Helical Magnetic Fields from Riemann Coupling Lead to Baryogenesis

PHYSICAL REVIEW D 104, 063502 (2021)

Helical magnetic fields from Riemann coupling lead to baryogenesis

† Ashu Kushwaha * and S. Shankaranarayanan Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India

(Received 15 March 2021; accepted 20 July 2021; published 1 September 2021)

The spectrum of energy density fluctuations, baryon asymmetry, and coherent large-scale magnetic fields are the three observables that provide crucial information on physics at very high energies. Inflation can only provide a mechanism to explain the density perturbations, and the origin of primordial magnetic fields and baryon asymmetry require physics beyond the standard models of cosmology and particle physics. In this work, we show that the mechanism that leads to primordial helical fields also leads to baryogenesis at the beginning of the radiation-dominated epoch. The model we consider here consists of mass dimension 6 operators that include Riemann coupling between gravity and electromagnetic field without extending the Standard Model of particle physics. We explicitly show that the generation of primordial helical magnetic fields leads to baryogenesis. We further show that the model predicts the observed amount of baryon asymmetry of the Universe for a range of reheating temperatures consistent with the observations.

DOI: 10.1103/PhysRevD.104.063502

I. INTRODUCTION The predictions of inflation are in good agreement with Understanding the physical processes in the very early the present-day observations of CMB anisotropies and Universe is a crucial ingredient for deciphering the physics polarization [18]. However, within the standard electrody- at energies that we cannot currently probe in terrestrial namics, inflation cannot provide a mechanism to generate experiments. While most observables have been washed large-scale B fields. This is because in 4-dimensions away by the thermal bath of the prerecombination era and do electromagnetic field is conformally invariant. Since FRW not have observational consequences, three observables models are conformally flat, the electromagnetic field provide crucial information of the physics at high-energies. vacuum in FRW is the same as the Minkowski space-time. These are the spectrum of energy density fluctuations [1–4], Hence, the standard electromagnetic fields generate negli- excess of baryons over antibaryons (baryon asymmetry) gible magnetic fields. More importantly, even if the baryon [5–11], and coherent large-scale magnetic fields [12–17]. asymmetry or cosmological magnetic fields existed before The inflationary paradigm provides an attractive mecha- the epoch of inflation, these would have been diluted by a −3N nism to generate the primordial density perturbations that factor of e , where N is the number of e-foldings of – lead to anisotropies in the cosmic microwave background inflation [19 21]. (CMB) and the formation of large-scale structures [1–4]. The present Universe is observed to contain essentially During inflation, the early Universe underwent an accel- only matter and no antimatter, except for the rare anti- erated expansion, stretching quantum fluctuations to super- particles produced by cosmic rays. The asymmetry between horizon scale density perturbations. Besides providing a baryons and antibaryons, referred to as baryon asymmetry causal mechanism to density perturbations, inflation also of the universe (BAU), can be expressed as [18,22] ’ solves the standard cosmological model s long-standing −10 − ¯ 5.8 − 6.6 × 10 fromBBN puzzles, such as the horizon, flatness, and monopole η nb nb ½ ð Þ B ¼ ¼ −10 problems. nγ ð6.09 0.06Þ × 10 ðfromCMBÞ ð1Þ

¯ *[email protected] where nb;nb;nγ refer to the density of baryons, antibaryons † [email protected] and photons, respectively. Magnetic fields permeate the Universe. Coherent magnetic fields in spiral galaxies and Published by the American Physical Society under the terms of clusters of galaxies have a magnitude of the order of μGauss the Creative Commons Attribution 4.0 International license. [12–15]. There is also indirect evidence of a lower limit of Further distribution of this work must maintain attribution to −16 the author(s) and the published article’s title, journal citation, order 10 G for the magnetic field contained in the voids and DOI. Funded by SCOAP3. between galaxies and clusters of galaxies [23].

2470-0010=2021=104(6)=063502(14) 063502-1 Published by the American Physical Society ASHU KUSHWAHA and S. SHANKARANARAYANAN PHYS. REV. D 104, 063502 (2021)

The origin of primordial magnetic fields and baryon possible link between the presence of magnetic fields to the asymmetry of the Universe are still unresolved issues and conditions required for baryogenesis [25]. require physics beyond the standard models of cosmology Davidson’s conditions are necessary but not sufficient. and particle physics. This leads to the following questions: One key missing ingredient, as we show, is the require- As the Universe cooled, from the early Universe to today, ment of primordial helical magnetic fields (details in what were the processes responsible for generating baryon Sec. II). Primordial helical magnetic fields are generated asymmetry and large-scale magnetic fields? Are these by the terms that break conformal invariance and parity processes cosmological or particle physics or both? symmetry [26–33]. If we could measure them, primordial Since both require physics beyond the standard model, helical magnetic fields provide evidence of CP violation there is a tantalizing possibility that the same new physics in the early Universe. Interestingly, the presence of can solve both. In this work, we consider such a possibility primordial helical fields leads to nonzero Chern-Simons and show that the mechanism that leads to primordial number [19,34,35] and, eventually, the change in the helical magnetic fields also leads to baryogenesis at the fermion number. beginning of the radiation-dominated epoch. Interestingly, Recently, the current authors constructed a simple model our mechanism also requires stretching of the primordial of inflationary magnetogenesis that couples the electro- helical magnetic fields to superhorizon scales during magnetic fields with the Riemann tensor [33]. We showed inflation—the same mechanism that leads to primordial that this model leads to a primordial helical magnetic field density perturbations. where one helical mode is enhanced while the other mode Before we discuss the model itself, it is necessary to is suppressed. The model has two key advantages over understand the key ingredients to generate baryon-asym- other models [28–32]: First, it does not require the coupling metry and magnetic fields and why the same new physics of the electromagnetic field with any scalar field. Hence, can potentially solve both these problems [24,25]. In 1967, unlike Ratra model [36–38], there is no strong-coupling Sakharov listed three necessary conditions for creating the problem caused by the extra degrees of freedom. Second, BAU [5,24]: (1) baryon number violation, (2) charge ðCÞ the model is free from backreaction for generic slow-roll and charge parity ðCPÞ violation, and (3) departure from inflation models [33]. In Ref. [29], authors have shown the thermal equilibrium. All three of the Sakharov conditions strong-coupling problem in Ratra model [36] can be are satisfied in the Standard Model; however, the electro- avoided by choosing a particular coupling function. weak phase transition is not sufficiently strong in the first In that work, the current authors used the general order [5–8]. The CP-violating effects are not sufficiently effective field theory of gravity coupled to the Standard pronounced to account for as large a BAU as we observe. Model of particle physics framework to obtain leading As a result, there must have been additional physics beyond order gravity terms that couple to the standard model the standard model to produce it. This physics could have bosons [39]. As we have done in the previous work, we been operating anywhere between the weak scale and the limit to mass dimension 6-operators coupling to the gauge GUT scale. Corresponding to out-of-equilibrium condi- field Lagrangian, specifically, to the electromagnetic field. tions, the baryogenesis scenarios are divided into two In this work also, we limit to mass dimension 6-operators categories: (a) by the universe expansion itself or (b) by coupling to the gauge field, specifically, to the electromag- fast phase transition and bubble nucleation. In particular, netic field. We show that the generation of primordial the latter concerns the electroweak baryogenesis schemes, helical magnetic fields from the above model leads to while the former is typical for a GUT type baryogenesis or baryogenesis. Since the model produces helical fields over leptogenesis [5–8]. large length scales, we show that the Chern-Simons (CS) More than two decades ago, Davidson pointed out an number density is nonzero (details in Sec. II). Considering interesting relation between the primordial magnetic field that the model generates primordial helical modes at all and Sakharov’s conditions [25]. She argued that the presence length scales, we focus on the last ten e-foldings of of background magnetic fields in the early Universe could inflation. This is because the modes that leave the lead to the breaking of C; CP; SOð3Þ symmetries and Hubble radius during the last 10e-foldings of inflation will thermal equilibrium. Specifically, she argued that the pres- reenter the Universe after reheating; these primordial ence of the magnetic fields leads to the following three helical modes will lead to baryogenesis just at the begin- conditions: (1) There should be some moderately out-of- ning of the radiation-dominated epoch. Furthermore, we thermal-equilibrium dynamics because in equilibrium, the show that the BAU is independent of inflation models and photon distribution is thermal, and there are no particle depends only on the energy scale at the exit of inflation and currents to sustain a “long-range” field, (2) Since B is odd reheating temperature. under C and CP, the presence of magnetic field will lead to In Sec. II, we discuss the relation between primordial CP violation, (3) Since the magnetic field is a vector helical magnetic fields and baryogenesis, in particular, the quantity, it chooses a particular direction hence breaks the chiral anomaly in the presence of the magnetic field, isotropy (rotational invariance). Thus, Davidson provided a and obtain the expression for Chern-Simon number density.

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2 2 2 i j In Sec. III, we discuss the generation of primordial helical ds ¼ a ðηÞðdη − δijdx dx Þð3Þ modes and show that primordial helical modes lead to a nonzero CS number density. Then we evaluate the baryon the contribution of the first term in the right-hand side (rhs) asymmetry parameter in Sec. IV. Section V contains the of Eq. (2) vanishes, i.e., implications of the results. Appendices contain the details μνρσ αβ of the calculations. ϵ RμναβR ρσ ¼ 0: ð4Þ In this work, we use ðþ; −; −; −Þ signature for the 4-D space-time metric. Greek alphabets denote the It can be shown that even at the first-order, the gravitational 4-dimensional space-time coordinates, and Latin alphabets contribution vanishes, and the nonzero contribution arises denote the 3-dimensional spatial coordinates. A prime only at second order [44]. Due to the presence of the stands for a derivative with respect to conformal time antisymmetric tensor, the gravitational fluctuations lead to η ð Þ and subscript ;i denotes a derivative with respect to gravitational birefringence and can lead to net chiral spatial coordinates. We use the Heaviside-Lorentz units current. ϵ μ 1 such that c ¼ kB ¼ 0 ¼ 0 ¼ . The reduced Planck mass In the flat FRW background, the second term in the rhs of 8π −1=2 is denoted by MP ¼ð GÞ . Eq. (2) is given by:

2 2 II. CONDITIONS ON BARYOGENESIS IN THE e μναβ e ϵ FμνFαβ ¼ ϵ ∂ A ∂0A : ð5Þ PRESENCE OF PRIMORDIAL MAGNETIC FIELD 16π2 4a4 ijk j k i ’ As we mentioned in the Introduction, Davidson s con- In the presence of the magnetic field, this term is nonzero ditions are necessary but not sufficient. One key missing and hence leads to a net chiral current. Thus, if we consider ingredient is the requirement of primordial helical mag- only up to the first-order in perturbations, only the second netic fields. In this section, we briefly discuss this. term in the rhs of Eq. (2) contributes and the chiral anomaly In the very early Universe, just after the exit of inflation, equation reduces to: the energy scale of the Universe was close to 1014 GeV. All particles, including fermions, are highly relativistic and can 2 pffiffiffiffiffiffi μ e μναβ be treated as massless. Although the massless Dirac ∂μ −gJ η FμνFαβ; ð AÞ¼16π2 ð6Þ equation is invariant under chiral transformations in the classical theory, the chiral symmetry is broken due to where we have used quantum mechanical effects in the presence of the external electromagnetic fields. This phenomenon, known as the μ 1 pffiffiffiffiffiffi μ μναβ 1 μναβ quantum axial anomaly, affects the transport properties of ∇μJ ¼ pffiffiffiffiffiffi ∂μð −gJ Þ; ϵ ¼ pffiffiffiffiffiffi η : A −g A −g the chiral medium, leading to experimentally accessible signatures such as the chiral magnetic effect [40] and the Note that during inflation, left-hand side (lhs) in Eq. (6) chiral separation effect [41]. is zero, and due to the exponential expansion, standard In the early Universe, the generation of the nonzero model particles are diluted. However, if we can generate primordial helical magnetic fields leads to a chiral anomaly nonzero primordial helical fields during inflation, then resulting from the imbalance between left and right-handed these nonzero primordial helical fields can lead to chiral fermions. In the presence of an electromagnetic field in current at the radiation-dominated epoch (or during reheat- curved space-time, the chiral anomaly is given by the ing when the standard model particles are created). following equation [42,43]: μναβ To see this, we rewrite Eq. (6) using η FμνFαβ ¼ μναβ 1 2 4∂μðη Aν∂αAβÞ, i.e., ∇ μ − ϵμνρσ αβ e ϵμναβ μJA ¼ 2 RμναβR ρσ þ 2 FμνFαβ ð2Þ 384π 16π 2 2 pffiffiffiffiffiffi μ e μναβ e pffiffiffiffiffiffi μ μ ∂μ −gJ ∂μ η Aν∂αAβ ∂μ −gK αβ ð AÞ¼4π2 ð Þ¼4π2 ð Þ where JA is the chiral current, Rρσ is the Riemann tensor and Aμ is the four-vector potential of the electromagnetic ð7Þ μνρσ 1 μνρσ ∇ − ∇ ϵ pffiffiffiffi η field, Fμν ¼ μAν νAμ. ¼ −g is a fully antisymmetric tensor, ημνρσ is Levi-Civita symbol whose where 0123 values are 1 and we set η ¼ 1 ¼ −η0123. It is easy to ημναβ see from the above equation that the anomaly contribution μ K ¼ pffiffiffiffiffiffi Aν∂αAβ from the electromagnetic field and the gravity act inde- −g pendently and, for most parts, can be treated independently. In the case of flat FRW background in conformal is the topological current. For FRW background, the time ðηÞ: components are given by

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0 −4 i −4 K ¼ a ðηÞϵijkAi∂jAk and K ¼ a ðηÞϵijkAj∂0Ak: where SGrav is the Einstein-Hilbert action Z ð8Þ 2 ffiffiffiffiffiffi − MP 4 p− Solving Eq. (7), we get, SGrav ¼ 2 d x gR; ð12Þ 2 μ e J Kμ: A ¼ 4π2 and Sϕ is the action for the minimally coupled, self- interacting canonical scalar field: − Thus, the net baryon number density, nB ¼ nb nb¯ ¼ Z η 0 0 0 ffiffiffiffiffiffi 1 að Þh jJAj i is related to Chern-Simon number density 4 p μ 0 Sϕ ¼ d x −g ∂μϕ∂ ϕ − VðϕÞ : ð13Þ nCS ¼h0jK j0i as [43], 2 2 ≡ e η S ;S refer to the standard electromagnetic (EM) and nB 2 að ÞnCS: ð9Þ EM CB 4π conformal breaking part of the electromagnetic terms, 0 respectively, which are given by: Note that nCS ¼ at the start of inflation, and due to the n 0 Z absence of standard model particles B ¼ during infla- 1 ffiffiffiffiffiffi tion. Using the expression for K0, we can write the Chern- − 4 p− μν SEM ¼ 4 d x gFμνF ; ð14Þ Simon number density as Z 1 1 ffiffiffiffiffiffi ϵ 0 ∂ 0 − 4 p− αβ ˜ ρσ nCS ¼ 4 ijkh jAi jAkj i SCB ¼ 2 d x gRρσ FαβF a Z M Z 1 Λ 4 1 ffiffiffiffiffiffi dk k 2 − 2 − 4 p− ˜ μναβ ¼ 4 2 ðjAþj jA−j Þ; ð10Þ ¼ 2 d x gR FαβFμν; ð15Þ a μ k 2π M where Λ, and μ set the possible energy range (or epoch) ˜ μναβ 1 μνρσ αβ where R ¼ 2 ϵ Rρσ is the dual of Riemann tensor during which baryon asymmetry is generated after infla- ˜ ρσ 1 μνρσ and F ¼ ϵ Fμν is the dual of Fμν. The standard tion, and A refer to the positive and negative helicity 2 electromagnetic action S is conformally invariant; how- modes of the electromagnetic field. The above expression is EM ever, the presence of Riemann curvature in SCB breaks the key in illuminating a useful relation between primordial conformal invariance. M is the energy scale, which sets the helical magnetic fields generated during inflation and scale for the breaking of conformal invariance. Note that baryogenesis: First, we see that the contribution to nCS the signs of SEM and SCB are chosen with respect to the is from all the modes that reenter the horizon at the positive electromagnetic energy density. beginning of the radiation-dominated epoch. Thus, In Ref. [39], the authors systematically showed that the first Λ the value of nCS depends on the upper cutoff . Second, gravity operators appear at mass dimension 6 in the series the expression corresponds to the total Chern-Simons expansion of the coupling between gravity and the standard number density generated from the modes in the energy model of particle physics. These operators only couple to the μ Λ — range ½ ; when these helical modes reenter during the standard model Bosons. They also showed that (i) no new radiation-dominated epoch. The helicity modes Aþ and A− −4 gravity operators appear at mass dimension 7, (ii) in mass are generated during inflation, and a ðηÞ is the dilution dimension 8, the standard model Fermions appear, and due to the expansion of the Universe during this epoch. (iii) coupling between the scalar (Higgs) field andthe standard Finally, nCS vanishes if the primordial magnetic fields are model gauge Bosons appear only at mass dimension 8. Since nonhelical, i.e., jAþj¼jA−j. Hence, as mentioned at the mass dimension 8 operators are highly suppressed, like in beginning of this section, the generation of nonhelical Ref. [33], we limit ourselves to mass dimension 6 operators. magnetic fields will not lead to baryogenesis. Thus, the key Due to Riemann coupling, M appears as a time-dependent ’ 1 ∼ missing ingredient of Davidson s argument is the require- coupling in the FRW background i.e., =Meff H=M.Atthe −42 ment of primordial helical magnetic fields. current epoch where H0 ≈ 10 GeV and assuming the 17 −59 In the following two sections, we explicitly evaluate the parameter M ≈ 10 GeV, we obtain H0=M ∼ 10 . Chern-Simons number for our model and show that it is not Therefore, the coupling (Riemann tensor) is tiny and the sensitive to inflationary and reheating dynamics. nonminimal coupling term in the electromagnetic action will have significant contribution only in the early universe. We III. THE MODEL AND THE PRIMORDIAL also would like to point that the coupling term ðSCBÞ is tiny HELICAL FIELDS near the Schwarzschild radius of a solar mass black-hole (for details, see Appendix D). We consider the following action [33]: We assume that the scalar field ðϕÞ dominates the energy density in the during inflation and leads to 60 − 70e- 14 S ¼ SGrav þ Sϕ þ SEM þ SCB ð11Þ ∼ 10 foldings of inflation with HInf GeV. Specifically,

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3=2 we consider power-law inflation in which the scale factor −1 M η0 jCj ≈ ς jC2j ≈ pffiffiffi ; (in conformal time) is [45]: 4η 1045 GeV3 qffiffiffiffiffiffiffiffiffiffiffi end −1 2 1 F ≈ jςj ≈ M η0; α ¼ − − ϵ ð20Þ η ðβþ1Þ 2 η − að Þ¼ η ð16Þ 0 For details, see Appendix B. Our model generates primordial magnetic fields through where, the constant η0 denotes the scale of inflation and the nonminimal coupling of the electromagnetic field. The β ≤−2. β ¼ −2 corresponds to exact de Sitter. During model requires inflation. Inflation generates density per- inflation, η ∈ ð−∞; 0Þ. For slow-roll inflation β ≈−2 − ϵ turbations at all scales and provides a causal mechanism to and H ≡ a0=a ≈−ð1 þ ϵÞ=η, where H is the Hubble generate the structure formation. Similarly, our model parameter in conformal time and ϵ is the slow roll generates magnetic fields at all length scales, including parameter. For our discussion below, we also assume that the current Horizon radius [12–16]. This has to be con- 10−3 ≤ ≤ 1 – trasted from the models where the magnetic field is ðHInf=MÞ [46 49]. Equation of motion of the gauge field can be obtained by generated during recombination. In these models, the varying the action (11) with respect to Aμ. In the Coulomb coherence scale of the generated fields cannot exceed gauge ðA0 ¼ 0; ∂ Ai ¼ 0Þ, we have: the size of the horizon radius at that time. i In Appendix B, we have plotted the power spectrum of the present-day helical magnetic field ðB0Þ as a function of 4ϵ 000 00 0 1017 00 ijl a a a k. Assuming M ¼ GeV, our model predicts the A þ − 3 ∂ A − ∂ ∂ A ¼ 0 ð17Þ −20 i M2 a3 a4 j l j j i primordial helical magnetic fields of strength 10 Gon Gpc scales at the current epoch. From Fig. 3 we can see that our model predicts the present-day helical magnetic field of ϵ −15 where ijl is the Levi-Civita symbol in the 3-D Euclidean strength 10 G on Mpc scales. The primordial fields space. The above equation is different from other models in generated from our model are within the upper bounds on the literature and leads to distinct evolution of the magnetic the strength of the seed magnetic fields needed to explain field fluctuations in comparison to nonminimally coupled the current galactic magnetic fields [50]. These primordial scalar field models [33]. In the helicity basis, the above fields are amplified by the dynamo mechanism and can lead equation reduces to (see Appendix A): to the observed magnetic fields; hence our model requires the dynamo mechanism. 4kh a000 a00a0 A00 þ k2 − − 3 A ¼ 0: ð18Þ IV. BARYON ASYMMETRY OF THE UNIVERSE h M2 a3 a4 h In this section, we compute the baryon asymmetry parameter due to the primordial helical magnetic fields. For the two helicity states ðh ¼Þ, the above expression Specifically, we compute it for the maximum helicity leads to two different evolution equations [cf. Eqs. (B6a), modes—one mode is enhanced compared to the other. (B6b)]. From Eq. (10) we see that to obtain appreciable Substituting Eq. (19) in Eq. (10), we obtain value of Chern-Simons number ðn Þ, the difference Z CS 1 Λ between the two helicity states should be nonzero, and it nCS ¼ 2 4 dk is maximum if one helicity mode is enhanced compared 2π a ðηÞ μ to other. −1 2 2 3 1 F 1 3− 1 −2 þ2α Γ 2ατ α In our previous work [33], we showed that for a range of jCj k þjC2 π 2α k : ð21Þ parameters of interest, negative helicity mode decays while the positive helicity mode is enhanced. Hence, negative Integrating the above expression, we get helicity mode ðA−Þ will have negligible contribution and 0 1 can be set to zero, i.e., jA−j¼ . Using the series expansion n ¼ of the Bessel functions, in the leading order, the positive CS 2π2a4ðηÞ 4 1 Λ −1 2 4− 1 Λ helicity mode takes the following form (B6a): þ2α F 1 2α 2 k k −2 C C2 Γ τ α : 22 j j 4 1 þ π 2α 4− 1 ð Þ þ 2α μ 2α μ −1 1 F 1 − 1 −1 τ 4α − Γ 4ατ α Aþð ;kÞ¼Ck C2 π 2α k ð19Þ We want to make the following remarks regarding the above expression: First, the BAU is generated similarly to the inflationary mechanism of the generation of density pertur- where, bation. During inflation, the primordial helical magnetic

063502-5 ASHU KUSHWAHA and S. SHANKARANARAYANAN PHYS. REV. D 104, 063502 (2021) field fluctuations are stretched exponentially and exit the zero) and is approximately constant only after the last horizon. The modes that reenter during the radiation- scattering surface. Since entropy density per comoving dominated epoch are responsible for the generation of volume is conserved, the quantity nB=s is better suited for baryon asymmetry. Second, the generation of baryon theoretical calculations [1]. Assuming that there was no asymmetry does not strongly depend on the reheating significant entropy production after reheating phase, dynamics since only the modes that reenter the Hubble entropy density in the radiation-dominated epoch is: radius during the radiation-dominated epoch are relevant. 2π2 Assuming a de Sitter (or approximately de Sitter) ≃ 3 −2 −2 s gTRH; ð25Þ Universe, from Eq. (20),wehaveτ α ¼ a ðηÞ. 45 Substituting this in the Eq. (22), we see that the second where TRH is the reheating temperature and the effective term in the rhs decays faster compared to the first term by relativistic degrees of freedom g ∼ 100 at reheating. From −2 η a ð Þ. Hence, we can neglect the second term. Substituting Eqs. (24), (25), we can define the following dimensionless the resulting form of nCS in Eq. (9) leads to: BAU parameter:

2 4 1 Λ 1 þ2α 2 2 e 2 k n jCj · e 45 n ¼ jCj : ð23Þ η B ≈ 10−24 Λ3 − μ3 B 4π2 2π2a3 η 4 1 B ¼ 24π4 ð Þ 2π2 3 ð Þ þ 2α μ s gTRH Λ3 Λ μ ≈ 10−29 2 To obtain the ranges of and ,weneedtoknowthemodes jCj 3 ð26Þ exited during inflation. For the density perturbations, the TRH largest scales observed in the CMB are produced around where in the last expression we have neglected μ3 i.e., 40−60e-foldings before the end of inflation [51].Thisis Λ3 − μ3 ≈ Λ3. Appendix C contains plots for different because the adiabatic quantum fluctuations responsible for Λ μ the density perturbations reenter the Hubble radius around values of and . From these plots, we infer that the results do not strongly depend on the exact value of μ. z ∼ 1500. Hence, in Ref. [33], the current authors only 2 looked at primordial helical fields generated around Finally, substituting the value of jCj (from Eq. (20) and 40−60e-foldings before the end of inflation. However, in using the values in Appendix B) in Eq. (26), we obtain: this case, we will concentrate on the primordial helical fields 10−29 η2 3Λ3 η ≈ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi· 0 M that renter the horizon very early (at the beginning of the B 3 η · 1045 GeV3 T radiation-dominated epoch) to generate the required BAU. end RH This means that the modes that left the horizon around the 3 Λ 3 ≈ 10−2 M last 5 to 10e-foldings of inflation are only relevant. Since ð27Þ MP TRH these modes have already left the Hubble radius during inflation, the reheating dynamics do not alter these primor- This is one of the crucial expressions in this work dial helical modes. Hence, the model is insensitive to the regarding which we would like to stress the following: reheating dynamics. First, the BAU parameter depends on three quantities—M Our focus now shifts to explicitly evaluating BAU for (the conformal invariance breaking scale), TRH (reheating our model. First step is to evaluate the dilution factor a−3 in temperature scale) and Λ (the largest helical mode that Eq. (23). To do this, we define aΛ (and aμ) as the scale catalyses baryogenesis). Second, the BAU parameter is factor at the time when the maximal helicity mode with inversely proportional to the reheating temperature. This energy Λ (and μ) left the Hubble radius during inflation. behavior is different from the results of Ref. [19,21, Assuming an instant reheating, and following the calcu- 43,44]. In some of these models, BAU is linearly depen- 6 lations given in Appendix C, we have aμ ¼ 10 aΛ. Taking dent on the reheating temperature. The difference in the into account that these modes exited the Hubble radius relationship is because the detailed reheating dynamics is during inflation in the last 5e-foldings, the dilution factor not required, only the information about the entropy [prefactor in Eq. (23)] becomes a−3 ∼ 10−24. production is required in our model. In other models, The second step is to obtain the constant C. As discussed the exact detailed reheating dynamics is required, which is in previous section, for slow-roll inflation, jCj is given by avoided in our approach. Third, the BAU parameter is Eq. (20). Thus, Eq. (23) reduces to: linearly proportional to M and Λ. For smaller M,the contribution of the conformal breaking term (15) will be 10−24 · jCj2 · e2 much larger, and hence, more primordial helical fields are n ≈ Λ3 − μ3 : B 24π4 ð Þ ð24Þ produced during inflation. However, for the same reheating temperature, Λ has to be larger to produce the same Third step is to compare the theoretically derived quantity amount of BAU. Fourth, to get a better understanding of ðnBÞ with observations Eq. (1). However, nγ is not constant the dependence of BAU on various parameters, we use the in the early Universe (since the photon chemical potential is following parametrization:

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where n, m, δ, γ are dimensionless parameters. The maximum reheating corresponds to the inflation scale [51]. With supersymmetry, the requirement that not too many gravitinos are produced after inflation provides a ∼ stringent constraint on the reheating temperature, TRH 1010–1011 GeV [52,53]. Hence, we consider the range of γ to be f10−2; 1000g. Since the value of M should be between the GUT and Planck scale, we consider the range of m to be f1; 1000g. We assume that the modes that reenter during radiation epoch is around 1012 GeV. Hence, we consider the range of δ to be f1; 100g.Using the above parametrization in Eq. (27),weget:

3 δ3 m × ≈ 107 γ3 n : ð29Þ FIG. 1. Plot of the rescaled reheating temperature TRH with the rescaled conformal symmetry breaking parameter M, for different 14 10 values of n. Here, we have set Λ ¼ 10 GeV; μ ¼ 10 GeV. Figures 1 and 2 contain the plots of γ versus m for different values of n and fixed δ. In Appendix C we have −10 14 plotted the same for other values of δ. From these plots, we ηB ¼ n × 10 ;M¼ m × 10 GeV; deduce the following: First, for a range of values of γ, δ, and Λ δ 1012 γ 1012 −10 −9 ¼ × GeV;TRH ¼ × GeV ð28Þ m, BAU can have values between 10 to 10 . Thus, the model can lead to the observed amount of baryon

(a) (b)

FIG. 2. Plots showing the behavior of reheating temperature TRH (vertical axis) with parameter M (horizontal axis), for lower energy scales of Λ and μ.

063502-7 ASHU KUSHWAHA and S. SHANKARANARAYANAN PHYS. REV. D 104, 063502 (2021) asymmetry of the Universe consistent with the Planck data temperature [51–53].Thismeansthatourmodeldoes not [18]. Second, the model does not depend on the nature of prefer a very low-energy reheating temperature [53]. the reheating dynamics. As can be seen from the plots, for a In the literature, various mechanisms have been discussed range of values of m, δ, the model can lead to BAU for a to solve the BAU problem using the primordial helical range of reheating temperatures. This has to be contrasted magnetic fields [19–21,34,43].InRef.[19], the authors with other models in the literature [19,21] which requires obtained the required BAU by assuming the presence of detailed knowledge of the reheating phase of the Universe. helical magnetic fields of present-day strength 10−14G< −12 Third, the unknown parameter in the model is M.In B0 < 10 G and coherence length 1 pc < λ < 1 Mpc, and Ref. [33], we showed that for the model to be consistent taking into account of the MHD effects. In Ref. [21], authors with the lower limit of 10−16 Gauss magnetic fields in the studied the generation of a primordial magnetic field in voids [23], then M ∼ 1017 GeV. The current analysis shows conjunction with the BAU generation through leptogenesis; that M ∼ 1017 GeV is consistent with baryogenesis. Thus, however, the predicted value of the present-day coherence the model is tantalizingly close to solving baryogenesis and length of such magnetic fields is very small ∼10 pc. magnetogenesis using the same causal mechanism that In Refs. [20,34], the authors consider pseudoscalar solves the origin of density perturbations. inflation (axion inflation) model with a dimension five couplings. In these models, the authors assumed the scale of the baryogenesis to be electroweak scale, and they V. CONCLUSIONS AND DISCUSSIONS obtained the required BAU assuming the scale of inflation 10 12 In this work, we have proposed a viable baryogenesis to be 10 GeV—10 GeV [20,34].InRef.[43],the scenario in the early Universe that does not require any authors considered the extension of the Standard Model extension to the Standard Model of particle physics. The with anomalous gauge symmetry. They obtained the 14 crucial ingredient is the generation of primordial helical required BAU for HInf ∼ 10 GeV and reheating tem- magnetic fields due to Riemann coupling. The advantage of perate at 1016 GeV. In Ref. [54], the authors argued that to the primordial helical fields is that the nonzero helicity generate the observed baryon asymmetry, some asymme- suggests a nonzero contribution in the CP violation term. try in the initial conditions of either B or scalar field ϕ is An interesting feature of our model is the stretching of the required, which can be induced from temperature- primordial helical magnetic fields to super-horizon scales dependent potential or asymmetry in quantum fluctua- during inflation—the same mechanism that leads to pri- tions. Our model is robust to inflationary/reheating mordial density perturbations. While the helical modes dynamics and uses the same success of inflationary generated around 40−60e-foldings before the end of perturbations to generate BAU. Thus, our model is inflation lead to the observed large-scale magnetic fields, tantalizingly close to solving baryogenesis and magneto- the helical modes that renter the horizon very early (at the genesis using the same causal mechanism that solves the beginning of the radiation-dominated epoch) lead to the origin of density perturbations. baryon asymmetry. Thus, our mechanism provides possible In this work, we did not consider the gravity contribution testable evidence for the entire inflationary epoch. to the chiral anomaly equation. In Ref. [44], the authors More than two decades ago, Davidson pointed out an considered the phenomenon of gravitational birefringence interesting relation between the primordial magnetic field to show that the gravitational fluctuations generated during and Sakharov’s conditions [25]. In this work, we have inflation can give the Universe’s observed amount of explicitly shown that Davidson’s conditions are necessary baryon asymmetry. However, as we showed in. Sec. II, but not sufficient. The key missing ingredient is the RR˜ contributes only in the second-order, and hence we have requirement of primordial helical magnetic fields. While ignored it in this analysis. It may be interesting to look at the helical and nonhelical fields break the isotropy and lead the second-order corrections and analyze the parameter to CP violation, only the modes with maximal helicity constraints. contribute significantly to the Chern-Simon number den- In this work, we have used the general effective field sity. We have shown that the BAU parameter predicted by theory of gravity coupled to the Standard Model of particle our model is independent of any specific inflation model physics framework to obtain leading order gravity terms and reheating dynamics; however, it depends on the scale at that couple to the standard model bosons [39].Wehave which inflation ends and reheating temperature. considered only the mass dimension 6-operators coupling The BAU parameter (27) obtained in our model is to the gauge field Lagrangian, specifically, to the electro- inversely proportional to reheating temperature. Assuming magnetic field. The coupling to the Fermions arises at the 14 the exit of inflation at 10 GeV, for the observed amount of mass dimension 8. Thus, coupling of Fermion-anti- η ∼ 10−10 baryon asymmetry B , we obtained that the reheat- Fermion with Uð1Þ field will play a role only at this order. ing temperature should be in the range 1012–1014 GeV, While these are expected to be suppressed compared to which is consistent with the constraints on the reheating mass-dimension 6 operators, they are relevant at Planck

063502-8 HELICAL MAGNETIC FIELDS FROM RIEMANN COUPLING … PHYS. REV. D 104, 063502 (2021) Z 3 X scale. We plan to look at the effects of mass dimension 8 d k ¯ ik·x ¯ † ⃗ −ik·x A x; η εˆ λ Aλbλ k e A b k e : operators on the baryogenesis. ið Þ¼ 2π 3 i ½ ð Þ þ λ λð Þ ð Þ λ 1 2 In this work, we focused on the electromagnetic fields ¼ ; and the effects of the helical fields on baryogenesis. It will ðA3Þ be interesting to extend the analysis to gluons and study the effects on the asymmetry generated in quarks and the Substituting Eq. (A3) in Eq. (17), we get: baryons. It is particularly important, and a study on this is X 4i currently in progress to acquire more stringent constraints εˆ ¯ 00 ϵ εˆ ¯ bλ iλAλ þ 2 ijlkj lλAλ on the parameters M and TRH [55]. λ 1 2 M ¼; 000 00 0 — a − 3 a a 2εˆ ¯ 0 Note added. As we were finalizing this manuscript, the 3 4 þ k iλAλ ¼ ðA4Þ article [56] appeared on the arXiv which also discusses a a baryogenesis from magnetic fields. However, the approach ∂ ∂ − 2 followed in the reference requires MHD amplification where we have used j j ¼ k . while our approach requires helical fields generated during Since the action (11) contains parity breaking term inflation. (helicity term), it is useful to work in the helicity basis. ε ε The helicity basis vectors þ and − corresponding to h ¼ 1 and h −1 are defined as ACKNOWLEDGMENTS þ ¼ The authors thank Joseph P. Johnson and Urjit A Yajnik 1 ε ¼ pffiffiffi ðεˆ1 iεˆ2Þ: ðA5Þ for comments on the earlier version of the manuscript. The 2 authors thank Kandaswamy Subramanian for useful discussion. The authors thank the anonymous referee for raising Assuming that the wave propagates in the z−direction, the some points which clarified important issues in the work. vector potential in the helicity basis is given by: The MHRD fellowship at IIT Bombay financially supports A¯ ¯ εˆ ¯ εˆ ε ε A. K. This work is supported by the ISRO-Respond grant. ¼ A1 1 þ A2 2 ¼ Aþ þ þ A− − ðA6Þ

APPENDIX A: QUANTIZATION IN THE where AþðA−Þ refer to the vector potential with positive HELICITY BASIS (negative) helicity. The ground state in the helicity basis is defined as In this section, we briefly discuss the evolution of the quantum fluctuations of the electromagnetic field in the bhðkÞj0i¼0 ðA7Þ helicity basis [29]. Decomposition of the vector potential in Fourier domain leads to: and satisfy the following commutation relations: Z 3 † 3 3 d k ½b ðkÞ;b 0 ðqÞ ¼ ð2πÞ δ ðk − qÞδ 0 ðA8Þ Aiðx;⃗ηÞ¼ h h hh ð2πÞ3 X † † † k q 0 k q i ⃗ ik·x ⃗ −ik·x ½bhð Þ;bh0 ð Þ ¼ ¼½bhð Þ;b 0 ð Þ: ðA9Þ ελ½Aλðk; ηÞbλðkÞe þ Aλðk; ηÞbλðkÞe h λ¼1;2 Rewriting (A4)P in the Helicity basis and replacing ðA1Þ ϵ ∂ → − ε ijl jAl k h¼1 hAh h, we have: b k b† k where ð Þ and ð Þ are the annihilation and creation 2 4kh k εi A00 þ k − ΓðηÞ A ¼ 0; ðA10Þ operators respectively for a given comoving mode , and λ h M2 h is the orthogonal basis vector which in right-handed coordinate system [29] is given by where, ˆi 000 00 0 μ 1 μ ελ a a a 1 ε ¼ ; 0 ; ε ¼ 0; ; ΓðηÞ¼ − 3 ¼ ðH00 − 2H3Þ: ðA11Þ a a a3 a4 a2 ˆ μ k ε ¼ 0; for λ ¼ 1; 2; ðA2Þ 3 a APPENDIX B: GENERATION AND EVOLUTION i ˆ OF HELICAL MODES 3-vectors εˆλ are unit vectors orthogonal to k and to each other. Substituting Eq. (A2) in Eq. (A1) and defining the Substituting the power-law inflation scale factor (16) in ¯ new variable Aλ ¼ aðηÞAλðk; ηÞ, we have: Eq. (18),wehave:

063502-9 ASHU KUSHWAHA and S. SHANKARANARAYANAN PHYS. REV. D 104, 063502 (2021) 2β 5 8kh β β 1 β 2 −η0 ð þ Þ constants for α ¼ −1. There are two reasons for this choice: 00 2 − ð þ Þð þ Þ 0 Ah þ k 2 3 Ah ¼ : τ ∝ η M η0 η First, in this special case, and the superhorizon modes can be written in terms of η using the linear relation. ðB1Þ Second, the constants C1, C2, C3, C4 have a weak α β −2 dependence of and, hence, finding the value for a given Helicity term vanishes for de-sitter case ð ¼ Þ, which is value of α will be accurate within an order [33]. Thus, consistent with the fact that the de Sitter symmetry will not matching the solutions and the derivatives at the horizon- be preserved in the presence of helicity terms. However, it exit, we get: will be nonzero for the approximately de Sitter universe β −2 − ϵ − kη ≫ 1 rffiffiffiffiffiffiffi i.e., ¼ . In subhorizon limit ðj j Þ, Eq. πη 1 pffiffiffiffi i 0 (B1) simplifies to: C1 ¼ −e pffiffiffiffi sin Θ þ i Θ cos Θ ; 2 Θ rffiffiffiffiffiffiffi A00 þ k2A ≈ 0 ðB2Þ πη 1 pffiffiffiffi h h i 0 C2 ¼ −ie pffiffiffiffi cos Θ − i Θ sin Θ 2 Θ and assuming that the quantum field is in the vacuum state rffiffiffiffiffiffiffi πη 1 pffiffiffiffiffiffi at asymptotic past (Bunch-Davies vacuum state), we have: i 0 C3 ¼ e pffiffiffiffiffiffi sinh Θ þ iΘ cosh Θ ; 2 iΘ 1 rffiffiffiffiffiffiffi ffiffiffi −ikη πη 1 pffiffiffiffiffiffi Ah ¼ p e : ðB3Þ i 0 k C4 ¼ −ie pffiffiffiffiffiffi cosh Θ þ iΘ sinh Θ : ðB7Þ 2 iΘ On super-Horizon scales ðj − kηj ≪ 1Þ, Eq. (B1) becomes: qffiffiffiffiffiffiffiffi 15η Θ where ¼ M2η3 is the dimensionless constant. d2A αðα þ 1Þ dA 0 α2 h h hkς2A 0 In Ref. [33], the current authors derived the magnetic τ2 þ τ τ þ h ¼ ðB4Þ d d field spectral energy density and is given by where 2−4α dρB 2 k 3 4α 1 ¼jCðk ; αÞj k þ þ2α 1 d ln k k ς2 ≡ − ð2α − 3Þð2α − 1Þð2α þ 1Þ; 2η 4−4α M 0 2 k 1 4α− 1 C α þ 2α η α 3 þj 2ðk; Þj k ðB8Þ τ − 0 α β k ¼ η ; ¼ þ 2 ðB5Þ C α C α where k is the pivot scale, and ðk; Þ and 2ðk; Þ are Note that τ (dimensionless variable 0 < τ < ∞) and η constants that depend on the inflationary energy scale (See (negative during inflation) are linearly related. [At the start Eq. (44) in Ref. [33]). For de Sitter inflation ðα ¼ −1=2Þ, of inflation, τ is large and vanishes at the end of inflation.] the present day magnetic field as a function of k is given by: α − 1 α ≤−1 Note that ¼ 2 corresponds to de-sitter and 2. The 2 solutions for the above Eq. (B4) are −20 k B0ðkÞ ∼ 10 G ðB9Þ pffiffiffi pffiffiffi k − 1 ς k − 1 ς k A ðτ;kÞ¼τ 2α J 1 τ C1 þ τ 2α Y 1 τ C2 þ 2α α 2α α where we have included only the leading order contribution and have discarded the subleading contribution. ðB6aÞ In Fig. 3 we have plotted the power spectrum of the pffiffiffi ς present day primordial helical magnetic field at different − 1 k 0 002 −1 A−ðτ;kÞ¼τ 2α J 1 −i τ C3 length scales for two pivot scales k ¼ . Mpc [57] 2α α −1 and k ¼ 0.05 Mpc [18]. One can see from Fig. 3(a) that pffiffiffi − 1 ς k for around Mpc scale, the value of present day magnetic þ τ 2α Y 1 −i τ C4; ðB6bÞ −15 2α α field is 10 G. This is consistent with the current observations [14–17,50]. where C1, C2, C3, C4 are arbitrary constants of dimension Using the fact that modes exit the horizon around 1=2 5 L . For the two helicity modes, we fix the constants C1, e-foldings, 0 C2 ðC3;C4Þ by matching Ah and Ah at the transition time of ∼ η−1 η η · 102 B10 subhorizon and superhorizon modes at k where ¼ end ð Þ refers to the quantities evaluated at the horizon-exit. −1 14 14 17 Although the analysis can be done for any general value and H ∼ η0 ∼ 10 GeV, for M ∼ 10 –10 GeV of α, to keep the calculations tractable, we obtain the [33,45], we obtain

063502-10 HELICAL MAGNETIC FIELDS FROM RIEMANN COUPLING … PHYS. REV. D 104, 063502 (2021)

(a) (b)

FIG. 3. Plot showing the strength of the present day primordial helical magnetic field at different length scales (from Mpc to Gpc), for two different pivot scales. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi modes with energy Λ and μ reentered the radiation η · 1045 GeV3 Θ ≈ end ; dominated Universe. The Friedmann equation is M2 2 2 1 π 4 which is very small value. Note also that H gΛ T : ¼ 3 2 30 Λ ðC2Þ MP − 1 e NInf η ¼ − ¼ − ≈ 10−41 GeV−1: end aðη ÞH H The baryogenesis occurs as soon as the helical modes end Inf reenter the Hubble radius during the radiation-dominated Using the fact that Θ is very small, we get epoch. For simplicity, we assume that baryogenesis occurs rffiffiffiffiffi at the start of radiation-dominated epoch, hence we take pffiffiffiffiffiffiffiffi η energy scale of Λ to be of order H. Substituting TΛ from ≈ ≈ Θη ≈ ≈ 0 jC1j jC3j 0; and jC2j jC4j Θ: ðB11Þ Eq. (C2) in Eq. (C1), we get, pffiffiffiffi 1=12 1 4 1=2 1=2 9 Hence, we obtain the following relations among the aμ g 90 = M Λ 10 Λ ¼ Λ P ≈ ; ðC3Þ coefficients jC1j ≈ jC3j ≪ jC2j ≈ jC4j. 1=3 2 aΛ gμ π Tμ μ

APPENDIX C: BAU PARAMETER FOR where gΛ ∼ 100 (during reheating) and gμ is of the order 10. ARBITRARY VALUES OF Λ AND μ Physically, we see that the Universe has expanded by a pffiffiffi 109 Λ Λ μ Following Ref. [15], we first evaluate the contribution of factor of μ when the helical modes of energy and dilution factor for arbitrary values of Λ and μ. Assuming reenter the radiation dominated epoch and hence, in instantaneous reheating, the universe transited to radiation Eq. (23), the inverse of this factor will act as the dilution domination after inflation. Using entropy conservation i.e., factor. ga3T3 ¼ constant during its evolution, where g is effective Setting Λ ¼ 1014 GeV, and μ ¼ 1010 GeV, in the above relativistic degrees of freedom, we have: expression, we have: 1=3 1=12 aμ gΛ TΛ 90 1=4 Λ 1=2 ¼ ðC1Þ aμ gf ðMP Þ 6 ¼ ≈ 10 : aΛ gμ Tμ 1=3 π2 μ aΛ g1 where aΛ;aμ are the scale factors at which the helical modes with energy Λ and μ reentered the radiation For the modes that exit the horizon during inflation at 5 dominated Universe, gΛ;gμ are the effective relativistic N ¼ ,wehave: degrees of freedom at which the helical modes with energy 1 μ3 Λ and μ reentered the radiation dominated Universe, and ∼ 10−33 3 3=2 : TΛ;Tμ are the Universe temperatures at which the helical a Λ

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Therefore, for generic scales of baryogenesis, the BAU APPENDIX D: EFFECT OF THE RIEMANN parameter (26) is given by COUPLING NEAR SCHWARZSCHILD BLACK HOLE 10−38 η2 3 Λ3μ3 − μ6 In this Appendix, we will show that S coupling is tiny η ≈ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi· 0 M ð Þ CB B 3=2 3 near the solar mass Schwarzschild black holes. η · 1045GeV3 Λ T end RH Specifically, we evaluate at the Schwarzschild radius of 3 Λ3μ3 − μ6 μ ≈ 10−11 M ð Þ a nonrotating spherically symmetric black hole of mass . 3=2 3 : ðC4Þ MP Λ T Since the calculation is order of magnitude, we calculate RH μναβ the Kretschmann scalar K ¼ RμναβR for the Schwarzs child black hole. For this case, the Kretschmann scalar K Using the parametrization in Eq. (28),wehave at a radial distance r from the black-hole center is given by: 3 3 3 12 6 48 2μ2 m ðδ Δ 10 − Δ Þ 22 G ≈ n10 ðC5Þ K ¼ 4 6 ðD1Þ δ3=2γ3 c r pffiffiffiffi which implies that the Riemann tensor ∼ K.Thecoupling term μ Δ108 Δ ∈ 1 100 where ¼ GeV, f ; g and ffiffiffiffi ffiffiffiffiffi −2 p p γ ∈ f10 ; 1000g. Fig. 4 shows the behavior of the K 48Gμ 1 ∼ : ðD2Þ reheating temperature as a function of M, for different M2 c2r3 M2 ranges of Λ, μ. From the plots we infer that the results μ 2 obtained in Sec. IV by neglecting are consistent with the For Schwarzschild radius rh ¼ 2Gμ=c , the coupling term results in this Appendix. becomes

(a) (b)

FIG. 4. Plot showing the behavior of reheating temperature TRH (vertical axis) with parameter M (horizontal axis), for different ranges of Λ, μ. In upper panel (and lower panel) Λ is fixed at 1014 GeV (and 1012 GeV) and lower energy scale is varied.

063502-12 HELICAL MAGNETIC FIELDS FROM RIEMANN COUPLING … PHYS. REV. D 104, 063502 (2021) pffiffiffiffi rffiffiffi K 3 c4 since Riemann tensor is coupled with gauge kinetic terms ∼ : ðD3Þ μν M2 4 G2μ2M2 FμνF , the coupling term contains the frequency of the electromagnetic waves, i.e., ω2, hence the coupling will 30 We set μ ¼ 1 M⊙ where M⊙ ≈ 10 Kg is the solar mass. have effects at very high frequency. The coupling constant Since the result might be interesting to various astrophysical (D3) is of the order of one or greater for black holes of mass μ ∼ 100 and cosmologicalffiffiffiffi phenomenon, we will calculate the value MP, i.e., p 2 of the coupling K=M (D3) in units which are preferred in pffiffiffiffi early universe cosmology (natural units) and gravity (geom- K ∼ 1; for μ ∼ 100M ðD9Þ etrized units). M2 P: Natural units are preferred in the early Universe physics, and all scales are rewritten in terms of GeV. For simplicity, Such size primordial black-holes form in the very early we write the following quantities in terms of mP as, Universe. For such black holes, Hawking temperature is [58] G ≈ M−2;c1;M≈ 10−2M ; P ¼ P 30 38 ℏc3 1 μ ¼ 10 Kg ≈ 10 MP ðD4Þ −3 TH ¼ ≈ 10 MP: ðD10Þ GkB 8πμ 19 −8 where we have used MP ≈ 10 GeV ≈ 10 Kg and 17 −2 M ¼ 10 GeV ≈ 10 MP. Using Eq. (D4) in Eq. (D3), Since the Hawking radiation is thermal, we can obtain the we get peak wavelength of the black-body spectrum from Wien’s λ λ pffiffiffiffi displacement law maxT ¼ constant. The wavelength max K corresponding to 100M mass black holes is given by: ≈ 10−72: ðD5Þ P M2 λ ∼ −1 ≈ 103 −1 max T M ðD11Þ To understand the effect in astrophysical phenomenon, we H P calculate the value of the coupling (D3) in geometrized Using the conversion 1 GeV ¼ 5.06 × 1015m−1 which units ðc ¼ G ¼ 1Þ. Since our model parameter M ≈ 34 −1 gives M ≈ 10 m , we get 1017 GeV has unit of energy, we need to substitute the P MG conversion M → 4 in Eq. (D3), which gives c λ ≈ 103 −1 ≈ 10−31 ⇒ ν c 1039 −1 max MP m; ¼ λ ¼ s : pffiffiffiffi rffiffiffi max K 3 c12 ≈ : ðD6Þ ðD12Þ M2 4 G4μ2M2

Using the following values, Assuming these PBHs are produced just after big bang, the above frequency will be redshifted by a factor 1020. Thus, 19 c ¼ G ¼ 1;M¼ 10−10 Kg; μ ≈ 1030 Kg ðD7Þ the redshifted frequency is ∼10 Hz (Gamma-ray) and can have potential signatures. However, it is important to note we obtain that at that scale, we need to include higher-order corrections. As mentioned earlier, in this analysis, we have pffiffiffiffi K ignored the mass dimension 8 operators [39]. At the ≈ 10−40 Kg−4: ðD8Þ M2 Planck scale, we need to include dimension 8 and beyond. Hence, to understand these effects of PBH of Planck mass Hence, from the above analysis, it is clear that the effect size, we need to include mass-dimension 8 operators and near the solar mass size black-hole is negligible. Note that beyond.

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