

6

1.5

0.5
z

Fig. 5. Reconstructions using one (top-left) to six (rightbottom) PCs for different choices of Hbase and considering
hf id as a free parameter, for our mock sample. The black(full) line represents the fiducial model ΛCDM, the blue
(dashed) line corresponds to the final reconstructions in thecase Hbase (z) = HΛCDM and the red (full) line corresponds tothe case Hbase (z) = cte > 0.

identical for non-evolving and ΛCDM models (figure (2)),
this agreement is a direct consequence of the best-fit values of the set of parameters {αi , hzmax } always arranging
themselves to more accurately describe the information in
the data.
We present in Fig. 6 the results of the reconstruction
using 1 to 5, and 10 PCs, and corresponding values of tM .
Comparing these results with those in Fig. 3, we can see
a huge improvement in the agreement between the reconstructed function and the fiducial model. The reconstruction with 10 PCs encloses the fiducial model within 2σconfidence levels in the whole redshift range covered by
the data. Considering that initially we had 34 parameters
βi , this represents a reduction of ≈ 70% in the parameter
space dimensionality.
We have so far demonstrated that PCA is an effective
method for determining the Hubble parameter behavior
with redshift. It provides a considerable reduction in theinitial parameter space dimensionality, without introducing any hypothesis about the energy content, cosmologicalmodel, or underlying gravity theory. The reconstruction
relies on the assumption of a homogeneous and isotropic
Universe, described by a FRW metric. The simulated data
set used above is composed of independent data points,

Ishida & de Souza: Hubble parameter reconstruction

2.5

t M '=1 =54.3 %

2.5

1 PC
H0

1.5
1.

Mock samplet M '=3 =83.10.5%

1.

2.5

H0

H rec

H0

H rec

t M '=4 =88.10.5%
4 PC

1.

1.5

1.

1.5

1.

1.5

z

2.

1.5

1.5

1.

1.

0.5

0.5

2.5

t M '=5 =91.2 0.5
%

1.

1.5
2.5

z

5 PC

t M '=10 =97.10.5
%
z

10 PC

1.5

H0

2.
H rec

2.
H0

0.5

1.5

z

3 PC

2.

H rec

1.5
1.

0.5

2.5

2 PC

2.
H rec

H0

H rec

2.

t M '=2 =73.9 %

1.5

1.

1.

0.5

0.5
0.5

1.

1.5

0.5

z

z

Fig. 6. Reconstruction of the Hubble parameter using equations (13) and (14). Hubble parameter reconstructed with 1(top-left) to 5 (bottom-left) and 10 (bottom-right) PCs, inunits of H0 . The color code is the same used in figure (3).

each one associated with a Gaussian probability density
function, in a flat ΛCDM Universe.
In what follows, we apply this procedure to a real (andconsequently less well behaved) data set. Our goal is tosee whether, in a realistic scenario, the effectiveness of theprocedure remains.

4. Results from current SNIa data
We applied this procedure to real supernova Ia data compiled by the Sloan Digital Sky Survey (SDSS) SN group,
hereafter real sample. This sample include measurementsfrom the SDSS (Kessler et al. 2009), the ESSENCE survey (Miknaitis et al. 2007; Wood-Vasey et al. 2007), theSupernova Legacy Survey (SNLS)(Astier et al. 2006), theHubble Space Telescope (HST) (Garnavich et al. 1998;
Knop et al. 2003; Riess et al. 2004, 2007), and a compilation of nearby SN Ia measurements (Jha et al. 2007).
The first eight PCs found from the real data set are shownin Fig. 7. We used 28 redshift bins of width ∆z = 0.05 andthe 287 data points from the aforementioned data set withz ≤ 1.4. The resulting reconstructions with one to six PCs
are shown in Fig. 8. The blue-dashed line corresponds tothe behavior of the Hubble parameter in a flat Universe
containing dark energy with an equation-of-state parameter for dark energy wdark = −0.76 and matter density
parameter Ωm = 0.30. This corresponds to the best-fit,

7

flat cosmology found by Kessler et al. (2009) in the context of the Multicolor Light Curve Shape (MLCS2k2, (Jhaet al. 2007)), hereafter fXCDM. It is shown here exclusively for comparison reasons, this model was not used inour calculations.
Comparing Figs. 6 and 8, we realize that the confidence
intervals are much larger in Fig. 8, as expected, becauseof the observational uncertainties that are present only inthe real sample. Beyond that, the contours corresponding
to 2σ confidence levels do not evolve in Fig. 8 as they
do in Fig. 6. This is a direct consequence of our choiceof introducing hzmax as a free parameter. In the simulated case, σhzmax is much smaller than the uncertainty
associated with the parameters αi . As a consequence, theevolution of the confidence levels is dominated by the uncertainties associated with the PCs. In the real case, theopposite situation occurs. For the six cases presented inFig. 8, σhzmax À σα , making the final uncertainty almost
independent of how many PCs are used in the reconstruction. This behavior is a consequence of the low number
and quality of data points at high redshift. However, it isalso related to a non-null correlation between the uncertainties in determining the parameters αi and hzmax . To
explore this method in the best case scenario, we need toensure not only that we have high number and quality ofdata points at high redshift, but also that the PCs are aswell determined as possible.
In comparing Figs. 6 and 8, the reader should be awarethat the blue dashed line means different things in eachfigure. Figure 6 is a simulation, and in such a case theblue dashed line corresponds to the fiducial model used
to generate our mock sample. Fig. 8 was created using
real data, in this case the blue dashed line represents thebest-fit flat ΛCDM model, as reported by Kessler et al.
(2009).
We can also see, in Fig. 8, that the reconstruction becomes more irregular when more than four PCs are used.
If we take the blue dashed line as a good representationof the “real” cosmological model, we could say that four
PCs are enough to enclose the desired behavior within
2σ confidence level. For the sake of completeness, we plot
reconstructions only up to six PCs.
To compare our results with other model-independent
determinations of H(z), we plot in the top panel of Fig. 9
the reconstruction with four PCs, superimposed on measurements of H(z) derived from red-envelope galaxies observations by Stern et al. (2010). The error bars associatedwith these data points are still pretty large, but they already provide important insights into the behavior of the
Hubble parameter in the redshift range covered by thereal data sample. We can see that the reconstruction encloses the fXCDM model, as well as agreeing with the red-Ishida & de Souza: Hubble parameter reconstruction

5. Conclusions
We have presented an alternative procedure for extracting
cosmological parameters from type Ia supernova data. Our
analysis is concentrated in the Hubble parameter, although
we emphasize that the same procedure can be applied toother quantities of interest. Our goal has been to be as
general as possible, so we have tried to avoid parametricforms or specific cosmological models by using PCA.
Writing H(z) according to equation (6) and considering type Ia supernova observations, we have shown that it
is possible to obtain analytical expressions for the Fisher
matrix. We used a mock sample formed by 34 redshiftbins of width ∆z = 0.05, with errors calculated following
the prescription proposed by Kim et al. (2004). This mocksample represents a simplification of future data sets, suchas the JDEM, and is not a realistic representation of current data. Our goal in using it was to check the consistentency of our procedure.
Our first attempt in reconstructing the Hubble parameter as a linear combination of the eigenvectors of F was
unsuccessful. In trying to fit high-redshift data with PCs
that go asymptotically to zero, the most oscillatory modes
propagate their behavior to the reconstructed H(z) in thewhole redshift range. As a consequence, the final result
barely resembles our fiducial model.

Real Sample
0.5
ei

envelope galaxy measurements. To compare our resultswith the predictions of a standard model-dependent procedure, we show in the bottom panel of Fig. 9, the 2σ confidence levels derived from the error propagation of statistical uncertainties in Ωm and w reported by Kessler et al.
(2009) assuming a fXCDM model and using MLCS2k2.
We again find good agreement between the two results.
In Figs. 8 and 9, we point out that the blue-dashed line
does not represent the behavior we are trying to achieve,
but is merely a representation of a model we are used todealing with. The purpose of showing it here is to provide
an idea of how far our results are from others presented
in the literature, although, in our particular analysis, noassumptions about the energy content of the Universe isnecessary.
The determination of what kind of physical and/or
systematic effect generates patterns seen in the two lower
panels of Fig. 8 is beyond the scope of this work. In our interpretation, these results confirm that fXCDM provides
a good first-order approximation of the real behavior ofH(z) within the current observational errors and assumptions underlying our procedure. However, a more realistic
simulation and detailed study of systematic errors are necessary in order to fully understand second-order effects.

0.

-0.5

1st PC
2nd PC0.2

0.4

3rd PC
4th PC

0.6

0.8

1.

1.2

z
0.5
ei

8

0.

-0.5

5th PC
6th PC0.2

0.4

7th PC
8th PC

0.6

0.8

1.

1.2

z

Fig. 7. Principal components obtained from real data sample. All PCs are plotted following the convention ei (z =0) > 0. Top: First (red-full), second (blue-dashed), third(green-dotdashed), and fourth (brown-dotted) PCs. Bottom:
Fifth (purple-full), sixth (cyan-dashed), seventh (magentadotdashed), and eight (darker blue-dotted) PCs.

To suppress the influence of the high-redshift behavior
present in all PCs of interest, we considered the value ofthe Hubble parameter at high redshift as an extra free parameter in our analysis. This simple modification provided
reliable results when used with simulated and real supernova data. Beyond that, our results are corroborated withmeasurements of red-envelope galaxies from Stern et al.
(2010).
As a final remark, we emphasize that PCA provides
a viable way of avoiding phenomenological parameterizations. It represents one of the few statistical methods thatallow us to obtain the behavior of a chosen quantity directly from the data. It has its own assumptions, such as
Gaussianity, independence of data points and in the specific case analyzed here, cosmologies that obey a FRW
metric. In the final reconstruction phase, it also exhibits
a bias in the upper redshift bound. On the other hand,the procedure proposed here can drastically suppress theinfluence of this bias. Beyond that, we show that in thecontext of this work, the Fisher matrix can be analyticallyobtained. This avoids all uncertainties related to numerical derivations of step functions and might be a goodalternative to standard statistical analyses applied to cosmological data.

Ishida & de Souza: Hubble parameter reconstruction

Real sample0.5
0.75

1.

2.

1.25
2.5

1.5

1.
0.5
0.75

1.

z

5 PC

2.
1.5

0.75

1.

1.5
1.

1.25

z

4PC

0.5

1.5

0.5
t M '=50.25=89.3 % 0.5

t M '=40.25=86.7 % 0.5

2.

1.

2.5

Hrec H0

0.5

z

2.

1.5
1.

3PC

2.5

2PC

2.

1.

0.25
t M '=3=83.1 %0.5

t M '=2 =77.2 %

1.25
2.5

t M '=60.25=90.9 % 0.5
6PC

0.75

1.

1.25

z

2.
1.5

1.

1.

0.5

0.5

0.25

0.5

0.75
z

1.

1.25

0.25

0.5

0.75
z

1.

1.25

2.5
2.
Hrec H0

H rec H 0

1.5

H rec H 0

H rec H 0

2.5

1PC

2.

2.5

H rec H 0

t M '=1 =63.8 %H rec H 0

H rec H 0

2.5

9

1.5
1.
0.5

0.25

0.5

0.75
z

1.

1.25

0.25

0.5

0.75

1.

1.25

z

Fig. 8. Reconstructions obtained from linear combinations of
the PCs shown in figure 7. Panels run from 1 (top-left) to 6
(bottom-right) PCs. The blue dashed line represents H(z) in aflat XCDM cosmological model with w = −0.76 and Ωm = 0.30
(best-fit results reported by Kessler et al. (2009)). Black curve
corresponds to the best-fit reconstruction and red lines are 2σ
confidence levels.

Acknowledgements. E.E.O.I. thanks the Brazilian agency
CAPES (1313-10-0) for financial support. R.S.S. thanks
the Brazilian agencies FAPESP (2009/05176-4) and CNPq
(200297/2010-4) for financial support. We also thank theanonymous referee for fruitful comments and suggestions. This
work was supported by World Premier International ResearchCenter Initiative (WPI Initiative), MEXT, Japan.

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