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Revista Mexicana de Astronomıa y Astrofısica, 52, 3–10 (2016)

A CROSS-CHECK FOR H0 FROM LYMAN-α FOREST AND BARYON

ACOUSTIC OSCILLATIONS

V. C. Busti,1,3 R. N. Guimaraes,2,4 and J. A. S. Lima3

Received September 1 2014; accepted October 13 2015

RESUMEN

Se propone un nuevo metodo para inferir el valor de la constante de Hubble,H0, mediante el flujo medio observado de cuasares de alto corrimiento al rojo y lasoscilaciones acusticas de los bariones (BAOs). Se adopta un modelo semianalıticopara la distribucion de densidad volumetrica independiente de la cosmologıa quenos permite obtener lımites para los parametros cosmologicos una vez que se cuentacon una aceptable estimacion de los parametros del medio intergalactico (IGM).Basado en dos muestras diferentes del bosque de Lyman-α y en la medicion delas BAOs, nuestro analisis restringe (h,Ωm) a los intervalos 0.19 ≤ Ωm ≤ 0.23 y0.53 ≤ h ≤ 0.82. Si bien estos lımites son menos estrictos que los que se obtienenpor otros metodos, senalamos que con muestras mas numerosas y con un mejorconocimiento del IGM nuestro metodo podrıa aportar resultados complemetariospara la determinacion de la constante de Hubble, independientemente de la escalacosmica de distancias.

ABSTRACT

A new method is proposed to infer the Hubble constant H0 through the ob-served mean transmitted flux from high-redshift quasars and the baryon acousticoscillations (BAOs). A semi-analytical model for the cosmological-independent vol-ume density distribution function was adopted; it allowed us to obtain constraintson the cosmological parameters once a moderate knowledge of the Inter GalacticMedium (IGM) parameters is assumed. Our analysis, based on two different sam-ples of Lyman-α forest and the BAO measurement, restricts (h,Ωm) to the intervals0.19 ≤ Ωm ≤ 0.23 and 0.53 ≤ h ≤ 0.82 (1σ). Although the constraints are weakercompared with other estimates, we point out that, with a bigger sample and abetter knowledge of the IGM, this method could provide complementary results tomeasure the Hubble constant independently of the cosmic distance ladder.

Key Words: cosmological parameters — distance scale — intergalactic medium

1. INTRODUCTION

Measurements of the Hubble constant H0 are cru-cial for the establishment of a cosmic concordancemodel in cosmology. This constant plays a role inmost of cosmic calculations, such as those about theage of the Universe, its size and energy density, pri-

1Astrophysics, Cosmology & Gravity Centre (ACGC), andDepartment of Mathematics & Applied Mathematics, Univer-sity of Cape Town, South Africa.

2CAPES Foundation, Ministry of Education of Brazil,BrasıliaDF, Brazil.

3Instituto de Astronomia, Geofısica e Ciencias At-mosfericas (IAG-USP), Universidade de Sao Paulo, Sao Paulo,Brazil.

4Biomedical Informatics Group (IBIME), ITACA Insti-tute, Technical University of Valencia, Valencia, Spain.

mordial nucleosynthesis, the physical distances be-tween objects, etc. Its importance will even in-crease in the next decade, since new missions andobservational projects are currently being designedto provide accurate measurements of the Hubble con-stant, e.g., improved parallax calibration for GalacticCepheids with Spitzer and GAIA, and 1% precisionof H0 with LISA based on the gravitational radiationfrom inspiraling massive black holes, which wouldalso increase the precision of the other cosmologicalparameters.

There are several ways to determine H0 by us-ing low and high-redshift sources. The most com-mon methods are based on the period-luminosity re-

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4 BUSTI, GUIMARAES, & LIMA

lation of cepheids, the tip of the red giant branch(TRGB) and the type Ia supernovae (SNe Ia) ob-served in the local Universe (Freedman & Madore2010). However, some alternative procedures, suchas the Sunyaev–Zel’dovich effect combined with X-ray emission from clusters, which allow a measure-ment of H0 from high-redshift objects, have also beendiscussed (Cunha et al. 2007; Holanda et al. 2012).

Actually, independent estimates are needed inorder to obtain a reliable value for H0 that is notplagued by systematic errors arising from astrophys-ical environments, as well as from calibrations re-lated to the cosmic distance ladder. In this regard,the importance and cosmological interest of differentestimates of H0 that are independent of the distanceladder has also been discussed by many authors (seeJackson 2007, and references therein for a recent re-view).

Recently, a tension was reported betweenthe value obtained by (Riess et al. 2011) ofH0 = 73.8 ± 2.4 km s−1 Mpc−1 (1σ), based onmeasurements of cepheids and SNe Ia (or al-ternatively, that of Freedman et al. 2012, whoobtained H0 = 74.3 ± 2.1 km s−1 Mpc−1 (1σ),and the value released by Planck ofH0 = 67.3 ± 1.2 km s−1 Mpc−1 (1σ), Ade et al.2014), obtained from measurements of theanisotropies of the cosmic microwave background(CMB). The former is based on local measurements,while the latter assumes a cosmological model toinfer H0. There are several possible explanations ofthis discrepancy, such as the following: we live in aHubble bubble and the local value of H0 is subjectto cosmic variance (Marra et al. 2013); or there is aproblem in the calibration of the cepheids and thetrue value of H0 is lower, as given by (Sandage etal. 2006); or with the TRGB and SNe Ia given by(Tammann & Reindl 2013); or an extension beyondthe standard model of cosmology is needed.

In fact, it is the last option which must be con-sidered, since the emergence of the ΛCDM model asthe concordance model in 1998, based on the SNeIa observations (Riess et al. 1998; Perlmutter et al.1999), brought with it the cosmological constant andthe coincidence problems. As a complete and con-vincing solution to these problems is still unknown,it is very important to test all the hypotheses of themodel. In addition, it is important to verify thatdifferent analyses give the same results, since severalkinds of data covering different cosmic epochs areavailable.

In this regard, a possible alternative procedure tomeasure H0 is to consider the Lyman-α (Lyα ) for-est data and the acoustic peak caused by the baryonsacoustic oscillations. The Lyα forest probes the lowdensity intergalactic medium (IGM) over a uniquerange of redshifts and environments, as seen in thespectra of high-redshift quasars. The observed meantransmitted flux depends on the local optical depth,which in turn depends on the expansion rate of theUniverse H(z) and some properties characterizingthe intervening medium along our line of sight to anyquasar. In principle, the values of some cosmologicalparameters, including the Hubble constant, could beconstrained if we had a moderate knowledge of theIGM properties, such as the hydrogen photoioniza-tion rate and the mean temperature, and combinedthis information with the BAO measurement.

In the last decade, several studies have been per-formed using Lyα forests as cosmological tools (Mc-Donald et al. 2000; Weinberg et al. 2003; Wyitheet al. 2008; Viel et al. 2009). Some works use theflux power spectrum of the Lyα forest to infer thecosmological parameters in two ways: inverting theflux power spectrum to estimate the underlying darkmatter power spectrum (Croft et al. 1998, 1999; Hui1999; Nusser & Haehnelt 1999), or using the powerspectrum directly (McDonald et al. 2000; Mandel-baum et al. 2003; McDonald et al. 2005). More re-cently, new possibilities for using Lyα forests for cos-mological purposes were created by the detection ofbaryon acoustic oscillations in the correlation func-tion of the transmitted flux fraction (Busca et al.2013; Slosar et al. 2013).

A simpler approach is to use a semi-analyticalmodel to describe the IGM as proposed by (Miralda-Escude et al. 2000), who derived a fitting formula forthe distribution function of volume density in agree-ment with simulations that yielded similar results fordifferent numerical methods and different cosmolo-gies (Rauch et al. 1997). This allowed to estimatea theoretical mean transmitted flux and to compareit with observational data in order to constrain thecosmological parameters.

Assuming a flat ΛCDM cosmology, we appliedthe above described procedure for deriving cross-check values for the Hubble constant H0 and thematter density parameter Ωm. Based on two in-dependent samples of Lyα forest data compiled by(Bergeron et al. 2004) and (Guimaraes et al. 2007),respectively, we performed two distinct statisticalanalyses. First, we considered only the Lyα forestdata set, while the second approach involved a jointanalysis combining the Lyα forest data with the

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CONSTRAINING H0 FROM LYMAN-α AND BAOS 5

measurements of the baryon acoustic peak from theWiggleZ survey (Blake et al. 2012). As we shall see,the degeneracy in the free parameters defining the(h,Ωm) plane is naturally broken when the Lyα dataare combined with standard ruler datasets as givenby the baryon acoustic oscillations, thereby provid-ing a new independent method to estimate the Hub-ble constant.

2. THE FOREST SAMPLES

As remarked earlier, in the present analysis weconsidered two different samples. The first sampleconsisted of 18 high-resolution high signal-to-noiseratio (S/N) spectra (2.2 ≤ zem ≤ 3.3). The spec-tra were obtained with the ultraviolet and VisibleEchelle Spectrograph (UVES) mounted on the ESOKUEYEN 8.2 m telescope at the Paranal observa-tory for the ESO-VLT Large Programme (LP) ’Cos-mological evolution of the Inter Galactic Medium’(Bergeron et al. 2004). The spectra were taken fromthe European Southern Observatory archive and arepublicly available. This program was devised forgathering a homogeneous sample of echelle spectraof 18 QSOs, with uniform spectral coverage, resolu-tion and signal-to-noise ratio suitable for studyingthe Lyα forest in the redshift range 1.8 - 3.1. Thespectra had a signal-to-noise ratio of 40 to 80 perpixel and a spectral resolution of λ/∆λ ∼ 45000 inthe Lyα forest region. The details of the procedureused for data reduction can be found in (Chand etal. 2004) and (Aracil et al. 2004). All possible metallines and Lyman systems were flagged and removedfrom the Lyman forest in this sample. We confirmedthat the mean transmitted flux values obtained wereconsistent with previous works that have used thesame data- set (Chand et al. 2004; Aracil et al. 2004;Rollinde et al. 2013).

The second sample consisted of 45 quasars in theredshift range 3.8 ≤ zem ≤ 4.6 (Guimaraes et al.2007), where the spectra were measured with theESI (Sheinis et al. 2002) mounted on the Keck II 10-m telescope. The spectral resolution was R ∼ 4300,with a signal-to-noise ratio usually larger than 15per 10 km s−1 pixel. This sample has also beenstudied by Prochaska and colleagues (Prochaska etal. 2003a,b) in their discussions related to the exis-tence of damped Lyα systems. In the present analy-sis, we used the Lyα forest over the rest-wavelenghtrange 1070-1170 A , corresponding to an interval ofabsorption redshift of 3.3 ≤ zem ≤ 4.4, in order toavoid contamination by the proximity effect closeto the Lyα emission line and possible absorbers as-sociated with OVI. We also carefully avoided re-gions flagged because of damped/sub-damped Lyα

absorption lines. We used only spectra with meanSNR ≥ 25. We rejected the broad absorption lineQSOs (BAL) and QSOs with more than one dampedLyα system (DLA) redshifted between the Ly-α andthe Ly-β emission lines, as their presence may over-pollute the Lyα forest. Metal absorptions were notsubtracted from the spectra. We estimate that thenumber of intervening CIV and MgII systems withWobs > 0.25 A is of the order of five along each ofthe lines of sight ( e.g. Aracil et al. 2004; Tytler etal. 2004). This means that we expect an error inthe determination of the mean absorption of the or-der of 1 %. This is at least five times smaller thanthe error expected from the placement of the contin-uum. Normalization of the spectra is known to bea crucial step in these studies. An automatic pro-cedure (Aracil et al. 2004) iteratively estimates thecontinuum by minimizing the sum of a regularizationterm (the effect of which is to smooth the continuum)and a χ2 term. In our case, the regulation matrixis simply the unit matrix. The procedure smoothsthe continuum to a fixed scale and the χ2 term iscomputed from the difference between the quasarspectrum and the continuum estimated during theprevious iteration. The effect of the procedure isthat the errors increase when the difference betweenthe spectrum and the continuum is large (e.g., whenabsorption occurs), so that the weights of the corre-sponding pixels in the χ2 minimization is reduced.A smoothing parameter allows to vary the scale ofthe estimates. This procedure is very successful inremoving the strong absorption lines from a local es-timate of the continuum and works very well at lowredshift. At high redshift, however, the method un-derestimates the overall absorption in the Lyman-αforest.

We estimated a median error of 5% in the contin-uum in the whole Lyα forest region at high-redshift.In some regions,the placement can be more uncer-tain due to the blend of strong lines. However, closeto the emission redshift of the QSO, and due to theproximity effect, the Lyα forest is not that dense andthe continuum is better defined. The pixels affectedby the largest errors are (mostly) those with largerdepths (blends of strong lines). Therefore they donot strongly influence the estimation of the trans-mitted flux.

The mean transmitted flux for the two sampleswas calculated as the mean flux of all pixels in a bin.At each redshift, we then averaged the transmittedfluxes over all spectra covering this redshift. Fig-ure 1 shows the mean transmitted flux for the twosamples. Notice that the errors were calculated us-

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6 BUSTI, GUIMARAES, & LIMA

2.0 2.5 3.0 3.5 4.0 4.50.0

0.2

0.4

0.6

0.8

1.0

<F>

z

h=0.5 h=0.6 h=0.7 h=0.8 h=0.9 h=1.0

Fig. 1. Mean transmitted flux as a function of redshiftfor Ωm = 0.23, Γ = 0.8×10−12 s−1, T0 = 2.5×104 K andsome selected values of the h parameter. The data pointscorrespond to the Lyα forest measurements obtained by(Rollinde et al. 2013) (red circles) and (Guimaraes et al.2007) (black squares). The color figure can be viewedonline.

ing a bootstrap estimator, even though we only givethe diagonal terms in the covariance matrix. Thelarge scatter in the results can be explained by er-rors in the placement of the continuum fluctuationsin the forest number density, cosmic variance, red-shift determination, etc. The evolution of the meantransmission with z in the two samples is consistentwith previous determinations (e.g. Rauch et al. 1997;Aracil et al. 2004; Tytler et al. 2004; Kirkman et al.2005; Faucher-Giguere et al. 2008b; Aghaee et al.2010).

3. BASIC EQUATIONS

The basic measurement obtained from theLyα forest is the mean transmitted flux at redshiftz defined as

〈F 〉(z) ≡ 〈exp (−τ(z))〉, (1)

where τ(z) is the local optical depth, and anglebrackets denote an average over the line of sight. Fol-lowing standard methods (Hui & Gnedin 1997), thispaper assumes photoionization equilibrium and apower law temperature-density relation, T = T0(1 +δ)β , for the low-density IGM, where δ stands forthe local overdensity and T0 for the IGM temper-ature at mean density (δ = 0). The local opticaldepth is given by (Peebles 1993; Padmanabhan 2002;Faucher-Giguere et al. 2008a)

τ(z) = A(z)(1 + δ)2−0.75β , (2)

with

A(z) ≡πe2fLyα

meνLyα

(

ρcritΩb

mp

)2 X(X + 0.5Y )

H(z)×

×R0T

−0.750

Γ(1 + z)6.

In the expression above, fLyα is the oscillatorstrength of the Lyα transition, νLyα is its fre-quency, me and mp are the electron and protonmasses respectively, Ωb is the baryon density pa-rameter, H(z) is the Hubble parameter, ρcrit isthe critical density, X and Y are the mass frac-tions of hydrogen and helium respectively, whichare taken to be 0.75 and 0.25 (Burles et al. 2001),R0 = 4.2 × 10−13 cm3 s−1/(104 K)−0.75, and Γ isthe hydrogen photoionization rate. The effect ofredshift-distortions due to thermal broadening andpeculiar velocities on the above equations was ne-glected since the mean optical depth is in first orderindependent on these distortions. The mean trans-mitted flux is a non-linear function of < τ >, butthe distortions are responsible for only a small effect(Faucher-Giguere et al. 2008b).

The mean transmitted flux is obtained by inte-grating the local optical depth through a volumedensity distribution function for the gas ∆ ≡ 1 + δ,

〈F 〉(z) =

∫ ∞

0

d∆P (∆; z) exp [−τ(z)]. (3)

As it is widely known, Miralda-Escude et al.(2000) derived an approximate analytical functionalform for the distribution function given by

P (∆; z) = A exp

[

−(∆−2/3 − C0)

2

2(2δ0/3)2

]

∆−b, (4)

where the parameters A and C0 are derived by re-quiring the total volume and mass to be normalizedto unity. They extrapolated the distribution func-tion and obtained δ0 = 7.61/(1+z) with an accuracybetter than 1%, from fits to a numerical simulationat z =2, 3, and 4 from Miralda-Escude et al. (1996).In order to apply this formalism to our data, the val-ues of b were derived from a cubic interpolation ofthe values in the simulation. The application of thisdistribution function to constrain cosmological pa-rameters is well justified, since simulations using dif-ferent numerical methods and different cosmologiesyielded very similar results (Rauch et al. 1997), eventhough the two simulations share approximately thesame amplitude of density fluctuations at the Jeans

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CONSTRAINING H0 FROM LYMAN-α AND BAOS 7

scale. However, one must be cautious about the lim-itations of this distribution function. Actually, thisfunction may be affected by several effects. To citea few: the numerical properties of the simulation,the thermal state of the gas, fluctuations in the UVbackground, inhomogeneous reionization of hydro-gen and helium (Faucher-Giguere et al. 2008b, andreferences therein).

From now on we will assume a flat ΛCDM cos-mology. In this case, the Hubble parameter is givenby

H(z) = H0

[

Ωm(1 + z)3 + (1 − Ωm)]1/2

, (5)

where we will adopt the conventionh = H0/100 km s−1 Mpc−1, which is the Hubbleconstant normalized in units of 100 km s−1 Mpc−1.

Figure 1 shows the effects of H0 on the meantransmitted flux, along with our observational data(red circles) and the data from (Guimaraes et al.2007) (black squares). From our sample we ob-tained a scattered data set, compared to the otherLyα sample, which is related to the sample size. It isalso evident that our data cover most values of the hparameter; thus, we do not expect tight constraintsfrom the Lyα data set alone.

4. ANALYSIS AND RESULTS

Let us now perform a statistical analysis tofind the constraints on the cosmological parame-ters. The full set of parameters are represented byp ≡ (h,Ωm,Γ, T0,Ωbh

2, β). We fixed the value ofΩbh

2 = 0.0218 using the latest observations of deu-terium (Pettini et al. 2008) from the Big Bang Nu-cleosynthesis (Simha & Steigman 2008). We chooseβ = 0.3 due to the photoheating during HeII reion-ization (McQuinn et al. 2009), which is expectedto occur in the redshifts covered by our samples.An early reionization model with β = 0.62 (Hui &Gnedin 1997) yielded similar results. The posteriorprobability of the parameters P (p|d) given the datad is

P (p|d) =P (d|p)P (p)

P (d), (6)

where P (d) is a normalization constant, P (p) is the

prior over the parameters and P (d|p) ∝ e−χ2/2 isthe likelihood, using the usual definition. For theLyα forest data,

χ2Lyα =

∑

i

(〈Fth〉(zi;p) − 〈F 〉(zi))2

σ2〈F 〉(zi)

, (7)

where 〈Fth〉(zi;p) is the theoretical mean transmit-ted flux, 〈F 〉(zi) is the observational mean transmit-ted flux and σ〈F 〉(zi) is their respective uncertainty.We treat the combination ΓT 0.75

0 as a nuisance pa-rameter with a flat prior. The ranges chosen are T0 =[2, 2.5]× 104 K, from estimates of (Zaldarriaga et al.2001), and Γ = [0.8, 1]×10−12 s−1 which cover somemeasurements reported in the literature (Rauch etal. 1997; McDonald & Miralda-Escude 2001; Meiksin& White 2004; Tytler et al. 2004; Bolton et al. 2005;Kirkman et al. 2005), although they are in disagree-ment with the values obtained by (Faucher-Giguereet al. 2008a,b), which used a redshift-dependent re-lation for T0 not favoured by our data. We definedthe 1, 2 and 3σ confidence intervals as the iso-χ2 re-gions given by ∆χ2 = χ2 − χ2

min equals to 2.3, 6.17and 11.8, respectively.

It is worth noting that there seems to be a ten-sion between the two Lyα samples according to Fig.1, which shows that the low redshift sample preferslower values of h. Therefore, we included two pa-rameters to multiply the errors, one for each sample,in order to achieve an acceptable fit. We consideredflat priors within the range [1.0, 1.1] and marginal-ized over them.

In what follows, we first consider the Lyα forestdata separately, and, then, we present a joint analy-sis including the BAO signature extracted from theWiggleZ survey (Blake et al. 2012).

4.1. Limits from the Lyα forest data set

Figure 2 shows the results of the statistical anal-ysis performed with the Lyα forest data. The datado not provide good constraints on both parame-ters. Several sanity checks were performed in or-der to validate the values obtained in the statisticalanalysis. The reduced χ2 obtained is 0.96. Differ-ent values for β, different intervals for ΓT 0.75

0 in themarginalization, or even a redshift-dependence forT0, have resulted in poorer fits compared to what isshown in Figure 2. In addition, even when consid-ering that cosmological constraints are affected bythe physical parameters of the IGM, the oppositeis also true, that is, that different cosmologies mayprovide results not compatible with the IGM param-eters. In this context, it is recommended to fit cos-mological and IGM parameters together in order toobtain more reliable results. The interval for ΓT 0.75

0 ,selected in the marginalization, may suffer from acircularity problem, i.e., the values obtained for T0

and Γ were obtained assuming a particular cosmolog-ical model. Nonetheless, such parameters should re-flect local physics, which are not expected to depend

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8 BUSTI, GUIMARAES, & LIMA

Fig. 2. Confidence regions (68.3%, 95.4% and 99.7%) inthe (Ωm, h) plane provided by Lyα forest data. The basicconclusion is that the Lyα forest alone cannot constrainboth parameters.

strongly on the assumed model. With this in mind,the effect of a broader interval on the marginalizationis a broader interval in the (Ωm, h) plane, but with awell-defined slope. Independent estimates of ΓT 0.75

0

are of great interest; Γ may be determined from theproximity effect (Guimaraes et al. 2007; Carswell etal. 1982; Murdoch et al. 1986; Rollinde et al. 2005)and T0 from linewidth measurements, providing animportant cross check for the method developed inthis work.

Despite the considerations discussed above, Fig-ure 2 shows that the slope in the (h,Ωm) plane sug-gests that a joint analysis with an independent testconstraining only Ωm could provide interesting limitsto the Hubble constant. Thus, a joint analysis withthe BAO data is presented in the next subsection.

4.2. The Lyα forest and BAO: A Joint Analysis

In order to obtain better constraints on the cos-mic parameters, we performed a joint analysis in-volving Lyα forest data and data from baryon acous-tic oscillations obtained from the WiggleZ survey(Blake et al. 2012). The BAO scale can be repre-sented by the parameter (Eisenstein 2005)

A ≡Ω

1/2m

H(z∗)1/3

[

1

z∗Γ(z∗)

]2/3

, (8)

where z∗ is the redshift at which the acoustic scalehas been measured, H(z) is H(z)/H0 and Γ(z∗) isthe dimensionless comoving distance to z∗.

Fig. 3. Contours in the Ωm − h plane provided by thejoint analysis combinig Lyα Forest and BAO. As before,the contours correspond to 68.3%, 95.4% and 99.7% con-fidence levels. Note that the best-fit model convergesto h = 0.66 and Ωm = 0.21. These results should becompared with the ones presented in Figure 2.

From equation (8), it is seen that the BAO scaleis independent of h, and thereby yielded constraintsonly for the matter density parameter. The statisti-cal analysis was performed with χ2 = χ2

Lyα + χ2BAO,

where

χ2BAO = (Ai −Aobs)

T C−1(Ai −Aobs), (9)

Ai is a vector of three theoretical values at three ef-fective redshifts, Aobs is a vector with the respectiveobserved values and C is the covariance matrix forthe observations. Here, we used the data provided inTable 2 by (Blake et al. 2012), where we marginal-ized over all other parameters.

Figure 3 shows the constraints on the pair of pa-rameters (h,Ωm) obtained from our joint analysisinvolving Lyα forest data and BAO. Within a 68.3%confidence level, we obtained 0.19 ≤ Ωm ≤ 0.23 and0.53 ≤ h ≤ 0.82. The reduced χ2 is now equals to1.03. The value for Ωm is lower than what was foundby other studies (Amanullah et al. 2010), but is stillconsistent within the 68.3% confidence level.

Table 1 shows some recent measurements of H0

using different techniques and data; we can see thatour H0 value cannot provide stringent limits thatwould allow us to make a decision about the cor-rect value of H0. The interesting aspect is that themethod discussed here provides an additional esti-mate, independent of the cosmic distance ladder and,covering a redshift range (2 < z < 5), that other

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CONSTRAINING H0 FROM LYMAN-α AND BAOS 9

TABLE 1

LIMITS TO H0 FROM SEVERAL METHODS. RAND. STANDS FOR RANDOM ERRORS WHILE SYST.FOR SYSTEMATIC ERRORS

Method Reference h

Cepheid Variables (Freedman 2001) (HST Project) 0.72 ± 0.08

Age Redshift (Jimenez et al. 2003) (SDSS) 0.69 ± 0.12

Age Redshift (Busti et al. 2014) 0.649 ± 0.042

SNe Ia/Cepheid (Sandage et al. 2006) 0.62 ± 0.013(rand.)±0.05(syst.)

SZE+BAO (Cunha et al. 2007) 0.74+0.04−0.03

Old Galaxies + BAO (Lima et al. 2009) 0.71 ± 0.04

SNe Ia/Cepheid (Riess et al. 2009) 0.742 ± 0.036

Time-delay lenses (Paraficz & Hjorth 2010) 0.76 ± 0.03

CMB (Komatsu et al. 2011) (WMAP7) 0.710 ± 0.025

SNe Ia/Cepheid (Riess et al. 2011) 0.738 ± 0.024

Median Statistics (Chen & Ratra 2011) 0.680 ± 0.028

SZE+BAO (Holanda et al. 2012) 0.74+0.05−0.04

CMB (Hinshaw et al. 2013) (WMAP9) 0.700 ± 0.022

SNe Ia/Cepheid (Freedman et al. 2012) 0.743 ± 0.021

SNe Ia/TRGB (Tammann & Reindl 2013) 0.64 ± 0.016(rand.)±0.02(syst.)

CMB (Ade et al. 2014) (Planck) 0.673 ± 0.012

techniques cannot provide. Therefore, it is a comple-mentary measurement that can be used as a cross-check with local and global measurements.

5. CONCLUSIONS

In this work, we used a cosmological indepen-dent semi-analytical model to describe the IGM andthe data from the Lyα forest and baryon acousticoscillations, in order to constrain cosmological pa-rameters. We provided a cross-check for the Hubbleconstant H0 and the matter density parameter Ωm,which were derived by assuming a flat ΛCDM model;we also established that local properties such as Γand T0 are only weekly dependent on the adoptedcosmology.

We did not obtain good constrains on both pa-rameters from the statistical analysis using only theLyα forest data. This happened also due to ourscarce knowledge about the local properties of theintergalactic medium. However, when we performeda joint analysis involving the Lyα forest data andBAO, we found interesting constraints with the pa-rameters restricted to the intervals 0.19 ≤ Ωm ≤ 0.23and 0.53 ≤ h ≤ 0.82 within a 68.3% confidence level.All these results are in agreement with recent mea-surements reported in the literature (Table 1); how-ever, due to the limited sample and our poor knowl-

edge of the IGM, they are weaker. We expect toobtain better and more reliable constraints on theHubble constant in the near future by using a biggersample and having a deeper understanding of theIGM (preferably with independent estimates for Γand T0). This will make our technique a true com-plementary of other estimates, with the advantageof being independent of the cosmic distance ladder.

The authors are grateful to J. V. Cunha and J. F.Jesus for their helpful discussions. VCB is supportedby CNPq and RNG is supported by CAPES. JASLis partially supported by CNPq and FAPESP (No.04/13668-0).

REFERENCES

Ade P. A. R. et al., 2014, A&A, 571, A16Aghaee, A., Petitjean, P., Srianand, R., Stalin, C. S.,

& Guimaraes, R. 2010, Journal of Astrophysics andAstronomy, 31, 59

Aracil B., Petitjean P., Pichon C., Bergeron J., 2004,A&A, 419, 811

Amanullah R. et al., 2010, ApJ, 716, 712Bergeron J. et al., 2004, The Messenger, 118, 40Blake C. et al., 2012, MNRAS, 425, 405Bolton J. S., Haehnelt M. G., Viel M., Springel V., 2005,

MNRAS, 357, 1178

© C

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gh

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01

6: In

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to d

e A

stro

no

mía

, U

niv

ers

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ac

ion

al A

utó

no

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de

Mé

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o

10 BUSTI, GUIMARAES, & LIMA

Burles S., Nollett K. M., Turner M., 2001, ApJ, 552, L1Busca N. G. et al., 2013, A&A, 552, A96Busti V. C., Clarkson C., Seikel M., 2014, MNRAS, 441,

L11Carswell R. F., Whelan J., Smith M., Boksenberg A.,

Tytler D., 1982, MNRAS, 198, 91Chand H., Srianand R., Petitjean P., Aracil B., 2004,

A&A, 417, 853Chen G., Ratra B., 2011, PASP, 123, 1127Croft R. A. C., Weinberg D. H., Katz N., Hernquist L.,

1998, ApJ, 495, 44Croft R. A. C., Weinberg D. H., Pettini M., Hernquist

L., Katz N., 1999, ApJ, 520, 1Cunha J. V., Marassi L., Lima J. A. S., 2007, MNRAS,

379, L1Eisenstein D. J. et al., 2005, ApJ, 633, 560Faucher-Giguere C.-A., Lidz A., Hernquist L., Zaldar-

riaga M., 2008, ApJ, 682, L9Faucher-Giguere C.-A., Lidz A., Hernquist L., Zaldar-

riaga M., 2008, ApJ, 688, 85Freedman W. L. et al., 2001, ApJ, 553, 47Freedman W. L., Madore B. F., 2010, ARA&A, 48, 673Freedman W. L. et al., 2012, ApJ, 758, 24Guimaraes R., Petitjean P., Rollinde E., de Carvalho R.

R., Djorgovski S. G., Srianand R., Aghaee A., CastroS., 2007, MNRAS, 377, 657

Hinshaw G. et al., 2013, ApJS, 208, 19Hui L., Gnedin N. Y., 1997, MNRAS, 292, 27Hui L., 1999, ApJ, 516, 519Jackson N., 2007, Living Rev. Relativ., 10, 4Jimenez R., Verdi L., Treu T., Stern D., 2003, ApJ, 593,

622Holanda R. F. L., Cunha J. V., Marassi L., Lima J. A.

S., 2012, J. Cosmology Astropart. Phys., 1202, 035Kirkman D. et al., 2005, MNRAS, 360, 1373Komatsu E. et al., 2011, ApJS, 192, 18Lima J. A. S., Jesus J. F., Cunha J. V., 2009, ApJ, 690,

L85Mandelbaum R., McDonald P., Seljak U., Cen R., 2005,

MNRAS, 344, 776Marra V., Amendola L., Sawicky I., Valkenburg W.,

2013, Phys. Rev. Lett., 110, 241305McDonald P., Miralda-Escude J., Rauch M., Sargent W.

L. W., Barlow T. A., Cen R., Ostriker J. P., 2000,ApJ, 543, 1

McDonald P., Miralda-Escude J., 2001, ApJ, 549, L11

McDonald P. et al., 2005, ApJ, 635, 761McQuinn M., Lidz A., Zaldarriaga M., Hernquist L.,

Hopkins P. F., Dutta S., Faucher-Giguere, C.-A.,2009, ApJ, 694, 842

V. C. Busti: Astrophysics, Cosmology & Gravity Centre (ACGC), and Department of Mathematics & AppliedMathematics, University of Cape Town, Rondebosch 7701, Cape Town, South Africa.

R. N. Guimaraes: CAPES Foundation, Ministry of Education of Brazil, Brasılia, DF, Brazil (rguimara@on.br).J. A. S. Lima: Instituto de Astronomia, Geofısica e Ciencias Atmosfericas (IAG-USP), Universidade de Sao

Paulo, 05508-900 Sao Paulo, SP, Brazil.

Meiksin A., White M., 2004, MNRAS, 350, 1107Miralda-Escude J., Cen R., Ostriker J. P., Rauch M.,

1996, ApJ, 471, 582Miralda-Escude J., Haehnelt M., Rees M. J., 2000, ApJ,

530, 1Murdoch H. S., Hunstead R. W., Pettini M., Blades J.

C., 1986, ApJ, 309, 19Nusser A., Haehnelt M., 1999, MNRAS, 303, 179Padmanabhan T., 2002, Theoretical Astrophysics. Vol-

ume III, Galaxies and Cosmology. Cambridge Univ.Press, Cambridge

Paraficz D., Hjorth J., 2010, ApJ, 712, 1378Peebles P. J. E., 1993, Principles of Physical Cosmology.

Princeton Univ. Press, PrincetonPerlmutter S. et al., 1999, ApJ, 517, 565Pettini M., Zych B. J., Murphy M. T., Lewis A., Steidel

C. C., 2008, MNRAS, 391, 1499Prochaska J. X., Gawiser E., Wolfe A. M., Castro S.,

Djorgovski S. G., 2003, ApJ, 595, 9Prochaska J. X., Castro S., Djorgovski S. G., 2003, ApJS,

148, 317Rauch M. et al., 1997, ApJ, 489, 7Riess A. G. et al., 1998, AJ, 116, 1009Riess A. G. et al., 2009, ApJ, 699, 539Riess A. G. et al., 2011, ApJ, 730, 119Rollinde E., Srianand R., Theuns T., Petitjean P., Chand

H., 2005, MNRAS, 361, 1015Rollinde E., Theuns T., Schaye J., Paris I., Petitjean P.,

2013, MNRAS, 428, 540Sandage A., Tammann G. A., Saha A., Reindl B., Mac-

chetto F. D., Panagia N., 2006, ApJ, 653, 843Sheinis A. I., Bolte M., Epps H. W., Kibric R. I., Miller

J. S., Radovan M. V., Bigelow B. C., Sutin B. M.,2002, PASP, 114, 851

Simha V., Steigman G., 2008, J. Cosmology Astropart.Phys., 0806, 16

Slosar A. et al., 2013, J. Cosmology Astropart. Phys.,1304, 026

Tammann G. A., Reindl B., 2013, A&A, 549, 136Tytler D. et al., 2004, ApJ, 617, 1Viel M., Bolton J.S., Haehnelt M.G., 2009, MNRAS, 399,

L39Weinberg D. H., Dave R., Katz N., Kollmeier J.A.,

2003, in Holt S. S., Reynolds C. S., eds, AIP Conf.Proc. Vol. 666, The Emergence of Cosmic Structure:Thirteenth Astrophysics Conference. Am. Inst. Phys.,New York, p. 157

Wyithe J. S. B., Bolton J. S., Haehnelt M. G., 2008,MNRAS, 383, 691

Zaldarriaga M., Hui L., Tegmark M., 2001, ApJ, 557, 519