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 Mehtapress 2013 Print-ISSN: 2320-6756 Online-ISSN: 2320-6764 J.Phy.Ast. Journal of FulPl hpaypseircs & Astronomy Full Paper WWW.MEHTAPRESS.COM

C.C.Onuchukwu1*, A.C.Ezeribe2 Constraining the Hubble parameter using dis-

1Department of Industrial Physics, tance modulus – relation Anambra State University, Uli, (NIGE- RIA) Abstract 2Department of Physics/Industrial Phys- ics, Nnamdi Azikiwe University, Awka, Using the relation between distance modulus (m – M), redshift (z) and deceleration parameter (NIGERIA) (q ), obtained using Friedman-Robertson-Walker (FRW) metric, we constrain the Hubble 0 E-mail: [email protected] parameter (h). Using the mean and the median values of our data obtained from NASA Extragalactic Database (NED), the value of Hubble parameter we estimated is 0.66  0.11 Received: September 26, 2013 and 0.84  0.11 for q = –1 and q = 0 respectively. In general, using the minimum and the 0 0 Accepted: November 10, 2013 maximum values of the parameter in our data, the value of Hubble parameter we estimated lies in the range 0.6  0.2  h  2.2  0.7, with an average value of h = 0.67  0.22 – 0.96 0.29, giving an average age for the as 11.4  2.4 – 14.3  2.2 Gyr. *Corresponding author’s Name & Address Key Words

C.C.Onuchukwu - Miscellaneous; Astronomical database - Miscellaneous; Method - Statistical; Data Department of Industrial Physics, Analysis. Anambra State University, Uli, (NIGE- RIA) E-mail: [email protected]

INTRODUCTION the evolution of the universe, the Theory is pres- ently the most acceptable model that described most of The universe is all of space, time, matter and energy the observational features in the evolution of the universe. that exist. The ultimate fate of the universe is determined, It explained the universe to have started from an extremely through its gravity, thus, the amount of matter/energy in hot dense phase called the epoch (all fundamental the universe is therefore a considerable importance in cos- forces are unified – the period of Theory Of Everything mology. The Wilkinson Microwave Anisotropy Probe (TOE)) at the end of which gravitational force separated (WMAP) in collaboration with National Aeronautics and from gauge forces, and passed through the Grand Unifi- Space Admiration (NASA) in one of their mission esti- cation Epoch and (at the end of which mated that the universe comprises of about 4.6% of vis- strong forces separated from electroweak force) and the ible matter, 24% of matter with gravity but do not emit (unification of and observable light (the ) and about 71.4% was weak nuclear interaction). Other phases include – attributed to (possibly anti-gravity) that may , , Lepton Era, Nucleosyn- be responsible for driving the acceleration of the observed thesis, Photon Era, Recombination Epoch and presently expansion of the universe[20]. The universe has been ob- Matter Dominated Era[7,14,23-25]. served to be expanding, first suggested by Einstein in his The generally acceptable mathematical theory for study- general theory of relativity and observed by ing the evolution of the universe is general relativity[9]. Hubble[17,21,29,30]. General relativity is the theory of gravitation, in which Amongst the different models proposed to explain gravitational effects between masses results from warping Full Paper JOPA, 2(4) 2013 of space-time by the masses. In a uniform universe, gen- deceleration parameter (q ) of the universe to constrain 0 eral relativity has a simple solution for the evolution of the the Hubble constant for a large database obtained from geometry of the universe (contraction or expansion) which NED and possibly trace the evolution of the Hubble pa- depends on its content and past history[29]. In the presence rameter as a function of redshift (H(z)). of enough matter, the expansion will slow or even be- come a contraction. On the other hand, the dark energy THEORETICAL RELATIONSHIPS (cosmological constant ()) drives the universe towards increasing expansion[29]. The current rate of expansion is The evolutionary trend of the universe has been mod- usually expressed as Hubble constant (H ) with an esti- eled by using the density parameter (), determine by the 0 mated value of H = 100h kms-1 Mpc-1, with the Hubble density of the universe, the deceleration parameter (q ) 0 0 parameter (h) having values of h = 0.5 – 1.0[24,25]. and the Hubble constant (H ). The Hubble constant is an 0 One of the fundamental goals of cosmology is to important parameter in cosmology as it not only deter- determine the expansion rate of the universe. Various mines its expansion rate, it also set limit to the possible methods that have been applied include: age, critical density and size of the . (1) The use of distant Type 1a supernovae (SN 1a) as The velocity (v) of the expansion of the universe has been standard candles[22,2,8,31]. The apparent peak magnitude defined by Hubble[16] as of these supernovae yields a relative luminosity dis- v(t) = H (t)d (1) tance d as a function of redshift from which the 0 L where d is the radius of the expanding universe. The size Hubble constant is estimated[27,28,32]. of the universe is unknown, yet it undergoes expansion or (2) Large galaxy surveys for mapping of cosmic distances contraction, thus, the evolution of the universe can be ex- and expansion, by using the large scale clustering pat- press in terms of cosmic scale factor (a(t)) as tern of galaxies which contains the signature of Baryon Acoustic Oscillation (BAOs). BAO refers to the regu- (2) lar periodic fluctuations in the density of the visible In terms of the cosmic expansion factor (R ), at the baryonic matter of the universe, caused by acoustic 0 time (t ), the Hubble constant is given by (e.g.[24]) waves which existed in the early universe. By looking 0 at large scale clustering of galaxies, a preferred length-

scale which was imprinted in the distribution of pho- (3) tons and baryons propagated by the sound waves in where is the first time derivative of R . The cosmic the relativistic plasma of the early universe, can be 0 scale factor affect all distances, thus, the wavelength ( ) calibrated by the observation of Cosmic Background e of a photon emitted at time of emission (t ) and observed Microwave Radiation (CMBR) and applied as cos- e [24] [1,5,19] at another time (t ), will be (e.g. ) mological standard rod (e.g. ). 0 (3) Other test include the redshift-angular size test, galaxy cluster gas mass fraction, strong gravitational lensing (4) test and test, these test generally where  is the wavelength of the observed photon at 0 constrain cosmological parameters as a function of time (t ). The cosmological redshift (z) is usually given by [26] 0 redshift (see review by Samushia & Rastra ). (e.g.[24]) Several decades have passed since Hubble published the correlation between distances to galaxies and their ex- (5) pansion velocities, but establishing an accurate cosmologi- cal distance scale and value for the Hubble constant (H ) where d is radius of the universe centered on the observer 0 at time t , and c is the speed of photons. Allowing for have proved challenging. The value of H has evolved 0 0 from H = 500 kms-1 Mps-1 recorded by Hubble[17] to a some form of time evolution of the expansion factor 0 [24] well-known range of range of H = 50 - 100 kms-1 Mps- (R(t)), we expand it using Taylor series. Following Roos 0 1[3]. The Hubble Space Telescope (HST) gave a more spe- we can for t in general write cific value of H = 70 - 80  10%[11], while the WAMP 0 (6) data give H = 72  5 kms-1 Mps-1,  = 0.3, and  = 0 m  0.7[30], where  is relative matter density (both luminous Making use of the definition of equation (3), equa- m and dark matter) and  is the relative dark energy den- tion (6) implies that to a second order expansion, the cos-  sity. mic scale factor can be written as In this article, we which to use the dependence of observed distance modulus (m – M) on redshift (z) and

FP 141 JOPA, 2(4) 2013 . Full Paper

(e.g.[24]) (7) (16) From equation (3), we can write For an expanding universe parameterized by the cos- (8) mic scale factor (a(t)), in which photons are redshifted and The deceleration parameter (q) which measure the rate suffer from energy effect, if the apparent brightness of a galaxy is B , then its proper distance is given by (e.g.[24]) of slowing down of the expansion factor is defined a (e.g.[24]) by (17) Equating B = B , implies that d = (1 + z)d . Thus, the a o L p (9) luminosity distance in terms of redshift is given by Makin use of equations (3), (4), (7) and (9) in equation

(5), the cosmological redshift can be expressed as (18) In terms of distance modulus (m – M), the luminosity (10) distance to a source is given by (e.g.[15]) making use of the series expansion

, equation (10) can be approxi- (19) mated to Substituting equation (18) into equation (19), we have (20) (11) where , and H = In obtaining equation (11), we made use of terms 0 only to the second order in t. Inverting equation (11), 100h kms-1 Mpc-1 (e.g.[24]) and in S.I is given by H = 3.241h 0 making use of equation (3), we have H in terms of red- x 10-18 S-1. For q = 0 (e.g.[18] – matter dominated uni- 0 0 shift (z) as verse)

(12) (21a) For q = –1 (e.g.[18] – vacuum energy dominated uni- For an isotropic and homogeneous universe, the 0 Robertson–Walker metric in Minkowiski space-time best verse) describe the geometry of space given by (e.g.[24]) f(z) = 5 log z + 5 log(1 + z) (21b) Equation (20) provides a way to constrain the Hubble

(13) parameter (h) for a given sample of sources with ob- where ds is the invariant line element,  is the dimension- served redshift and distance modulus. less comoving coordinate, k is the curvature parameter,  and  are spherical coordinate points (polar and azimuthal ANALYSIS AND RESULTS angle respectively). In such homogenous isotropic universe, photons propagates along null geodesic given by ds2 = 0, The data used in this analysis were sourced from the and along the line of sight of an observer,  and  are NASA Extragalactic Database (NED). We selected sources kept constant. Thus, the observed proper distance (d ) to with observed distance modulus (m - M) and redshift (z). p a galaxy for a flat universe with k = 0 is given by (e.g.[24]) The sample consists of 11 585 sources which is spread over 0.00016  z  8.26. Due to large volume of data, we (14) binned the sources into various redshift bin widths to en- able fair representation of sources in each bin. Using equation (12) in equation (14), the proper dis- The binning ranges, the mean, median values of each tance in terms of redshift (z) to the lowest order in (t – t ) 0 e bin size and the estimated Hubble parameter and esti- is given by mated from Hubble parameter based on equation (21) are shown in Table 1. The plot of m - M (15) against f(z) for all the sources are shown in figures 1 and 2 The first term on the right of equation (15) gives the for q = 0 and q = 1 respectively, the error bars are errors 0 0 Hubble linear law, while the second term measures the associated with the distance modulus. A linear fit to the deviation from linearity to lowest order depending on the plot gives: for q = 0, we have m - M = (1.02  0.07) f(z) + 0 value of q . 43.23  0.25, which gives h = 0.93  0.1; for q = –1, we 0 0 The luminosity distance (d ) to a galaxy of absolute have m - M = (0.95  0.07) f(z) + 42.76  0.20, which gives L luminosity (L) with observed brightness (B ) is given by h = 0.75  0.07. In general, using the minimum and the o

FP 142 Full Paper JOPA, 2(4) 2013 maximum values of the parameter in our data, the value against different redshift (z) bins to check the dependence of Hubble parameter we estimated lies in the range 0.6  of the Hubble parameter on different epoch (shown in 0.2  h  2.2  0.7, with an average value of h = 0.67  figure 3). The plot showed an exponential dependence of 0.22 - 0.96  0.29, giving an average age for the universe h on z, with the best fit being h = (0.71  0.36)e(0.21  0.09)z as 11.4  2.4 - 14.3  2.2 Gyr. and h = (0.65  0.27)e(0.11  0.07)z for q = – 1 and q = 0 0 0 We also plotted the estimated Hubble parameter (h) respectively.

Figure 1 : Plot of m - M against f(z) for q = 0 0

Figure 2 : Plot of m - M against f(z) for q = -1 0

Figure 3 : The plot of estimated hubble parameter (h) against redshift (z)

FP 143 JOPA, 2(4) 2013 . Full Paper DISCUSSION AND CONCLUSION Series in High Energy Physics, Cosmology and Gravitation, 253, (2002). The value of Hubble constant (H ) we estimated (i.e. [8] A.Conley, M.Sullivan; Astrophysics Source Code Library, 0 converting the Hubble parameter (h) to Hubble constant) 111, 38 (2011). [9] A.Einstein; 1916 in ‘The principle of relativity’ with notes lies in the range of 60  20 kms-1 Mpc-1  H  220  70 0 by A.Sommerfeld; Dover Publication INC, USA, 216 (1923). kms-1 Mpc-1 with an average value of H = 67  22 - 96  0 [10] O.Farooq, B.Rastra; Modern Physics D., 21, 1 (2012). 29 kms-1 Mpc-1. The mean value of H we estimated is in 0 [11] W.L.Freedman; ASP Conference, 245, 542 (2012). reasonable agreement with the HST value of H = 70 - 0 [12] W.L.Freedman, B.F.Madore, V.Scowcroft, C.Burns, 82  10% kms-1 Mpc-1 recorded by Freedman[11]. Our re- A.Monson, S.E.Persson, M.Seibert, J.Rigby; ApJ, 758, 10 sult in general is also in agreement with results obtained (2012). from other works in literature e.g. H = 50 - 100 kms-1 [13] A.H.Guth; Physical Review D., 23, 347 (1981). 0 Mpc-1 by Assis et al.[3], H = 72  5 kms-1 Mpc-1 from WMAP [14] E.R.Harrison; Masks of the Universe, Cambridge Univer- 0 data by Spergel et al.[30], by sity Press, UK, Second Edition, 351 (2003). Aviles et al.[4], H = 79  30% kms-1 Mpc-1 by Blinnikov et [15] D.W.Hogg; Distance Measure in Cosmology, Institute for 0 al.[6], H = 74  2.5 kms-1 Mpc-1 by Lima et al.[18] similar to Advance Study, Olden Lane, Princeton Press, 98 (1998). 0 [16] E.P.Hubble; Cepheid in Spiral Nebulae, American Astro- H = 74  2.5 kms-1 Mpc-1 by Freedman et al.[12] and H = 0 0 nomical Society, 33, 139 (1925). 68.9  7.1 kms-1 Mpc-1 by Reid et al.[20]. The limit of the [17] E.P.Hubble; Proceeding of the National Academy of Sci- range 60 kms-1 Mpc-1  H  220 kms-1 Mpc-1 is also in agree- 0 ence of USA, 15, 168 (1929). ment with H = 60 - 220 kms-1 Mpc-1 obtained by Farooq 0 [18] J.A.S.Lima, J.F.Jesus, R.C.Santos, M.S.S.Gill; A&A, 4, 46 & Rastra[10]. (2012). Our plot of Hubble parameter (h) against redshift (z) [19] N.Padmanabhan, X.Xu, D.J.Eisenstein, R.Scalzo, indicates that h depends stronger on z for q = -1 than for A.J.Cuesta, K.T.Mehta, E.Kazin; MNRAS, 427, 2132 (2012). 0 q = 0, with an exponential dependence on redshift, an [20] M.J.Reid, J.A.Braatz, J.J.Condon, K.Y.Lo, C.Y.Kuo, 0 indication that the Hubble constant (H ) is a function of C.M.V.Impwllizzeri, C.Henkel; APJ, 767, 154 (2013). 0 time an indication that the earlier universe expanded faster [21] A.G.Reiss, L.Marci, S.Casertano, H.Lampelt, H.C.Ferguson, (assuming redshift indicates the time evolution of the uni- A.V.Fillipenko, S.W.Jha, W.Li, R.Chornock, J.M.Silverman; verse). The generally acceptable standard theory of the APJ, 732, 129 (2011). [22] A.G.Reiss, A.V.Fillipenko, P.Challis, A.Clocchiattia, early universe – inflationary theory[13], incorporates an ex- A.Dierrcks, P.M.Garnavich, R.L.Gilliland, C.J.Hogan, ponential increase in the very early evolution of the uni- S.W.Jha, R.P.Kirshner, B.Leibundgut, M.M.Phillips, D.Reiss, verse. B.P.Schmidt, R.A.Schommer, R.C.Smith, J.Spyromilio, In conclusion, using the observed distance modulus C.Stubbs, N.B.Suntzeff, J.Torny; APJ, 116, 1009 (1998). and redshift, we estimated Hubble parameter, for differ- [23] I.Robson; Active Galactic Nuclei, Praxis Publishing Ltd, The ent assumed energy densities of the universe represented White House Eastergate, Chichester, West Sussex, England, by the deceleration parameter (q ). The age of the uni- 310 (1996). 0 verse estimated from the Hubble parameter suggests that [24] M.Roos; Introduction to Cosmology, Third Edition, John the universe is ~11.4  2.4 - 14.3  2.2 Gyr old with the Wiley & Sons, Chichester, West Sussex, England, 287 (2003). possible limit to the edge of the observable universe of d [25] B.Ryden; Introduction to Cosmology, Ohio State Univer- ~ 5000 Mpc. sity Press, 163 (2006). [26] L.Samushia, B.Rastra; APJ, 650, L5 (2006). [27] A.Shafieloo, C.Clarkson; Physical Review D., 81, 83 (2010). REFERENCES [28] J.Sollerman, M.Ergon, C.Inserra, S.Valenti, P.A.Wilson, S.Jon Juliusson, H.Holma, M.Ingemyr, O.Saxen, [1] L.Anderson, E.Aubourg, S.Bailey, D.Bizyaev, et al., (75 co- L.Haukanes; Central Bureau Electronic Telegrams, 2068(1), authors); MNRAS, 427, 3435 (2012). 32 (2009). [2] R.Amanuallah, C.Lidman, D.Rubin, et al., (42 co-authors); [29] D.N.Spergel, M.Bolte, W.Freedman; Proceeding of National APJ, 716, 712 (2012). Academy of Science, USA, 94(13), 6579 (1997). [3] A.K.T.Assis, M.C.D.Neves, D.S.L.Soares; ASP Conf., 413(2), [30] D.N.Spergel, L.Verde, H.V.Peiris, E.Komatsu, M.R.Nolta, 225 (2009). C.L.Bennett, M.Halpern, G.Hinshaw, N.Jarosik, A.Kogut, [4] A.Aviles, C.Gruber, O.Luongo, H.Quevedo; Physical Re- M.Limon, S.S.Meyer, L.Page, G.S.Tucker, J.L.Weiland, view D., 86, 123516 (2012). E.Wollack, E.L.Wright; APJSS, 148, 175 (2003). [5] C.Blake, S.Brough, M.Colles, et al., (22 co-authors); MNRAS, [31] K.Suzuki, H.Nagai, M.Kino, J.Kataoka, K.Asada, A.Doi, 425, 405 (2012). M.Inoue, M.Orienti, G.Giovanni, M.Giroletti, [6] S.Blinnikov, M.Potashov, P.Baklanov, A.Dolgov; Journal A.Lähteenmäki, M.Tornikoski, J.León-Tavares, U.Bach, of Experimental and Theoretical Physics, 96, 153 (2012). S.Kameno, H.Kobayashi; APJ, 18, 251 (2011). [7] S.Bonometto, V.Gorini, U.Moschella; Modern Cosmology [32] Y.Wang, M.Tergmark; Physical Review D., 71, 167 (2005).

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