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 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

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). The present temperature of the CMB is that of the blackbody radiation in (12). The plots cross at T ¼ 2.725K. Note that the blue plot is between the red ω=T ¼ 0 360 CMB CMB . (not shown) with the higher values of the blue and the green, which will always be the case. The dot in the plot over the black plot representing the enhancement of photons ω=T ¼ 0 0251 T green plot at CMB . represents the upper bound over that of the blackbody radiation with temperature CMB. T ¼ 1 060T (6) for the 21 cm photons for RJ . CMB. Note that the observation of the absorption signal of 21 cm photons spectrum with temperature T2 in (13) is negligible in will determine the maximum possible value of T . RJ comparison to the overall internal energy per unit volume. Figure 2 shows plots for our example in (13) and the Figure 3 shows the region of maximum difference Planckian blackbody spectrum in (12) (black). The plots between our example (13) and the blackbody radiation cross at ω=T ¼ 0.360 (not shown) with the higher CMB in inequality (12). The difference between the two plots for values of the blue plot over the black plot representing the T ¼ 1.060T is 0.03% at x ¼ 2.82, where the black- enhancement of photons over that of the blackbody RJ CMB body radiation attains its maximum value. Therefore, the radiation. The enhancement of photons in the RJ tail does model of interacting DR quanta that are much softer, but not give rise to an increase in the internal energy of our more numerous than CMB photons [5] may result in example (13) over that of the blackbody radiation given “ ” T > actually cooling the CMB radiation even though RJ in (12). In fact, the internal energy per unit volume of both T spectra are about the same since the contribution of the CMB implies the enhancement of photons in the RJ tail, viz., the “heating-up” of low-frequency photons. It is important to remark that CðtÞ¼1 is in total disagreement with the data for the CMB. The lower bound T ¼ 1 060T (4) implies for RJ . CMB, that is, a 6% increase of

FIG. 1. Plots of the upper bound given in (6) (green), the lower bound in (4) (red), our example in (13) (blue) and the Planckian blackbody spectrum in (12) (black) at the temperature of the CMB as observed in the present day, viz., TCMB ¼ 2.725K. The ω=T ¼ 0 0251 dot in the green plot at CMB . represents the upper FIG. 3. Region of maximum difference between our example T ¼ 1 060T bound (6) for the 21 cm photons for RJ . CMB. (13) and the blackbody radiation in inequality (12).

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T T x ¼ 2 82 the temperature RJ over CMB, that at . , which is representation (2). One obtains an upper bound (6) to near the peak of the blackbody radiation, the lower bound the spectrum given by the Rayleigh-Jeans law with CðTÞ¼1 T (4) with is 19% higher than the blackbody temperature RJ and a lower bound given by (4) radiation and so in total disagreement with the data. [10,11]. Observations of the absorption signal of the CðtÞ > 1 T This, of course, implies that , which is the case 21 cm photons set an upper bound to the value of RJ. T >T CðtÞ > 1 for our example (13) of the two-temperature approximation Also, if RJ CMB, then we must have that in to the integral (2) and indicates an additional cosmological (5) which requires additional cosmological sources which contribution to the present CMB besides the contribution of when added to the remnant radiation from the big bang the original remnant of the big bang. give us the present CMB. We believe that the integral representation (2) for nonthermal radiation may be con- V. SUMMARY AND CONCLUSION sidered as an additional constraint to other cosmological or We have established several constraints on the astrophysical constraints in the study of the distortions of spectrum of the CMB that follow from the integral the CMB brought about by new physics particles or fields.

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