In the days before fast com puter programs for calculating the transfer function a popu lar approximation to was the BBKS Bardeen Bond Kaiser and Szalay formula ln1 0 171 171 1 0 284 1 18 0 399 0 490 where kk eq and eq 0 073 m0 Mpc a Plot ID: 30177
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TheCDMtransferfunctionandpowerspectrumThepresent-daypowerspectrumofCDMperturbationscanbewrittenasP(k)=22Hkns Hns+30T2(k);whereHisanormalizationconstant,nsisthespectralindexofprimordialperturbationsandT(k)isthetransferfunction.Inthedaysbeforefastcom-puterprogramsforcalculatingthetransferfunctionapopularapproximationtoT(k)wastheBBKS(Bardeen,Bond,KaiserandSzalay)formulaT(x)=ln(1+0:171x) 0:171xh1+0:284x+(1:18x)2+(0:399x)3+(0:490x)4i 1=4;wherex=k=keqandkeq=(0:073\nm0h)hMpc 1.a)PlotT(x)for\nm0=1:0,h=0:5,andfor\nm0=0:3,h=0:7.b)Makealog-logplotofthepowerspectrumforthesametwomodels.Usens=1,andH=1:910 5for\nm0=1,H=4:610 5for\nm0=0:3.c)HowwelldoesT(x)agreewiththesimple-mindedresultswederivedinthelecturesinthelimitsx 1andx 1?ApopularmeasureoftheamplitudeofthedensityperturbationsistheRMSoverdensityinasphereofradiusR,de nedas2R=h2R(x)i;withR(x)=Zd3x0(x0)WR(x x0):whereWR(x)isequalto1forxRandvanishesotherwise.Onecanshow(youdon'thaveto!)that2R=1 22Z10dkk2P(k)W2(kR);whereW(x)=3 x3(sinx xcosx):1 d)Writeaprogramthatcalculates8,thatisRwithR=8h 1Mpcforthetwomodelsyoulookedatina)andb).e)GototheLAMBDAarchiveontheinternetandlookatthetablesofderivedcosmologicalparametersfromtheWMAPsatellite.Whichofthetwomodelsagreesbestwiththevaluesfor8you ndthere?2