Department of Physical Sciences Embry-Riddle Aeronautical University 600 S. Clyde Morris Boulevard Daytona Beach, FL 32114, USA

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1 Journl of Applied Mthemtics nd Computtion (JAMC), 017, 1(1), ISSN Online: ISSN Print: Using Hubble Prmeter Mesurements to Find Constrints on Drk Energy Bsed on Different Cosmologicl Models Willim Wilson, Muhmmd Omer Frooq Deprtment of Physicl Sciences Embry-Riddle Aeronuticl University 600 S. Clyde Morris Boulevrd Dyton Bech, FL 3114, USA How to cite this pper: Willim Wilson, Muhmmd Omer Frooq (017) Using Hubble Prmeter Mesurements to Find Constrints on Drk Energy Bsed on Different Cos mologicl Models. Journl of Applied Mthemtics nd Computtion, 1(1), 1-7. Abstrct In this pper, Hubble prmeter versus redshift dt, collected from multiple resources, is used to plce constrints on the prmeters of two current Cosmologicl drk energy models. The first drk energy model considered is the Stndrd Model of cosmology, lso known s ΛCDM with sptil curvture, which is primrily bsed on Einstein s Generl Theory of Reltivity with sptilly homogeneous time-independent cosmologicl constnt, Λ. The second is the XCDM model which prmeterize drk energy s fluid whose density cn vry with time. The H(z) dt collected through different experimentl sources ws used to put constrints on the prmeters of these models. The constrints obtined re then compred with the previously obtined constrints using different probes like type-1 supernove, distnce modulus, CMB nisotropy, nd bryonic coustic oscilltions pek length scle. The results of nlyzing the Hubble prmeter vs redshift dt is consistent with previous conclusions tht we live in n pproximtely flt, ccelerting Universe. However, in order to deduct tighter constrints on cosmolo g- icl models prmeter, like the geometry of the Universe, more nd better-qulity dt will be needed. Keywords Hubble prmeter, Drk Energy, Cosmologicl Models 1. Introduction Since the erly 1900s, it ws widely ccepted in the scientific community tht the Universe ws expnding. The ide ws first introduced by Georges Lemître in 197, who suggested n expnding Universe by solving Einstein s field equtions. Then, in 199, using redshift observtions, Hubble ws ble to mesure this vlue, relizing tht glxies were receding from Erth (Li et l). This recession cused redshift in ech glxy s light spectrum, nd Hubble expressed this redshift mthemticlly s: v = H 0 d (1) where v is the rdil velocity of the glxy, d is the glxy s distnce, nd H 0 is constnt of proportionlity tht ws lter coined Hubble s constnt. Over the decdes, Hubble s constnt hs been refined by mny new nd improved probes in the cosmos (Liddle). In this pper, two different constnts were used on both drk energy models. H 0 = 68 ±.8 kms 1 Mpc 1, from sttisticlly nlyzing 553 mesurements of the constnt, nd H 0 = 73.8 ±.4 kms 1 Mpc 1, bsed on recent mesurements by the Hubble Spce Telescope. DOI: /jmc Journl of Applied Mthemtics nd Computtion(JAMC)

2 It wsn t until , however, tht two independent reserch groups discovered tht the Universe ws ctully ccelerting (Li et l). Before this, it ws thought tht the expnsion of the Universe ws decelerting under the grvittionl ttrctio n present between glxies of the Universe, so this discovery cme s big surprise. Since then, multiple models hve been proposed to try nd explin this phenomenon. This reserch ttempts to fit empiricl dt to two of these models. The first is t he ΛCDM model nd stems from Albert Einstein s Generl Theory of Reltivity. In this, Einstein introduced time-independent energy density (the cosmologicl constnt, Λ), which ws lso considered sptilly homogeneous. He included this in his field equtions to help support the ide tht the universe ws sttic. He lter retrcted this term, clling it his biggest blunder. The fct is, it wsn t blunder t ll, Einstein ws just looking t the universe s sttic insted of expnding (Liddle). The second is the XCDM model. This model insted considers the drk energy density to be time dependent or chnging s time goes on (Frooq et l, 013). The dt used in this reserch is the mesurements of the Hubble prmeter s function of redshift, H(z). Specificlly, H(z) mesurements were used to constrin nd nlyze the ΛCDM nd XCDM models. Trditionlly, constrints on drk energy models were mde with dt from SNI, CMB nisotropy, nd bryonic coustic oscilltions, so using these Hubble prmeter vs redshift mesurements offer us new pproch when considering wht drk energy model best fits our cosmos. The results gthered further support the ide tht we live in n ccelerting Universe. Some firly common questions rise when discussing drk energy nd the cosmos. For instnce, wht is the shpe or curvture of our Universe? Curvture, K, shows up mthemticlly in the Friedmnn eqution (3) nd is very importnt when discussing the pst, present, nd future of our Universe. There re three distinct possibilities of K, it cn either hve closed, open or flt geometry. A closed Universe hs sphericl (K>0) curvture nd hs defined surfce re, while still keeping homogeneity nd isotropy. An open Universe is considered to be hyperbolic (K<0) nd is infinite in ll directions in order to keep the homogeneity nd isotropy. A Universe with flt (K=0) geometry hs no curvture to it nd is similr to the open Universe in the sense tht is infinite in ll directions while being homogeneous nd isotropic (Peebles). This pper uses Hubble prmeter vs redshift dt to find constrints on drk energy bsed on two different cosmologicl models. Previous studies hve introduced the ide of using this type of dt to constrin the prmeters of the drk energy models. However, this reserch used new updted set of H(z) dt, compiled in Frooq et l, 017 to reexmine the estimtes of the prmeters of the two commonly known models of drk energy. In section, we discuss the drk energy models in detil nd lso mention the prmeters used in these models. Section 3 lys out the H(z) dt used in this reserch nd how the dt ws collected. Section 4 discusses in detil the sttistics used to nlyze this dt nd how it cn be interpreted in ech model. Finlly, the lst section concludes the reserch.. Drk Energy Models nd Prmeters In this pper, two drk energy models were studied, the ΛCDM nd XCDM models. To determine how the H(z) influences these two models, we strt with the Einstein eqution of generl reltivity: R μν 1 g μν R = 8πGT μν Λg μν, () where g μν is the metric tensor, R μν is the Ricci tensor, R is the Ricci sclr, nd T μν is the energy-momentum tensor of ny mtter present. Λ is the cosmologicl constnt nd G is Newtonin s grvittionl constnt. In homogeneous spce, Einstein s eqution simplifies to two independent Friedmnn equtions: = 8πGρ = 4πG Λ 3 K, (3) ρ + 3p + Λ 3, (4) where (t) nd K is the time-dependent scle fctor nd curvture of the Universe, respectively. When considering the DOI: /jmc Journl of Applied Mthemtics nd Computtion

3 evolution of the scle fctor nd mtter densities of the Universe, it is lso convenient to use the eqution of stte prmeter: p = p ρ = ωρ, (5) where p is the pressure of the fluid, ρ is the energy density, nd ω is the dimensionless eqution-of-stte prmeter. Tking the derivtive of Eqution (3) with respect to time nd combining it with Eqution (4) nd (5) provides the energy conservtion eqution: ρ = 3 ρ + p = 3ρ 1 + ω. (6) For non-reltivistic mtter, ω = ω m = 0 nd ρ m is proportionl to -3, nd for cosmologicl constnt, ω = ω Λ = 1 nd ρ Λ = Λ 8πG, with ρ Λ = 0. Setting up nd solving Eqution (6) gives us time-dependent energy density: ρ t = ρ 0 o 3 1+ω, (7) where ρ 0 nd 0 re the current energy density of prticulr type of energy nd scle fctor vlues, respectively. In the ΛCDM model, rewriting Eqution (3) using the present density prmeter vlues: Ω m 0 = 8πGρ 0 3H 0, Ω Λ = Λ 3H 0, Ω K0 = K H 0 0, (8) where Ω m 0 is the non-reltivistic mtter energy density prmeter, Ω Λ is the drk energy density prmeter, nd Ω K0 is the curvture density prmeter. These prmeters, long with the redshift eqution: gives us: z = 0 1, (9) H z;h 0, p = H 0 Ω m z 3 + Ω Λ + 1 Ω m 0 Ω Λ 1 + z, (10) where the model prmeters re p = (Ω m 0,Ω Λ ), H(z) is the Hubble prmeter, lso expressed t t, nd H 0 is the present vlue of the Hubble prmeter, lso known s the Hubble constnt. Here, we used the Friedmnn eqution of the ΛCDM model with specil curvture: Ω K0 = 1 Ω m 0 Ω Λ. (11) With the XCDM model, we tret drk energy s time-vrying nd sptilly homogenous, with n eqution of stte prmeter: nd so the Friedmnn eqution becomes: ω X < 1 3, (1) H z,h 0, p = H 0 Ω m z Ω m z 3 1+ω X, (13) where the model prmeters for this model is p = (Ω m 0,ω X ). We only consider the XCDM model in flt sptil hypersurfces, s it breks down in the evolution of energy density inhomogeneities (Frooq et l, 013). 3. H(z) Dt The dt collected used to plce the constrints on these cosmologicl models were collected from Frooq et l, 017. These mesurements of the Hubble prmeter H(z) between redshifts were compiled using multiple references nd re used in current work to plce constrints on the ΛCDM nd XCDM models. DOI: /jmc Journl of Applied Mthemtics nd Computtion

4 Tble 1. Hubble Prmeter versus Redshift Dt z H z (kms 1 Mpc 1 ) σ H (kms 1 Mpc 1 ) Sttisticl Anlysis of Models In this reserch, H(z) dt points were used to constrin the drk energy model prmeters. Ech dt point contined three pieces of informtion, the Hubble prmeter, H obs (z i ), t the redshift of z i, nd the corresponding one stndrd devition of uncertinty, in the Hubble prmeter vlue t given redshift z. We hve considered tht the dt points re independent. To plce constrints on ech models prmeters, we used sttisticl nlysis, computting χ H function: χ H H 0,p = H th z i ;H 0,p H obs z i, (14) where H th z i ; H 0,p is the theoreticl Hubble prmeter vlue. Keeping in mind tht H th z i ; H 0, p = H 0 E z; p, eqution (14) becomes: DOI: /jmc Journl of Applied Mthemtics nd Computtion

5 χ H H 0,p = H E Z i ;p 0 ς H H obs z i E Z i;p 0 i ς + H obs i ς. (15) i We know tht χ H depends on the prmeters p nd H 0, but H 0 is tricky since the vlue is uncertin. To fix this, we ssume tht the distribution of H 0 is Gussin with one stndrd devition width ς H0 nd men H 0. Assuming this llows us to crete the posterior likelihood function: L H p = 1 πς H 0 Simplifying the integrl nd defining: α = 1 ς + E z i, p H 0 gives us the integrl: L H p = 1 e χ H H 0,p / e H 0 H 0 /(ς H ) 0 dh 0 0., β = H 0 + H obs z i E z i, p ς H0 γ = H 0 ς + H obs H0 πς H 0 z i, z i, (16) e 1γ e 1 α H 0 βh 0 dh 0 0 (17) Finlly, completing the squre in the exponent inside the integrl nd simplifying, we get: L H p = 1 ας H 0 e 1 γ β α 1 + erf β α (18) where erf x = x e t dt. Since the likelihood function nd the χ π 0 H function re relted by the eqution χ H p = lnl H p, mximizing the likelihood function mens finding the minimum χ H vlue, with respect to the prmeters p to find the best-fit prmeter vlues p 0. Just like in ny sttistics, we lso need to define stndrd devitions, or in this cse, 1ς, ς, nd 3ς confidence contours. So, we set these confidence intervls s two-dimensionl prmeter sets bounded by χ H p = χ H p 0 +.3,χ H p = χ H p , nd χ H p = χ H p As mentioned erlier, two different Hubble constnt mesurements were used for both models, with the higher vlue more loclly obtined by the Hubble Spce Telescope nd the lower vlue obtined by medin sttistics nlysis in 001. Furthermore, the uncertinties in the Hubble constnt still gretly ffect the prmeter estimtions (Frooq et l, 013). Still, the Hubble prmeter vs redshift dt ws comprble to other methods (SNI, CMB nisotropy, nd bryonic coustic oscilltion mesurements) when plcing constrints on these drk energy models nd cn be very helpful in future experiments involving joint constrints (Huterer et l). The plots of the ΛCDM model in figure 1, when compred to the results obtined by Frooq et l, 013, shows tighter constrints on the prmeters of models by the shrinking of the sigm intervls. We cn now sy tht within 3 sigm vlues, using the new H(z) dt set, tht we live in n ccelerting Universe. Figure 1 lso gives better insight into whether we live in n open or closed geometry. With H 0 = 68 ±.8 kms 1 Mpc 1, we cn deduct tht within 1 sigm vlue, tht we live in n Universe tht hs n open geometry. DOI: /jmc Journl of Applied Mthemtics nd Computtion

6 No Big Bng No Big Bng Closed Open Accelertion Decelertion Closed Open Accelertion Decelertion Results for ΛCDM Model Figure 1: Contour plot for the ΛCDM model from the H(z) dt. Three ellipses represent 1σ, σ, nd 3σ confidence intervls surrounding the minimum χ H vlue. The left contour plot is with the Hubble prmeter, H 0 = 68 ±.8 kms 1 Mpc 1 nd the right plot is with the Hubble prmeter, H 0 = 73.8 ±.4 kms 1 Mpc 1. Flt ΛCDM Line Decelertion Accelertion Flt ΛCDM Line Decelertion Accelertion Results for XCDM Model Figure : Contour plot for the XCDM model from the H(z) dt. Three ellipses represent 1σ, σ, nd 3σ confidence intervls surrounding the minimum χ H vlue. The left contour plot is with the Hubble prmeter, H 0 = 68 ±. 8 kms 1 Mpc 1 nd the right plot is with the Hubble prmeter, H 0 = ±. 4 kms 1 Mpc 1. The plots of the XCDM model in figure lso show tightening of the sigm intervls when compred with the results by Frooq et l, 013. These plots both point to n ccelerting Universe to within 3 sigm vlues. Also, both minimum χ H vlues re very close to the flt ΛCDM line, which further strengthens the ide tht we re in n ccelerting, flt Universe with n open geometry. In order to deduce more informtion from the plots, however, more H(z) dt with better precision will be needed in the future. DOI: /jmc Journl of Applied Mthemtics nd Computtion

7 5. Conclusion In summry, the results of sttisticlly nlyzing the H(z) dt is consistent with n ccelerting Universe. The ΛCDM model with Hubble constnt, H 0, of 68 ±.8 kms 1 Mpc 1 shows tht, within 1σ vlue, or with bout 68% ccurcy, we live in n open, ccelerting Universe. In the XCDM model, both Hubble constnt vlues point towrds flt, ccelerting cosmos. The cosmologicl Principle sttes tht the Universe, s whole, is homogeneous nd isotropic. So, in order for this to be true, while still mintins flt geometry, spce must be infinite in ll directions. In order to deduct more results, better qulity dt will be needed to reduce the size of the sigm intervls nd nrrow down the uncertinty in the model prmeters. Acknowledgements This work is supported by the McNir Scholr s progrm nd the Physicl Science deprtment t Embry-Riddle Aeronuticl University Dyton Bech cmpus. We would like to thnk Dr. Ler nd Ms. Reed for their support nd generosity in the completion of this work. We re lso grteful to Dr. Form Mdiyr for her insight nd expertise tht ssisted the reserch, nd for providing the empiricl dt used in this reserch. We would lso like to show our grtitude to Jime Rmirez for ssistnce in the coding necessry to compute the dt. Finlly, we thnk the referee for comments tht gretly improved the mnuscript. References Frooq, Omer, Dt Mni, nd Bhrt Rtr. "Hubble Prmeter Mesurement Constrints on Drk Energy." The Astrophysics Journl, 0 Feb Frooq, Omer, Form Mdiyr, Sr Crndll, nd Bhrt Rtr. Hubble Prmeter Mesurement Constrints on the Redshift of the Decelertion-Accelertion Trnsition, Dynmicl Drk Energy, nd Spce Curvture. The Astrophysics Journl, 835(1): 6, 017. Huterer, Drgn, nd Michel S. Turner. "Prospects for Probing the Drk Energy vi Supernov Distnce Mesurements." Physicl Review D 60.8 (1999) Li, Mio, Xio-Dong Li, Shung Wng, nd Yi Wng. "Drk Energy." Drk Energy Liddle, Andrew. An Introduction to Modern Cosmology. John Wiley nd Sons, 015. Peebles, P. J. E., nd Bhrt Rtr. "The Cosmologicl Constnt nd Drk Energy." Reviews of Modern Physics 75. (003): DOI: /jmc Journl of Applied Mthemtics nd Computtion(JAMC)