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Integral Representation of the Cosmic Microwave Background Spectrum PHYSICAL REVIEW D 99, 063014 (2019) Integral representation of the cosmic microwave background spectrum Moorad Alexanian* Department of Physics and Physical Oceanography, University of North Carolina Wilmington, Wilmington, North Carolina 28403-5606, USA (Received 3 October 2018; revised manuscript received 7 February 2019; published 22 March 2019) We use an integral representation for nonthermal radiation, which is bounded from below and above, to describe the spectrum of the cosmic microwave background (CMB). The upper bound is given by the T Rayleigh-Jeans law with a temperature RJ that can be determined by the absorption signal of 21 cm T T >T photons, where RJ represents the equilibrium temperature of photons in the RJ tail. If RJ CMB, then the lower bound allows us to conclude that photons, additional to the remnant of the big bang, are needed to explain the present CMB. These constraints are additional to other cosmological or astrophysical constraints in the study of the distortions of the CMB brought about by new physics particles or fields. DOI: 10.1103/PhysRevD.99.063014 n ≪ n I. INTRODUCTION energy than typical CMB photons, DR CMB and ωDR ≫ ω . The interest in an enhancement of the The lambda cold dark matter (ΛCDM) model has been CMB CMB is based on recent tentative observation of a established as the standard cosmological model to describe stronger-than-expected absorption signal of 21 cm photons the expansion history and the growth of the large-scale 0 [9], which can be explained by resonant A → A oscilla- structure of the Universe [1]. Assuming the ΛCDM model tions of dark photons into regular photons in the interval of [2], cosmological parameters have been measured within redshifts 20 <z<1700 [5]. percent-level uncertainties by a combination of observa- This paper is arranged as follows. In Sec. II, we review tions such as the cosmic microwave background (CMB) the integral representation for nonthermal or nonequili- experiments [3,4]. Additional measurements include type brium radiation, where the low-frequency photons are in Ia supernovae and baryon acoustic oscillations (see refer- T thermal equilibrium with temperature RJ owing to brems- ences in [2]). Despite the success of the model, we are strahlung and the spectrum is bounded from below [10,11]. challenged by a fundamental lack of physical understand- In Sec. III, we indicate the upper bound determined by the ing of the main components of the Universe, dark matter, RJ law with temperature T and obtain the constraints on Λ RJ and cosmological constant or more generally dark the photon number per unit volume and the internal energy energy. In order to understand these dark components, it per unit volume for the CMB that follow from the lower Λ is of great importance to test the CDM model at high bound of Sec. II. Section IV gives a simple illustrative precision using a variety of cosmological probes [2]. example that shows that the “heating” of the CMB in the Recently, Pospelov et al. [5] considered modifications T >T “ ” RJ tail, viz., RJ CMB, does actually lead to cooling of of the CMB, within its Rayleigh-Jeans (RJ) end of the the CMB. Finally, Sec. V summarizes our results. ω ≪ T spectrum, CMB, owing to dark matter (DM) and the interaction of nonthermal dark radiation (DR) A0 with 0 II. NONTHERMAL RADIATION ordinary photons A via the interaction eFμνFμν [6]. The DR quanta are much softer, but more numerous than CMB The distortions of the CMB have been studied with the photons, aid of the integral representation for the photon number density ω ≪ ω ;n>n ; ω n ≪ ρ ; ð Þ Z DR CMB DR RJ DR DR tot 1 1 ω2 ∞ σðT;tÞ n ðω;tÞ¼ dT ; ð Þ CMB 2 3 ℏω=kT 2 ρ π c 0 e − 1 where tot is the total energy density of radiation and DM, n n DR is the number density of DR quanta, and RJ represents where the spectral function σðT;tÞ is positive definite and, the low-energy RJ tail of the standard CMB. Recent in general, is a function of time or the cosmological redshift papers [7,8] examined interacting DR in the regime where z, where z ≥ 0 and z ¼ 0 corresponds to our present time the individual quanta are fewer in number but harder in [10,11]. Such integral representations were obtained by unifying the notions of the “approach-to-equilibrium” in *[email protected] quantum statistical mechanics and that of the asymptotic 2470-0010=2019=99(6)=063014(4) 063014-1 © 2019 American Physical Society MOORAD ALEXANIAN PHYS. REV. D 99, 063014 (2019) T ðtÞ condition in axiomatic quantum field theory in order to value of RJ from obtained data of the cosmological describe nonthermal radiation whereby the equilibrium, excess at 21 cm emission or absorption signal [9]. thermal states form a basis for nonthermal or nonequili- One obtains the following inequalities for the number of n ðtÞ brium states [12]. photons per unit volume CMB and the internal energy u ðtÞ It is interesting that the distribution (2) appears in an per unit volume CMB with the aid of inequality (4): integral equation of Laplace [13] by replacing the Maxwell Z ∞ 2ζð3Þ ðkT ðtÞÞ3 distribution by that of Planck. Paley and Wiener [13] n ðtÞ¼ dωn ðω;tÞ ≥ RJ ; ð Þ CMB CMB 2 3 2 7 described radiation from a source in approximate local 0 π ðcℏÞ ðCðtÞÞ equilibrium by (2), where σðT;tÞ denotes the “amount” of radiation coming from blackbodies at temperature T; where ζð3Þ is the Riemann’s zeta function and consequently, σðT;tÞ must be positive definite and Z σðT;tÞdT ∞ π2 ðkT ðtÞÞ4 gives the amount of radiation coming from the u ðtÞ¼ dωℏωn ðω;tÞ ≥ RJ ; T T þ dT CMB CMB 3 3 temperature range to . 0 15ðcℏÞ ðCðtÞÞ The case of the integral equation of Laplace [13], viz. ð Þ Laplace transforms, is the application of the completeness 8 of the equilibrium thermal states in the description of respectively. nonthermal or nonequilibrium states in classical statistical The ratio of the internal energy per unit volume u ðtÞ mechanics. The classical integral transform was applied CMB to the internal energy per unit volume uPðtÞ of a Planckian successfully in the study of the approach to equilibrium of spectrum with temperature T ðtÞ is Maxwell molecules [14], an exact (similarity) solution of CMB the nonlinear Boltzmann equation [15,16]. The application u ðtÞ 1 3 T ðtÞ 4 of the classical transform and consequently that for non- CMB ≥ RJ ; ð Þ u ðtÞ CðtÞ T ðtÞ 9 thermal radiation were characterized as the temperature P CMB integral transform [17]. where The low-frequency photons comprising the RJ tail of the nonthermal radiation (2) are in thermal equilibrium with 1 ω2 1 n ðω;tÞ¼ ð Þ temperature, P 2 3 ℏω=kT ðtÞ 10 π c e CMB − 1 Z ∞ T ðtÞ¼ dTTσðT;tÞ; ð Þ RJ 3 and 0 Z ∞ which arises from bremsstrahlung processes that are always uPðtÞ¼ dωℏωnPðω;tÞ: ð11Þ present. 0 Spectrum (2) is bounded from below [11] by CðtÞ¼1 T ðtÞ >T ðtÞ Note, in particular, that if and RJ CMB , “ ” 1 ω2 CðtÞ then the present CMB is hotter than a blackbody n ðω;tÞ ≥ ; ð Þ T ¼ 2 725K CMB 2 3 4 radiation with temperature CMB . , viz., π c exp ½ℏωCðtÞ=kT ðtÞ − 1 u ðtÞ >u ðtÞ T ðtÞ >T ðtÞ RJ CMB P . Therefore, if RJ CMB , then one must have CðtÞ > 1 to allow the present CMB to be where “cooler” than a blackbody radiation with temperature Z T ∞ CMB. CðtÞ¼ dTσðT;tÞ; ð5Þ Actually, from (4) one has 0 1 ω2 1 and CðtÞ and T ðtÞ are, in general, functions of time, viz., n ðω;tÞ > ; ð Þ RJ CMB π2 c3 ½ℏω=kT ðtÞ − 1 12 redshift [11]. exp CMB CðtÞ¼1 T ðtÞ >T ðtÞ if and RJ CMB and so there would be a III. CMB SPECTRUM CONSTRAINTS photon number enhancement for all values of ω, which n ðω;tÞ contradicts present CMB data. One obtains an upper bound to CMB with the aid of the inequality ex − 1 − x ≥ 0 and so (2) implies IV. ILLUSTRATIVE EXAMPLE 1 kω T ðtÞ ≥ n ðω;tÞ; ð Þ π2 ℏc3 RJ CMB 6 Recent works suggest a CMB that is very close to a Planckian spectrum but with a significant increase of which bounds the CMB spectrum by the Rayleigh-Jeans photon counts in the RJ tail [5]. We present a simple T ðtÞ law with temperature RJ . This places a constraint on the example of the distortions of the CMB, from that of a pure 063014-2 INTEGRAL REPRESENTATION OF THE COSMIC MICROWAVE … PHYS. REV. D 99, 063014 (2019) T Planckian spectrum with temperature CMB, that follows from the integral representation (2). Consider ω3 C C u ðωÞ¼ 1 þ 2 ; ð Þ CMB 2 13 π eω=T1 − 1 eω=T2 − 1 where we have chosen ℏ ¼ c ¼ k ¼ 1. One has from (3) and (5) that C ¼ C þ C T ¼ C T þ C T : ð Þ 1 2 and RJ 1 1 2 2 14 C ¼C ¼1 T ¼0 999T T ¼1 060T Let 1 2 and 1 . CMB and RJ . CMB, T ¼ 0 061T which implies that 2 . CMB. Figure 1 shows the plots for the upper bound given in (6) (green), the lower bound in (4) (red), our example in (13) (blue), and the Planckian blackbody spectrum FIG. 2. The blue plot represents our example (13) and the black in (12) (black).
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