Universes without the Weak Force: Astrophysical Processes with Stable Neutrons

E. Grohs1, Alex R. Howe2, and Fred C. Adams1,2 1Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA and 2Department of Astronomy, University of Michigan, Ann Arbor, Michigan 48109, USA (Dated: January 19, 2018) We investigate a class of universes in which the weak interaction is not in operation. We consider how astrophysical processes are altered in the absence of weak forces, including Big Bang Nucle- osynthesis (BBN), galaxy formation, molecular cloud assembly, star formation, and stellar evolution. Without weak interactions, neutrons no longer decay, and the universe emerges from its early epochs with a mixture of protons, neutrons, deuterium, and helium. The baryon-to-photon ratio must be smaller than the canonical value in our universe to allow free nucleons to survive the BBN epoch without being incorporated into heavier nuclei. At later times, the free neutrons readily combine with protons to make deuterium in sufficiently dense parts of the interstellar medium, and provide a power source before they are incorporated into stars. Almost all of the neutrons are incorporated into deuterium nuclei before stars are formed. As a result, stellar evolution proceeds primarily through strong interactions, with deuterium first burning into helium, and then helium fusing into carbon. Low-mass deuterium-burning stars can be long-lived, and higher mass stars can synthesize the heavier elements necessary for life. Although somewhat different from our own, such universes remain potentially habitable. Keywords: big bang nucleosynthesis, galaxy formation, stellar nucleosynthesis, multiverse, habitability

I. INTRODUCTION interactions, core-collapse supernova will fail to explode and will simply collapse to a degenerate remnant, thereby The fundamental constants that describe the laws of hampering the dispersal of heavy elements. physics appear to have arbitrary values that cannot be These effects would appear to compromise the habit- explained by current theory. One possible — and partial ability of universes where the weak force is significantly — explanation is that other universes exist in which the weaker than in our own. However, this standard argu- fundamental constants have different values, so that they ment has been challenged with the concept of a “weak- are drawn from an as-yet-unknown probability distribu- less universe” [9], namely, a universe without any weak tion. Many authors have argued that significant changes interaction at all. Neutrons would be stable in such a in these constants could render the universe uninhabit- universe, and would therefore most likely have an equal able to life as we know it, and as a result our universe ap- abundance to protons. A universe with the same baryon pears to be “fine-tuned” for life [1–5]. On the other hand, density as ours would convert virtually all baryons to recent work suggests that when multiple constants are al- helium during BBN, leaving behind no free protons (no lowed to vary, large regions of the parameter space that hydrogen to make water). However, if the baryon den- result in habitable universes can be found [6–8]. This sity is lower (more properly the baryon-to-photon ratio paper continues the exploration of alternate possibilities η), significant amounts of protons and deuterium can sur- for the fundamental constants with a focus on the weak vive the BBN epoch [9] and would be available later for force. deuterium burning in stars (which takes place through The strength of the weak interaction is a fundamen- the strong interaction). Note that stars in such a weak- tal part of the standard model of particle physics and less universe can also produce heavier elements by strong represents one parameter that could vary from region reactions. Although core-collapse supernova might not to region. The weak interaction governs the rate of ra- function, at least not in the same manner as in our uni- dioactive decay, the rate of conversion of hydrogen into verse, these heavy elements could still be dispersed by + helium via the p(p, νee )D stage of the pp-chain in low- Type Ia supernova and classical novae, allowing the pos- arXiv:1801.06081v1 [astro-ph.GA] 17 Jan 2018 mass stars, and the cross section for neutrino interac- sibility of planet formation and life. tions. The latter two effects are crucial to determin- The opposite case, where the weak interaction is sig- ing whether or not a universe can produce life. If the nificantly stronger than in our universe, would resemble weak force is too weak, or absent altogether, long-lived our universe more closely. A stronger weak interaction stars fueled by weak reactions could not exist. In the ab- increases the rate of radioactive decay, most notably the sence of weak interactions, helium can still be synthesized neutron lifetime, but it does not affect the stability of through strong interactions during Big Bang Nucleosyn- stable nuclei, which is governed primarily by the strong thesis (BBN), as free neutrons are present, but helium interaction. If the neutron lifetime is less than 30 sec- production is suppressed in stellar interiors (which do onds, BBN is suppressed because most neutrons decay not have neutrons). Once synthesized, helium can later before BBN begins. However, this complication is not fuse into heavier elements via the triple alpha process an impediment because stars in this all-hydrogen uni- in sufficiently massive stars. However, without neutrino verse will be more efficient at converting hydrogen to he- 2 lium. In addition, core-collapse supernovae will be more Iron peak elements in the ISM will capture an excess of efficient at dispersing heavy elements due to the larger free neutrons, allowing stars to form with a small excess neutrino interaction cross sections. The primary concern of protons, which can then form oxygen via the reaction would be the shortened lifetimes of these more efficient 12C(2p, γ)14O. Similar reactions may lead to a deficit of stars. (Note that if the weak interaction is strong enough nitrogen and an excess of neon compared with our uni- to approach the strength of the electromagnetic interac- verse, and likewise, the lack of core-collapse supernovae tion, non-linear effects will likely render these concerns will likely lead to a deficit of non-alpha-process elements. moot.) However, the effect on the chemical evolution of the weak- In this paper, we update and expand upon this idea less universe is neither negligible nor dominant, and the of a weakless universe in Refs. [10–12], and in particu- necessary elements for both planet formation and organic lar Ref. [9]. An important parameter in this problem is chemistry will still be present, implying that such a uni- the baryon-to-photon ratio η, which impacts the compo- verse could still be hospitable to life. sition of the universe after BBN. For the value in our The organization of this paper is as follows. Section II universe, BBN with an equal amount of neutrons and details the outcome of BBN in universes with a range of protons would result in a composition where the 4He η and neutron-to-proton ratios. We examine the impact abundance is greater than 90%. The high abundance of weakless physics on: galaxy formation in Sec. III; the of 4He yields short-lived stars and is problematic for the ISM in Sec. IV; and stellar evolution in Sec. V. Section development of life. For a judicious choice of η, how- VI contains our study of the habitability of the weakless ever, BBN can result in a richer composition with large universe. We summarize and discuss our results in Sec. fractions of deuterium, free protons, and free neutrons. VII. Throughout this paper, we use cgs units with the We determine the range of η for which weakless universes exception of BBN in Sec. II. We use natural units in Sec. could support life, using models of galaxy formation, star II to be consistent with the BBN literature. formation, and subsequent stellar evolution. One crucial issue not addressed in the original proposal II. BIG BANG NUCLEOSYNTHESIS [9] is the abundance of free neutrons left over from BBN, which is comparable to the abundance of free protons. These neutrons can capture onto protons at zero tem- A. Standard versus weakless BBN perature, forming deuterium via the n(p, γ)D reaction. As a result, nuclear fusion can occur in the interstellar In this section, we give a brief overview of the role the medium and could potentially halt the collapse of a gas weak interaction plays in Standard BBN (SBBN) and cloud. This paper considers the effects of neutron cap- then discuss the modifications to SBBN in a weakless ture reactions on four scales of formation: [A] from the universe. We will use the ratio of neutrons to protons intergalactic medium down to the size scale of galaxies; (denoted the n/p ratio) extensively throughout this work [B] from the neutral interstellar medium (ISM) to the nn formation of giant molecular clouds; [C] from the cloud n/p . (1) ≡ n to molecular cloud cores – the sites of individual star p formation events; and finally [D] from the cloud cores Note that the number densities ni in Eq. (1) are the to- to the production of protostars themselves. Neutron fu- tal particle number densities (free particles and nucleons sion becomes significant during the latter stages of this bound in nuclei) and not solely the free particle number hierarchy, and most of the free neutrons are processed densities. Reference [13] showed that n/p would evolve into deuterium before the onset of stellar nuclear burn- due to six weak reactions during primordial nucleosyn- ing. Although gas cooling is delayed, these processes do thesis. We schematically write the six reactions as not disrupt star formation entirely. − We next consider stellar evolution and nuclear burning νe + n p + e , (2) ↔ + in a weakless universe and the chemical evolution of the e + n p + νe, (3) ↔ universe through successive generations of stars. Long- − n p + e + νe. (4) lived stars can exist in a weakless universe if the deu- ↔ terium abundance is high enough for deuterium fusion to and colloquially refer to them as the n p rates. In the continue on Gyr timescales. With the main sources of approximation that the nucleon rest masses↔ are much heavy element dispersal being red giant winds and Type heavier than both the neutrino and electron masses, Ref. Ia supernovae, the composition of the interstellar medium [14] gives the prescription for calculating the rates for will be different from our universe, relatively enriched in each of the six reactions listed in Eqs. 2 – 4. At high carbon and iron peak elements and depleted in oxygen. temperature T , the weak interaction rates are fast and A low oxygen abundance could cause a dearth of water maintain the n/p ratio in weak equilibrium as a function which would impose a problem to life – assuming water of T is an essential ingredient to life like it is on our planet.   However, later generations of stars will undergo different δmnp + µe− µν n/p (T ) = exp − − e , (5) reactions that will likely mitigate this problem. T 3 where δmnp = 1.293 MeV is the mass difference between network includes 34 strong, electromagnetic, and weak a neutron and proton, µe− is the chemical potential of interactions from Refs. [17] and [18]. For more precise the electrons, and µνe is the chemical potential of the calculations of SBBN, one can evolve the neutrino spectra electron neutrinos. In SBBN, the chemical potentials as they go out of equilibrium which can lead to changes at of the electron and electron neutrino are small and so the 1% level [19]. The present work does not require this the mass term dominates. Therefore, n/p 1 at high level of precision, so we always assume neutrinos are in temperature. Figure 6 of Ref. [14] shows how' n/p ini- a Fermi-Dirac distribution with zero chemical potential tially follows an equilibrium track for temperatures above and a temperature given by the comoving temperature 1 MeV, and diverges from equilibrium at lower tempera- parameter Tcm as a function of scale factor a ture. Free neutron decay [the forward process in Eq. (4)] ain would eventually transmute all neutrons into protons if Tcm(a) Tin . (9) there were no other nuclear processes. n/p would go to ≡ a zero and the universe would emerge from BBN with a The subscript “in” on the temperature and scale factor pure 1H composition. Such a scenario does not transpire symbols denotes an initial epoch when the neutrinos are as a chain of reactions assembles free protons and neu- in thermal equilibrium with the electromagnetic compo- trons into 4He and some other low atomic mass nuclides. nents of the plasma. We must use T because it is man- Nuclear freeze-out of 4He occurs when the temperature cm ifestly different from T and has ramifications on the n/p has reached T 100 keV and n/p 1/7. ratio [20]. In a weakless∼ universe, there are∼ no weak interactions A wBBN calculation proceeds in the same way as a and Eqs. (2) – (4) do not apply. This implies that neu- SBBN one. We evolve T , a, φ , and the nine nuclides trons are stable to decay. The neutron to proton ratio is e as a function of time. We maintain all strong and elec- set at a high temperature and is a function of the ratio tromagnetic cross sections unaltered from SBBN. There of up quarks to down quarks, u/d are two key differences between wBBN and SBBN. In 2 u/d a weakless universe, there are no weak nuclear inter- n/p = − . (6) 2u/d 1 actions. We remove the n p rates, i.e., Eqs. (2) – − (4), and any other β-decay rates,↔ most notably that of u/d ranges from 1/2 for a pure neutron universe to 2 the three-nucleon hydrogen isotope tritium (denoted as 3 − for a pure proton universe. Assuming baryon number T), T He + e + νe. Tritium has a mean lifetime → 3 conservation, n/p is fixed and does not evolve. Weakless τ3 20 years and will not decay into He until well after BBN (wBBN) proceeds with a fixed n/p ratio. If there SBBN∼ concludes. If doing precision SBBN calculations, are multiple families of quarks with nonzero densities, we would add the abundance of T to that of 3He to then there would be other hadrons present. We do not compare to observation [21]. In a weakless universe how- consider such scenarios in this work. ever, the difference in atomic number between T and 3He could be important in the initial stages of stellar evolu- tion. As a result, we delineate the freeze-out T and 3He B. Description of calculations abundances in order to use them in our stellar calcula- tions. We employ the code burst [15] to do both SBBN and The other difference between SBBN and wBBN is the wBBN calculations. burst is based off of the work in existence of neutrinos. Our model of SBBN in Ref. [15] Refs. [16] and [17]. In a SBBN calculation, there are four assumes that neutrinos only interact via the weak in- sets of quantities to evolve as a function of time. There teraction and gravity. We do not know the mechanism are three thermodynamic variables: the plasma temper- which populates the neutrino states at early times in our ature T ; the scale factor a; and the electron-degeneracy universe. In this work, we assume that if there is no parameter φe where weak interaction, then there would be no relic neutrino seas. We note that if there were neutrinos in a weakless µe− φe . (7) universe, then the neutrino energy states would not nec- ≡ T essarily be thermally populated. Alternatively, one could All thermodynamic quantities of the plasma, such as en- consider the Cosmic Microwave Background (CMB) ob- ergy density and pressure, are functions of T , a, and φe. servable Neff. Neff is the “effective number of neutrino The last set of evolution quantities are the abundances species” [see Eq. (3.11) in Ref. [22]]. In our universe, Yi. Abundances are related to the mass fractions Xi via Neff 3 for the three flavors of neutrino species [23]. In ∼ a weakless universe where neutrinos were present, Neff Xi Yi = , (8) would most likely be smaller than 3. The exact num- A i ber would depend on the mechanism which created the where Ai is the atomic mass number for species i. To neutrinos. The neutrinos would behave as a dark ra- compare SBBN and wBBN calculations, we will employ diation component [24] and wBBN would proceed with a network of nine nuclides which include all bound nu- an expansion rate different than SBBN. For the wBBN clides up to mass number A = 7. The nuclear reaction calculations in this work, Neff = 0. 4

C. Mass fraction evolution 1 10− 3 10− In this section we discuss the evolution of the mass 5 10− fractions as a function of the Tcm variable from Eq. (9). 7 10− For our model of SBBN, the baryon number is the only 9 10− cosmological input. We do not consider other cosmolog- 11 10− 13 ical inputs such as Neff or neutrino chemical potentials. X 10− 15 There are many equivalent ways to represent baryon 10− 17 solid : SBBN number in BBN. We adopt the nomenclature of Ref. [17] 10− 19 dashed : wBBN and use the baryon-to-photon ratio η 10− 21 n T 10− p 3He nb 23 η , (10) 10− D 4He ≡ nγ 25 10− 1 0 1 2 3 10 10 10− 10− 10− where n and n are the proper number densities of b γ Tcm (MeV) baryons and photons, respectively. Based on the strict definition in Eq. (10), η decreases as electrons and positrons annihilate to become photons for T . 100 keV. 1 10− Therefore, we will refer to numerical values of η after the 3 10− epoch of electron-positron annihilation. In wBBN, η and 5 10− the n/p ratio are both inputs. When comparing SBBN 7 10− to wBBN, we will use the same value of η. There is no 9 10− 11 way to compare SBBN to wBBN using the same n/p ra- 10− 13 tio as n/p evolves in SBBN according to the n p rates X 10− ↔ 15 schematically shown in Eqs. (2) – (4). We will pick a 10− 17 solid : SBBN value of the n/p ratio to elucidate specific points on how 10− 19 dashed : wBBN wBBN differs from SBBN. 10− 21 n T Figures 1 and 2 show mass fractions versus Tcm in a 10− p 3He 23 10− D 4He number of BBN cases. Solid lines are from a SBBN calcu- 25 lation with an evolving n/p ratio and dashed lines from 10− 1 0 1 2 3 10 10 10− 10− 10− a wBBN calculation with a constant n/p ratio. Both T (MeV) calculations, for a given figure, use the same value of η. cm For ease in reading, we only plot the mass fractions of free neutrons (n), free protons (p), deuterium (D), tri- tium (T), helium-3 ( 3He), and helium-4 ( 4He). We note 6 7 7 FIG. 1. Mass fractions, X, as a function of Tcm. Each color that the mass fractions of Li, Li, and Be are all sub- represents a different isotope. (Top) Solid lines are a SBBN sidiary to that of the hydrogen and helium isotopes in calculation, and dashed lines are a wBBN calculation where both SBBN and wBBN. We do not include other possi- n/p = 1. The dashed green line for n is coincident with the bly stable isotopes in a weakless universe, i.e., 6He. dashed red line for p. η = 6.075×10−10 for both calculations. Figure 1 uses η = 6.075 10−10 which is consistent (Bottom) Same as the top panel except n/p = 0.14 for the with the value in our universe× [23]. n/p = 1 for the wBBN calculation. wBBN calculation in the top panel. At high tempera- tures, the abundances are in Nuclear Statistical Equi- librium (NSE). NSE abundances are functions of plasma in wBBN) in the top panel in Fig. 1. The wBBN mass temperature, η, Xn, Xp, and nuclear properties [25]. The fractions for n, p, and D have frozen out at the end of mass fractions of D, T, and 3He are all following NSE the horizontal plotting axis. Although neutrons and pro- 4 tracks above T & 1 MeV ( He also follows a NSE track tons can interact to form deuterons, the baryon density below the scale of the vertical axis). For temperatures is so low that all three mass fractions have frozen out at T > δmnp in a SBBN calculation, Eq. (5) shows that roughly the 1% level. n/p 1, implying that the SBBN and wBBN scenarios For SBBN at η = 6.075 10−10, 4He freezes-out with ∼ × have the same conditions for NSE. As a result, the mass a mass fraction X 4 YP 0.25, where we adopt the He ≡ ' fractions evolve identically for both SBBN and wBBN cosmological notation of YP to denote the mass fraction at high temperatures in the top panel. Although the of 4He. The top panel of Fig. 1 shows that the number of mass fractions remain in NSE, the SBBN curves begin baryons in D and 3He is only a few in 105, implying that to diverge from the wBBN curves once T becomes com- the vast majority of neutrons are in 4He nuclei. There- parable to δmnp. At even lower temperatures, n/p for fore, n/p 1/7 = 0.14 at the conclusion of SBBN. The SBBN is well below unity and the mass fractions diverge bottom panel' of Fig. 1 shows the same SBBN calcula- for the two scenarios. Note that the red dashed curve tion as the top panel, but the wBBN calculation now has (Xp in wBBN) is on top of the green dashed curve (Xn n/p = 0.14. In the NSE regime at high temperatures, 5 there is a clear difference between the SBBN and wBBN 1 calculations. As the temperature decreases and the n/p 10− 3 ratios converge to one another for the two scenarios, the 10− 5 abundances begin to come into agreement, especially for 10− 7 p 4 10− and He. 9 10− Nuclear reactions which involve the capture of a neu- 11 10− 13 tron are not subject to the Coulomb repulsion unavoid- X 10− able in proton capture reactions. At low temperatures, 15 10− 17 solid : SBBN neutrons can continue to capture on heavier nuclei or free 10− 19 dashed : wBBN protons. It is possible for BBN to occur over a long pe- 10− 21 n T riod if free neutrons are present and the proper baryon 10− p 3He 23 number density is large. For the wBBN calculation in 10− D 4He 25 the top panel of Fig. 1, there is a preponderance of free 10− 101 100 10 1 10 2 10 3 neutrons at Tcm 10 keV, although the number den- − − − sity of baryons is' low enough that there is no late-time Tcm (MeV) rise in D. However, the flux of neutrons is large com- pared to the number of 3He targets. The principal reac- 3 1 tions which utilize neutrons and He nuclei as reactants, 10− 3 namely 3He(n, p)T and 3He(n, γ) 4He, have not frozen- 10− 5 3 10− out at the end of the plotting axis. As a result, the He 7 10− mass fraction continues to decrease at the end of the sim- 9 6 10− ulation. Similarly, the flux of neutrons compared to Li, 11 7 7 10− Li, and Be targets is also large in the weakless sce- 13 X 10− nario. If we had plotted the mass fractions of the Li and 15 10− Be isotopes, they would also be decreasing at the end 17 solid : SBBN 10− dashed : wBBN of the horizontal plotting axis in much the same man- 19 10− ner as the mass fraction for 3He does. We verified our 21 n T 10− p 3He 23 hypotheses by extending the wBBN calculation down to 10− D 4He temperature of T = 100 eV. However, our library for the 25 10− 1 0 1 2 3 integrated cross sections (based off of that in Ref. [17]) 10 10 10− 10− 10− is not accurate at such low temperatures. Furthermore, Tcm (MeV) the error in not finishing the wBBN calculation would be small as the mass fractions of 3He and the heavier isotopes are orders of magnitude smaller than the more abundant species. As we consider the composition of the FIG. 2. Mass fractions, X, as a function of Tcm. Each color later weakless universe, we will take the mass fractions represents a different isotope. (Top) Solid lines are a SBBN from wBBN at the T 1 keV epoch and ignore the small calculation, and dashed lines are a weakless calculation where 3 ∼ n/p = 1. The dashed green line for n is coincident with the error present in He. dashed red line for p. η = 4.0 × 10−12 for both calculations. −12 In Fig. 2, we use a value of η = 4.0 10 . This (Bottom) Same as the top panel except n/p = 0.026 for the specific value is in line with the value chosen× in Ref. [9]. wBBN calculation. The top panel of Fig. 2 shows the evolution of the mass fractions for SBBN and wBBN with n/p = 1. For SBBN, 3 the mass fractions of D, T, and He are all increased over fraction of D as compared to 4He. With fewer neutrons, −10 the freeze-out mass fractions when η = 6.075 10 in the NSE abundances are lower. We observe this by using 4 × Fig. 1. He is significantly reduced for the low value the top and bottom panels of Fig. 2 to compare the lo- of η in Fig. 2. Conversely, in wBBN there are roughly cations of the dashed lines of T and 4He with respect to 4 equal amounts of n, p, D, and He. The freeze-out mass the solid SBBN lines. The initial conditions for out-of- fractions are in close agreement with Fig. [1] in Ref. [9]. equilibrium nucleosynthesis occur when the n/p ratio is As compared to the top panel of Fig. 1, neutrons are still evolving in SBBN, i.e., n/p > 0.026 when the mass 4 not predominantly incorporated into He nuclei for the fractions go out of equilibrium. As a result, the sum of lower value of η. In both SBBN and wBBN, the D mass XD and YP is larger in SBBN than wBBN. For the n/p 4 fraction is comparable to that of He. SBBN makes less ratios to be equal at freeze-out, the deficit of neutrons 4 He (and less D) as the n/p ratio evolves to 0.026. in wBBN must be incorporated into a different nuclide. The bottom panel of Fig. 2 is identical to the top panel Indeed, the bottom panel of Fig. 2 shows a mass fraction except we run wBBN with n/p = 0.026. Although the of free neutrons on order of 1%. This is a key difference n/p ratios are identical between SBBN and wBBN, the of SBBN compared to wBBN which was absent in Fig. 1: final freeze-out mass fractions of D and 4He differ. Fur- identical n/p ratios at freeze-out do not imply identical thermore, in the weakless scenario, there is a larger mass mass fractions of D and 4He. If the baryon number is low 6 enough, out-of-equilibrium nucleosynthesis begins during X weak freeze-out. The n/p ratio alone is not enough to 101 p predict the final mass fractions of 4He.

10 0 10 . To conclude this section, we note neither SBBN nor − 01 − 10 0 5 wBBN possesses D and T peaks in Fig. 2. The peaks .1 T are quite visible in the bottom panel of Fig. 1 at cm 0 50 keV for both SBBN and wBBN. The peaks stem from∼ .25 0 synthesis of D and T into larger nuclei, most notably 4He, 10 n/p 0 and no production channels of equivalent strength. For .5 the lower value of η, the nuclear reaction rates freeze-out at an earlier time and the mass fractions plateau. 0.75

1 10− 12 11 10 9 8 7 6 D. Parameter space scan 10− 10− 10− 10− 10− 10− 10− η The previous section detailed a comparison between SBBN and wBBN at various values of η. In this sub- section, we explore the n/p versus η parameter space of FIG. 3. The neutron-to-proton ratio, n/p, versus the baryon- wBBN. We cannot directly compare with SBBN because to-photon ratio, η, at contours of constant p mass fraction. In there is no dual parameter space for cosmological inputs a weakless universe, n/p is fixed at the start of BBN and does [see Fig. (2) in Ref. [26] for SBBN mass fractions as a not evolve through nuclear freeze-out. In our universe, weak function of η]. As an alternative, we place a red star on interactions change n/p until 4He formation when n/p ≈ 1/7. the contour plots at the values η = 6.075 10−10 and The red star indicates the point in parameter space where n/p n/p = 0.14 of our universe. The red star comes× from a terminates its evolution in our universe. This point should simulation that models our universe with the weak in- only be taken for illustrative purposes and not labeled a mem- teraction, namely the SBBN calculation in Fig. 1. n/p ber of the weakless class of universes. changes with time in such a scenario. In addition, our universe contains neutrino energy density so the Hubble expansion rate is different. We emphasize that the red X 101 n star is only for illustrative purposes and does not belong to the class of weakless universes. It is only meant as a .75 guide and not for direct comparison. 0 Figure 3 shows contours of constant p mass fraction in .5 the n/p – η plane. The red star is located close to the 0 100 75% contour, which is in agreement with the value from 25

n/p . 0

SBBN as seen on the bottom panel of Fig. 1. For the .1 0

η 01 5 range of plotted, a significant fraction of the baryons . 0 −

10 10 are free protons if n/p is below unity. Once n/p becomes − larger than unity, few free protons remain. The isospin 10 mirror of Fig. 3 is Fig. 4: contours of constant Xn in 1 10− 12 11 10 9 8 7 6 the n/p – η plane. For n/p above unity, there exists a 10− 10− 10− 10− 10− 10− 10− significant fraction of free neutrons. Conversely, for n/p η below unity, few free neutrons remain. Figures 3 and 4 show a remarkable degree of symme- try about the line n/p = 1. The symmetry is broken, albeit slightly, in the mass fraction of D. Figure 5 shows FIG. 4. n/p versus η at contours of constant n mass fraction. contours of constant XD in the n/p – η plane. At con- The red star indicates the point in parameter space where n/p stant η, the mass fraction of D increase as n/p approaches terminates its evolution in our universe. unity from either direction. The rate of increase is larger as n/p decreases towards unity. This is only evident for large η in Fig. 5. The contours of X at 10−5 and 10−10 n/p = 1 and η = 5 10−12. D × are closer in the parameter space for n/p > 1. The value The symmetry about n/p = 1 is restored in Fig. 6: of the contours indicates that the degree of asymmetry contours of constant YP in the n/p – η plane. For the 5 is small – a few parts in 10 . parameter space shown in Fig. 6, the smallest value of YP −5 −3 The red star in Fig. 5 is located on the XD = 2.7 10 is 7 10 . This value occurs for small η and either contour. For comparison, the D mass fraction of SBBN× is large∼ or× small n/p. The mass fraction value is above our 4.0 10−5. wBBN is able to produce a large mass fraction asymmetry limit of 10−5, so we cannot detect any visible × of D. The largest mass fraction of D is XD = 0.14 at asymmetry in Fig. 6. For small η, Fig. 6 shows that not 7

X Y 101 D 101 P

10 10 0 0 − − . . 10 1 01 5

1 5

25 75

. 0 0 .

. .

0 0

10 10 0 0 n/p n/p

1 1 10− 12 11 10 9 8 7 6 10− 12 11 10 9 8 7 6 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− η η

4 FIG. 5. n/p versus η at contours of constant D mass fraction. FIG. 6. n/p versus η at contours of constant YP , the He The red star indicates the point in parameter space where n/p mass fraction. The red star indicates the point in parameter terminates its evolution in our universe. space where n/p terminates its evolution in our universe. all neutrons are incorporated into 4He nuclei. In fact, over that of free neutrons even though the universe is the mass fraction of D is larger than that of 4He for a isospin symmetric, i.e., n/p = 1. This is due to the high certain range of η if n/p = 1. Furthermore, Fig. 4 shows reactivity of neutrons, or equivalently, a Coulomb barrier that most neutrons are free at small η. For large η, the in proton capture. mass fraction of 4He is independent of η in the range of n/p we employ in Fig. 6. The contours of constant mass fraction are symmetrical about n/p = 1 and horizontal. We can succinctly capture the relationship between the III. GALAXY FORMATION mass fraction of 4He and n/p in this range

min(1, n/p) This section considers how nuclear reactions involving YP 2 . (11) ' 1 + n/p free neutrons can potentially affect the galaxy formation process. Here we assume that structure formation in- If n/p < 1, Eq. (11) reduces to the familiar YP = volves processes that are analogous to those acting in our 2(n/p)/(1 + n/p) [27]. Finally, we note that the loca- universe. Without weak interactions, a universe can in tion of the contours in Fig. 6 (or, more specifically, the principle still produce dark matter [9]. In this case, the rounded ends at low η) depends on the nuclear reaction timing of structure formation would remain the same, rates versus the Hubble expansion rate. We do not in- and this is the case considered here. clude neutrinos or any other form of dark radiation in For completeness we note that even in the absence our model of wBBN. If we had, that would increase the of dark matter, a purely baryonic universe can produce Hubble rate and decrease YP . The result would be a shift structure. In the present context, however, the baryon- of the contours in Fig. 6 in the horizontal direction to- to-photon ratio η must be smaller due to BBN constraints wards higher η. There would be no change in the vertical (see Sec. II D) so that the epoch of matter domination oc- direction of Fig. 6 at the level of precision presented in curs at lower redshift. As a result, galaxy formation takes the parameter space. place at later epochs when the universe is more diffuse, Table I gives the freeze-out mass fractions at numerous so that galaxies are less dense for a fixed value of the values of η which we will use in sections III – V. n/p = 1 amplitude Q of density fluctuations. For the particular for all values of η. It appears that the D mass fraction value Q = 10−5 realized in our universe, the resulting gets closer to the n and p mass fractions for increasing densities of galaxies could be so low that baryons have η. For n/p = 1, D gets closest to n when η 10−7 difficulty cooling and condensing [28, 29]. This issue can ' at a value XD/Xn = 0.95. D gets closest to p when be alleviated with larger values of the fluctuation ampli- −8 3 η 5 10 at a value XD/Xp = 0.86. The He mass tude Q [8, 29]. With no dark matter (which has 6 fraction' × is much lower than the mass fractions for the times the density of baryons in our universe) and lower∼ η other nuclides, as explained in Figs. 1 and 2 by the lack (by a factor of 50), the total matter density is smaller of a Coulomb barrier in neutron capture. By the same by a factor of ∼ 300. The corresponding value of the logic, Table I shows a slight excess of the p mass fraction fluctuation amplitude∼ Q 3 10−3. ∼ × 8

η 10−11 10−10 10−9 n 0.2136 3.420 × 10−2 3.535 × 10−3 p 0.2139 3.428 × 10−2 3.545 × 10−3 D 0.1197 2.615 × 10−2 2.766 × 10−3 T 9.764 × 10−4 2.320 × 10−4 2.469 × 10−5 3He 8.844 × 10−10 2.605 × 10−10 2.819 × 10−11 4He 0.4518 0.9051 0.9901

TABLE I. Mass fractions at different values of η. All values for n/p = 1.

A. Properties of Dark Matter Halos gaseous component and subsequent shocks heat up the material to a temperature given by Simulations of structure formation show that dark GMm matter halos asymptotically approach a nearly univer- 3kT = p , (16) sal form. The density profile of the halo is thus taken to r be a Hernquist profile of the form where M = M(r) is the enclosed mass of the dark matter ρ halo at radius r. For the galaxy profiles considered here, ρ(r) = 0 , (12) ξ(1 + ξ)3 this expression becomes GM m ξ where the dimensionless coordinate ξ is defined via 3kT = T p . (17) r (1 + ξ)2 r 0 ξ = . (13) r0 For the Milky Way values, the benchmark temperature scale is given by Note that we use a slightly steeper power-law for the den- −4 sity distribution (ρ ξ for ξ 1) compared to the GMT mp 6 often-used NFW profile∼ [30] (ρ ξ−3). Although only T = 2.7 10 K . (18) 3kr0 ≈ × the inner part of the potential well∼ matters for the con- siderations of this paper, we use the form (12) because This estimate is approximate and does not require that it has a finite mass and because numerical simulations the gaseous density and temperature profiles approach an show that halo density profiles become steeper at later equilibrium condition. We thus generalize the treatment epochs [31, 32]. Given the form of equation (12), the cor- in the following discussion. responding potential and enclosed mass have the forms For a given dark matter halo, we can determine the density and temperature profile of the gas under the as-  2 Ψ0 ξ sumptions that: (1) the dark matter halo dominates the Ψ = and M(r) = MT . (14) 1 + ξ 1 + ξ gravity of the system; (2) the gas can be considered in hydrostatic equilibrium; and (3) the equation of state for We can take ρ0 and r0 to be the defining parameters of the gas is polytropic the halo. The scale Ψ0 for the potential and the total Γ 1+1/n mass MT are then given by P = Kρ Kρ , (19) ≡ 2 3 Ψ0 = 2πGρ0r0 and MT = 2πρ0r0 . (15) where n is the usual polytropic index and K is a scaling constant. Hydrostatic equilibrium thus implies For the example of the Milky Way galaxy, we can model the halo with a profile of the form given by equa- dρ Ψ KΓρΓ−2 = 0 , (20) tion (12) if we take the scale length r0 = 65 kpc and dξ −(1 + ξ)2 −25 −3 the density scale ρ0 = 10 g cm (see Ref. [33] and references therein). so that the density profile has the form

 1/(Γ−1) (Γ 1)Ψ0 ρ(ξ) = − (1 + ξ)−1/(Γ−1) . (21) B. Hydrostatic Equilibrium KΓ

In order to assess the possible effects of nuclear reac- To fix ideas, consider the standard case of an adiabatic tions on galaxy formation, we need to determine the tem- equation of state for a monoatomic gas, where Γ = 5/3. perature and density distributions of the baryonic gas. The density profile then becomes We start with an order of magnitude estimate: the usual  3/2 assumption is that dark matter collapses first, and then 2Ψ0 −3/2 −3/2 ρ(ξ) = (1 + ξ) ρX (1 + ξ) , (22) gas falls into the dark matter halo. The collapse of the 5K ≡ 9 where the second equality defines a benchmark density We can use the hydrostatic profiles from the previous scale. The total mass in baryons is given by the integral section to obtain the column density of the gaseous com- ponent. For simplicity, we use the solutions for Γ = 5/3, Z ξmax ξ2 M πr3ρ dξ πr3ρ I ξ , and find a column density in gas b = 4 0 X 3/2 4 0 X ( max) 0 (1 + ξ) ≡ −1/2 −1/2 (23) Ngas(ξ) = 2ρX r0(1 + ξ) 2fbρ0r0(1 + ξ) , (31) where the final equality defines the dimensionless integral ≈ I. Note that we must invoke a finite boundary to the where we have used Eq. (26). The benchmark value τ0 of density distribution of the baryons in order to keep the the optical depth of the halo to its radiation field – that mass finite (for this profile). If we define fb to be the generated by nuclear reactions — can be defined as baryonic fraction of the total mass, then σT 3 τ0 = 2fbρ0r0 , (32) Mb = fbMT = fb2πρ0r0 . (24) mp Equating Eqs. (23) and (24) implies where we assume that the cross section for interactions between the gamma rays from the nuclear reactions and fbρ0 2ρX I = fbρ0 or ρX = . (25) the remaining gas is given by the Thomson cross sec- 2I tion σT . For values in our universe, this optical depth The integral I is of order unity. For example, if ξmax = 1, τ 10−3. As a result, the halo does not have a photo- the integral I 1/5. In general, we want to consider sphere∼ – it is optically thin to the radiation it generates, ∼ somewhat larger values of ξmax, so we can take 2I 1 so that photons freely stream outwards. In the inner re- ≈ and hence use the ansatz gions of the galaxy, the optical depth approaches τ0. In the outer regions, the the optical depth has spatial de- ρ = f ρ , (26) X b 0 pendence given by where fb 1/6 in our universe. Next we note that since −1/2 ∼ τ(ξ) τ0(1 + ξ) . (33) kT ≈ P = KρΓ = ρ , (27) mp D. Heating and Cooling Rates where mp is the proton mass, we find that Γ−1 2/3 2/3 −1 The cooling rate per unit volume can be generically kT = mpKρ = mpKρ = mpKρX (1 + ξ) . (28) written in the form We can thus write dE = nenpΛ(T ) = nenp σv coolcool , (34) TX mpK 2/3 2mp dtdV T = where TX ρ = Ψ0 . cool h i 1 + ξ ≡ k X 5k (29) where ne and np are the number densities of electrons The temperature in a potential well is typically assumed and free protons, respectively. For sufficiently high gas to be of order kT mpΨ. For the particular case of temperature, the cooling process is primarily due to ∼ an n = 2/3 polytrope considered here, the last equal- bremsstrahlung scattering [2], so the cross section is close ity expresses the particular realization of this expected to the Thomson cross section σT and the speed is given 1/2 relation. For halos with properties of the Milky Way, by the thermal speed of the electrons vs = (kT/me) . 6 the temperature scale TX 8 10 K, which is roughly The energy lost per scattering  can be written in the ≈ × cool comparable to the original estimate from equation (18). form 4e2 h cool = 2.37 keV where λ = . (35) C. Column Density λ ≈ mec Note that these expressions are approximate – an ac- The discussion thus far has assumed that the radia- curate treatment requires integration of the interactions tion produced by any possible nuclear reactions escapes over the thermal distribution of particles and results in the halo and does not affect its structure. To verify the corrections given by factors of order unity. We can thus validity of this assumption, we need to estimate the op- write the cooling rate per unit volume in the form tical depth of the halo of its internal radiation. The first step is to determine the column density. To wit, the col-  1/2 2 dE kT 4e mec umn density of the dark matter in the halo, integrated = AnenpσT , (36) from spatial infinity to a radial location ξ, is given by the dtdV cool me h expression where A is a dimensionless parameter of order unity. Al-  1 + ξ  2ξ + 3  though Eq. (36) is highly simplified, the resulting cooling Ndm(ξ) = ρ0r0 log . (30) ξ − 2(ξ + 1)2 times are comparable to those found previously [34, 35]. 10

This cooling treatment assumes fully ionized gas and E. Time Scales hence high temperatures. As the temperature falls to 4 about T 10 K, this process becomes ineffective, and The collapse time is given by the cooling∼ rate becomes much smaller. As a result, halo 4 −1/2 gas tends to cool down to 10 K and then stay at that −1/2  n  tG = (Gρ) 95 Myr (39) temperature. ≈ 1 cm−3 The corresponding heating rate due to nuclear reac- The cooling time is given by tions can be written in the form 3kT 3kT tcool = = dE 2nΛ 2Anσ v  n n σv  , T s cool = p n np nuke (37) 1/2 dtdV heat h i  n −1  T  5 Myr . (40) ≈ 1 cm−3 106 K where σv 10−19 cm3 s−1 and  is the energy np nuke The heating time due to nuclear reactions is given by depositedh ini the∼ gas due to the reaction. Most of the energy from the reaction is contained in 3kT tnuke = gamma rays, which are (mostly) optically thin and hence 2n σv npnuke h i tend to leave the system. The deuterium nucleus itself −1   −1  n  T  nuke  experiences a recoil energy of about 1 keV, so that the 41 Gyr . ≈ 1 cm−3 106K 1 keV deposited energy has a lower bound  > 1 keV. nuke (41) We can find the requirements for the heating rate (36) and the cooling rate (34) to be in balance: The cooling time is much shorter than the heating time over the entire range of applicability of the cooling func- tion used here. Once the gas cools down to T 104  1/2 kT K, the cooling processes become much less effective,∼ and AσT cool = χn σv npnuke me h i cooling processes cease to operate.  2 We can also find a benchmark density scale where the me σv npnuke = T = h i χ2 . (38) collapse time is equal to the heating time. This scale is k Aσ  n ⇒ T cool given by

 2 2 The parameter χn = nn/np is the relative abundance −3 T  nuke  n? 20 cm . (42) of free neutrons with respect to free protons [and not ≈ 104 K 1 keV equivalent to n/p from Eq. (1)]. Note that we have also assumed that the gas is fully ionized so that ne = np. This number density is larger than the typical mean den- sity of the interstellar medium in the Galaxy, but smaller For optically thin conditions, where the gamma rays than the density of molecular clouds (n 102 103 from the nuclear reactions escape, the two energy scales cm−3), where star formation takes place. ∼ − are comparable (  ) and the equilibrium tem- nuke cool In summary, nuclear reactions in this scenario do not perature would be T <≈ 1 K. The approximations used affect structure formation for scales larger than molec- in the cooling function break down well before this tem- ular clouds. Cloud formation and subsequent evolution perature is reached, so that we expect the gas to stay at of star forming regions, however, can be affected and are T 104 K. In the other limit, where all of the energy discussed in subsequent sections. from∼ the nuclear reactions is contained within the gas, then nuke 1 MeV and the equilibrium temperature would be T ∼ 8 104 K. ∼ × F. Scaling with Amplitude of the Density As a result, under optically thin conditions, the heat- Fluctuations ing due to nuclear reactions is ineffective and the heating and cooling of gas on galactic scales proceeds in the usual Our universe has initial density fluctuations, inferred fashion. The heating due to nuclear processes would only from the observed inhomogeneities in the CMB radia- become important if the temperature from Eq. (38) ex- tion, with amplitude Q 10−5. In other universes, these 4 ceeds T 10 K. This requirement, in turn, implies that fluctuations could be larger,≈ with the consequence that ∼ > the energy deposited per reaction nuke 1 MeV. In or- galaxies can form earlier. This difference in timing, in der for this much energy to be retained∼ in the gas, the turn, results in galaxies that are denser. Here we define optical depth must be close to unity, but Eq. (33) im- the relative amplitude plies that τ reaches a maximum value of τ0 1. As a result, the galaxy remains optically thin to the radiation Q q , (43) generated by nuclear reactions. ≡ Q0 11 where Q0 is the value in our universe. The parameters ρ0 which in turn support the process of star formation. −3 and r0 of the dark matter halos vary with the fluctuation Molecular clouds have densities of order n 100 cm on amplitude that specifies the initial conditions. For dark their largest scales, with much denser internal∼ structure. matter halos with density profiles given by Eq. (12), the At these densities, the nuclear reactions from stable free dependence of ρ0 and r0 on the amplitude Q has been neutrons can act to prevent the cooling of cloud material derived previously [8], where these results are based on and hence delay star formation. the standard paradigm for galaxy formation [34]. The To start, we consider the simplest possible model of a resulting scaling laws can be written in the form molecular cloud: The structure is assumed to have con- n −3 3 −1 stant density with 100 300 cm . Like clouds in our ρ0 q and r0 q . (44) ≈ − ∝ ∝ universe, the thermal pressure is much smaller than that provided by both magnetic fields and turbulence. These The temperature scale T is determined by the depth of X latter sources of pressure thus support the cloud, so that the gravitational potential well of the halo. Since Ψ 0 we can assume that its mechanical structure is largely Gρ r2, the temperature scales according to ∼ 0 0 independent of the thermal evolution of the constituent

TX q . (45) gas. ∝ The cooling processes for the gas are different from The fluctuation amplitude can be larger by a factor of those of the previous section. In this case, the gas is 1000 [8]. largely neutral (not ionized) and the cooling processes ∼ Now we consider how the time scales vary with changes become inefficient. In our universe, the cooling processes in the fluctuation amplitude. Using Eqs. (44) and (45), become dominated by line emission from heavy elements, we find the scaling laws in spite of their low relative abundances. In this context, −3/2 −5/2 −2 however, we consider the gas to be composed only of hy- tG q , tcool q , and tnuke q . ∝ ∝ ∝ drogen, free neutrons, and helium. The cooling function (46) will thus be similar to that applicable for the formation We first consider the case where the gas in the halo of the first stellar generation in our universe. In the limit remains optically thin, so that the deposited energy of high density n 103 cm−3, the gas can maintain local nuke 1 keV. For the larger starting temperature in- thermodynamic equilibrium→ (LTE) and the cooling func- duced∼ by larger q, the cooling rate dominates even more tion Λ nH [35]. Moreover, we can model the cooling over the heating rate due to nuclear reactions. The gas function∝ with the from in these denser galaxies will readily cool down to the 4  3 temperature T 10 K where these cooling processes dE T ∼ = nH C , (47) become ineffective. However, as the galaxies create their dtdV 104 K substructures, they do so at higher densities, which can cool be larger than the threshold where nuclear reactions play where the constant a role [see Eq. (42)]. 2 −19 −1 Conversely, the optical depth scales as τ ρ0r0 q . C = 10 erg s , (48) For q > 30, the halos become optically thick∝ and retain∝ most of∼ the energy generated by the nuclear reactions. and where the functional form is approximate. In the The energy scale nuke thus increases from 1 keV to approximation of constant cloud density, which holds in 1 MeV for sufficiently dense galaxies. ∼ the absence of expansion or contraction of the gas, the ∼ time evolution of the cloud material is governed by the equation IV. PROCESSES IN THE INTERSTELLAR MEDIUM 3 d 3 n (kT ) = nCT + npnn σv npnuke , (49) 2 dt − 4 h i

A. Molecular Clouds 4 where T4 = T/10 K. The equilibrium temperature cor- responds to both sides of the equation vanishing, and has The considerations of the previous section show that the value the additional heating due to nuclear reactions does not greatly inhibit the cooling of gas on galactic scales.† In  n 1/3 Teq 78 K , (50) this section, we thus assume that galaxies form in an ≈ 300 cm−3 analogous fashion to those in our universe and have the same basic internal structures. In the next level of struc- where we have taken nuke = 1 keV. ture formation, the galaxy assembles molecular clouds, In the discussion thus far, the nuclear reactions took place on sufficiently long time scales that we did not need to consider the time evolution of np and nn due to de- pletion of protons and neutrons. If the reaction n(p, γ)D † We are considering universes with the same amplitude Q of the occurs when np = nn, then the less abundant species will initial density fluctuations. exponentially vanish,6 while the more abundant species 12 will asymptotically approach a nonzero constant value. The first term gives us the simple form For simplicity, we assume that the starting densities of neutrons and protons are equal and evolve in the same 1 T4(τ) = , (58) manner. We denote the initial number density of either (1 + 2τ)1/2 species as n0. The characteristic time scale for the pop- ulations of nuclei to change is the inverse of the rate which describes the initial phase of evolution. In con- trast, the long term evolution is given by assuming a

−17 −1  n0  1 quasi-steady-state solution for Eq. (54), which implies γ = n0 σv np 3 10 s −3 . h i ≈ × 300 cm ∼ Gyr 1/3 (51) B T4(τ) = . (59) The population of each nuclear species will thus decrease (1 + Γτ)1/3 with time according to the expression These solutions indicate that the gas can cool relatively n0 quickly (over time scales measured in years) from an ini- n(t) = . (52) 4 1 + γt tial temperature of T0 = 10 K down to temperatures T Teq 80 K. Subsequent evolution and cooling re- The full differential equation for the time evolution of the lies∼ on the∼ depletion of neutrons, which provide a nuclear (constant density) cloud thus has the approximate form heating source. The relevant depletion time is 1 Gyr [see Eq. (51)]. ∼ 3 d 3 γnuke These results suggest that the formation of molecular (kT ) = CT4 + . (53) 2 dt − 1 + γt clouds, and the subsequent onset of star formation, will be only modestly affected by the presence of nuclear reac- Next we can write this evolution equation in dimension- tions due to free neutrons. Heating due to these reactions less form by defining a time scale tc = (3/2)kT0/C, where will slow the cooling of the gas material and thus delay 4 T0 = 10 K (note that tc 0.67 yr). In terms of the ∼ star formation. The depletion time is of order 1 Gyr, dimensionless time τ = t/tc, the evolution equation be- so that, after a delay of this order, star formation could comes proceed unimpeded. The nuclear reactions produce deu- terium, so that the stars that form will be enriched in dT4 B = T 3 + , (54) deuterium relative to those in our universe. dτ − 4 1 + Γτ where B = γ /C 4.8 10−7 and where Γ = γt nuke c B. Star Formation 6 10−10. Since the≈ second× term evolves on a much∼ longer× time scale than the dimensionless time τ in the differential equation, we can find an approximate solu- As the next stage of structure formation, molecu- tion by fixing the value of the second term when solving lar clouds produce small centrally concentrated regions the equation, and putting the time dependence back in which constitute the actual formation sites for individual afterwards. We thus define stars [36]. These structures, called molecular cloud cores, slowly condense out of the larger cloud as they lose pres-  B 1/3 sure support from both turbulence and magnetic fields. a = a(τ) , (55) ≡ 1 + Γτ After the core regions reach a sufficiently concentrated configuration, they undergo dynamic collapse, with a star and integrate the differential equation to find the implicit forming at the center of the collapse flow. A circumstel- solution lar disk forms around the star and serves as a reservoir of angular momentum. This section shows that the con- Z 1 dT densation phase of this sequence leads to the processing τ = T 3 a3 of most of the free neutrons into deuterium. T4 −   2    During the condensation phase, the density distribu- 1 a + a + 1 1 a tion of the molecular cloud core can be described by a = 2 log 2 2 + 2 log − 6a a + aT4 + T4 T4 a profile of the form [37]     − 2√3 a + 2 a + 2T4 + tan−1 tan−1 . Λ 1 r 6a2 √3a − √3a ρ(r, t) = where ξ = . (60) 2πGt2 1 + ξ2 a t (56) | | In this expression, time t is defined as the time before This form is rather cumbersome. We can also write the dynamic collapse begins, so that t decreases as the core integrand as a series and integrate term by term to obtain grows more concentrated. The dimensionless parameter

∞ 3n Λ specifies the initial overdensity of the region and is of X a h −(3n+2) i τ = T 1 . (57) order unity. The parameter a is the effective transport 3n + 2 4 n=0 − speed in the gas. 13

With the density distribution of Eq. (60), molecular relatively few free neutrons left, and to begin their evo- cloud cores will remain optically thin until extremely lution with a large deuterium composition. short times before the onset of collapse. These short The discussion thus far assumes that the energy pro- times are not realized in practice, as the solution break duced by the nuclear reactions has a negligible effect on down once the time is shorter than 104 years. At this the evolution of the condensing cloud core. Equation time before collapse, the core experiences∼ a rapid transi- (63) specifies the number of reactions per unit time inte- tion into its collapse state, where this transient phase is grated over the entire core structure. The total luminos- no longer described by Eq. (60). The total optical depth, ity (power) generated by the core is thus given by τ, of the core is given by dN L =  n , (66)  −1 nuke σT Λa t dt τ = 1.4 10−3 | | , (61) 4mpG t ≈ × 1 Myr where nuke is the energy per reaction that is retained | | by the gas. Equation (61) shows that the core remains where mp is the proton mass and we have taken Λa = optically thin to the radiation produced by the reactions, 0.3 km/s. The core thus remains optically thin until a implying most of the energy (2.2 MeV per reaction) is time t 1400 yr before its dynamic collapse, and hence lost. Only the recoil energy is retained so that we expect ∼ for the entire evolutionary time of interest. nuke = (1 keV). The luminosity is thus given by The rate per unit volume at which neutrons are syn- O 2 3 thesized into deuterium has the form nuke σv npΛ a L = h 2 i 2 4G mp t dNn 2 | | = n σv np . (62)  3  −1 dV dt h i  nuke  a t 0.7L . (67) ≈ 1 keV 0.3 km/s 1 Myr The total number of nucleons processed is larger by a factor of 2. The total conversion rate is determined by For comparison, during the subsequent stage of dynamic integrating over the volume of the cloud core that will be- collapse, the protostellar luminosities are L several L , come the star. In this case, the reaction rate is a steeply and are not large enough to affect the dynamics.∼ † decreasing function of radius, so that we can ignore the The above considerations indicate that even though outer boundary of the core and integrate out to infinity, nuclear reactions can process most of the free neutrons into deuterium during the phase of core condensation, Z ∞ 2 3 dNn σv np 2 2 σv npΛ a the energy generated has little effect on the evolution. = h i2 4πr drρ = h i2 2 . (63) dt mp 0 4G mp t This finding may seem counterintuitive: Nuclear reac- | | tions in our universe can power the Sun for 10 Gyr, Including the factor of 2 to account for both the neutrons whereas core evolution takes place on the much∼ shorter and the protons that are processed, the total number of time scale of 1 Myr, but both systems burn up com- nucleons burned is given by the time integral parable amounts∼ of nuclear fuel. Even though the core processes essentially all of its nuclei at this faster rate (by tf 2 3 Z dN σv Λ a  t  4 ∆N = 2 n dt = np log 0 . (64) a factor of 10 ), each reaction provides only 1 keV h i 2 2 ∼ ∼ t0 dt 2G mp tf of usable energy, compared to 28 MeV for each he- lium nucleus produced in the Sun.∼ The effective energy The solution for the condensing core only holds up to 4 resources are thus also different by a comparable factor a time tf 10 yr before dynamic collapse. After this ( 104) so that the object has about the same power time, the core≈ undergoes a transition before rapidly ap- ∼ (L 1 L ). This result holds only because the core re- proaching a well-defined collapsing state [38]. The initial mains∼ optically thin to the gamma rays produced by the time t0 1 10 Myr is determined by the time when reactions. the solution≈ of− equation (60) first holds. The logarithmic factor is thus log 103 7, and number of converted ∼ ∼ nucleons is of order V. STELLAR EVOLUTION  3 56 a ∆N 8 10 0.64N , (65) A. Changes to the MESA package ≈ × 0.3 km/s ≈ where N is the number of nucleons in a solar mass star. In this section we detail the changes we made to mesa Given that roughly one third of the mass is already in [39, 40] to compute stellar evolution in the absence of the deuterium (from BBN), this result indicates that most of the remaining nucleons will experience nuclear reactions † during the contraction phase of the molecular clouds core Specifically, the luminosity is a fraction of the power scale L0 = that forms stars. Any remaining free neutrons are then GM∗M/R˙ ∗, where M∗ and R∗ are the mass and radius of the likely to be burned during the subsequent dynamic col- forming star at the given time and M˙ ∼ a3/G is the rate at lapse phase. As a result, we expect stars to form with which mass falls into the central region. 14 weak interaction. The primary reaction chain for 4He convert to deuterium during star formation. The evo- synthesis in our universe is the pp chain, schematically lution of these stars would be similar to the η = 10−11 given as case, but their lifetimes would be roughly twice as long. η + At higher values of , BBN produces a universe composed p + p D + νe + e (68) 4 → almost entirely of He. p + D 3He + γ (69) Notably, the only scenario that produces long-lived ↔ −11 3 3 4 stars with Gyr lifetimes is η = 10 . Therefore, we He + He He + 2p. (70) ↔ adopt this as the “weakless universe” where not otherwise The reactions in Eqs. (69) and (70) are electromagnetic specified, including the discussion on habitability in Sec. and strong, respectively, and would be in operation in a VI. We compute three other stellar main sequences for weakless universe. Eq. (68) is a weak interaction and by comparison to this weakless universe model. The weak- definition is no longer applicable. As a result, we remove less universe with metals is a model for stars in a weak- + p(p, νee )D from the nuclear reaction network in mesa, less universe that has undergone chemical evolution as while preserving D(p, γ) 3He and 3He( 3He, 2p) 4He. described in Section VI. This model adopts mass frac- The BBN calculations in Table I show that the pri- tions of Xp = 0.01 for free protons, XD = 0.53 for deu- 4 mordial composition can have significant contributions terium, Y = 0.45 for He, and Z = 0.01 for metals. “Our from D. Free protons can capture on the ambient D to universe” is simply the main sequence in our universe, form 3He, in line with Eq. (69). Additionally, two D also computed with metals. We compute this because it nuclei can interact with each other to form larger nuclei is most recognizable on the Hertzsprung-Russell (H-R) through three channels diagram. Finally, we define a “weak analog universe,” which has a weak interaction, but it has the same helium D + D 3He + n (71) abundance as the weakless universe (Y = 0.4569) and ↔ D + D T + p (72) no metals to make the closest possible comparison to the ↔ metal-free weakless universe. D + D 4He + γ. (73) ↔ We are unable to compute the very bottom end of the All three of the reactions are in operation in a weakless main sequence in a weakless universe because the min- universe. Reactions (71) and (72) are the principal means imum stellar mass is 0.013 M , the deuterium-burning of D destruction whereas reaction (73) is subsidiary. To limit, which the minimum mass computed by MESA is properly compute the nucleosynthesis in stars, we must 0.03 M . There are also gaps in some of our computed include all three reactions in our nuclear reaction net- main sequences because MESA failed to converge. work, and in addition, we must include n and tritium, T, in our isotope list. With the inclusion of free neutrons, we need to in- clude other nuclear reactions, for example, T(D, n) 4He and T(T, 2n) 4He. Our final nuclear reaction network in- cludes all of the BBN reactions which involve A < 5 from Ref. [17]. In addition, we include other reactions which are not important in BBN but could be important with a high D mass fraction, e.g., D(D, γ) 4He from Eq. (73). Finally, our nuclear reaction network includes reactions which synthesize 12C and 16O for completeness. 12C is the catalyst for the CNO cycle which burns free protons into 4He. The CNO cycle relies on β decays which are not in operation in a weakless universe. We do not in- clude any part of the CNO cycle in our calculations.

B. Weakless stars

We compute the deuterium-burning main sequence at FIG. 7. The stellar main sequence in our universe (black zero metallicity for three cases of the weakless universe −9 −10 −11 triangles), a “weak analog” universe (red triangles), and the as shown in Table I: η = 10 , 10 , 10 . For all cases weakless universe (circles) plotted on the H-R diagram. For of η, we fix n/p = 1. The range of η includes the value weak universes, this is a pp-burning main sequence, while −10 for our universe of 6 10 [23] and the value adopted in the weakless universe, it is a deuterium-burning main se- −×12 by Ref. [9] of 4 10 , plus an intermediate value. quence. For the weakless universe, we plot the main sequence At still lower× values of η, a negligible amount of pri- with and without metals (black and red, respectively). The mordial nucleosynthesis occurs, and the universe consists weak analog universe has no metals and the same helium frac- almost entirely of free protons and free neutrons, which tion as the weakless universe. 15

We plot the stellar Zero-Age Main Sequence (ZAMS) cooler and takes 10% of the stars’ main sequence life- from all four of these models in Fig. 7 on the H-R dia- time to reach the∼ redmost point. Returning to the ZAMS gram. The start of hydrogen burning defines the ZAMS. position takes a further 70% of the main sequence life- Our universe is represented with black triangles, the weak time, but the stars have∼ grown significantly brighter by analog universe with red triangles, our adopted weakless that point. For the smallest stars that are of most inter- universe with red circles, and the weakless universe with est for habitability, their brightness increases by a factor metals with black circles. Note that the main sequence of 5 over the main sequence lifetime, as opposed to in weak universes is a pp-burning main sequence, while 2∼ for Sun-like stars in our universe. The blueward ex- in weakless universes, it is a deuterium-burning main se- cursion∼ takes the remaining 20% of the main sequence quence. lifetime and stops at about∼ 104 K for a wide range of Our universe produces the familiar main sequence on masses as the deuterium fuel is exhausted, and the star the H-R diagram, while a more helium-rich universe pro- enters a new contraction phase ahead of helium burning. duces hotter stars. In contrast, the main sequence in a Stars in cases with a higher value of η have a much weakless universe falls mostly along the Hayashi track, lower deuterium fraction. This means that they must lying vertically over much of its length, staying near have a higher core temperature and contract further to 3 Teff = 4.5 10 K for a wide range of masses. Stars reach an equilibrium state, and they will also have a × more massive than about 50 M are bluer, and stars less higher surface temperature due to the higher helium frac- massive than 0.25 M are redder. This occurs because tion. Thus, the main sequence is shifted blueward while deuterium burning begins at a much lower central tem- the endpoint of deuterium burning remains roughly the perature and thus a much earlier time than the pp-chain. same: a nearly pure-helium star with an effective tem- The protostars descend down the Hayashi track as in our perature of 104 K. The stars will follow similar, but ∼ −10 universe, until deuterium burning reaches an equilibrium much shorter-lived evolutionary tracks. For η = 10 , level, which occurs while the star is still large and red. As the longest lived stars live a few hundred Myr and typ- a result, most stars in a weakless universe appear as red ical effective temperatures are around 6 103 K. For −9 × giants. The lower metallicity and higher helium fraction η = 10 , stars live only a few Myr at most and have results in them being more orange, but they do become typical effective temperatures around 9 103 K, in both × redder by 500-1000 K as they age. cases moving blueward to the same endpoint. Figure 8 shows evolutionary tracks on the H-R diagram The limiting case of pure helium stars will not occur for our stellar main sequences for all three of our weakless in a weakless universe even for very high baryon densi- −6 universe models through the end of deuterium burning as ties of η = 10 because the helium fraction produced by well as a fourth model computed with 100% 4He, which BBN appears to approach a limit of 98%. However, ∼ represents the limit as η increases. We begin plotting the we still include them here for comparison purposes. The tracks at an age of 105 years, reflecting the fact that it evolutionary tracks for these stars represent the helium takes that long to make a protostar in our universe – the burning phases. These helium stars do not “stall” at a time to build up a protostar to its final mass and reach deuterium-burning main sequence and instead continue the birth line. This is an aspect of history that is not contracting until they reach a helium-burning main se- captured by mesa, which computes a pre-main sequence quence (the point at which the power output from helium model at the final mass and a very large radius, then lets burning overwhelms that of hydrogen/deuterium burn- 4 4 it contract. Nonetheless, we are confident in our results ing) with temperatures of 3 10 – 15 10 K, while × × at the 10% level because models with different initial con- stars smaller than 0.3 M fail to start helium burning ∼ ditions tend to converge quickly and because our models at all. 4 5 look very similar at 10 years and 10 years, indicating Of the cases we study, only small stars (. 0.1M ) in they have reached such a quasi-equilibrium state. A red the η = 10−11 case have the multi-Gyr lifetimes needed to dot on each track denotes the location in the H-R dia- have a large chance of supporting habitable planets. For gram when the star reaches the ZAMS. Some red dots comparison purposes, we adopt a “weakless sun” with a do not appear to be on the tracks which indicates that mass of 0.056 M and a lifetime of 8.3 Gyr, and we plot those stars burn deuterium during protostar formation. its temperature, luminosity, and radius evolution com- We also put a red dot on the tracks for the 100% 4He pared with our sun in Figure 9. models which denote the start of 4He burning. Unlike our sun, the weakless sun grows significantly For all of our weakless models, but especially for the fainter and redder over the first Gyr of its life, decreas- η = 10−11 case, the evolutionary tracks look significantly ing in luminosity by a factor of 3. By itself, this is not different from our universe. Instead of moving upward too different from M-dwarfs in our universe, but the red- and slightly blueward on the H-R diagram as they age, ward excursion that occurs on the main sequence (top- weakless stars of less than about 5 M make a large left panel of Fig. 8) is still unusual. The weakless sun re- redward excursion and grow significantly fainter early in sembles a bright M-dwarf with a temperature of 2.5 103 × their lives, then move back to near their ZAMS position, K, a luminosity of 0.1 L , and a radius 0.15 R during followed by a much larger blueward excursion near the the period when its properties are most stable. ends of their lives. The redward excursion is 500-1000 K The other striking difference between the evolution of 16

FIG. 8. Evolutionary tracks of stars plotted on the H-R diagram in our three weakless universe cases and pure helium stars. Masses of individual tracks are labeled in solar masses. The tracks begin at an age of 105 years and end at the end of deuterium burning (helium burning for pure helium stars). The red dots denote the ZAMS for the weakless models, and the start of helium burning for the 100% helium models. the weakless sun and our sun is that despite its main L M 2 compared with our universe, where the relation sequence lifespan of 8.3 Gyr, the weakless sun begins to is∝ closer to L M 3−4. ∝ take a sharp upturn in temperature and luminosity at Because the temperature of weakless stars is nearly an age of 3.6 Gyr, increasing in brightness by a factor constant over a wide range of masses, we expect to find of 10 by the end of deuterium burning, whereas a solar- a mass-radius relation of R M, such that radius scales type star brightens much more slowly and more steadily linearly with mass. This is borne∝ out by our plot of radius over its main sequence life. This shift corresponds to the versus mass in the bottom-right panel. This results in the blueward excursion on the H-R diagram. smallest stars being brighter, while the largest stars have Finally, we plot important ZAMS stellar parameters a similar luminosity to those in our universe on the order 6 versus mass in Fig. 10 for all three of our weakless uni- of 10 L . verse cases, here computing a much greater number of The bottom-left panel of Fig. 10 plots the central tem- individual models. The top two panels, showing effective perature versus mass for our three weakless universe temperature and luminosity, mirror the results we see in cases and shows significant variation with mass. The the H-R diagram. We note that the mass luminosity re- deuterium-burning temperature is usually described at lation for weakless stars have a much shallower slope of 106 K. However, the lowest-mass stars in our η = 10−11 17

to the intermediate slope of the mass-luminosity rela- tionship. This finding can also be understood through the following approximate derivation. Using order of magnitude scaling laws [41, 42], we can write the central pressure of the star in terms of the stel- lar mass and radius,

GM 2 Pc , (74) ∼ R4 where dimensionless constants of order unity are sup- pressed. Using the ideal gas law to evaluate the pressure, we find GMµ kT , (75) ∼ R where µ is the mean molecular weight and all quantities are evaluated at the stellar surface. In writing Eq. (75), we ignore dimensionless constants so that we can explore the scaling relationships. The dimensionless constants could be orders of magnitude different between the center in Eq. (74) and the surface in (75). Next, we note that the photospheric temperature is nearly constant across the entire stellar mass range, as indicated by the nearly vertical main sequence in Fig. 7 and the effective tem- perature plotted in the upper left panel of Fig. 10. This trend of a low, nearly constant surface temperature is much like the behavior of stars ascending the red giant branch in our universe (as noted earlier, these weakless stars have much in common with red giants). In the case of red giants, as the stellar envelope expands and the FIG. 9. Comparison of the time evolution of temperature, surface temperature decreases, the opacity of the photo- − luminosity, and radius of our sun and our adopted “weakless sphere eventually increases due to contributions from H −11 sun” with a mass of 0.056 M and η = 10 . ions. In addition, when the photosphere reaches a mini- mum temperature of T 5 103 K, the outer layers of the star become fully convective.∼ × Luminosity is efficiently carried out of the star and prevents further lowering of case have central temperatures approaching 3 105 K, the surface temperature. As a result, red giants move al- much lower than expected. It may be that the× much most vertically up the H-R diagram (at nearly constant higher deuterium concentration and the higher central temperature). The behavior in the top right panel of Fig. density make up for the lower temperature in the reac- 10 is thus the weakless analog of the well-known Hayashi tion rate, or that following the stellar evolution over Gyr forbidden zone, which arises in pre-main-sequence evolu- time scales implies that lower reaction rates can be sig- tion [43] and in red giants [44]. Weakless stars behave nificant where they would not be in our universe. In any in a similar manner to these two stellar states from our case, the central temperature also has a very consistent 0.14 universe. As a result, Eq. (75) implies that M R. The power law relation with stellar mass of Tc M . ∝ ∝ stellar luminosity is given by The mass-luminosity relationship shown in the upper right panel of Fig. 10 has the power-law form L M 2. L = 4πR2σT 4 R2 M 2 , (76) Although a detailed derivation of this relation is beyond∝ ∼ ∼ the scope of this paper, we can understand this finding where σ is the Steffan-Boltzmann constant and the final in approximate terms. Firstly, we note that this scaling approximate equalities assume that the surface temper- law is intermediate between the mass-luminosity relation ature is constant. found for low mass stars in our universe (L M 4) and that for high mass stars (L M). The weakless∝ stars under consideration here have∝ properties in common with VI. CHEMICAL EVOLUTION AND both low-mass and high-mass stars. Weakless stars of low HABITABILITY mass are brighter than those in our universe, but objects at the high mass end of the distribution are somewhat Of the cases we consider, only the η = 10−11 case dimmer. This compression of the luminosity range leads produces stars with Gyr lifetimes. In the η = 10−10 case, 18

FIG. 10. Plots of ZAMS stellar parameters versus mass for our three weakless universe models. Top-left: effective temperature. Top-right: luminosity. Bottom-left: core temperature. Bottom-right: radius. we can extrapolate from our results that a star at the compounds. Chemistry in a weakless universe would be deuterium-burning limit would have a lifetime of 0.6 nearly identical to our own, so we postulate that life Gyr. However, in the η = 10−11 case, stars of 0.013-∼ would require, at a minimum, carbon and oxygen. Ad- 0.10 M would be similar to M-dwarfs and would have ditionally, life requires materials that can form planets, long lifetimes of 3-30 Gyr, long enough for complex life although the possibility of water worlds means that this to develop on orbiting planets with liquid water. does not necessarily require new elements. There are a few differences in weakless stars that im- The chemical evolution of a weakless universe is much pact habitability. Most significantly, the weakless sun more speculative once later-generation stars form with only remains in conditions stable enough to support life a nonzero metallicity because the nuclear reactions in- for about 30% of its main sequence lifetime, compared volved are not well-studied. However, the initial gen- with 60% or more for a solar-type star, making them sig- eration of stars will disperse their heavy elements via nificantly less hospitable to life. Additionally, the habit- two primary mechanisms: red giant winds and Type Ia able zone of our weakless sun model would be at roughly supernovae. Core-collapse supernovae (notably, the pri- 0.3-0.5 AU, which is outside the tidal locking radius [45], mary source of oxygen in our universe) fail to explode so tidal locking is not a concern. because of the lack of neutrinos, with the entire star col- The habitability of the weakless universe also depends lapsing to a degenerate remnant. If the star has under- on the presence of the elements needed to make organic gone dramatic mass loss, it could collapse to a “nucleon 19 star” analogous to a neutron star in our universe. The formation. Furthermore, the same is true of calcium; ar- maximum mass of a neutron star is computed as 2.01 – gon atoms will absorb at least five neutrons each, and 12 16 28 2.16 M [46]. However, a nucleon star is composed of sulfur atoms two or three. C, O, and Si will un- both protons and neutrons, giving the degenerate matter dergo very little such processing due to small neutron twice as many degrees of freedom, so the maximum mass capture cross sections. Because the n/p ratio is identical of a nucleon star could potentially be as high as 4.32 M . to one, this will result in second-generation stars forming Nonetheless, most massive stars will collapse directly to with a slight overabundance of free protons. Based on a black hole since they will not undergo enough mass loss solar abundances [52], the abundance of free protons will to form such a nucleon star. probably be on the order of 1%. Asymptotic Giant Branch (AGB) winds from high- Weakless stars will not have a CNO cycle, which de- mass stars dredge up triple-alpha products from the cores pends on beta decay. Instead, any free protons will con- of stars and disperse the largest amount of carbon in tribute to what Ref. [9] call “proton-clumping” reactions, our universe [47, 48]. The metals in these winds con- in which protons are added to carbon and oxygen nuclei, sist mostly of carbon, but they also include some oxy- which are stable up to the proton drip line, essentially gen. Reference [49] estimates that the triple-alpha pro- resulting in an “sp-process” in the star. For 12C, the cess produces 16O at a rate of 7%. proton drip line limits these reactions to ∼ Meanwhile, Ref. [50] estimates yields of Type Ia super- p + 12C 13N + γ, (77) novae in eight different models, resulting in 56Ni yields → between 35% and 90%, most likely near the upper end of p + 13N 14O + γ, (78) → that range. Most of the remaining ejecta is composed of 16 54Fe and 58Ni. Other alpha-process elements, 28Si, 32S, and similarly for O 36Ar, and 40Ca are also produced in significant amounts p + 16O 17F + γ, (79) at the same order of magnitude as their solar abundances. → Without beta decay and with a n/p ratio near one, the p + 17F 18Ne + γ. (80) → most important products of Type Ia supernovae in a weakless universe are most likely 56Ni and 52Fe, both However, if conditions are hot enough to overcome the of which will be stable. Thus, the second generation of Coulomb repulsion between protons and larger nuclei, stars will form with “nickel peak elements” and carbon other reactions are possible and oxygen, the necessary elements to form terrestrial 4p + 28Si 32Ar, (81) planets and life. These planets will have a very iron-rich, → 4p + 32S 36Ca, (82) Mercury-like composition, but will also have carbon and → oxygen. 2p + 36Ar 38Ca, (83) One other problem for habitability is the very high → 4p + 40Ca 44Cr. (84) carbon-to-oxygen ratio, which would seem to suppress → the formation of water. However, Ref. [51] suggests that Interestingly, there is no reaction chain that has nitrogen in the reducing environment present at high C/O ratios, or fluorine as an endpoint (every stable isotope of nitro- as much as 10% of the oxygen could still bind into water gen and fluorine can become a stable isotope of oxygen instead of carbon monoxide, so this is also not necessarily and neon, respectively, by adding a proton), or indeed a barrier to life forming. most odd-atomic-number elements. A weakless universe Many metals produced in stars will undergo further might therefore have a nitrogen abundance an order of processing in the ISM. Just as free protons combine with magnitude lower than our universe, like other odd-atomic free neutrons to form deuterium during star formation, number elements, and a greater neon abundance, similar these metals will also combine with free neutrons to form to carbon and oxygen in our universe. The lower nitro- neutron-rich isotopes. Any neutron capture reactions gen abundance could have implications for life, but on with a cross section similar to or greater than that of the other hand, these reactions do increase the abun- free protons (σv = 7.3 10−20 cm3 s−1) will occur in dance of oxygen, potentially solving the problem of the × significant amounts, essentially resulting in an s-process high C/O ratio. Further analysis of proton-clumping re- during star formation. Without beta decay, this neutron actions would be needed to determine how these abun- process could theoretically continue all the way to the dances evolve over time. neutron drip line, but neutron capture cross sections be- If carbon-based life arises in a weakless universe, it come small compared with free protons for neutron-rich will have one further difference from our universe in that isotopes, so this is unlikely in practice. nearly all of the hydrogen is in the form of deuterium, and The neutron capture cross sections have not been mea- thus, nearly all of the water will be heavy water, D2O. On sured for all of the isotopes in question, but it is known Earth, most plants and animals will die if roughly 50% of that they are several times greater than those for free pro- the water in their bodies is replaced with D2O [53]. The tons from 58Ni through 64Ni and for 54Fe through 58Fe. first place to look for the reason for this effect would be Thus, each atom of nickel peak elements in a star-forming cellular respiration. All eukaryotic life on Earth, includ- core will absorb on the order of ten neutrons during star ing humans, derives energy by pumping protons across a 20 membrane via proton pump proteins to create an electro- BBN epoch with roughly comparable abundances of pro- chemical potential which then powers adenosine triphos- tons, free neutrons, deuterium, and helium. phate (ATP) synthase proteins to form energy-storing The main difference between a weakless universe and ATP molecules. However, this process is not significantly ours is the presence of a substantial admixture of both affected by the introduction of heavy water; the proton free neutrons and deuterium. However, the formation pumps appear to work just as well as deuteron pumps of galaxies (Section III) is largely unchanged. On these [53]. large spatial (and mass) scales, the densities are too low The exact cause of the toxicity of heavy water remains for the nuclear composition to play a role. The formation uncertain, but it is believed to be due to the altered of substructure within galactic disks, such as molecular strength of hydrogen bonds (intermolecular forces be- cloud complexes where stars form (Section IV A), is only tween polar molecules) involving deuterium atoms [54]. modestly affected. Some nuclear reactions of the free The hydrogen bonds between adjacent water molecules neutrons can occur at cloud densities, and the resulting have a bond energy of 21 kJ mol−1, and deuteration in- heating can delay the onset of star formation, but the creases the strength of these bonds on the order of 10% energy injection is not sufficient to destroy the clouds. As [55]. Because protein folding is determined in large part substructures within the clouds condense further (Section by hydrogen bonds, any change in their strength can dra- IV B), nuclear reactions involving the free neutrons and matically impair cellular processes. It is believed that the protons process essentially all of the free neutrons into most important toxic effect resulting from this is damage deuterium. As a result, the free neutrons are used up to fast-dividing cells, similar to the symptoms of cyto- before they are incorporated into stars. toxic poisoning and radiation poisoning [54]. Finally, we have considered stars and stellar evolution In a weakless universe, where the majority of hydrogen in universes without the weak interaction (Sec. V). For is deuterium, enzymes and biochemical reactions would the low value of η we primarily consider here, stars begin have to adapt to the strength of deuterium-based hydro- their evolution with much greater amounts of deuterium gen bonds and the other quantum chemical properties of than in our universe. In the absence of the weak interac- deuterium. But this is no greater an evolutionary chal- tion, the standard pp-chain and CNO cycle for hydrogen lenge than producing biochemistry based on light hydro- fusion are no longer operative, and stars are powered by gen, so we do not consider it an impairment to life. the strong interaction via deuterium burning (roughly One other factor deserves note: the abundance of many analogous to the scenario where diprotons are stable and elements that are crucial to biochemistry on Earth in stars must burn exclusively through the strong interac- ionic form is much lower in a weakless universe because tion [56]). These stars resemble red giants and have pro- they are primarily produced by core-collapse supernovae. portionately greater luminosities and shorter lifetimes, Sodium in particular would be nearly nonexistent, and but the smaller minimum mass of deuterium-burning chlorine would also be depleted by two orders of magni- stars means that stars with Gyr lifetimes are still pos- tude relative to our universe. Based on the yields of Type sible. The lightest possible stars more closely resemble Ia supernovae, the most common salt that is soluble in late-K or early-M dwarfs and can live for up to 30 Gyr. water is likely to be magnesium sulfate (Epsom salt in These stars also show more variation in luminosity∼ over its hydrate form), which is the second largest component their main sequence lifetimes than stars in our universe. of sea salt on Earth. After deuterium is processed into helium, the subse- quent nuclear reactions that take place via the strong interaction proceed in much the same way as in our uni- VII. CONCLUSION verse. The weakless universe is similar to our universe with large abundances of alpha-elements. However, the A. Summary of Results dispersal mechanisms in a weakless universe are limited to AGB winds for 12C and Type Ia supernovae for the The overarching result of this work is that universes other alpha-elements. Projecting the chemical evolution in which the weak interaction is absent can remain vi- of a weakless universe over multiple generations of stars able. This builds upon the original proposal of a weakless suggests some differences from our universe, such as an universe [9] and is largely consistent with that scenario. overabundance of carbon and neon and an underabun- More specifically, our main results can be summarized as dance of nitrogen and elements heavier than nickel, but follows. these differences are not enough to preclude either planet We have studied the epoch of Big Bang Nucleosynthe- formation or the development of organic chemistry. sis in detail, exploring a wide range of baryon-to-photon ratio η and over the full range of possible initial neutron- to-proton ratios (Section II). In order for the universe to B. Discussion avoid the overproduction of helium, which would lead to a shortage of hydrogen, the value of η must be smaller than In addition to indicating that hypothetical universes that of our universe by a factor of 100. For the work- without weak interactions remain viable, the results of ing range of parameters space, universes∼ emerge from the this work provide insight into the workings of our own 21 universe. By considering such scenarios, we can under- While a weakless universe produces all of the elements stand how far trends in our universe can be taken. We necessary for life, the relative habitability of such a uni- understand BBN in our universe such that we can pre- verse compared with our own depends on several factors dict the primordial abundances to high precision (with such as the C/O ratio that are a result of chemical evo- the exception of 7Li [57]). Much current research done lution. In this paper, we have examined the most impor- on SBBN is in computing higher-order effects detectable tant effect on stellar evolution by replacing the pp-chain in future experiments [22, 58–60]. However, the rela- and CNO cycle with a strong deuterium-burning reac- tionship between YP and n/p is approximated to high tion. The next step would be to determine the reaction accuracy as YP 2(n/p)/(1 + n/p). This expression re- rates of proton-clumping reactions and incorporate them sults from the assumption' that all of the neutrons (to into the nuclear network of mesa, which would make it one part in 105) are incorporated into 4He. It has tra- possible to determine the stellar yields of carbon, nitro- ditionally been used in the context of an evolving n/p gen, oxygen, neon, and possibly several heavier elements. ratio [given by the reactions in Eqs. (2) – (4)] and a These yields would provide clearer insights into the range baryon number within an order of magnitude of that of possible chemical makeup of life in a weakless universe. given by the CMB temperature power spectrum. In this Finally, the results of this paper suggest that the inclu- work, we expanded the parameter space to encompass sion of nuclear processing due to the presence of free neu- both a larger portion of the range 0 < n/p < 1 and the trons alters the evolution of galaxies, star formation, and new range n/p > 1. We have found that the expression stellar evolution to a moderate degree. The changes are YP 2(n/p)/(1 + n/p) is valid over the larger range of neither negligible nor dominant. In particular, the pres- ' 0 < n/p < 1, and furthermore can be generalized with ence of free neutrons does not prevent a universe from be- Eq. (11) to extend to n/p > 1. Although it is physically coming habitable. In some sense, this intermediate result reasonable that with fewer than 50% protons, there will can be understood on energetic grounds. The presence of be less 4He, what is surprising is the degree of symme- stable neutrons allows for fusion to take place in the inter- try in Fig. 6. This indicates that our weakless nuclear stellar medium (rather than having all nuclear reactions reaction network — identical to that of SBBN sans the take place in stellar interiors) and produce deuterons, weak interaction rates — isolates the baryons into 4He which have a binding energy of 2.2 MeV. For compari- and free nucleons independent of isospin. An interesting son, the production of helium-4, with a binding energy scenario would be to extend the BBN network to include of 28 MeV, provides much of the energy for galaxies, so heavier nuclei and more neutron capture reactions in the that the new energy source represents an 8% effect. case of n/p > 1. Such an environment (high entropy and n/p > 1) could be conducive to r-process nucleosynthe- sis, where there is no analog for n/p < 1 and hence an asymmetry. The r-process relies on β-decays of neutron- ACKNOWLEDGMENTS rich nuclei, so our argument would have to be applied to a class of universes where the weak interaction is indeed We thank Juliette Becker, George Fuller, Lillian present. This thought experiment would test whether Huang, and Michael Meyer for useful conversations. This the symmetry in Fig. 6 is the result of a limited network, work was supported by the John Templeton founda- or something more fundamental in the nuclear proper- tion through Grant ID55112: Astrophysical Structures ties and interactions of the light nuclides. We note that in other Universes, and by the University of Michi- this discussion is predicated on η being large enough so gan. Computational resources and services were provided that out-of-equilibrium nuclear reactions can occur at low by Advanced Research Computing at the University of enough temperatures. Michigan. We thank the referees for their comments.

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