id3621125 pdfMachine by Broadgun Software - a great PDF writer! - a great PDF creator! - http://www.pdfmachine.com http://www.broadgun.com 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 – Redshift 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 universe as 11.4 2.4 – 14.3 2.2 Gyr. *Corresponding author’s Name & Address Key Words C.C.Onuchukwu Cosmology - 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 Big Bang 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 Planck 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 Inflationary Epoch (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 Electroweak Epoch (unification of electromagnetism and observable light (the dark matter) and about 71.4% was weak nuclear interaction). Other phases include – attributed to dark energy (possibly anti-gravity) that may Baryogenesis, Hadron Epoch, 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 observable universe. (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 structure formation 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.