On the Scale factor of the Universe and Redshift.

Transcription

1 On the Sle ftor of the Universe nd Redshift. J. M. unter. ABSTRACT It is proposed tht there hs been longstnding misunderstnding of the reltionship between sle ftor of the universe nd redshift. It is shown how vlue of omeg(mtter) of one qurter of the true vlue, (hene the pprent drk energy phenomenon) n result from suh misoneption. Preditions for the mgnitudes of supernove ginst redshift re mde nd found to be in good greement with supernove dt, without drk energy.. Key words: Cosmology: distne sle, osmologil prmeters, drk energy INTRODUCTION Modern osmology relies on the hypothetil entities of drk energy nd drk mtter. The onlusion tht drk energy must exist is from two min lines of evidene. Due to the observtions of distnt supernove (Riess et l 7), osmologists hve onluded tht the expnsion of the universe is speeding up. Thus drk energy hs been inferred, the nture of whih is poorly understood. There is lk of n understnding of physil mehnism, by whih drk energy uses n elerting expnsion of the universe. Mesurements from WMAP show flt universe, with vlue of omeg(mtter) of pproximtely.5. This hs led to vlue for omeg(lmbd) of bout.75. X-ry mesurements of glxy lusters mesures the bryon frtion, from whih omeg(mtter) is dedued. owever the vlue obtined relies on the vlue of omeg(bryons) from WMAP dt, nd so is not independent. These two methods seem to support eh other, so it is understndble tht mny osmologists (some relutntly!) support the drk energy onlusion. owever, simple hnge to our notions of how sle ftor reltes to redshift n remove these two min rguments in fvour of drk energy, s shown below.. A NEW RELATIONSIP BETWEEN REDSIFT AND SCALE FACTOR Trditionlly, in osmology, the rtio of the wvelength of light is proportionl to the rtio of the sle ftor of the universe. z () With the new proposl there is different reltion

2 z n () For given rtio of sle ftor, the redshifts for the two osmologies re relted by z zn (3) ubbles onstnt,, is ssumed onstnt. d dt (4) The observed vlue of (wht is trditionlly ssumed to be) ubbles onstnt of pprox 7km/s/Mp is not the true vlue. The vlue s defined by (4) is hlf, i.e pprox 36km/s/Mp. From (), for smll hnges in time, the redshift depends on twie the vlue of ubbles onstnt, nd still mthes observtions. z n ( dt ) dt (5) Why () should be true is not disussed (lthough the uthor hs resons for expeting it to be the se). The onsequenes of suh reltion, is the subjet of this pper, prtiulrly how it n led to the onlusion of the drk energy phenomenon.. A SOLUTION OF EINSTEINS EQUATIONS. For onstnt, exp( t ), where is the sle ftor of the universe, Einsteins equtions of Generl Reltivity redue to 8 3 3k G (6) 8 p 3 k G (7) 4 so for flt universe with k =, nd p (i.e. ) (8) nd 3 (9) 8 G therefore the trditionlly inferred vlue of omeg(mtter) would be

3 3 8 G m.5 () 3( ) 8 G Mesurements from WMAP5, (Komtsu et l, 8), led to n inferred vlue for omeg(mtter) of.58 (.3). Their preferred model is flt CDM model with k =, nd n eqution of stte prmeter,, of -. A vlue for the mximum likelihood for omeg(mtter) is given s.49. In relity m, s the denomintor of () should ontin not, nd. It is not neessry to ssume drk energy, s the universe is nturlly t ritil density. The oinidene problem tht the vlues of omeg(lmbd) nd omeg(mtter) re similr only t the time we live in, is voided with this pproh. At ll times omeg(mtter) is nd omeg(lmbd) is zero. 3. TE SUPERNOVAE DATA The flux F due to distnt supernove is given by L L F () 4 ( z) d 4 d L p d L is the luminosity distne, L is luminosity nd d p is the proper distne. Trditionlly, for flt universe d p z ' ' h( z ) () With the new pproh is hlf of the trditionl vlue nd d p ( z ) (3) (there is derivtion in Appendix A), so d L ( z)( z ) (4) the distne modulus is 5 5log d L (5) 3

4 Using (4) in (5), there is good mth to the supernove dt (Riess 7), gold set. The hi-squred fit is 83.8 for 8 degrees of freedom. This lose mth is with, onstnt, with no requirement for drk energy omponent of the universe. Figure 3 shows omprison between the new pproh nd the drk energy model, The urves re very similr. The top urve is for = 65.kms - Mp -. The bottom urve is for the best flt drk energy model with = 63.8kms - Mp - (Wright 7). The drk energy model hs vrible prmeter, the mtter density, for the urve shown omeg(mtter) =.7. The new pproh uses no extr vrible prmeter. With the new pproh, the deelertion prmeter q(z) = - (onstnt), for drk energy q(z) vries, in wy tht is not understood (Shpiro & Turner, 6). Figure 3 Supernov moduli with redshift, for resling nd drk energy models. 4. CONCLUSIONS AND PREDICTIONS. There hs been serious nd long-stnding misinterprettion of the reltionship between the rtio of sle ftor nd redshift. The reltion should be s in (). If this is indeed the se, then we would expet the following. i) The inferred vlue of omeg(mtter) will be extly.5. ii) The distne moduli (5) for supernove, will be ording (4) 4

5 ACKNOWLEDGEMENTS With thnks to my wife Amnd for her enourgement nd Professor Lidsey of Queen Mry College London, for leture notes used in Appendix A. REFERENCES Riess A. G et l., 7, ApJ, stro-ph/657 Shpiro C. A., Turner M. S., 6, stro-ph/5586 Komtsu E et l., 8, stro-ph/83.547v Wright E. L., 7, stro-ph/7584 APPENDIX A DERIVATION OF (3) AND (4) Strting from the Robertson-Wlker metri dr ds dt ( t) r ( d sin d ) (A) kr in terms of the o-moving o-ordintes, hs the role of the rdil o-ordinte r sin if k = + r if k = r sinh if k = - For ry of light moving long rdil pth with nd onstnt, for flt universe, ds dt ( t) d (A) so d dt dt d d d. d (A3) for the new reltion () z (A4) t d (A5) 3 5

6 so from (A3) d (A6) d (A7) from () d (A8) z d p z ( z) (A9) d p ( z ) (A) whih is (3), nd is bout 36km/s/Mp, nd from () d L ( z)( z ) (A) whih is (4). 6