aa r X i v : . [ phy s i c s . g e n - ph ] A ug CMB in non-standard cosmology: A first look
G¨unter Scharf ∗ Physics Institute, University of Z¨urich,Winterthurerstr. 190 , CH-8057 Z¨urich, Switzerland
Abstract
We study CMB in the nonstandard background cosmology recently investigated. Using thepreviously calculated first order metric perturbations we discuss the Sachs-Wolfe and the integratedSachs-Wolfe effects. We show how small-multipole CMB data can be used to determine the matterdensity of the Universe.
Keyword: Cosmology ∗ e-mail: [email protected] Introduction
The cosmological standard FLRW model is a high-density model. On the contrary nonstandardcosmology is a low-density model where only the few percent of visible matter contribute to theenergy density [1]. As far as observations can decide between the two, the magnitude-redshift data areexcellently reproduced by both models (see next section). The next step is the analysis of CMB whichconsists of two parts: (i) Understanding the early universe so far that one gets initial conditions at thetime of last scattering, for: (ii) Propagating CMB in the cosmic gravitational field from last scatteringto the present. The standard model solves both problems successfully. For the nonstandard modelwe solve (ii) in this paper, that is our first look. Problem (i) is much harder because the nonstandardearly Universe is quite different from the standard one.The paper is organized as follows. In the next two sections we review the nonstandard backgroundand its first order perturbations. In sect.4 we calculate the CMB temperature anisotropy by applyinga formula of Tomita [2]. In the discussion we point out how data of small multipoles l can be usedto determine the matter density in the Universe. This is complementary to standard cosmologywhere small l -values are neglected because of foreground effects. As a preparation we discuss thetransformation of CMB data to the cosmic rest frame in the appendix. The nonstandard background is defined by the line element ds = dt − X ( t ) dr − R ( t ) ( dϑ + sin ϑdφ ) (2 . R ( t ) and X ( t ) isgiven in parametric form by [1] R ( t ) = T L sin w, (2 . X ( t ) = cot w (2 . t = T L ( w − sin w cos w ) . (2 . T L determines the lifetime of the Universe. From (2.2) it follows˙ R ( t ) = 2 T L sin w cos w dwdt . (2 . dtdw = 2 T L sin w. (2 . R ( t ) = cos w sin w = cot w = X. (2 . k µ =(1 /X, − /X , , z = X obs X em = cot w obs cot w em (2 . z = ∞ , that means w em = π/
2, and π/ < w obs < π , because z (2.8) must be positive. From dz = cot w obs cot w dw sin w = cot w obs cos w dt T L sin w (2 . dzdt (cid:12)(cid:12)(cid:12) z =0 = − H = (1 + cot w obs ) T L cot w obs . (2 . drdz = drdt dtdz = 2 T L cot wX cot w obs = 2 T L (1 + z ) [(1 + z ) + cot w obs ] from z = 0 to z >
0. With the new variable of integration x = 1 / (1 + z ) we get r ( z ) = 2 T L Z / (1+ z ) dxx (1 + cot w obs x ) . (2 . α = 1 | cot w obs | (2 . c = dr/dt = 1 / | X | . Using the Hubble constant (2.10) wefinally obtain r ( z ) = c H (1 + α ) Z / (1+ z ) dxx ( α + x ) . (2 . z ) r ( z ). The rational integral in (2.11) is elementary so that d L ( z ) = c H (1 + z ) (1 + α ) α h
11 + α − (1 + z ) α (1 + z ) ++ 1 α log 1 + α (1 + z ) α i . (2 . m ( z ) is defined by m ( z ) = 5 log d L + M + 25 (2 . M is the absolute magnitude of the supernova standard candle. In the Hubble diagram oneplots the distance modulus µ ( z ) = m ( z ) − M. (2 . α will now be determined by the Hubble data. The measured Hubble diagram isnicely represented by the standard FLRW luminosity distance˜ d L ( z ) = c (1 + z ) H z Z dx √ Ω M x + Ω Λ . (2 . M = 0 . Λ = 0 . H = 72 km/(s Mpc). This is the best fit in [3]. In the table the corresponding distancemodulus ˜ µ ( z ) is listed in the second column. We have taken the value ˜ µ (1) = 44 .
08 at z = 1 as ameasured value and have determined the free parameter α in (2.14) such that this value is reproduced.The result is α = 6 . . (2 . z = 10 is excellently represented by (2.14-16) ascan be seen in the third column of the table. The last two columns show the look-back times in thestandard and nonstandard models [1]. z ˜ µ ( z )(mag) µ ( z )(mag) ˜ t ( z )(10 Y ) t ( z )(10 Y )0.01 33.12 33.12 0.1349 0.13480.02 34.64 34.64 0.2678 0.26760.03 35.53 35.54 0.3990 0.39850.04 36.17 36.18 0.5283 0.52750.05 36.67 36.68 0.6558 0.65460.06 37.08 37.09 0.7816 0.77990.07 37.43 37.44 0.9057 0.90340.08 37.74 37.75 1.0281 1.0250.09 38.01 38.02 1.1488 1.1450.1 38.25 38.26 1.2679 1.2630.2 39.89 39.91 2.3756 2.3600.3 40.89 40.91 3.3443 3.3170.4 41.62 41.64 4.1969 4.1580.5 42.20 42.22 4.9489 4.9030.6 42.69 42.71 5.6145 5.5650.7 43.10 43.12 6.2054 6.1570.8 43.46 43.48 6.7317 6.6900.9 43.79 43.80 7.2020 7.1711.0 44.08 44.08 7.0236 7.6082.0 46.05 45.96 10.181 10.453.0 47.22 47.03 11.318 11.904.0 48.05 47.78 11.928 12.795.0 48.70 48.36 12.300 13.386.0 49.22 48.82 12.541 13.807.0 49.67 49.21 12.711 14.128.0 50.05 49.55 12.836 14.379.0 50.38 49.84 12.93 14.5710.0 50.68 50.10 13.033 14.73 In the following it is convenient to use the variable x = 1 X ( t ) (3 . R = T L x x + 1 . (3 . x is directly related to the redshift (2.8) x = α (1 + z ) (3 . α is the parameter (2.12). Due to (2.10) T L is proportional to the Hubble time: T L = ( α + 1) H α . (3 . g µν = diag(1 , − /X , − /R , − /R sin ϑ ) . (3 . = X ˙ X = − ˙ xx (3 . = R ˙ R = 2 T L x ( x + 1) ˙ x (3 . = 2 T L x ( x + 1) ˙ x sin ϑ (3 . = ˙ XX = − ˙ xx (3 . = Γ = ˙ RR = 2 ˙ xx ( x + 1) (3 . = − sin ϑ cos ϑ, Γ = cot ϑ. (3 . h µν dx µ dx ν = h AB ( x C ) Y ( ϑ, φ ) dx A dx B + h A ( x C ) Y, a ( dx A dx a + dx a dx A )++ R [ KY ( ϑ, φ ) γ ab + G ( Y, a | b + l l + 1) Y ( ϑ, φ ) γ ab )] dx a dx b . (3 . x = t and x = r and small Latin indices refer to the angles ϑ and φ , Y ( ϑ, φ ) are the spherical harmonics where the indices l and m = − l, . . . + l are always omittedbecause they are decoupled on the spherically symmetric background. The comma always meanspartial derivatives and the vertical bar denotes covariant derivatives on M (spanned by x C , C = 0 , ϑ, φ with metric tensor γ ab ), respectively [4]. Furthermore wehave shown in [1] that only the following components are different from zero h = − x H , h = H h = − x H , R K = T L (cid:16) x x + 1 (cid:17) K (3 . h µν coincides with the gauge invariant perturbations k µν which are needed for the CMB calculations [2]. After Fourier transform in the radial coordinateˆ f ( x, q ) = (2 π ) − / Z f ( x, r ) e iqr dr (3 . H , ˆ H and ˆ K are solutions of the differential equations [1]( x + x ) ˆ K ′ = − ( x + 3 x ) ˆ K − x ˆ H − l ( l + 1)( x + 1) ˆ H ( x +2 x + x ) ˆ H ′ = − ( x +4 x +3 x ) ˆ K +(2 x +6 x +4 x ) ˆ H − [ Q x + l ( l +1)( x +2 x +1)] ˆ H (3 . x + 2 x + x ) ˆ H ′ = 2 x ( ˆ K + ˆ H ) + 2( x + 2 x + 1) ˆ H . x and we have introduced the quantitiesˆ H = ˆ H iqT L (3 . Q = 2 q T L (3 . q is the wave number.To calculate the metric perturbations one has to integrate the system (3.15) from last scattering z = 1090 to present time z = 0, that means from x = 1091 α to x = α = 2 .
59. It is important tonotice that (i) there is no m -dependence in (3.15) and (ii) the dependence on l appears in the form l ( l + 1). This has two consequences: (i) If the initial condition for (3.15) were axially symmetric thenCMB must be axisymmetric around the direction of the dipole anisotropy. (ii) One should studyCMB observables as functions of l ( l + 1) instead of simply l . We shall return to this point in Sect.5.Regarding (i) we do not know whether the degree of axisymmetry of CMB has been analyzed. Weinvestigate this in the Appendix by transforming CMB data to the cosmic rest frame. We assume that the observer of CMB sits at the origin r = 0 in the cosmic rest frame. That means itscomoving coordinates coincide with the cosmic rest frame. Measurements on Earth must be correctedfor the motion of the Earth. Then CMB radiation arrives on radial null geodesics. Let K µ = dx µ dλ (4 . λ is an affine parameter. The radial wave vector has only twonon-vanishing components A = 0 , K A K A = ( K ) − X ( K ) = 0 (4 . K A | B K B = ( K A , B +Γ ABC K C ) K B = 0 . (4 . A = 0 reads K , K + K , K + x ˙ X ( K ) = 0 . Here we insert (4.2) K = ± K X (4 . K , ± X K , + ˙ XX K = 0 . (4 . A = 1 component gives the same equation. Since X ( t ) does not depend on r , the general solutionis K = CX , K = ± Cx . (4 . dλ = 1 K dx + 1 k dx = Cx dt − Cx dr. dr = − xdt we have dλ = 2 Cx dt. (4 . C = 1 / K A = x , − x ) . (4 . λ we shall use the physical variable x (3.3). The relation between thetwo is given by the auxiliary variable w x = | tan w | . (4 . w is connected with the comoving time t by (2.4) t = T L ( w − sin w cos w ) . (4 . dt = 2 T L sin wdw (4 . w by tan w which gives x (4.9) and insert dw = dx/ ( x + 1). Then we obtain dλ = 2 T L x ( x + 1) dx. (4 . λ can be identified with R ( t ) (3.2). For the Christoffel symbols (3.6) we also need˙ x = 12 T L sin w cos w = ( x + 1) T L x . (4 . T = X lm a lm Y ml ( ϑ, φ ) . (4 . a lm as an integral over the light path from last scattering λ e to the present λ f : a lm = − λ f Z λ e dλK h g D ( k DB | C + k DC | B − k BC | D ) −− k BC K | + Γ D K D K i K B K C . (4 . k = h = − x H ( x, r ) (4 . k = h = iQ √ H ( x, r ) (4 . k = h = − x H ( x, r ) . (4 . ∂ = ∂/∂r produces a factor − iq = − i Q √ T L (4 . ∂ = 12 T L sin w ∂∂w = ( x + 1) T L x ∂∂x . (4 . k | = ∂ k = − ( x + 1) T L x ∂ x (cid:16) H x (cid:17) k | = ∂ k + 2 ˙ xx k = iQ √ T L x (cid:16) H + ( x + 1) x H (cid:17) (4 . k | = ∂ k + ˙ xx k = iQ √ x + 1) T L x (cid:16) H ′ + H x (cid:17) k | = ∂ k + ˙ xx k + ˙ xx k = Q T L H − ( x + 1) T L x H (cid:16) x (cid:17) k | = ∂ k + 2 ˙ xx k = − ( x + 1) T L (cid:16) H ′ + 4 x H (cid:17) k | = ∂ k + 2 ˙ xx k = iQ √ T L (cid:16) x H + ( x + 1) x H (cid:17) . We note that the third function K does not contribute, consequently, a glance to (3.12) shows thatthe angular dependence is really given by the simple spherical harmonics Y ml ( ϑ, φ ) as in (4.14). Forsimplicity we have omitted the hats for radial Fourier transform.Now using (4.8) the first term under the integral (4.15) becomes x k | − x k | + x k | + x k | − x k | == H ′ ( x + 1) T L (cid:16) x − x (cid:17) + H ( x + 1) T L x − iQx √ T L H ++ H ′ ( x + 1) √ T L iQx + H T L (cid:16) Q x − iQ √ x + 1) (cid:17) . (4 . K | + Γ D K D = ˙ xx = ( x + 1) T L x which must be divided by K = − g x / / k BC K B K C = − H x − − H iQx √ . In the final formula we must also include the Fourier transformation, the integral dλ is transformedby (4.12) into an integral over x . Then we obtain the following formula a lm = (2 π ) − / x f Z x e dx Z dq e − iqr x ( x + 1) n ˆ H h iQ √ x + 1) (cid:16) x x (cid:17)i −− ˆ H ′ x + 1) (cid:16) x − x (cid:17) + ˆ H h iQ √ x + 1) x − Q x i − ˆ H ′ iQ √ x + 1) o . (4 . r in the exponential must be derived from drdt = − X = − x Using (4.13) we have drdx = − T L x ( x + 1) (4 . r = x Z α T L x ′ ( x ′ + 1) dx ′ == T L (cid:16) log x + 1 α + 1 + 1 x + 1 − α + 1 (cid:17) . (4 . H ′ and ˆ H ′ in (4.23) can be integrated by parts in x . Since x e = 1091 α is verybig, the main contribution comes from the term a SW lm = (2 π ) − / x e ∞ Z dq cos qr e ˆ H ( x e )2 (4 . H is even in q . Thiscontribution is conventionally called the Sachs-Wolfe effect because it only depends on the initialconditions at last scattering x e . The contribution of ˆ H ′ is smaller, and all other integral terms givethe so-called integrated Sachs-Wolfe effect. The functions ˆ H , ˆ H and ˆ K appearing in (4.23) have been calculated in [1] as the following powerseries in x − : ˆ H = σ ml ( q ) h x (cid:16) Q + 9 − (cid:17) + 1 x (cid:16) l ( l + 1) + 2102 Q + 25 − l ( l + 1) + 542 Q + 9 − (cid:17)i . (5 . H = σ ml ( q ) h − x Q + 9 + 1 x (cid:16) l ( l + 1) + 362 Q + 9 − l ( l + 1) + 842 Q + 25 (cid:17)i (5 . K = σ ml ( q ) h x + 1 x (cid:16) − l ( l + 1) + 122 Q + 9 (cid:17)i (5 . σ ml ( q ) is the so-called spectral function which specifies some normalization and initial conditionfor the ordinary differential equations (3.15). These power series must be used as initial conditionsfor big enough x . Then the functions can be calculated for small x by numerical integration ofthe ordinary differential equations (3.15). The main problem that remains is the determination ofthe spectral function σ ml ( q ). In standard cosmology the corresponding quantity is assumed to be astochastic variable with a simple covariance like h σ ml ( q ) σ m ′ l ′ ( q ′ ) i = δ ll ′ δ mm ′ δ ( q − q ′ ) . (5 . σ ml is not stochastic. Instead it is a distinguished function which contains information about theinitial conditions at the early Universe and the history of the Universe, and about our special placein the Milky Way, and also about the motion of the earth. Then σ ml cannot be determined by theoryalone, it must be derived from other observations. For this purpose the matter density is well suited.By equ. (4.21) of [1] its Fourier transform ˆ ̺ ( q ) directly gives the spectral function in (5.1-3):16 πGR ˆ ̺ ml ( q ) = H (cid:16) x + ( l − l + 2) (cid:17) − K h ( l − l + 2) − Q x ( x + 1) + ( x + 1) (cid:16)
12 + 32 x (cid:17)i − + H h Q x x + 1 − l ( l + 1) ( x + 1) x i . (5 . l only. This is no harm because the surveys as theSloan Digital Sky Survey give the matter distribution for small redshift only which corresponds tosmall l values. So in non-standard cosmology the small l region is most interesting where standardcosmology has nothing to say. The comparison of CMB with matter density measurements can help tosolve the problem of the dark stuff. In all fairness we see that the more difficult density measurement(5.5) is more valuable than the CMB data (4.23), because the latter is only an integrated quantity.However, for (5.5) the density must be measured in more detail as a function of redshift and angles. Weemphasize that lensing and rotation curves in galaxies test the gravitational fields only ([5], Sect.6.4), not its sources . Until now the only real tests of dark matter are the direct searches with undergroundparticle detectors. Then, until now, dark matter does not exist. As a consequence, on galactic scalesthe gravitational field strongly deviates from Newtonian gravity [5]. But there is no need for modifyingEinstein’s theory. On the other hand the density (5.5) is intimately connected with the non-standardbackground (3.1), not as a source but as a first order anisotropic perturbation. The smallness of thebaryonic matter density and the small CMB multipole coefficients for l ≤ A Appendix: Transformation of CMB to the cosmic rest frame
In the analysis of CMB data this transformation is simply carried out by subtracting the dipoleanisotropy. This is a good approximation because the motion of the earth is slow. The exact trans-formation requires the transformation of spherical harmonics by means of rotation matrices. Thetemperature anisotropy in galactic coordinates ( ϑ, φ ) is given by∆ T l = l X m = − l a lm Y ml ( ϑ, φ ) . ( A. ϑ ′ , φ ′ ) according to Y ml ( ϑ, φ ) = X m ′ R mlm ′ ( α, β, γ ) + Y m ′ l ( ϑ ′ , φ ′ ) ( A. R mlm ′ ( α, β, γ ) = e im ′ α d mlm ′ ( β ) e imγ ( A. α, β, γ ) the Euler angles of the rotation between the two systems. The(2 l + 1) × (2 l + 1) matrix d is given by [6] d mlm ′ ( β ) = h ( l + m ′ )!( l − m ′ )!( l + m )!( l − m )! i / (cos β m ′ + m (sin β m ′ − m × P ( m ′ − m,m ′ + m ) l − m ′ (cos β ) ( A. P are the Jacobi polynomials. The upper plus in (A.2) denotes the hermitian conjugate rotationmatrix which transforms the galactic multipole moments a lm into a ′ lm ′ in the cosmic rest frame. By(A.1-2) the multipole moments a ′ lm ′ in the cosmic rest frame are given by a ′ lm ′ = X m R m ∗ lm ′ a lm . ( A. a lm for l = 1 , p (2 l + 1) / π . As a consequenceour moments a lm are obtained from the WMAP values by a lm = 1 √ a lm + i ˜ a l − m . ) ( A. m , for negative m we use a l − m = ( − ) m a ∗ lm . ( A. µ K)( x, y, z ) = ( − . , − . , .
0) ( A. ~n = ( − . , − . , . . ( A. z ′ -axis. The Euler angle β is the angle between the old and thenew z -axis, hence cos β = n = 0 . , β = 0 . . ( A. α remains arbitrary which is the freedom of defining the meridian φ ′ = 0 in the new frame.We choose α = 0. But γ is fixed by the requirement that the dipole has vanishing x ′ , y ′ componentsin the cosmic rest frame.For the dipole l = 1 the rotation matrix (A.3-4) is given by R mm ′ = (1 + cos β ) e iγ − √ βe iγ (1 − cos β ) e iγ √ β cos β − √ β (1 − cos β ) e − iγ √ βe − iγ (1 + cos β ) e − iγ ( A. a = − √ . i .
6) = a ∗− , a = 2505 . . ( A. a ′− = 0 = a ′ gives the two equations a
12 (1 + cos β ) e iγ + a √ β + a −
12 (1 − cos β ) e − iγ = 0 a
12 (1 + cos β ) e − iγ − a √ β + a −
12 (1 − cos β ) e iγ = 0 ( A. γ γ = 4 . . ( A. m ′ = 0 component a ′ = −√ β Re (cid:16) e − iγ a (cid:17) + a cos β = 3358 . . ( A. T = a ′ Y = a ′ cos ϑ ′ . ( A. v /c = β gives rise to a Lorentz boostwhich changes the temperature according to [8] T ′ = Tγ (1 + β cos ϑ ′ )where T = 2 . K is the CMB mean temperature and γ = (1 − β ) − / . Expanding this in powersof β gives an anisotropy.∆ T = T ′ − T = T h − β cos ϑ ′ − β β P (cos ϑ ′ ) + O ( β ) i . ( A. β = 0 . , v = 369km / s ( A. × l = 2 is equal to e iγ β ) ; e iγ β (1 + cos β ); √ e iγ β ; e iγ β (1 − cos β ); e iγ − cos β ) e iγ − β (1 + cos β ); e iγ β − β ); r e iγ sin β cos β ; e iγ β + 1)(1 − cos β ); e iγ β (1 − cos β ) √
64 sin β ; − r
32 sin β cos β ; 32 cos β −
12 ; r
32 sin β cos β ; √
64 sin βe − iγ − β (1 − cos β ); e − iγ − cos β )(1 + 2 cos β ); − r e − iγ sin β cos β ; e − iγ β − β ); e − iγ β (1 + cos β ) e − iγ − cos β ) ; e − iγ − β (1 − cos β ); √ e − iγ β ; e − iγ − β (1 + cos β ); e − iγ β ) ! . ( A. a − = 1 √ − .
41 + i .
80) = a ∗ a − = 1 √ .
05 + i .
86) = − a ∗ a = 11 . . ( A. a ′ = a ′− = 14 .
76 + i . , a ′ = − a , ∗− = − . i . , a ′ = 6 . . ( A. a ′ gets modified by the contribution P (cos ϑ ′ ) = Y from the Lorentz boost (A.17), we mustsubtract the quantity T β = 2 . µ K . ( A. × l = 3 is equal to (cos β = c , sin β = s ) e iγ c ) ; e iγ − √ s (1 + c ) ; √ e iγ s (1 + c ); e iγ − √ s ; e iγ √ s (1 − c ); e iγ √ s (1 − c ) : e iγ − c ) e iγ √ s (1 + c ) ; e iγ c − c ) ; r e iγ s (1 + c )(1 − c ); e iγ √ c (1 − c ); e iγ r s (1 − c )(3 c + 1); e iγ − c ) (3 c + 2); e iγ √ s (1 − c ) √ e iγ s ( c + 1); r e iγ s (1 + c )(3 c − e iγ c )(15 c − c − √ e iγ s (1 − c ); e iγ − c )(15 c + 10 c − r e iγ s (1 − c )(3 c + 1); √ e iγ s (1 − c ) √ s ; √
304 (1 − c ) c ; √ s (5 c − c − c ; √ s (1 − c ); √
304 (1 − c ) c ; − √ s √ e − iγ s (1 − c ); r e − iγ − s ( c − c ); e − iγ − c )(15 c + 10 c − √ e − iγ s (5 c − e − iγ c )(15 c − c − r e − iγ − s (1 + c )(1 − c ); √ e − iγ s (1 + c ) √ e − iγ s (1 − c ) ; e − iγ − c ) (3 c + 2); r e − iγ − s ( c − c + 1); √ e − iγ − c ) c ; r e − iγ − s (1 + c )(3 c − e − iγ c ) (3 c − √ e − iγ s (1 + c ) e − iγ − c ) ; √ e − iγ s (1 − c ) ; √ e − iγ s (1 − c ); √ e − iγ s e − iγ s (1 + c ); √ e − iγ − s (1 + c ) ; e − iγ c ) ! . ( A. a − = 1 √ .
24 + i .
46) = − a ∗ , a − = 1 √ . − i .
70) = a ∗ a − = 1 √ .
05 + i .
45) = − a ∗ , a = − . . Again to get the value in the cosmic rest frame we have to multiply this by the columns of (A.23)complex conjugated yielding a ′− = − .