Considering light-matter interactions in Friedmann equations
CConsidering light-matter interactions in Friedmann equations
V. Vavryˇcuk
The Czech Academy of SciencesBoˇcn´ı II 1401, 141 00 Praha 4 [email protected]
ABSTRACT
The Friedmann equations valid for the transparent universe are modified for the universe withopacity caused by absorption of light by ambient cosmic dust in intergalactic space. The modifiedequations lead to a cosmological model, in which cosmic opacity produces radiation pressure thatcounterbalances gravitational forces. The proposed model predicts a cyclic expansion/contractionevolution of the Universe within a limited range of scale factors with no initial singularity. Themaximum redshift, at which the contraction of the Universe stops, is z ≈ Subject headings: early universe – cosmic background radiation – dust, extinction – universe opacity –dark energy
1. Introduction
Dust is an important component of the inter-stellar and intergalactic medium, which interactswith the stellar radiation. Dust grains absorb andscatter the starlight and reemit the absorbed en-ergy at infrared, far-infrared and microwave wave-lengths (Mathis 1990; Schlegel et al. 1998; Calzettiet al. 2000; Draine 2003, 2011; Vavryˇcuk 2018).Since galaxies contain interstellar dust, they losetheir transparency and become opaque (Calzetti2001; Holwerda et al. 2005, 2007; Finkelman et al.2008; Lisenfeld et al. 2008). Similarly, the Uni-verse is not transparent but partially opaque dueto ambient cosmic dust. The cosmic opacity isvery low in the local Universe (Chelouche et al.2007; Muller et al. 2008), but it might steeplyincrease with redshift (M´enard et al. 2010b; Xieet al. 2015; Vavryˇcuk 2017b).The fact that the Universe is not transpar-ent but partially opaque might have fundamentalcosmological consequences, because the commonlyaccepted cosmological model was developed for thetransparent universe. Neglecting cosmic opacityproduced by intergalactic dust may lead to dis-torting the observed evolution of the luminositydensity and the global stellar mass density with redshift (Vavryˇcuk 2017b). Non-zero cosmic opac-ity may invalidate the interpretation of the TypeIa supernova (SNe Ia) dimming as a result of darkenergy and the accelerating expansion of the Uni-verse (Aguirre 1999a,b; Aguirre & Haiman 2000;M´enard et al. 2010a; Vavryˇcuk 2019). Intergalac-tic dust can partly or fully produce the cosmic mi-crowave background (CMB) (Wright 1982; Bondet al. 1991; Narlikar et al. 2003). For example,Vavryˇcuk (2018) showed that thermal radiation ofdust is capable to explain the spectrum, intensityand temperature of the CMB including the CMBtemperature/polarization anisotropies.If cosmic opacity and light-matter interactionsare considered, the Friedmann equations must bemodified and the radiation pressure caused by ab-sorption of photons by dust grains must be incor-porated. Based on numerical modeling and obser-vations of basic cosmological parameters, I showthat the modified Friedmann equations avoid theinitial singularity and lead to a cyclic model ofthe Universe with expansion/contraction epochswithin a limited range of scale factors.1 a r X i v : . [ phy s i c s . g e n - ph ] J un . Theory2.1. Friedmann equations for the trans-parent universe The standard Friedmann equations read (Pea-cock 1999, p. 665) (cid:18) ˙ aa (cid:19) = 8 πG ρ − kc a , (1)¨ aa = − πG (cid:18) ρ + 3 pc (cid:19) , (2)where a = R/R = (1 + z ) − is the relative scalefactor, G is the gravitational constant, ρ is themass density, k/a is the spatial curvature of theuniverse, p is the pressure, and c is the speed oflight. Considering the mass density ρ as a sum ofmatter and radiation contributions and includingthe vacuum contribution, we get8 πG ρ = H (cid:2) Ω m a − + Ω r a − + Ω Λ (cid:3) . (3)Eq. (1) is then rewritten as H ( a ) = H (cid:2) Ω m a − + Ω r a − + Ω Λ + Ω k a − (cid:3) , (4)with the conditionΩ m + Ω r + Ω Λ + Ω k = 1 , (5)where H ( a ) = ˙ a/a is the Hubble parameter, H is the Hubble constant, and Ω m , Ω r , Ω Λ and Ω k are the normalized matter, radiation, vacuum andcurvature terms. Assuming Ω r = 0 and Ω k = 0 inEq. (4), we get the ΛCDM model H ( a ) = H (cid:2) Ω m a − + Ω Λ (cid:3) , (6)which describes a flat, matter-dominated universe.The universe is transparent, because any interac-tion of radiation with matter is neglected. Thevacuum term Ω Λ is called dark energy and it isresponsible for the accelerating expansion of theUniverse. The dark energy is introduced into Eqs(3-5) to fit the ΛCDM model with observations ofthe Type Ia supernova dimming. The basic drawback of the ΛCDM model is itsassumption of transparency of the Universe and neglect of the universe opacity caused by interac-tion of light with intergalactic dust. Absorptionof light by cosmic dust produces radiation pres-sure acting against the gravity, but this pressureis ignored in the ΛCDM model.Let us assume a space filled by light and cos-mic dust formed by uniformly distributed spheri-cal dust grains. The dust grains absorb photonsand reemit them in the form of thermal radiation.The total force produced by absorption of pho-tons, which acts on dust in a unit volume of theUniverse, is M D ¨ R = S D p D , (7)where M D and S D are the mass and surface of alldust grains in the spherical volume of radius R ,and p D is the radiation pressure caused by dustabsorption of the extragalactic background light(EBL) present in the cosmic space p D = λc I EBL , (8)where λ is the bolometric cosmic opacity (definedas attenuation per unit raypath), and I EBL is thebolometric intensity of the EBL, which dependson redshift as (Vavryˇcuk 2018, his eq. 5) I EBL = I EBL0 (1 + z ) , (9)where subscript ’0’ means the quantity at z = 0.Since the production and absorption of photonsshould be in balance, the EBL intensity I EBL0 isrelated to the luminosity density j at z = 0 as(Vavryˇcuk 2018, his eq. 7) λ I EBL0 = j π . (10)If the comoving number density of dust grains isconstant, the opacity λ in Eq. (8) is redshift in-dependent, λ = λ (the proper attenuation coeffi-cient per unit ray path increases with z , but theproper length of a ray decreases with z ). Hence,the pressure p D in Eq. (8) reads p D = j πc (1 + z ) . (11)Inserting Eq. (11) into Eq. (7) and substituting R by the relative scale factor a = R/R , we obtain¨ a = S D M D j πc a , (12)2here R = 1. Integrating Eq. (12) in time (cid:18) ˙ aa (cid:19) = S D M D j πc a , (13)and including absorption terms defined in Eqs (12-13) into Eqs (1-2), we get a new form of the Fried-mann equations valid for a model of the opaqueuniverse (cid:18) ˙ aa (cid:19) = 8 πG ρ − S D M D j πc a − kc a , (14)¨ aa = − πG (cid:18) ρ + 3 p D c (cid:19) + S D M D j πc a , (15)which read for dust formed by spherical grains as (cid:18) ˙ aa (cid:19) = 8 πG ρ − πc j ρ D R D a − kc a , (16)¨ aa = − πG ρ + 34 πc j ρ D R D a , (17)where R D and ρ D are the radius and the spe-cific density of dust grains. In Eq. (17), we omitgravity forces produced by pressure p D , becausethey are negligible with respect to the other terms.Consequently, the Hubble parameter reads H ( a ) = H (cid:2) Ω m a − + Ω r a − + Ω a a − + Ω k a − (cid:3) , (18)which simplifies for a matter-dominated opaqueuniverse (Ω r = 0) as H ( a ) = H (cid:2) Ω m a − + Ω a a − + Ω k a − (cid:3) , (19)with the conditionΩ m + Ω a + Ω k = 1 , (20)where Ω m , Ω a and Ω k are the normalized gravity,absorption and curvature terms, respectively,Ω m = 1 H (cid:18) πGρ (cid:19) , (21)Ω a = − H (cid:18) πc j ρ D R D (cid:19) , (22)Ω k = − kc H . (23)The minus sign in Eq. (22) means that the radi-ation pressure due to absorption acts against the gravity. The dark energy is missing in Eqs (18-20), because the Type Ia supernova dimming cansuccessfully be explained by cosmic opacity, as dis-cussed in Vavryˇcuk (2019).Eq. (19) shows that the increase of the ab-sorption term Ω a with redshift is enormously high.The reasons for such a steep rise of Ω a with z are,however, straightforward. The steep rise combinesthe three following effects: (1) the increase of pho-ton density with (1 + z ) due to the space contrac-tion, (2) the increase of absorption of photons with(1 + z ) due to the shorter distance between dustgrains, and (3) the increase of rate of absorbedphotons by dust grains with (1 + z ) due to timedilation. The scale factor a of the Universe with the zeroexpansion rate is defined by the zero Hubble pa-rameter in Eq. (19), which yields a cubic equationin a Ω k a + Ω m a + Ω a = 0 . (24)Taking into account that Ω m > a <
0, Eq.(24) has two distinct real positive roots for (cid:18) Ω m (cid:19) > (cid:18) Ω k (cid:19) | Ω a | and Ω k < . (25)Negative Ω a and Ω k imply thatΩ m > ρ > ρ c = 8 πG H . (26)Under these conditions, Eq. (19) describes a uni-verse with a cyclic expansion/contraction historyand the two real positive roots a min and a max de-fine the minimum and maximum scale factors ofthe Universe. For Ω a (cid:28)
1, the scale factors a min and a max read approximately a min ∼ = (cid:115)(cid:12)(cid:12)(cid:12)(cid:12) Ω a Ω m (cid:12)(cid:12)(cid:12)(cid:12) and a max ∼ = (cid:12)(cid:12)(cid:12)(cid:12) Ω m Ω k (cid:12)(cid:12)(cid:12)(cid:12) , (27)and the maximum redshift is z max = 1 a min − . (28)The scale factors a of the Universe with themaximum expansion/contraction rates are definedby dda H ( a ) = 0 , (29)3hich yields a cubic equation in a k a + 3Ω m a + 5Ω a = 0 . (30)Taking into account Eq. (17) and Eqs (21-23),the deceleration of the expansion reads¨ a = − H (cid:2) Ω m a − + 3Ω a a − (cid:3) . (31)Hence, the zero deceleration is for the scale factor a = (cid:115)(cid:12)(cid:12)(cid:12)(cid:12) a Ω m (cid:12)(cid:12)(cid:12)(cid:12) . (32)Finally, the comoving distance as a function ofredshift is expressed from Eq. (19) as follows dr = cH dz (cid:113) Ω m (1 + z ) + Ω a (1 + z ) + Ω k (1 + z ) . (33)
3. Modeling3.1. Parameters for modeling
For calculating the expansion history and cos-mic dynamics of the Universe, we need observa-tions of intergalactic dust grains, the galaxy lu-minosity density, the mean mass density, and theexpansion rate and curvature of the Universe atthe present time.The size a of dust grains is in the range of0 . − . µ m with a power-law distribution a − q with q = 3 . a ≈ . µ m(Draine & Fraisse 2009; Draine 2011). The grainsof size 0 . µ m ≤ a ≤ . µ m are also ejectedto the IGM most effectively (Davies et al. 1998;Bianchi & Ferrara 2005). The grains form com-plicate fluffy aggregates, which are often elon-gated or needle-shaped (Wright 1982, 1987). Con-sidering that the density of carbonaceous mate-rial is ρ ≈ . − , and the silicate density is ρ ≈ . − (Draine 2011), the average densityof porous dust grains is ≈ − or less (Flynn1994; Kocifaj et al. 1999; Kohout et al. 2014).The galaxy luminosity density is determinedfrom the Schechter function (Schechter 1976).It has been measured by large surveys 2dFGRS(Cross et al. 2001), SDSS (Blanton et al. 2001, 2003) or CS (Brown et al. 2001). The lumi-nosity function in the R-band was estimated at z = 0 to be (1 . ± . × h L (cid:12) Mpc − for the SDSS data (Blanton et al. 2003) and(1 . ± . × h L (cid:12) Mpc − for the CS data(Brown et al. 2001). The bolometric luminositydensity is estimated by considering the spectral en-ergy distribution (SED) of galaxies averaged overdifferent galaxy types, being thus 1 . − . j = 2 . − . × h L (cid:12) Mpc − .The Hubble constant H is measured bymethods based on the Sunyaev-Zel’dovich effect(Birkinshaw 1999; Bonamente et al. 2006) or grav-itational lensing (Suyu et al. 2013; Bonvin et al.2017), gravitational waves (Vitale & Chen 2018;Howlett & Davis 2020) or acoustic peaks in theCMB spectrum provided by Planck Collaborationet al. (2016), and they yield values mostly rang-ing between 66 and 74 km s − Mpc − . Here I usean estimate H . ± . − Mpc − of H obtained by Freedman et al. (2019) using the SNeIa with a red giant branch calibration.Assuming the ΛCDM model, the CMB andBAO observations indicate a nearly flat Universe(Planck Collaboration et al. 2016). This methodis not, however, model independent and ignoresan impact of cosmic dust on the CMB. A model-independent method proposed by Clarkson et al.(2007) is based on reconstructing the comovingdistances by Hubble parameter data and compar-ing with the luminosity distances (Li et al. 2016;Wei & Wu 2017) or the angular diameter distances(Yu & Wang 2016). The cosmic curvature can alsobe constrained using strongly gravitational lensedSNe Ia (Qi et al. 2019) and using lensing time de-lays and gravitational waves (Liao 2019). The au-thors report the curvature term Ω k ranging be-tween -0.3 to 0 indicating a closed universe, notsignificantly departing from flat geometry. Estimating the required cosmological param-eters from observations (see Table 1), I calcu-late the upper and lower limits of the volumeof the Universe and the evolution of the Hubbleparameter with time. The mass density of theUniverse higher than the critical density is con-sidered, and subsequently Ω m is higher than 1.The Hubble constant is H = 69 . − Mpc − ,4able 1: Maximum redshift and scale factor in the cyclic model of the opaque universeModel ε c S / M j Ω m Ω a Ω k a max z max (10 h L (cid:12) Mpc − )A 20 4.0 3.8 1.2 − . × − − .
192 6.2 11.8B 5 1.9 2.5 1.2 − . × − − .
198 6.1 21.8C 15 3.3 3.1 1.2 − . × − − .
195 6.1 14.7D 15 3.3 3.1 1.1 − . × − − .
095 11.6 14.1E 15 3.3 3.1 1.3 − . × − − .
295 4.4 15.3Parameter ε is the ratio of the major to minor axis of the prolate spheroidal dust grains, c S / M is thecorrection for the S/M ratio of the spheroidal to spherical dust grains, j is the bolometric luminositydensity at z = 0, Ω m , Ω a , and Ω k are the matter, absorption and curvature terms, and a max and z max arethe maximum scale factor and redshift, respectively. Models A, B and C predict low, high and optimumvalues of z max . Models E, D and C predict low, high and optimum values of a max . z Fig. 1.— Maximum redshift as a function of Ω m and Ω a .5 ) E C D a) Redshift z Hubble parameter - past
A C B
Fig. 2.— The evolution of the Hubble parameter with redshift in the past and with the scale factor in thefuture (in km s − Mpc − ). (a) The blue dashed, dotted and solid lines show Models A, B and C in Tab. 2.(b) The blue solid, dashed, and dotted lines show Models C, D and E in Tab. 2. The black dotted lines markthe predicted maximum redshifts (a) and maximum scale factors (b) for the models considered. The blackdot denotes the state in C when the deceleration of the expansion is zero. The dot is not at the maximumof H ( z ) because the zero deceleration is with respect to time but not with respect to z . The red solid lineshows the flat ΛCDM model with H = 69 . − Mpc − , taken from Freedman et al. (2019), and withΩ m = 0 . Λ = 0 . A C B
Fig. 3.— Comoving distance as a function of redshift z . The blue dashed, dotted and solid lines showModels A, B and C in Tab. 2. The black dotted lines mark the predicted maximum redshifts for the modelsconsidered. The red solid line shows the flat ΛCDM model with H = 69 . − Mpc − , taken fromFreedman et al. (2019), and with Ω m = 0 . Λ = 0 . − . Parameter Ω a varies from − . × − to − . × − depending on the lumi-nosity density j and the spheroidal shape of thedust grains (see Eq. (22) and Table 1).As seen in Fig. 1, the maximum redshift of theUniverse depends mostly on Ω a , and ranges from11.5 to 21.3. The maximum redshift z max calcu-lated approximately by Eqs (26-27) has an accu-racy higher than 1% compared to the exact solu-tion of Eq. (24). In contrast to a min dependingmostly on Ω a , the maximum scale factor a max ofthe Universe depends primarily on Ω m . The lim-iting value is Ω m = 1, when a max is infinite. ForΩ m = 1 .
1, 1.2, 13 and 1.5, the scale factor a max is11.6, 6.5, 4.4 and 3.0, respectively.The history of the Hubble parameter H ( z ) andits evolution in the future H ( a ) calculated by Eq.(19) is shown in Fig. 2 for five scenarios summa-rized in Table 1. As mentioned, the form of H ( z )is controlled by Ω a (Fig. 2a), while the form of H ( a ) is controlled by Ω m (Fig. 2b). The Hub-ble parameter H ( z ) increases with redshift up toits maximum. After that the function rapidly de-creases to zero. The drop of H ( z ) is due to a fastincrease of light attenuation producing strong re-pulsive forces at high redshift. For future epochs,function H ( a ) is predicted to monotonously de-crease to zero. The rate of decrease is controlledjust by gravitational forces; the repulsive forcesoriginating in light attenuation are negligible. Fora comparison, Fig. 2 (red line) shows the Hubbleparameter H ( a ) for the standard ΛCDM model(Planck Collaboration et al. 2016), which is de-scribed by Eq. (6) with Ω m = 0 . Λ = 0 . H ( z ) attains its maximum (see the blackdot in Fig. 2a). The redshift of the zero decel-eration is about 2/3 of the maximum achievableredshift.The distance-redshift relation for the proposedcyclic model of the Universe is quite different fromthe standard ΛCDM model (see Fig. 3). In bothmodels, the comoving distance monotonously in-creases with redshift, but the redshift can go pos-sibly to 1000 or more in the standard model, whilethe maximum redshift is likely 14-15 in the cyclicmodel. The increase of distance with redshift is remarkably steeper for the ΛCDM model than forthe cyclic model. The ratio between distances inthe cyclic and ΛCDM models is about 0.54.
4. Other supporting evidence
The cyclic cosmological model of the opaqueuniverse successfully removes some tensions of thestandard ΛCDM model: • The model does not limit the age of starsin the Universe. For example, observationsof a nearby star HD 140283 (Bond et al.2013) with age of 14 . ± .
31 Gyr are in con-flict with the age of the Universe, 13 . ± .
02 Gyr, determined from the interpreta-tion of the CMB as relic radiation of the BigBang (Planck Collaboration et al. 2016). • The model predicts the existence of veryold mature galaxies at high redshifts. Theexistence of mature galaxies in the earlyUniverse was confirmed, for example, byWatson et al. (2015) who analyzed observa-tions of the Atacama Large Millimetre Ar-ray (ALMA) and revealed a galaxy at z > z ≈ ≈ × M (cid:12) and a dustmass of ≈ × M (cid:12) . A large amount ofdust is reported by Venemans et al. (2017)for a quasar at z = 7 . z ≈
11 wasfound by Oesch et al. (2016) and a signifi-cant increase in the number of galaxies for8 . < z <
12 was reported by Ellis et al.(2013). Note that the number of papers re-porting discoveries of galaxies at z ≈
10 orhigher is growing rapidly (Hashimoto et al.2018; Hoag et al. 2018; Oesch et al. 2018;Salmon et al. 2018). • The model is capable to explain the SNe Iadimming discovered by Riess et al. (1998)and Perlmutter et al. (1999) without intro-ducing dark energy as the hypothetical en-ergy of vacuum (Vavryˇcuk 2019), which isdifficult to explain under the quantum fieldtheory (Weinberg et al. 2013). Moreover, thespeed of gravitational waves and the speed7f light differ for most of dark energy mod-els (Sakstein & Jain 2017; Ezquiaga & Zu-malac´arregui 2017), but observations of thebinary neutron star merger GW170817 andits electromagnetic counterparts proved thatboth speeds coincide with a high accuracy. • The model avoids a puzzle, how the CMB asrelic radiation could survive the whole his-tory of the Universe without any distortion(Vavryˇcuk 2017a) and why several unex-pected features at large angular scales suchas non-Gaussianity (Vielva et al. 2004; Cruzet al. 2005; Planck Collaboration et al. 2014)and a violation of statistical isotropy andscale invariance are observed in the CMB.
5. Discussion and conclusions
The radiation pressure as a cosmological forceacting against the gravity has not been proposedyet, even though its role is well known in the stel-lar dynamics (Kippenhahn et al. 2012). The ra-diation pressure is important in the evolution ofmassive stars (Zinnecker & Yorke 2007), in super-novae stellar winds and in galactic wind dynamics(Aguirre 1999b; Martin 2005; Hopkins et al. 2012;Hirashita & Inoue 2019). Apparently, the radia-tion pressure in the evolution of the Universe wasoverlooked, because the Universe was assumed tobe transparent. By contrast, the role of radiationpressure is essential in the opaque universe model,because it is produced by absorption of photonsby cosmic dust. Since the cosmic opacity and theintensity of the EBL steeply rise with redshift, theradiation pressure, negligible at present, becomessignificant at high redshifts and can fully eliminategravity and stop the universe contraction.Hence, the expansion/contraction evolution ofthe Universe might be a result of imbalance ofgravitational forces and radiation pressure. Sincethe comoving global stellar and dust masses arebasically independent of time with minor fluctua-tions only, the evolution of the Universe is station-ary. Obviously, the recycling processes of stars andgalaxies (Segers et al. 2016; Angl´es-Alc´azar et al.2017) play a more important role in this modelthan in the standard cosmology.The age of the Universe in the cyclic model isunconstrained and galaxies can be observed at anyredshift less than the maximum redshift z max . The only limitation is high cosmic opacity, which canprevent observations of the most distant galaxies.Hypothetically, it is possible to observe galaxiesfrom the previous cycle/cycles, if their distance ishigher than that corresponding to z max ≈ − REFERENCES
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