Cosmological constraints on the radiation released during structure formation
EEPJ manuscript No. (will be inserted by the editor)
Cosmological constraints on the radiation released duringstructure formation
David Camarena and Valerio Marra
Departamento de Física, Universidade Federal do Espírito SantoAv. F. Ferrari, 514, 29075-910, Vitória, ES, BrazilReceived: date / Revised version: date
Abstract
During the process of structure formation in the universe matter is converted into radiationthrough a variety of processes such as light from stars, infrared radiation from cosmic dust and gravi-tational waves from binary black holes/neutron stars and supernova explosions. The production of thisastrophysical radiation background (ARB) could affect the expansion rate of the universe and the growthof perturbations. Here, we aim at understanding to which level one can constraint the ARB using futurecosmological observations. We model the energy transfer from matter to radiation through an effectiveinteraction between matter and astrophysical radiation. Using future supernova data from LSST andgrowth-rate data from Euclid we find that the ARB density parameter is constrained, at the 95% confi-dence level, to be Ω ar < . Ω ar ∼ − . Therefore, we conclude that cosmological observations will only beable to constrain exotic or not-well understood sources of radiation. The process of structure formation in the universe un-avoidably leads to the production of radiation. Electro-magnetic radiation is emitted during star formation, whichstarted in small halos at redshifts of order z ∼
20 andthen peaked at z ∼ The (twice contracted) Bianchi identity, within GeneralRelativity, implies that the total energy-momentum tensoris conserved. Hence, a possible interaction between mat-ter and astrophysical radiation can be modeled throughan interaction current Q β which transfers energy and mo-mentum from one source to the other with opposite direc-tion: ∇ α T αβm = Q β ∇ α T αβar = − Q β . (1)We will consider a phenomenological description for theeffective interaction between matter and radiation. In par-ticular, we will consider the following simple interaction: Q β = Γ T m u βm , (2) This kind of interaction (and its many variations) has beenstudied in order to model a possible interaction between darkmatter and dark energy (see e.g. [12] and references therein). a r X i v : . [ a s t r o - ph . C O ] N ov David Camarena, Valerio Marra: Cosmological constraints on the radiation released during structure formation where T m = − ρ m is the trace of the matter energy-momentumtensor so that the temporal component is Q = Γ ρ m . Forthe interaction rate we will consider Γ = α H , that is, weuse the Hubble function H = ˙ a/a in order to parametrizethe time dependence of the interaction. Since the energygoes from ρ m to ρ ar we set α >
0. Moreover, as the pro-duction of astrophysical radiation is a recent phenomenonwe will demand that α = 0 for z ≥ ¯ z where ¯ z ∼ The dynamical equations for the background are (a dotdenotes a derivative with respect to cosmic time t ): H = 8 πG ρ m + ρ γ + ρ ar ) + Λ , (3)˙ ρ m + 3 Hρ m = − α H ρ m , (4)˙ ρ ar + 4 Hρ ar = α H ρ m , (5)˙ ρ γ + 4 Hρ γ = 0 , (6)where equations (4-6) are the conservations equations formatter, astrophysical radiation and CMB photons, respec-tively, Λ is the cosmological constant and we have assumedspatial flatness, in agreement with recent cosmological ob-servations (see [14] and references therein).It is clear that this model is but a rough approximationto the actual process of production of radiation. An impor-tant approximation is the use of an overall coupling con-stant to describe different processes which may or may notinvolve both dark matter and baryons. Another approxi-mation comes from the fact that the time dependence ofthe interaction rate Γ is parametrized using the Hubblerate – i.e. according to a cosmological time scale – whileastrophysical processes could evolve on a shorter timescales. For example, the interaction rate could be mod-eled as being proportional to the matter density contrast, Γ = α δ m H , as proposed in [15]. However, the aim ofthis work is to understand if future observations can con-strain the effective coupling α or, equivalently, the energydensity of ARB. More precisely, we would like to know ifcosmological observations can constrain astrophysical ra-diation to the level predicted by astrophysical models ofstar formation, IR background and gravitational wave pro-duction. For such a goal the simple model above shouldbe adequate.As we will confirm a posteriori , for the forecasted ob-servations we consider it is ρ ar (cid:29) ρ γ . Consequently, wewill neglect the well understood CMB photons from theremaining of this analysis. For the same reason we neglectthe contribution from a possibly massless neutrino. The interaction here considered is formally similar to theone taking place at the end of inflation in an out-of-equilibriumdecay (see [13], equations (5.62) and (5.67)).
The equations (3-5) can be solved analytically: ρ m = ρ m (1 + z ) α , (7) ρ ar = ρ ar (1 + z ) + αα − ρ m (cid:2) (1 + z ) − (1 + z ) α (cid:3) , (8) E ( z ) = (cid:20) Ω ar + αα − Ω m (cid:21) (1 + z ) + 11 − α Ω m (1 + z ) α + Ω Λ , (9)where ρ m and ρ ar are the matter and astrophysical radi-ation energy densities today, respectively, E ( z ) = H ( z ) /H ,and Ω Λ = 1 − Ω m − Ω ar . The corresponding density pa-rameters are: Ω m ( z ) = Ω m (1 + z ) (3+ α ) E ( z ) , (10) Ω ar ( z ) = Ω ar (1 + z ) + αα − Ω m (cid:2) (1 + z ) − (1 + z ) α (cid:3) E ( z ) , (11)so that one has Ω m + Ω ar + Ω Λ = 1.As discussed earlier, the energy exchange from ρ m to ρ ar is a recent phenomena which we model as starting at aredshift ¯ z ∼ z ≥ ¯ z it is α = ρ ar =0. Using this initial condition and equations (8) one thenfinds the present-day astrophysical density parameter asa function of initial redshift and coupling: Ω ar = (cid:18) αα − (cid:19) (cid:2) (1 + ¯ z ) α − − (cid:3) Ω m . (12)As expected, Ω ar is proportional to both the couplingparameter and the matter density. One expects a small α and so Ω ar (cid:28) Ω m . For illustration purposes, the leftpanel of Figure 1 shows the evolution of the backgroundenergy densities for the case α = 0 . In equation (2) u βm is the four-velocity of the matter com-ponent. As discussed in [16] this kind of interaction doesnot alter the Euler equation as there is no momentumtransfer in the matter rest frame. This choice should bereasonable as one does not expect a fifth force for thephenomenology discussed in this paper. The perturbationequation for sub-horizon scales is then:¨ δ m + 2 H ˙ δ m + H (cid:18) α − (cid:19) Ω m δ m ≈ , (13)where we made the approximation Ω ar θ ar (cid:28) Ω m θ m and Ω ar δ ar (cid:28) Ω m δ m , and we neglected terms quadratic (orhigher) in combinations of α and Ω ar /Ω m . This meansthat θ tot ≈ θ m . In the previous equations δ is the densitycontrast, θ = ∇ i v i is the divergence of the velocity field,and the total divergence is given by: θ tot = 3 Ω m θ m + 3 Ω ar θ ar Ω m + 4 Ω ar . (14) avid Camarena, Valerio Marra: Cosmological constraints on the radiation released during structure formation 3 Ω m Ω ar Ω Λ f ( z ) f Λ ( z ) Figure 1.
Top: evolution of the background energy densitiesfor α = 0 .
2. At ¯ z the interaction is turned off and Ω ar = 0.Bottom: growth rate for the model of this paper with α = 0 . Λ CDM with the same present-day matter density.In both plots the interaction induces larger changes for red-shifts between 0 and ¯ z = 5 (vertical line). Indeed for z ≥ ¯ z the interaction is switched off and for redshifts close to zerothe cosmological constant dominates. See Section 2 for moredetails. We will denote with G ( t ) = δ m ( t ) /δ m ( t ) the growth func-tion normalized to unity at the present time. In obtaining(13) we have perturbed the expansion rate in Γ = α H asdone in [17,18] in order to preserve gauge invariance. Theright panel of Figure 1 shows the evolution of the growthrate f = d ln δ m d ln a for the case α = 0 .
2. For the sake ofcomparison, the growth rate of the Λ CDM with the samepresent-day matter density is also shown.
At the background level, we will use the forecasted super-nova Ia sample relative to ten years of observations by theLarge Synoptic Survey Telescope. This dataset features atotal of 10 supernovae with intrinsic dispersion of 0.12mag in the redshift range z = 0 . − . d L by: m ( z ) = 5 log d L ( z )10 pc , (15)which is computed under the assumption of spatial flat-ness: d L ( z ) = (1 + z ) Z z d˜ zH (˜ z ) . (16)The χ function is then: χ = X i [ m i − m ( z i ) + ξ ] σ , (17)where the index i labels the forecasted supernovae and σ = 0 .
12 mag. The parameter ξ is an unknown offsetsum of the supernova absolute magnitude and other pos-sible systematics. As usual, we marginalize the likelihood L SNe = exp( − χ /
2) over ξ , such that L SNe = R d ξ L SNe ,leading to a new marginalized χ function: χ = S − S S , (18)where we neglected a cosmology-independent normalizingconstant, and the auxiliary quantities S n are defined as: S n ≡ X i [ m i − m ( z i )] n σ i . (19)Note that, as ξ is degenerate with log H , we are effec-tively marginalizing also over the Hubble constant. At the linear perturbation level, we will build the growth-rate likelihood using the forecasted accuracy of a futureEuclid-like mission as obtained in [19]. Growth-rate dataare given as a set of values d i where d = f σ ( z ) = f ( z ) G ( z ) σ . (20)The χ function is then: χ fσ = X i [ d i − d ( z i )] σ i , (21)where the index i labels the redshift bins which span theredshift range 0 . < z < .
1. The uncertainties σ i are asgiven in [19] (Table II). David Camarena, Valerio Marra: Cosmological constraints on the radiation released during structure formation
The χ functions above depend on three parameters: α , Ω m and σ . It is useful to consider a prior on the lattertwo parameters in order to reduce possible degeneracies.As the interaction we are considering is absent at earliertimes – and so the cosmology is unchanged for z ≥ ¯ z – wecan use a prior on Ω m and σ from the CMB. However, asthe evolution for z < ¯ z is different, we have to adopt effec-tive present-day parameters in building the prior. Specif-ically, by demanding Ω Λm (¯ z ) = Ω m (¯ z ) and σ Λ (¯ z ) = σ (¯ z )we find that the effective parameters that we have to useare: Ω Λm = Ω m (1 + ¯ z ) α (cid:20) E Λ (¯ z ) E (¯ z ) (cid:21) , (22) σ Λ = σ G Λ (¯ z ) G (¯ z ) , (23)where E Λ ( z ) and G Λ ( z ) are the corresponding functionsin the Λ CDM case (i.e. with α = 0).From Figure 19 (TT, TE, EE+lowP) of Planck 2015XIII [14] one can deduce the covariance matrix between Ω m and σ (we approximate the posterior as Gaussian): σ Ω m = 0 . σ σ = 0 .
014 and ρ ’
0. Consequently, the χ function of the CMB prior is: χ cmb = " Ω fid m − Ω Λm σ Ω m + (cid:20) σ fid8 − σ Λ σ σ (cid:21) . (24) The full likelihood is based on the total χ which is: χ = χ SNe + χ fσ + χ cmb . (25)Furthermore, since energy is transferred from ρ m to ρ ar ,we adopt the following flat prior on the coupling constant: α ≥
0. Our fiducial model is specified by the followingvalues of the parameters: α = 0, Ω m = 0 . σ = 0 . σ , for which one has to extend thetheory of nonlinear structure formation (mass function,bias, etc) to the case of the interaction here considered. Figure 2 shows marginalized 1-, 2- and 3 σ constraints andcorrelations on the parameters α , Ω m and σ using the to-tal likelihood built from the χ function of equation (25).The left panel of Figure 3 shows the marginalized 1- and2 σ constraints on α and Ω m for each of the three indi-vidual likelihoods (LSST supernovae, Euclid growth-rate Figure 3.
Top: marginalized 1- and 2 σ constraints on α and Ω m for the SN likelihood of (18) (yellow contours), the growthrate likelihood of (21) (green contours) and the CMB priorof (24) (blue contours). The combination of these constraintsgive the corresponding panel of Figure 2. Bottom: the dashedempty contours show the constraints on α and Ω m for thecase ¯ z = 10. For comparison-sake, the red-to-orange contoursfrom Figure 2 corresponding to the case ¯ z = 5 are also shown.See Section 4 for more details. data and Planck prior). It is clear that the strong degen-eracy between α and Ω m comes from the supernova like-lihood, and that growth data and CMB prior marginallyhelp at constraining Ω m . At 95% confidence level we findthat α < . z = 5 has been adopted. In the right panelof Figure 3 we present constraints on α and Ω m for thecase ¯ z = 10. The 95% confidence level constraint on thecoupling is now slightly tighter: α < .
03. As one can see,the results do not depend strongly on ¯ z . As the processescontributing to the ARB start and take place at differentredshifts, the fact that our results do not depend stronglyon ¯ z means that our modeling and approximations areconsistent. avid Camarena, Valerio Marra: Cosmological constraints on the radiation released during structure formation 5 Figure 2.
Marginalized 1-, 2- and 3 σ constraints and correlations on the parameters α , Ω m and σ using the total likelihoodbuilt from the χ function of equation (25). The starting redshift of ¯ z = 5 has been adopted. See Section 4 for more details. In order to make contact with astrophysical bounds onthe ARB it is useful to express our results not with re-spect to the coupling constant α but with respect to thepresent-day density of astrophysical radiation Ω ar . Usingequation (12) it is easy to make this change of variableand obtain from the results of the previous Section that,at the 95% confidence level, it is Ω ar < . z .The total extra galactic background light (EBL) has adensity parameter of the order of Ω ebl ∼ · − , roughly afactor 10 times smaller than the density parameter relativeto the CMB photons [2].Regarding the stochastic gravitational-wave background,one usually defines the density parameter Ω gw ( f ) withinthe logarithmic frequency interval between f and f + d f .For f < Ω gw ( f ) ∝ f / , while for f > Ω gw ( f = 25Hz) ∼ − [20] one can then integrate Ω gw ( f ) and obtain Ω gw ∼ · − . The latter estimate only considers gravitationalwaves from binary black holes. Therefore, the total energydensity in the stochastic gravitational-wave background issomewhat larger, see [21,22] for a comprehensive reviews.If a fraction of dark matter is made of primordial blackholes, one would expect an additional gravitational wavebackground coming from their stochastic mergers [5,6,7].However, this background is supposed to be subdominantas compared to the one produced by black holes whichwere the result of star formation and evolution [23]. More-over, the merger rate of primordial black holes is not neg-ligible in the past, contrary to our assumption that α = 0for z ≥ ¯ z .A sizable contribution to the ARB comes from the dif-fuse supernova neutrino background (DSNB). The cosmicenergy density in neutrinos from core-collapse supernovaeis expected to be comparable to that in photons from David Camarena, Valerio Marra: Cosmological constraints on the radiation released during structure formation stars [3]. Indeed, one single core-collapse supernova pro-duces ∼ · erg in MeV neutrinos and such supernovaeoccur approximately every 100 years in the Milky Way.One can then conclude that the average neutrino powerof our Galaxy is ∼ erg/s, similar to its IR-opticalluminosity. One can then estimate that Ω dsnb ∼ · − .Finally, as discussed in the Introduction, backreactioncould contribute to the ARB at not well understood rates [9].Summing up, one expects the ARB to have a density pa-rameter of the order of Ω ar ∼ − . We have computed how well future cosmological observa-tions can constrain the energy density of the astrophysicalradiation background (ARB). ARB is the (to a first ap-proximation) uniform radiation density produced duringthe process of structure formation in the recent universe.We modeled the energy transfer from matter to radia-tion through an effective interaction between matter andastrophysical radiation, which is set to be zero at a red-shift of about 5–10. Using forecasted supernovas from theLSST and growth rate data from Euclid we found thatthe coupling constant α is constrained to be α < . Ω ar < .
008 (both at the 95% confidence level).Estimates of the energy density produced by well-knownastrophysical processes give roughly Ω ar ∼ − , almostthree orders of magnitude smaller than the upper limitthat can be obtained with LSST and Euclid. Therefore,we conclude that cosmological observations will be ableto constrain only exotic not-well understood sources ofradiation such as the backreaction of small-scale inhomo-geneities on the dynamics of the universe. It is a pleasure to thank Júlio Fabris, Oliver Piattella,Davi Rodrigues and Winfried Zimdahl for useful discus-sions. DCT is supported by the Brazilian research agen-cies CAPES. VM is supported by the Brazilian researchagency CNPq.
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