Exoplanets as Sub-GeV Dark Matter Detectors
MMIT-CTP/5230SLAC-PUB-17556
Exoplanets as New Sub-GeV Dark Matter Detectors
Rebecca K. Leane
1, 2, ∗ and Juri Smirnov
3, 4, † Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94039, USA Center for Cosmology and AstroParticle Physics (CCAPP),The Ohio State University, Columbus, OH 43210, USA Department of Physics, The Ohio State University, Columbus, OH 43210, USA (Dated: October 2, 2020)We present exoplanets as new targets to discover Dark Matter (DM), with advantages due to theirlarge expected abundance, low temperatures, and large sizes. Throughout the Milky Way, DM canscatter, become captured, deposit annihilation energy, and increase the heat flow within exoplanets.We estimate upcoming infrared telescope sensitivity to this scenario, finding actionable discoveryor exclusion searches. We find that DM with masses above about an MeV can be probed withexoplanets, with DM-proton and DM-electron scattering cross sections down to about 10 − cm ,stronger than existing limits by up to six orders of magnitude. Supporting evidence of a DM origincan be identified through DM-induced exoplanet heating correlated with Galactic position, andhence DM density. This also allows a potential tracer of DM overdensities. Our results providenew motivation to measure the temperature of the billions of brown dwarfs, rogue planets, and gasgiants peppered throughout our Galaxy. I. INTRODUCTION
Are we alone in the Universe? This question has drivenwide-reaching interest in discovering a planet like ourown. Regardless of whether or not we ever find alienlife, the scientific advances from finding and understand-ing other planets will be enormous. From a particlephysics perspective, new celestial bodies provide a vastplayground to discover new physics.Astrophysical systems have already been broadly usedto probe new physics, including investigating the effectsof gravitationally captured Dark Matter (DM). This canoccur if DM scatters with the system, loses energy, andbecomes gravitationally bound. If there is sufficient grav-itational force, deposited DM kinetic energy can notice-ably increase the temperature of the system. Regard-less of gravitational strength, DM annihilation can alsoinduce heating. This has been investigated in the con-text of neutron stars and white dwarfs [1–39]. Alter-natively, the DM-related heat flow in other moons andplanets has been considered, including Earth [40–42],Uranus [43, 44], Neptune and Jupiter [44, 45], Mars [42],Earth’s Luna [46, 47], Jupiter’s Ganymede [48], as wellas hot Jupiters [44].We explore the potential to discover DM using exo-planets – planets outside our solar system. We will usethe term “exoplanets” to refer to the broader class of allextra-solar planets (including rogue planets), as well asbrown dwarfs, which exist at the planet-star boundary.There are many advantages of using exoplanets to searchfor DM over other celestial bodies. These include:
A rapidly accelerating research program:
Un-til 1992, we didn’t even know if exoplanets existed. Al- ∗ Email : [email protected];
ORCID : 0000-0002-1287-8780 † Email : [email protected];
ORCID : 0000-0002-3082-0929 T e m p e r a t u r e [ K ] E a r t h P o s i t i o n JWSTNo DM D M H e a t i n g Exoplanet Temperatures
Figure 1. Mock temperature distribution of old example exo-planets with 20 −
50 Jupiter masses, as a function of distancefrom the center of our Galaxy. Black dots are DM-heated ex-oplanets, assuming a gNFW DM profile. The magenta trian-gles are the same set of planets, without DM heating. JWSTis the estimated minimum telescope sensitivity (see text). most all exoplanets we now know were only discoveredin the last decade, with the majority found in the lastfive years [49]. The exoplanet program is clearly rapidlygrowing. Telescopes such as the James Webb SpaceTelescope (JWST), Transiting Exoplanets Survey Satel-lite (TESS), the Vera C. Rubin Observatory (Rubin,previously LSST), and the Nancy Grace Roman Space a r X i v : . [ h e p - ph ] S e p Telescope (Roman, previously WFIRST), and the GaiaSpacecraft have or will have targeted programs to dis-cover as many exoplanets as possible. There are alsomany surveys such as the Optical Gravitational Lens-ing Experiment (OGLE), Two Micron All Sky Survey(2MASS), and the Wide-field Infrared Survey Explorer(WISE), which peer deep into our Galaxy. Further onthe horizon, new telescopes are being planned or con-sidered such as the Thirty Meter Telescope (TMT), theExtremely Large Telescope (ELT), Gaia Near Infra-Red(GaiaNIR), the Large Ultraviolet Optical Infrared Sur-veyor (LUVOIR), the Habitable Exoplanet Imaging Mis-sion (HabEx), and the Origins Space Telescope (OST).These current and potentially upcoming telescopes canalso observe exoplanets in new ways alongside other ex-periments, revealing even higher quality and more precisedata. This provides ample motivation to consider newways this exploding research area can be used to probenew physics.
Enormous number of expected exoplanets:
It isestimated that there is at least one planet per star in ourGalaxy, and about one cold planet per star [50]. Thismeans that there should be about 300 billion exoplanetsawaiting discovery. While of course these won’t all beimmediately found, even a small percentage of this num-ber leads to an enormous statistical advantage for under-standing potential signals. This makes exoplanets poten-tially more decisive than planets in our own solar system.It also allows ample room for growth with new discover-ies and possible surprises in observations. To date, thereare 4,284 confirmed exoplanets, and an additional 5,515candidates are currently under investigation [49].
Presence in non-local DM densities:
Exoplanetsalso abundantly exist in parts of the Milky Way wherethe local DM density is much larger, such as towardsthe Galactic Center (GC). This would provide a largerDM heating signal than a planet in our local DM den-sity. The DM heating signal will then be correlated withthe DM density, providing an additional handle on theDM distribution in our Galaxy. Provided sufficient sen-sitivity, it would also then be possible to confirm areasof DM overdensity, where the local density departs fromexpectations from DM density profile models.
Much larger surface area than neutron stars:
The other key proposed search using upcoming infraredtelescopes on DM-heated astrophysical bodies is with old,cold neutron stars [25]. However, while neutron stars aremuch more dense, and allow for higher heating rates inpart due to enhancements from kinetic heating, exoplan-ets and brown dwarfs are much larger than neutron stars.A typical neutron star has a radius of about 10 km, whileexoplanets of interest to us have radii of about 50,000 –200,000 km. This means that exoplanet temperaturescan be measured much further into the GC, as the spec-tral flux scales with the squared ratio of the radius of theobject to the distance away. Neutron stars therefore donot have the advantage of potentially providing a DM-density dependent heating signal, as cold neutron stars in enhanced DM density locations are too small to see atsuch distances (though see e.g. Ref. [51]). Inversely, thisalso means that closer-by exoplanets can be imaged tomuch higher significance, and with less exposure time.
Easier to find than neutron stars:
The infraredneutron star search requires that a sufficiently cold neu-tron star candidate at a distance < ∼
100 pc from Earth isfound [25]. While pulsars have been found at distancesof ∼
100 pc [52], it is possible that a sufficiently cold andsufficiently close-by neutron star may not ever be found,or cannot be measured with sufficient exposure time. Onthe other hand, exoplanets outnumber neutron stars inour galaxy by at least about a factor of a thousand [53],and are already known to exist in close enough proximityfor DM searches, as we will show in this work.
Low temperatures:
Lastly, exoplanets can be verycold, as they do not undergo nuclear fusion, and canexist very far in large orbits from any host star to whichthey may be bound. They can even go rogue , floatingfree from any parent star. As the low temperaturesallow for a clearer signal over background for DMheating, exoplanets are advantageous over fuel-burningstars. Furthermore, their low core temperatures inpart prevent DM evaporation compared to evapora-tion in these stars, providing new sensitivity to MeV DM.In this work, we exploit all these features to identifynew searches for DM in exoplanets. We establish twodifferent searches: one for distant exoplanets and one forlocal exoplanets.Figure 1 demonstrates these searches. We show an ex-ample distribution of exoplanets with masses of about20 −
50 Jupiters, with and without DM heating. Distantexoplanets can be used to map the Galactic DM den-sity, given sufficient telescope sensitivity. This is seen bythe uptick of many hot exoplanets, scaling with the DMdensity. As well as broadly searching for DM signals, lo-cal exoplanets can be used to test the hypothesis thatDM contributes to internal heat of the gas giants in ourown solar system, which are not well understood [44, 45].In both cases, DM-heated exoplanets can be potentiallymeasured when the infrared telescope JWST comes on-line. Both our suggested searches target new DM pa-rameter space, probing both the DM-proton and DM-electron scattering cross sections to unprecedented sensi-tivities. While the distant exoplanet search is certainlymore challenging, the local exoplanet study can be ex-pected to yield short-term results.We organize our paper as follows. We begin in Sec-tion II by briefly outlining expected properties of exo-planets, as well as explicitly identifying known exoplan-ets as candidates for our proposed searches, and futureprospects for exoplanet candidate discovery. We thenquantify DM heating in exoplanets in Section III. InSection IV, we estimate the sensitivity with upcominginfrared telescopes, and discuss both opportunities andchallenges for new DM-exoplanet searches. We then de-tail the reach these searches have on the DM parameterspace in Section V, determining the DM-proton and DM-electron scattering sensitivities across DM masses. Wesummarize and present our conclusions in Section VI.
II. OVERVIEW OF EXOPLANETSA. Abundance and Properties
It is expected that on average, all stars have at leastone planet [50]. Given we know there are about 300 bil-lion stars in our Galaxy, this amounts to about 300 billionor more exoplanets in the Milky Way. Of these, there is asmorgasbord of exoplanet types, with diverse propertiesand sizes, which we briefly outline below.
1. Earth-like Planets
The most popular exoplanet type for finding aliens are,of course, Earth-like planets. Earth-like planets haverocky interiors, and relatively small masses and radii rela-tive to all other exoplanets. They extend into the “Super-Earths” category, which usually have radii of about thatof Earth, but have up to a factor 10 higher in mass.These are not ideal for our DM searches, as their radii aresmaller than other exoplanets, leading to limitations intelescope sensitivity. Interestingly however, it has beenpointed out that DM annihilation heating of Earth-likeexoplanets can lead to liquid water, and therefore a hab-itable planet, when otherwise the planet would have beentoo cold [54].
2. Jupiters
The next-largest category is the gas giants, also called“Jovian planets” or “Jupiters”. Jupiters have radiiroughly comparable to that of Jupiter, and generallyhave masses about comparable to Jupiter, though theycan have up to about 10 times higher masses, becoming“Super Jupiters” (any higher, and they begin to transi-tion into brown dwarf classification). Jupiters are oneideal class of exoplanets for our searches: they have largeradii, and due to their lower mass compared to the nextclass, their internal heat flow can be very low. The min-imum temperature expectation for Jupiters with massesand radii comparable to Jupiter, after 1 Gyr, is about160 K [55]. After 10 Gyr, Jupiters are about 80 K [55].Note that large cold gas giant planets are expected to becommon [50, 56].
3. Brown Dwarfs
Larger again are brown dwarfs, which were only dis-covered in 1995. Brown dwarfs are what lurk in the gapbetween gas giant planets and the least massive stars,placing them with masses of about 14 −
75 Jupiters. Theygenerally have about the same radius as Jupiter, mak-ing them immensely dense . This makes brown dwarfs an ideal candidate for our searches; they are large and dense.However, for our scenarios of interest we do not want tosolely consider very massive brown dwarfs; too massiveand they take longer to cool. This leads to a large heatbackground that might obscure DM heating signals. Theminimum temperature expectation for brown dwarfs withmasses of about 14 −
75, after 1 Gyr, is about 200 − −
750 K respectively [55, 57, 58]. This meansthat while brown dwarfs of all masses can be relevantfor our study, internal heat from the heavier dwarfs mayoutpower DM heating in some DM densities/locations,or would only be relevant for our study if their age ap-proached 10 Gyr. It will therefore depend on the candi-date in question, whether its individual heat backgroundis acceptable or not. In any case, the abundance of coldbrown dwarfs is expected to be very high [50, 59]. Morebroadly, about 20% of stars are expected to have Jupiter-sized to brown dwarf sized planets [50]. Interestingly,brown dwarfs can have very exotic atmospheres, withsome experiencing iron rain [60, 61]. Talk about a stormyday!Brown dwarfs have been previously considered along-side DM, although in the context of asymmetric DM,which does not feature an annihilation heating signal,but rather a departure from the expected stellar evolu-tion curve [62]. This is a similar approach to studyingDM effects on stars in larger DM densities, such as infor example Refs. [63–80]. However, these have differentobservables to those pointed out in this work.
4. Lost in Space: Rogue Planets
Not all planets have a home. A class of planetscalled “rogue planets” or “free-floating planets” have beenejected from their planetary nursery, damned to aimlesslywander, alone, through dark and empty space. While allplanet types listed above can be rogue planets, Jupitersand brown dwarfs are by far the most common rogues.This lonely class of exoplanets is ideal for our searches.This is because they are free from light and heat pollu-tion from any host star, allowing them to be more easilyresolved. Similarly, at closer distances to Earth, Jupiterson larger orbits can be easier to distinguish than thoseclosely bound to their star, for the same reason.While rogue planets are currently thought to be lesscommon than bound planets, they can still be extremelyplentiful. The OGLE survey estimates that there is up toabout one rogue for every 4 stars – that amounts to upto about 100 billion rogues in the galaxy [81]. Even morerecently, a simulation of planetary systems in the OrionTrapezium Cluster showed that about 15% of all planetsended up ejected from orbit around their parent star [82].(Interestingly, about 0 .
1% ended up being later welcomedinto a new family, captured by another distant parentstar.) Extrapolating this system, it implies that therecould be about 50 billion rogue planets in our Galaxy [82].Alternatively, brown dwarfs can have never had a hoststar – they can form in molecular clouds like stars, andsimply be all alone from the very beginning.
B. Candidates and Further Discovery Potential
1. Local Planets
Table I lists some known Jovian planets within 100 pc,which are potential candidates for the local exoplanetsearch. These are chosen as examples based on theirproximity, radii, masses, and orbital sizes. JWST maybe able to image these planets, and probe new DM pa-rameter space. Note however that some may turn outto have too much atmospheric cloud cover, or may beheated or obscured for other reasons. Regardless, thereare many more potential candidates, which can be foundin Ref. [49].In addition to known candidates, many current and fu-ture telescopes will study our local neighborhood to iden-tify and measure more candidate planets for DM heating.In particular, Gaia is expected to find 21 , ± , , ± ,
000 new exoplanets of in-terest [83]. This will substantially increase candidatesand statistics for this search.
2. Distant Planets
The furthest planets ever found are SWEEPS-4 andSWEEPS-11 [113], which are about 8.5 kpc away (furtherthan the Earth-GC distance). However, these planets areclose to their host star, so are expected to be very hot(and therefore not ideal for DM heating searches). Manyother planets are already known to exist, over varyingdistances from the GC. However, many of these plan-ets are bound to a star. While this is helpful for dis-covery techniques (i.e. more techniques are available todiscover planets bound to a star), this is not helpful forour searches in the galactic bulge. This is because, evenwhile they still may have very large orbits, at the veryfar distances into the GC that we want to measure, theycan be outshone by their bound host star, making tem-perature measurements impossible. We therefore focuson rogue planets when examining potential DM signalsat large distances.While rogue planets are harder to find, some have al-ready been found, and it is expected that many morecan be found soon. Such searches require use of gravita-tional microlensing, which can be aided especially withsimultaneous use of telescopes, allowing for more decisiveconfirmation of planetary status. For example, this hasbeen achieved with Roman and Euclid [114].OGLE has been operating since 1992, focusing onsearches in the stellar bulge. It has already identifiedmany distant exoplanets and exoplanet candidates. For rogue planet candidates, this includes e.g. OGLE-2019-BLG-0551 [115], and a brown dwarf candidate OGLE-2015-BLG-1268, with 50 Jupiter masses and at 5 . ± . C. Temperature and Density Profiles
To study Jupiters, we use the profile models for ourJupiter, as per Ref. [121]. This features a core tem-perature of T c = 1 . × K, an average density of ρ jup = 1 . / cm , and a radius of R jup = 6 . × m.We set our benchmark Jupiters to all have the same ra-dius as Jupiter. We also check if results vary with twodifferent Jupiter density profile hypotheses, one with acore and the other without [121]. For our parameters ofinterest, there is no noticeable effect.To study brown dwarfs, we use the analytical modelfrom Ref. [122]. The brown dwarf radius, core densityand core temperature can be expressed as a function ofmass and electron degeneracy, R = 2 . × (cid:18) M (cid:12) M (cid:19) µ − / e (1 + γ + αψ ) cm , (1) ρ c = 1 . × (cid:18) M (cid:12) M (cid:19) µ e (1 + γ + αψ ) g / cm , (2) T c = 7 . × (cid:18) M (cid:12) M (cid:19) / µ / e ψ (1 + γ + αψ ) K . (3)Here, µ e is the number of electrons per baryon, ψ theelectron degeneracy parameter and γ is a higher-ordercorrection factor (see Ref. [122] for a detailed discus-sion). There is a particular electron degeneracy at whichthe core temperature reaches its maximum, and dropsto smaller values if the degeneracy is further increased.Once the brown dwarf passes this point in the cool-ing process, its core temperature decreases significantly,while its density grows. The relatively low core temper-atures and high densities make old brown dwarfs effi-cient accumulators for light DM. For our benchmark, thebrown dwarf (BD) radius is taken to be R BD = R jup , and Planet Radius ( R jup ) Mass ( M jup ) Distance Orbit Temp (No DM) Temp (with DM) RefEpsilon Eridani b 1.21 1.55 3 pc 3.4 au < ∼
200 K < ∼
650 K [84]Epsilon Indi A b 1.17 3.25 3.7 pc 11.6 au < ∼
200 K < ∼
650 K [85]Gliese 832 b 1.25 0.68 4.9 pc 3.6 au < ∼
200 K < ∼
650 K [86]Gliese 849 b 1.23 1.0 8.8 pc 2.4 au < ∼
200 K < ∼
650 K [87]Thestias 1.19 2.3 10 pc 1.6 au < ∼
200 K < ∼
650 K [88]Lipperhey 1.16 3.9 12.5 pc 5.5 au < ∼
200 K < ∼
650 K [89]HD 147513 b 1.22 1.21 12.8 pc 1.3 au < ∼
200 K < ∼
650 K [90]Gamma Cephei b 1.2 1.85 13.5 pc 2.0 au < ∼
200 K < ∼
650 K [91]Majriti 1.16 4.1 13.5 pc 2.5 au ∼
218 K < ∼
650 K [92]47 Ursae Majoris d 1.2 1.64 14 pc 11.6 au < ∼
200 K < ∼
650 K [93]Taphao Thong 1.2 2.5 14 pc 2.1 au < ∼
200 K < ∼
650 K [93]Gliese 777 b 1.21 1.54 15.9 pc 4.0 au < ∼
200 K < ∼
650 K [94]Gliese 317 c 1.21 1.54 15.0 pc 25.0 au < ∼
200 K < ∼
650 K [95]q Eridani b 1.23 0.94 17.5 pc 2.0 au < ∼
200 K < ∼
650 K [87]HD 87883 b 1.21 1.54 18.4 pc 3.6 au < ∼
200 K < ∼
650 K [96] ν Canis Majoris c 1.24 0.87 19.9 pc 2.2 au < ∼
200 K < ∼
650 K [97]Psi Draconis B b 1.21 1.53 22.0 pc 4.4 au < ∼
200 K < ∼
650 K [98]HD 70642 b 1.19 1.99 29.4 pc 3.3 au < ∼
200 K < ∼
650 K [99]HD 29021 b 1.2 2.4 31 pc 2.3 au < ∼
200 K < ∼
650 K [100]HD 117207 b 1.2 1.9 32.5 pc 4.1 au < ∼
200 K < ∼
650 K [101]Xolotlan 1.2 0.9 34.0 pc 1.7 au < ∼
200 K < ∼
650 K [102]HAT-P-11 c 1.2 1.6 38.0 pc 4.1 au < ∼
200 K < ∼
650 K [103]HD 187123 c 1.2 2.0 46.0 pc 4.9 au < ∼
200 K < ∼
650 K [104]HD 50499 b 1.2 1.6 46.3 pc 3.8 au < ∼
200 K < ∼
650 K [101]Pirx 1.2 1.1 49.4 pc 0.8 au ∼
200 K < ∼
650 K [105]HD 27631 b 1.2 1.5 50.3 pc 3.2 au < ∼
200 K < ∼
650 K [106]HD 6718 b 1.2 1.7 51.5 pc 3.6 au < ∼
200 K < ∼
650 K [107]HD 72659 b 1.2 3.9 52.1 pc 4.8 au < ∼
200 K < ∼
650 K [108]HD 4732 c 1.2 2.4 54.9 pc 4.6 au < ∼
200 K < ∼
650 K [109]HD 290327 b 1.2 2.4 56.4 pc 3.4 au < ∼
200 K < ∼
650 K [107]HD 154857 c 1.2 2.6 63.5 pc 5.4 au < ∼
200 K < ∼
650 K [110]Drukyul 1.2 1.6 83.4 pc 2.9 au < ∼
200 K < ∼
650 K [111]Kepler-539 c 1.18 2.4 92 pc 2.7 au < ∼
200 K < ∼
650 K [112]Table I. List of some candidate Jupiters within 100 pc, to use in a near-Earth search. Distance is quoted as from the Earth. Thepredicted temperature ranges include generic estimates for emissivity and planetary mass. Masses, radii, orbits, and distancesfrom Earth are estimates taken from the NASA exoplanet catalog [49]. the mass M BD = 75 M jup = 0 . M (cid:12) . This results in anaverage density of ρ BD = 103 g / cm , a core density of ρ c = 500 g / cm and a core temperature T c = 2 × K.Note that in the analytic model, fusion heating is notincluded. Our calculations are therefore only relevant inthe regime where there is no fusion, which is appropriateto obtain lower internal heat backgrounds.
III. DARK HEAT FLOW IN EXOPLANETSA. Dark Matter Densities
To calculate the DM-heating rate in exoplanets, weconsider different DM profiles, which control the amountof DM available for heating at a given location in ourGalaxy. We consider a Navarro-Frenk-White (NFW) pro-file, a generalized NFW (gNFW) profile, and a Burkertprofile. The NFW profile is defined as a density as afunction of galactic radius [123] ρ χ ( r ) = ρ ( r/r s ) γ (1 + ( r/r s )) − γ (4)where we take a scale radius of r s = 8 kpc [130] (largerchoices of e.g. r s = 20 kpc do not significantly change theresults). The standard NFW profile has γ = 1, while thegeneralized NFW profile is taken to have a steeper innerslope of γ = 1 . ρ χ ( r ) = ρ (1 + r/r sb )(1 + ( r/r sb ) ) . (5)For this profile, we will take a smaller core radius, suchthat r sb = 0 . .
42 GeV/cm [127]. While we only considervariations of these profiles in our calculations, it is ex-pected that overdensities – localized regions of increasedDM – likely exist, and would be potentially detectableas hot exoplanets would deviate from the expectations ofthe profiles above, which we will also briefly investigate. B. Heating Rates
1. Total Heat Flow
The total heat flow of the exoplanet Γ totheat can be de-termined by combining potential heat power sources, in-cluding internal heat Γ intheat , external heat Γ extheat , and DM heat Γ
DMheat :Γ totheat = Γ extheat + Γ intheat + Γ DMheat = 4 πR σ SB T (cid:15), (6)where R is the exoplanet radius, T is the exoplanettemperature (without other heat sources), σ SB is theStefan-Boltzmann constant, and (cid:15) is for emissivity of theplanet, and external heating is taken to be negligible forwide-orbit or free-floating planets. Emissivity captureshow effective the planet is at radiating heat, and rangesfrom 0 to 1. A more dense atmosphere often leads toa lower emissivity value; this can be caused, for exam-ple, by greenhouse effects on the planet. The emissivitycan in principle be determined from spectroscopy stud-ies. For reference, Earth has an emissivity value of 0.6,our Jupiter is about 0.9, and Venus is about 0.004 [54].Typically, larger or more dense planets have lower emis-sivities, though this can largely vary depending on thecandidate planet. The internal heat flow is a conservedquantity, and so decreasing emissivity below one will leadto higher planetary temperatures, balancing Eq. 6. Wenow consider the individual heating components.
2. Internal Heat Flow (Without DM)
We compute the internal heat flow for our range ofbenchmark brown dwarfs and Jupiters without DM.As the minimum temperature for heavy brown dwarfs(with 75 M jup ) and benchmark Jupiters (with M jup ) af-ter about 10 Gyr is about 750 K and 80 K respectively, wecan determine the internal heat flow required to producethese temperatures,Γ intheat = 4 πR σ SB T (cid:15), (7)we get internal heat flow values of about 1 . × TW and1 . × TW for brown dwarfs and Jupiters, respectively.This assumes that no DM heating is present, and a planetis sufficiently far away from any external heating sourcesuch as a host star. It will serve as our non-DM baselinefor comparing with a potential DM signal.
3. Dark Matter Heat Flow
We now consider contributions from DM heating. Thiscan proceed as DM scatters on exoplanet protons, and be-comes captured. The captured DM then can annihilate,producing heat that can be absorbed by the exoplanet.We assume that the DM scattering and annihilation pro-cesses are in equilibrium, which is expected for such largerates (see Sec. V C). Unlike the internal heat flow fromSM processes, the DM heat flow depends on how manyexternal DM particles are captured from the incomingDM flux reservoir. The amount of captured DM will de-pend on the DM masses and cross sections (see Sec. V A). H e a t F l o w [ T W ] No DM,75M jup
No DM, M jup E a r t h P o s i t i o n N F W g N F W B u r k e r t DM Induced Exoplanet Heat Flow
Figure 2. Exoplanet heat power as a function of distancefrom the center of our Galaxy. Solid lines shown are the DM-heat power of exoplanets with radius R jup , with a range ofmasses from M jup (lower line) up to 75 M jup (upper line). Theshaded region corresponds to DM-heat power for intermediateexoplanet masses. The dotted lines show the range of heatpower for the heaviest dwarfs (with 75 M jup ) down to thelighter benchmark Jupiters (with M jup ) in the absence of DMor external heat. The maximal capture rate (also known as geometric cap-ture rate) of DM is given by [128] C max = π R n χ ( r ) v (cid:18) v v d ( r ) (cid:19) ξ ( v p , v d ( r )) , (8)where n χ ( r ) is the DM number density at distance r from the GC, the average speed in the DM rest frame v is related to the velocity dispersion v d ( r ) as v = (cid:112) / (3 π ) v d ( r ) at distance r from the GC, and R is theexoplanet radius. The factor 1+3 v / v d is the result ofgravitational focusing, with v = 2 G N M/R being theescape velocity, M the exoplanet mass, and G N the gravi-tational constant. The motion of the planet with velocity v p with respect to the DM halo is taken into account by ξ ( v p , v d ( r )). In the scenarios we are interested in, the DMvelocity, the planetary velocity and the escape velocitiesare of similar order and the function ξ ( v d ( r ) , v p ) ∼ v c ( r ) in the galaxy are relatedto the DM velocity dispersion by v d ( r ) = (cid:112) / v c ( r ).We extract the circular velocities at different radii in theMilky Way by combining the data for the gas, bulge, anddisk components, as well as the analytic expressions forDM contributions to the total velocity from Ref. [130].The heat power produced by DM is given by the prod-uct of the DM mass m χ , the fraction of the captured DMparticles that have passed through the object f , and the maximal capture rate, such thatΓ DMheat = m χ f C max . (9)Using n χ ( r ) = ρ χ ( r ) /m χ , approximating ξ ( v d ( r ) , v p ) ∼
1, and combining with Eq. 8, the DM heat power can bewritten asΓ
DMheat = f πR ρ χ ( r ) v (cid:18) v v d ( r ) (cid:19) . (10)We see that the DM heat flow is independent of DM mass.This is because the heat flux scales as 1 /m χ , and eachDM particle releases an amount of energy equal to m χ once annihilating.Another type of potential DM heating is kinetic DMheating. This arises when astrophysical systems havesteep gravitational wells, causing DM to be acceleratedto speeds near that of light. However, even for densebrown dwarfs, the escape velocity is only about 0 . c ,rendering any DM kinetic heating negligible.Figure 2 shows the calculated heat flow from DM or in-ternal heat, as a function of galactic radius. DM-heatingarising due to several different DM profiles is shown, forNFW, gNFW, and Burkert profiles. The lower line showsthe heat power prediction for exoplanets of mass M jup ,while the upper line shows DM-heating for heavy browndwarfs (75 M jup ). All planetary radii R are taken to be R jup , which is the radius for all old brown dwarfs orJupiters. Here, it is assumed that all of the DM pass-ing through the planet is captured ( f = 1). Any sub-maximal DM capture would lead to a heat power sim-ply rescaled linearly with f . The shaded region for agiven DM profile shows the intermediate temperaturesfor any Super Jupiter or lighter brown dwarf. The dot-ted lines show the internal heat for the two benchmarkcases ( M jup vs 75 M jup ) without DM heating, after 10Gyr. Intermediate exoplanet masses without DM heat-ing will fall between these lines. We see that for Jupiters(lower solid line), the DM heat will outperform the inter-nal heat at all radii, making them ideal candidates for allsearches. Brown dwarfs, on the other hand, being moredense, have higher internal heat, and DM heating willonly clearly outperform their internal heat for some DMdensities and radii.The shape of the curves in Fig. 2 as a function of galac-tic radius is due to an interplay of the DM density profile,the DM velocity profile, and the effective capture radiusof the exoplanet, which varies as a function of the DMvelocity. That is, while the DM density increases closerto the GC, the DM velocity substantially decreases, whilethe effect of gravitational focusing at low velocities booststhe heating rate. IV. SEARCHES AND INFRARED TELESCOPESENSITIVITY
We now discuss two different searches: a local search,and a distant search. Exoplanets may first be identi-fied by e.g. Doppler spectroscopy (radial-velocity) meth-ods, transit photometry, direct imaging, or gravitationallensing. Once their location is found, infrared telescopessuch as JWST may be able to measure their temperature.This temperature will depend on the DM density to dif-ferent amounts, which provides a DM-correlated signalfor the two searches we detail below.
A. Fluxes and JWST Exoplanet Sensitivity
The general sensitivity of JWST to exoplanet heatingcan be found by considering the spectral flux density, f ν = πB ( ν, T ) × πR πd , (11)where d is the distance from the telescope to the exo-planet, R is the radius of the exoplanet, and B ( ν, T ) = 2 ν (cid:15) exp (cid:16) πν k b T (cid:17) − , (12)where ν is the wavelength, T is the temperature, k b isthe Boltzmann constant, and (cid:15) is the atmospheric emis-sivity. While (cid:15) = 1 provides the usual blackbody spectralflux density, deviations from a blackbody occur when theemissivity value is smaller than one; see Appendix A formore details on the impact this has on telescope sensitiv-ity. The spectral flux density for a given exoplanet can becompared to a variety of instruments and filters as part ofJWST, to determine the optimal sensitivities. Note thatto be conservative, we will not add the DM-heated spec-trum on top of the existing spectrum of dwarfs (whichcould be relevant at low temperatures), and rather willassume that the DM-heated temperature is the peak set-ting the sensitivity limit.Figure 3 shows the expected exoplanet temperatureas a function of distance from the center of the Galaxy,for DM-heating arising due to several different DM pro-files: NFW, gNFW and a Burkert profile. We distin-guish between Jovian exoplanets with masses between1 − M jup and brown dwarfs with masses in the rangeof 14 − M jup . The brown dwarfs are again sub-dividedinto three mass ranges. All exoplanets shown have a ra-dius of R jup , as all these exoplanets are expected to con-verge to this radius after 10 Gy. Each panel has emissiv-ity equal to one (i.e. a blackbody spectrum is assumed),which is the most conservative case (see Appendix A fordetails on how other emissivity choices can further in-crease the temperature due to DM). The shaded regionfor a given DM profile represents the range of heating pos-sibilities for the indicated mass range, with the heaviestexoplanets lying at the upper boundary and the lightestexoplanets lying at the lower boundary. The shape of thecurves as a function of galactic radius is due to an inter-play of the DM density profile, the DM velocity profile,and the effective capture radius of the exoplanet, whichvaries as a function of the DM velocity. That is, while the DM density increases closer to the GC, the DM ve-locity substantially decreases. Furthermore, the effect ofgravitational focusing at low velocities boosts the heatingrate, especially for larger planetary masses.The dotted lines in Fig. 3 are the temperatures for thesame ranges of exoplanets, without DM heating. Exo-planets without DM heating that have masses betweenthe masses labeled, would have temperatures betweenthese dotted lines. As heavier exoplanets have higher lev-els of internal heating, the total heat at higher masses ap-proaches the internal heat value when there is not muchDM heating available (e.g. at the local position). Thetop-left panel shows that, for Jupiters, for the lightermasses (lower line) the DM heating can outperform theinternal heat at all locations, as the internal heat is solow.We show in Fig. 3 the optimal JWST sensitivity, whichis found using Eq. 11 above, with the benchmark dwar-f/Jupiter radius of R jup . This is calculated using differ-ent JWST instruments: the Near-Infrared Imagerand Slitless Spectrograph (NIRISS) in Imaging mode for temperatures above about 500 K, and the
Mid-Infrared Instrument (MIRI) in Imaging or Medium-Resolution Spectroscopy mode for tem-peratures from about 100 −
500 K. As different JWSTinstrument filters are optimized for different flux densi-ties/temperatures, we use several different filters whilescanning over the minimal temperature measurable, toobtain the optimal sensitivity. In Fig. 3, the dashed lineis for JWST to obtain about 10 seconds of exposureto achieve a signal-to-noise ratio (SNR) of 2. 10 SNRcan be achieved at about 10 seconds of exposure at themost of the temperatures shown (though depending onthe filter it can approach a factor of few times 10 for10 SNR). Note however that these exposure times are forthe minimum temperatures on the dashed line; highertemperatures generally require less exposure time. Ex-posure times of up to around 10 seconds are achiev-able with deep field survey; a survey of the GC is verywell-motivated for numerous reasons even separate to ourwork. Significantly less time is required to achieve 10SNR in the local region; more detailed JWST sensitiv-ity estimates are discussed in the context of the searchesbelow.From Fig. 3, it is clear that different types of exoplan-ets have different regimes where they are most useful asa DM heating target. The lower mass Jupiters are idealfor local searches, as they outperform their internal heatat the Earth position. For higher mass brown dwarfs,their internal heat is too high to reveal a DM heating sig-nal at the Earth position. However, their larger massesare advantageous at larger distances and/or in more DMdense areas. This is because gravitational focusing al-lows them to collect more DM. This allows us to iden-tify two different searches: one for DM in local Jupiters,and another for all exoplanets (but especially those withhigher-masses) at larger distances, in DM dense regions. T e m p e r a t u r e [ K ]
14 M jup M jup E a r t h P o s i t i o n N F W g N F W B u r k e r t JWSTSuper Jupiters 10 T e m p e r a t u r e [ K ]
30 M jup
14 M jup E a r t h P o s i t i o n N F W g N F W B u r k e r t JWSTLow-Mass Brown Dwarfs10 T e m p e r a t u r e [ K ]
50 M jup
30 M jup E a r t h P o s i t i o n N F W g N F W B u r k e r t JWSTIntermediate-Mass Brown Dwarfs 10 T e m p e r a t u r e [ K ] jup jup E a r t h P o s i t i o n N F W g N F W B u r k e r t JWSTHigh-Mass Brown Dwarfs
Figure 3. Exoplanet temperatures as a function of distance from the center of our Galaxy, with variations due to DM for labeleddensity profiles (assuming no DM overdensities). Each panel represents our classification of different exoplanet types: SuperJupiters ( M jup − M jup ), low-mass brown dwarfs (14 M jup − M jup ), intermediate-mass brown dwarfs (30 M jup − M jup ) andhigh-mass brown dwarfs (50 M jup − M jup ). Any exoplanet within the indicated mass range will have temperatures betweenthese lines in the shaded region, with the heaviest exoplanets at the upper boundary and the lightest exoplanets at the lowerboundary. The dotted lines show the range of minimum temperatures for a 10 Giga-year-old exoplanet without DM; similarlyany exoplanet within the indicated mass interval without DM heating has temperatures between these dotted lines. Blackdashed line is the optimal minimum JWST sensitivity; anything above the line can potentially be probed (see text). B. First Search: DM in Local Jupiters
Our first search is for local Jupiters. As the Jupiterscan have lower internal temperatures compared to abrown dwarf at the same age, they are more likely to be found at lower temperatures. This means that closerto Earth, where the DM heating is lower, they are idealsearch candidates to ensure the DM heating outperformsthe background internal heating. This is clear from ex-amining the JWST dip in Fig. 3 at the Terra position,0
300 200 100 0Distance from Earth [pc]10 T e m p e r a t u r e [ K ] No DM, Jupiters J W S T T J W S T . T J W S T . T J W S T . T Local Jovian Sensitivity
Figure 4. Expected temperatures of local Jovian planets withmasses of M jup . Increased temperatures due to DM-heatingand varying emissivity values (cid:15) are shown as T (cid:15) (solid lines),where distances shown are towards the GC. Correspondingbest JWST sensitivities for each emissivity value are shownas JWST (cid:15) , as a function of the target-earth distance (dashedlines). We also show the temperature of an (cid:15) = 1 Jupiterwithout DM heating (dotted). and comparing the temperatures from internal heat alonefrom the types of exoplanets.Nearby Jupiters, especially if DM heated, are withinreach of direct imaging with JWST. This is an interest-ing possibility, as the gas giants of our own solar systemare not well understood, and it could corroborate anypotential DM contributions to their internal heat. Al-ternatively, measurement of a few sufficiently cold exo-Jupiters would exclude this hypothesis, and allow for aDM scattering constraint to be set (see Sec. V).Figure 4 shows the expected exoplanet temperature asa function of distance from the center of the Galaxy, forJupiters with different emissivities. We also show the op-timal minimal JWST temperature sensitivity, for varyingemissivity values. The JWST lines are found using the MIRI: Imaging instrument, with 2 SNR in 10 seconds.Anything above this line requires comparable or shorterexposure times, for the given emissivity. For example, fora Jupiter within 10 pc, only about 50 seconds of exposuretime is needed in the maximally DM-heated scenario (650K, emissivity 0.001) to achieve 10 SNR (using NIRISS in Imaging mode). At 100 pc, 10 SNR can be achievedin about 10 seconds. The weakened JWST sensitivitiesfor decreasing emissivity values are due to the spectralflux having a lower normalization proportional to emis-sivity; see Appendix A for more details. The dotted line shows the temperature of a 10 Gyr Jupiter with no DMheating, for emissivity equal to one. The non-DM tem-perature for smaller emissivities may scale proportionallyas T /(cid:15) / , however, the emissivity value may be affectedby feedback effects in the cooling process. Note that evencold (non-DM heated) Jovian planets have been found tobe possible to detect, in a more detailed JWST potentialsensitivity analysis [131].While the brown dwarf internal heating overpowersDM heating given local DM density, making them non-optimal targets for local searches, their high internal heatcan be advantageous for another search strategy. The lo-cal relative abundance of dwarfs compared to other stellarpopulations, and the age/temperature distribution of thedwarfs can be determined, and can be potentially usedto extrapolate to expected temperature/abundances to-wards the GC. This will be relevant for the distant searchwe now discuss below, as this would generate an apparentoverabundance of younger brown dwarfs. C. Second Search: DM Density Correlated HotDistant Exoplanets
Our second, more ambitious search, is for distant ex-oplanets. As shown in Fig. 3, at further radii, exoplan-ets increasingly are heated by DM, proportionally withthe increase in DM density. This means that exoplan-ets can in principle to be used to trace the DM densityin our Galaxy. This also means that, given a partic-ularly large statistical sample, DM overdensities couldalso be revealed by too many hot exoplanets in a givenregion. Compared to the local search above, the distantsearch is relevant for both Jupiters and brown dwarfs, asold Jupiters and brown dwarfs can both be DM-heatedpotentially well above their expected internal heat. AsFig. 3 shows, the smoking-gun signal of this search is arising exoplanet temperature towards the GC, consistentwith DM profile expectations.To demonstrate sensitivity to an example DM-heatedlarge-distance exoplanet, we consider an exoplanet withradius R jup at a distance from Earth of 8.2 kpc, which sitsoff the plane by 0.1 kpc. Given its location, for a massof about 14 M jup , a DM-heated temperature of roughly800 K would be obtained (assuming the gNFW profile),leading to a wavelength of ν − = 3 . f ν = 1 . NIRISS: Imag-ing mode, and the F356W filter, at 4 SNR with about 10 seconds of exposure. With about 10 seconds of exposureand the same filter, this can be detected at 10 SNR. Incomparison, without DM heating, this exoplanet wouldhave a temperature of around 220 K if sufficiently oldand isolated, and so such a large increase in temperaturewould present as evidence for DM heating.Figure 5 demonstrates more generally how DM densi-ties may be identified or excluded with exoplanets. Forthe Jupiter benchmark shown, no planets should be found1 DM Density [GeV / cm ]10 T e m p e r a t u r e [ K ] No DM-Heated Planets DM-Heated Planets Exoplanet Temperatures
Figure 5. Minimum DM-heated exoplanet temperature as afunction of DM density, for a Jupiter-like planet with mass M jup and radius R jup . A positive signal would lead to allplanets being found in the “DM-Heated Planets” region, andnone in the “No DM-Heated Planets” region. in the DM densities given with the low temperatures asshown. On the other hand, all planets measured wouldbe found in the DM-heating region. Any measurementcontrary to this prediction would lead to an exclusion.In this figure, the DM velocity is set to 230 km/s – thelocal DM velocity around a given candidate would needto be scaled in, along with gravitational focusing if theexoplanet is present in a low-DM velocity environment.The emissivity of the exoplanets in this plot it set to1; this is a conservative choice, as lower emissivity val-ues simply lead to higher DM-heating values. Note thatoverdensities of DM are expected from N -body simula-tions, such as Via Lactea [132, 133],
Ghalo [134], and
Aquarius [135]. It has been shown that the dense coresof many of the merging haloes that made our Milky Waysurvive as DM subhaloes, leading to e.g. DM streamsor clumping. These overdensities or subhaloes can po-tentially be independently identified by gravitational mi-crolensing (see e.g. Ref. [136]).
D. Challenges and Opportunities
We have presented optimal estimates for JWST sensi-tivity, which take into account some effects that degradethe sensitivity. We now briefly discuss assumptions andestimates of the impact of these effects, and discuss di-rections to overcome some of these challenges.
1. Planet-Star Separation
Bound planets, at large distances, will invariably betoo small to resolve with JWST, and will be outshone bytheir parent star. As such, we only consider free-floatingplanets for the distant search. However, for the localsearch, it is possible with JWST to observe bound plan-ets, as they can be close enough to be resolved. Regard-less, planets on wider orbits are still preferable candidateseven for the local search, as they are easier to disentanglefrom their star. An additional bonus for targeting plan-ets on wider orbits, is that they are likely to have lowertemperatures (as they receive less heat from their star).
2. Planet Emissivities
In Fig. 4, we display a range of emissivities. Thesecorrespond to how efficiently the planet radiates heat; alower emissivity will lead to more heat becoming trapped.While Fig. 4 showed fixed emissivities for each plot, ina true search, a given planet can have different emissiv-ity values at different temperatures. To know where theplanet would sit on an emissivity curve, it may be neces-sary to use complementary methods such as spectroscopyon the planet, to determine the atmospheric content, andtherefore an estimate of the emissivity value.In a similar vein, potential atmospheric clouds on anexoplanet can be important. Clouds may obscure thevisibility of the planet, as was shown in simulations forJWST in Ref. [131]. This size of this effect will dependon the candidate being observed, and can substantiallyvary from planet to planet.
3. Determining Exoplanet Ages
An independent age measurement of the exoplanet cancertainly be a challenge in some circumstances. Forbrown dwarfs and Jupiters, while their temperature evo-lution curves are well known for a given mass and ra-dius [58], it may be difficult to differentiate between a hotyounger exoplanet, and an older DM-heated exoplanet.The age of a given exoplanet can in principle be estimatedfrom its surroundings. For example, if it is in a boundsystem, it can be calibrated from its parent star. Browndwarfs can often be in a binary system with another typeof star, which can allow the age of brown dwarfs to beestimated from their companion star [137]. If it is a free-floating exoplanet, calibration is substantially more diffi-cult, but can in principle be deduced from the age of anynearby systems in which it may have originated.We expect overall that it will be difficult to resolve theage of all candidates, especially for planets at large dis-tances. This could be due to, for example, no nearbysystem(s) that are easily enough identified as a rogueplanet’s ejecting host, or due to the age uncertainties sim-ply being too large. We expect, however, that the large2statistics that can be provided by these searches can forma sufficiently large dataset, which appears anomalous onaverage, given the expectations for the numerous systemsstudied. Conversely, and perhaps an easier task, is to finda sufficiently cold exoplanet, in contrast with the expec-tations of DM heating. This would allow a constraint onDM properties to be set.
4. How Far Away Can We See DM-Heated Exoplanets?
The main source of signal degradation towards the GCis the number of stars present per pixel. This is becausethe number of stars increases dramatically, and these canovercrowd and outshine the exoplanet’s heat signal, mak-ing it impossible to detect. We now estimate how far intothe GC we can observe a DM-heated exoplanet by com-paring the expected stars-per-pixel with our regions ofinterest.Comparing with known stellar densities [138], and tak-ing a line of sight about 1 degree above the GC, the stellarmass is about 2 × M (cid:12) per square degree. To convertthis into a number of stars per square degree, we breakthe mass up into the fraction of expected mass in differenttypes of stars. The stellar mass function is dominated byM-type stars contributing about 76% of the stellar pop-ulation. We therefore expect about 5 − × stars persquare degree. Now, considering JWST’s NIRISS instru-ment, which provides our leading sensitivity towards theGC, we note that the field of view is 2.2 by 2.2 arcmin .The NIRISS instrument has a single 2048 by 2048 pixeldetector array with 65 milliarcsec pixels. This meansthere would be about 5 − × stars in the field of viewof JWST, and therefore that with NIRISS, we expectabout 0.15 - 0.2 stars per pixel when observing about 1degree above the GC. This means that, about 85% ofthe time, an exoplanet candidate at this distance couldpotentially be observed without any stars contaminatingits pixel. We therefore expect that about 1 degree off theplane, that is, out to about 0.1 kpc, is required to collectlarge statistics for observing this signal. In principle, itmay be possible to push this sensitivity further, at thecost of sacrificing statistics due to even further increasedstellar crowding. The absolute sensitivity will be limitedby the extremely dense region around the central blackhole, with a radius of about 30 pc [139] – at this point,any observations are likely completely hopeless. We how-ever present only 0.1 kpc in our results, as a more realisticsensitivity cutoff that may collect enough statistics.Another important source of signal degradation can bedust extinction. This occurs when more dust is present,as it may absorb the light emitted from the exoplanet,or any other background star. Dust extinction is mostprevalent for shorter wavelengths, where the scatter withcosmic dust is more likely. 2MASS has studied extinctionin the K -band (near infrared) within 10 degrees of theGC [140]. The inner 0.5 degrees have very high dustextinction. For within 1 − K -band can be as low as A K =0 .
1, which will not significantly affect a signal in infrared,particularly not for temperatures between about 500 −
800 K.We therefore expect, considering both the stars-per-pixel and dust extinction, above about 1 degree off theplane, that is, out to about 0.1 kpc, provides us withour maximum optimal expected JWST sensitivity. Ofcourse, depending on the location of the candidate exo-planet, and the properties of the planet itself, elementsof this search will vary. We expect that exoplanet experi-mentalists will be able to determine much more accurateresults than the estimates we have presented here. How-ever, we find that leading factors such as dust extinctionand the background stellar numbers do not appear tocompletely conceal the potential DM-heating, by aimingfor candidates off the plane, out to least 0.1 kpc. Thisis why we truncate our optimal sensitivity estimates inFig. 1 and Fig. 3 at 0.1 kpc.
5. Beyond JWST
It is possible that other telescopes may prove morefruitful than JWST in the future. While the current de-sign of Roman has a red cutoff at 2.0 microns, it has beenargued that extending the wavelength sensitivity furtherinto the IR, by adding an K -filter to Roman, could allowinfrared imaging of distant free-floating exoplanets [120],which due to its larger field of view and possible largersurvey times, could potentially outperform JWST. It isalso possible that Gaia Near Infra-Red (GaiaNIR), a pro-posed successor of Gaia in the near infrared, may im-prove this sensitivity, along with other potential futuretelescopes LUVOIR or OST. V. DARK MATTER PARAMETER SPACE
We now consider the implications of DM-heated ex-oplanets for particle DM models. Mostly, limits fromplanetary heat flow investigate the parameter space ofstrongly interacting heavy DM candidates, so calledSIMPs, see for example Refs. [40, 42]. We find thatJupiters and brown dwarfs are ideal laboratories to studylight (sub-GeV) DM models with cross sections as smallas a few picobarn. These exoplanets are advantageousover earth-like planets, as they can have large radii, highdensities, and low core temperatures.3
A. DM Scattering and Capture
We parameterize the DM parameter space in terms ofthe DM mass and its elastic scattering cross sections withStandard Model (SM) particles. We will consider sensi-tivity to both DM-nucleon and DM-electron interactions.For interactions with nucleons, we further distinguishtwo effective scenarios, a spin-independent scatteringcross section with nucleons, and a spin-dependent scat-tering cross section. In the spin-independent cross sec-tion regime, given that our cross sections are smallenough [142], we can write: σ SI χA = σ SI χN (cid:18) µ ( m A ) µ ( m N ) (cid:19) (cid:20) Z + a n a p ( A − Z ) (cid:21) , (13)where m A is the mass of the nucleus, m N the nucleonmass, A the atomic number, µ ( m A ) and µ ( m N ) the DM-nucleus and DM-nucleon reduced masses respectively,and σ SI χN the DM-nucleon scattering cross section.For the spin-dependent scattering cross section we have σ SD χA = σ SD χN (cid:18) µ ( m A ) µ ( m N ) (cid:19) J + 1)3 J [ a p (cid:104) S p (cid:105) + a n (cid:104) S n (cid:105) ] , (14)where J is the total nuclear spin, (cid:104) S p (cid:105) and (cid:104) S n (cid:105) the effec-tive proton and neutron spins of the nucleus respectively,and a p and a n the model dependent DM-proton and DM-neutron coupling strength respectively. For our scenarioonly the couplings to protons will be relevant, since ourtargets are dominantly made of hydrogen and helium,and the latter has zero total nuclear spin. We assume a p = 1.To relate the DM heat flow in the previous sectionswith scattering cross sections, we need to find the rangeof parameters for which a fraction f of the DM parti-cles passing through the planet is gravitationally cap-tured. In order to determine the capture cross section,we adopt the formalism in Refs. [143, 144], that can takeinto account multiple DM scatterings inside an object.This formalism extends the calculations in Ref. [40] andis also valid in a regime where the escape velocity of theobject exceeds the average DM velocity.Normalizing to the maximal DM capture rate, we ob-tain the captured DM fraction f = C cap C max = ∞ (cid:88) N =1 f N , (15)with the capture fraction for a given number of scatter-ings being f N = p ( N, τ ) (cid:34) − κ exp (cid:32) − (cid:0) v − v (cid:1) v d (cid:33)(cid:35) , (16)with κ = (cid:18) v v d (cid:19) (cid:18) v v d (cid:19) − . (17) Here v d is the velocity dispersion, v N = v esc (1 − (cid:104) z (cid:105) β ) − N/ where the average scatteringangle is (cid:104) z (cid:105) = 1 / β = 4 m χ m A / ( m χ + m A ) , and m A is the mass of the target particle. The probabilitythat the DM particle scatters N times is p ( N, τ ) = 2 τ (cid:18) N s + 1 − Γ( N s + 2 , τ ) N s ! (cid:19) , (18)where Γ( a, b ) is the incomplete gamma function. Thisscattering probability is a function of the optical depth, τ = 32 σσ sat , (19)where σ sat = πR /N SM is the saturation cross section, R the planetary radius, N SM is the target particle number,and σ is the DM-target cross section.In order to set sensitivity limits on DM scattering inJupiters and brown dwarfs, we assume spheres of hydro-gen with constant density. For reference, Jupiter con-tains about 84 % hydrogen, and 16 % helium [145]. Tobe conservative, we only consider the 84 % hydrogen. Asgas giants are expected to be dominated by these ele-ments, we expect a hydrogen sphere (like Jupiter’s com-position) to be approximately representative. For thespin-independent limits, we conservatively neglect thepotential enhancement by coherent scattering on molecu-lar hydrogen, which would increase the cross section sen-sitivities by a factor 2 in some of the parameter space.To obtain the limits on DM-electron scattering in ex-oplanets, we also assume a hydrogen sphere for the exo-planets. As the chemical composition is dominantly hy-drogen, this allows the assumption that the proton num-ber density is identical to the electron number density.A subdominant correction comes from the helium abun-dance, which we neglect to be conservative. Note thatgiven the hydrogen target, relativistic shell effects playno role in the considered processes. We assume a mo-mentum independent DM-electron cross section σ χ e , i.e.the electron form factor is F = 1.We note that recently a refined simulation, taking thedetailed propagation effects into account, showed howEarth and Mars’ heat flow bounds improve for the multi-scatter regime [42]. We emphasize that in light of this im-provement our cross section sensitivities are conservative;we do not perform such a simulation, which would simplystrengthen our sensitivities. However, once specific can-didates are known, the sensitivities may be refined withsimulations incorporating the planetary composition ingreater detail, which are outside the scope of this work. B. DM Evaporation
DM evaporation from the exoplanet truncates the lowDM mass sensitivity. This is because if the kinetic energyof the thermalized DM particle exceeds the gravitational4potential, the DM will become unbounded again. Thissets the orbit condition to remain bound, as E kinDM = 32 T ( r ) < G N M ( r ) m χ r , (20)where T ( r ) is the interior temperature of the exoplanetas a function of internal radius, G N is the gravitationalconstant, and M ( r ) is the mass of the exoplanet enclosedwithin a given radius r . As we require that DM particlescan accumulate in the cores of the objects in order toefficiently annihilate, we therefore require that the DMis bound to the exoplanet.To compare with evaporation in Earth, we first takeEarth density and temperature profiles [146]. For Earth,we find a minimum bounded DM mass of m min χ = O (100) MeV. Now, for Jupiter, with a core tempera-ture of T c = 1 . × K, an average density of ρ jup =1 . / cm , and a radius of R jup = 6 . × cm, wehave found the minimum bounded DM mass is m min χ ∼
30 MeV. This result applies to the exoplanet Jupiters, ofcomparable size. Note that we have checked two Jupiterdensity profile hypothesis, one with a core and the otherwithout [121] and find no significant effect on the minimalDM mass.For our brown dwarf benchmark point we use an an-alytical model as detailed in Sec. II. Compared to ourcross section sensitivity estimates, where we assumed ahydrogen sphere, the evaporation limits require a modelfor the exoplanet core. The relatively low core temper-atures and high densities make old brown dwarfs effi-cient accumulators for light DM. For our benchmark, thebrown dwarf radius is taken to be R BD = R jup , and themass M BD = 75 M jup = 0 . M (cid:12) . This results in anaverage density of ρ BD = 108 g / cm , a core density of ρ c = 500 g / cm and a core temperature T c = 2 × K.The resulting minimal DM mass that does not evaporatein the brown dwarf is m min χ ∼ M jup ), and thebrown dwarf benchmark (planetary mass 75 M jup ). TheDM mass where evaporation occurs will depend on wherethe mass of the exoplanet in question sits relevant to thesebenchmarks. C. DM Equilibration
DM capture and annihilation must reach equilibriumin order for the heating process to be maximally effec-tive [147]. In this subsection, we show equilibrium canbe expected to be reached in our exoplanets of interest.As our sensitivity extends into the sub-GeV regime (asshown above with lower evaporation masses), we considerboth the standard 2 → → χχ → SM + SM Annihilation Processes
For a DM candidate that annihilates via a 2 → → = (cid:90) dV n χ (cid:104) σ ann v rel (cid:105) , (21)where n χ is the DM number density, and (cid:104) σ ann v rel (cid:105) isthe thermal averaged cross section, with σ ann the anni-hilation cross section, and v rel the relative DM velocity.The equilibrium number of DM particles in the objectis found from the solution of the differential equation˙ N χ = C cap − C evap N χ − C → N χ , (22)where N χ is the DM number, C cap is the capture rate, C evap the evaporation rate, and the annihilation coeffi-cient is given by C → = (cid:104) σ ann v rel (cid:105) /V → . (23)The annihilation volume is V → = V /V , with the vol-ume for a given species j being V j = 4 π (cid:90) R e − jm χ φ ( r ) /T ( r ) r dr. (24)Here, R is the radius of the exoplanet, T ( r ) is the plan-tary interior temperature as a function of radius, φ ( r ) isthe gravitational potential, and r is the radius of the vol-ume within the exoplanet. The equilibration time scaleis then given by τ = ( C ann C cap ) − / . (25)This can be converted into a lower bound on the annihi-lation cross section, (cid:104) σ ann v rel (cid:105) ≥ V → / ( C cap τ ) . (26)Compared to Earth, Jupiters have an effective volumeof the order of V Jupiterseff ∼ O (100) V Eartheff , and equilibra-tion times of about τ ∼
10 Gyr are feasible, in con-trast to Earth, where τ ∼ (cid:104) σ ann v rel (cid:105) ≥ − ( m χ / GeV) − cm / s, which is expected to be satis-fied in models with a thermal freezeout. χ + χ + χ → χ + χ Annihilation Processes
As our searches focus on the sub-GeV regime, now wediscuss models of light, thermally produced DM, whichare based on 3 → χ + χ + χ → χ + χ . The freezeout con-dition for the interaction rate factor is then (cid:104) σ → v (cid:105) =10 ( m χ / GeV) − GeV − [149]. The corresponding anni-hilation rate in planets is given by the volume integralΓ → = (cid:90) dV n χ (cid:104) σ → v (cid:105) , (27)resulting in an annihilation rate of C → = (cid:104) σ → v (cid:105) / ( V → ) , (28)with an annihilation volume of V → = V (cid:112) V /V . Theequilibrium condition on the rate factor then reads (cid:104) σ → v (cid:105) ≥ (cid:0) V → (cid:1) / ( C cap τ ) . (29)Given a Jupiter-like planet with M = M jup , R = R jup and τ = 10 Gyr, this gives (cid:104) σ → v (cid:105) ≥ ( m χ / GeV) GeV − , (30)which is many orders of magnitude larger than the valueexpected from the thermal freezeout. This means thatthe χ + χ + χ → χ + χ process does not reach equilibrium insufficient time. There is however, another 3 → χ + χ + SM → χ + SM Annihilation Processes
Recently, a different number changing interaction hasbeen proposed in order to produce light, thermal DM,called the Co-SIMP [155]. In this scenario, the DMfreeze-out is assisted by SM particles, in the process χ + χ + SM → χ + SM. Since the number density ofSM particles in a planet is by many orders of magni-tude larger than the accumulated DM number density,this interaction rate is significantly more efficient. Thisleads to a prediction for the rate factor, (cid:104) σ → v (cid:105) =10 ( m χ / GeV) − GeV − . The annihilation rate in exo-planets is given byΓ → = (cid:90) dV n χ n SM (cid:104) σ → v (cid:105) , (31)resulting in an annihilation rate of C → = (cid:104) σ → v (cid:105) n SM /V → , (32)with the condition that (cid:104) σ → v (cid:105) ≥ V → / ( τ C cap n SM ) . (33)For a Jupiter-like planet, this gives a minimum rate toreach equilibrium, (cid:104) σ → v (cid:105) ≥ − ( m χ / GeV) GeV − . (34)This is well below the thermally expected rate, such thatcaptured Co-SIMP particles always reach equilibrium. As we consider elastic cross sections that lead to all theoutgoing particles becoming trapped in the planet, thismeans that the entire mass energy released in the Co-SIMP process will be converted to the planetary heatflow.In fact, this process is even more broadly applicable.In Ref. [148], it is shown that in the SIMP model the Co-SIMP process exists, however, the rate is suppressed bya factor (cid:15) . Experimentally this quantity is constrainedto be in the range of (cid:15) ∼ − − − , and thereforethe Co-SIMP process will be subdominant to the ther-mal SIMP rate. Regardless, the subdominant Co-SIMPprocess will still be the process that brings the particlesinto annihilation equilibrium (as opposed to kinetic equi-librium that only equilibrates the temperatures), due tothe larger number density. We therefore expect that theJupiters and brown dwarf searches will probe new terri-tory of the number changing, thermal DM models. D. DM Cross Section Sensitivities
Figure 6 shows our sensitivity estimates for Jupiter-like planets and brown dwarfs to the DM parameterspace for the spin-dependent and spin-independent DM-nucleon scattering. Note that the scattering sensitivityarises predominately from DM-proton interactions. Thisis because gas giants and brown dwarfs are predomi-nately hydrogen and helium; hydrogen only has a proton,and helium has zero total nuclear spin, thus DM-neutroninteractions are not significant. For both Jupiters andbrown dwarfs, we show sensitivity to the both the max-imum capture rate (for which all DM is captured, andplanets are maximally heated), as well as a 10% DM cap-ture rate (which causes less heating, but still may be de-tectable). These sensitivities do not specifically dependon the DM density profile or DM density value; sensi-tivity to the cross sections shown only requires that theDM-heating temperature has exceeded the internal heat-ing temperature for the relevant target exoplanet. Weshow the earth heat flow bounds from Ref. [42] for com-parison, and direct detection bounds [156, 157]. In thecase of spin-independent scattering, the sensitivity of di-rect detection experiments to light DM can be enhancedby taking into account boosts from collisions with cosmicrays [154, 158–160], which are shown in the top regionsof the plots.Figure 7 shows the DM-electron scattering sensitiv-ity estimates, alongside existing limits from direct de-tection [161–172] and solar reflection [173, 174]. Whilethe region of sensitivity of Jovian planets is already con-strained by direct detection experiments, brown dwarfswill have some sensitivity to new DM-electron scatter-ing parameter space. We show the sensitivity for when100% and 10% of DM is captured. Electron-dominatedinteractions may be found in for example leptophilic DMmodels [175–179].For both Fig. 6 and Fig. 7, the sensitivity region would6 m [GeV]10 S I N [ c m ] J u p i t e r s , C m a x J u p i t e r s , . C m a x B r o w n D w a r f s , C m a x B r o w n D w a r f s , . C m a x E a r t h X e n o n T ( C R ) DDSpin-Independent, Local DM Velocity 10 m [GeV]10 S I N [ c m ] J u p i t e r s , C m a x J u p i t e r s , . C m a x B r o w n D w a r f s , C m a x B r o w n D w a r f s , . C m a x E a r t h X e n o n T ( C R ) DDSpin-Independent, GC DM Velocity B o r e x i n o ( C R ) m [GeV]10 S D p [ c m ] J u p i t e r s , C m a x J u p i t e r s , . C m a x B r o w n D w a r f s , C m a x B r o w n D w a r f s , . C m a x DD , a n = DD , a n = a p E a r t h Spin-Dependent, Local DM Velocity B o r e x i n o ( C R ) m [GeV]10 S D p [ c m ] J u p i t e r s , C m a x J u p i t e r s , . C m a x B r o w n D w a r f s , C m a x B r o w n D w a r f s , . C m a x DD , a n = DD , a n = a p E a r t h Spin-Dependent, GC DM Velocity
Figure 6. Spin-independent (top row) and spin-dependent (bottom row) DM-nucleon scattering cross section sensitivity esti-mates for Jupiters and brown dwarfs, for exoplanets in a local DM velocity (left column) or GC DM velocity (right column)calculated in this work. The solid lines show cross sections assuming 100% of incoming DM is captured ( C max ), and the dottedlines show cross sections for when 10% of DM is captured (0 . C max ). Complementary constraints are also shown; Earth isthe limit on Earth DM-heat flow [42], DD is a collection of direct detection experiments [150–153], Xenon1T (CR) [154] andBorexino (CR) [154] correspond to cosmic-ray boosted DM signals. For spin-dependent scattering, two different DD boundsare shown; if the proton a p and neutron a n couplings are equal, the light pink line would be filled, if the neutron coupling a n is zero, the magenta shaded region is the DD limit (the exoplanet limits are not affected by this choice). be all filled in as a constraint if a sufficiently cold Jupiteror brown dwarf were measured. For instead discovery ofa DM-heating signal, the DM parameters would lie abovethe dashed lines shown. As both these figures show 100%and 10% values of the DM capture rate, in principle evenstronger sensitivity to DM cross sections can be reachedif an even smaller DM capture fraction can be probed.However, given the JWST optimal sensitivity, about a10% DM capture fraction is likely the smallest capturefraction that can be probed in the near future. Note that there is a ceiling for the cross sections abovewhich the DM does not drift fast enough into the planet’score [180]. However, even in the case of a dense browndwarf, and sub-GeV DM masses, we find that this ceilingis of the order of σ max ∼ − cm (for the sub-GeV DMmass range). Such cross section values are at the thresh-old where a point-like DM description is barely valid, andanother physical description for DM must be used. Im-portantly, brown dwarfs provide complementary sensitiv-ity to parameter space that can be tested by CR boosted7 m [MeV]10 e [ c m ] J u p i t e r s , C m a x J u p i t e r s , . C m a x B r o w n D w a r f s , C m a x B r o w n D w a r f s , . C m a x R e f l e c t i o n DDElectron Scattering, Local DM Velocity 1 10 100 m [MeV]10 e [ c m ] J u p i t e r s , C m a x J u p i t e r s , . C m a x B r o w n D w a r f s , C m a x B r o w n D w a r f s , . C m a x R e f l e c t i o n DDElectron Scattering, GC DM Velocity
Figure 7. DM-electron scattering cross section sensitivity for brown dwarfs and Jupiters, for exoplanets in a local DM velocity(left) or GC DM velocity (right) calculated in this work. The solid lines show sensitivity assuming 100% of incoming DM iscaptured ( C max ), and the dotted lines show sensitivity for when 10% of DM is captured (0 . C max ). Complementary constraintsfrom direct detection (DD) [161–172] and solar reflection [173] are shown. DM, which can be difficult to interpret owing to no-energy dependence being used, despite being high-energyprocesses. Note that while we have cast our sensitivityin terms of one DM particle with one interaction type, inprinciple several particle processes may be present in thedark sector, which can alter the expected phenomenol-ogy (see e.g. Refs [181–186]). Detailed model-dependentstudies would need to be performed to determine the fullrange of particle physics possibilities. Importantly, notethat direct detection or other competing bounds may beweakened or removed in some DM models, while the ex-oplanet sensitivities would remain present. This can betrue, for example, in inelastic DM models.In order to contribute significantly to the heat flow,the DM population should have a dominant symmetriccomponent (and not dominantly annihilate into invisi-ble final states), which is a natural outcome in scenarioswith thermally produced DM. Thermally produced DMcandidates, with sub-GeV masses are known to exist inmodels with light mediators [187] and production mech-anisms with number changing interactions [148, 155]. Asdiscussed in detail in Ref. [187], DM models with domi-nant interactions with nucleons, and m χ < GeV face se-vere experimental constraints if their thermal abundanceis set by a 2 → → VI. SUMMARY AND CONCLUSIONS
The exoplanet program is rapidly accelerating.Amongst the billions of new worlds in our Galaxy, manyare waiting to reveal their surprises. Unexpected discov-eries are inevitable, and numerous new telescopes withcutting-edge technology are ready to make them. In thiswork, we have examined how exoplanets can be used todiscover DM or other new physics. For the first time, wehave pointed out the broad applicability of exoplanetsto be used as DM detectors, with actionable discoveryor exclusion searches using new infrared telescopes. Wetarget old, cold, Jupiter-like planets and brown dwarfs,which are particularly advantageous due to their largesizes, densities, and low core temperatures.Our first suggested search can be expected to bringshorter-term results. There are hundreds of knownJupiters in our local neighborhood, and Gaia is expectedto identify tens of thousands of potential candidatesin the next few years. These collectively provide anenormous statistical sample for a DM-heated exoplanetsearch. We identified numerous known exoplanets as can-didates for this search, and estimated the JWST sensi-tivity to their potential planetary temperatures. We con-cluded that local searches show strong promise to dis-cover DM-heated planets. If no DM-heating signal isfound, new constraints can be set on the DM mass and8scattering rate.Our second suggested search is more ambitious, butmay be fruitful simply due to the enormous number ofexoplanets in our Galaxy. We pointed out that the ex-oplanet temperature is expected to be correlated withDM density, rising sharply towards the GC. This leadsto a new DM search; by brute statistical force, we may beable to discover DM by measuring DM-density-correlatedheated exoplanets. The presence of DM overdensities orsubstructure may also be confirmed with exoplanets, witha pocket of even hotter DM-heated exoplanets. Alterna-tively, the larger expected DM-heating signal in theseDM overdensities can lead to even stronger constraints,if exoplanets are measured to be sufficiently cold.We estimated the impact of difficulties in seeing a DM-heated exoplanet at such large distances into the GC.We found that stellar crowding was the main limitingfactor, and that dust extinction was minimal at the in-frared wavelengths and galactic locations of interest. Wedetermined that to minimize both dust extinction andstellar crowding, and maximize a potential DM-heatingsignature, the optimal distance for a candidate exoplanetis about 0.1 kpc off the plane. We also noted that, inorder to measure exoplanet temperatures at such largedistances, the exoplanet must be a rogue planet. Other-wise, it cannot be resolved from a parent star, and wouldbe greatly outshone. We concluded that, at an estimate,JWST may have sensitivity to exoplanet temperaturesabove about 650 K, for exoplanets all the way into about0.1 kpc of the inner Galaxy (for more local searches,the minimum temperature sensitivity is of course lower).While this search may be challenging, we again empha-size the large statistics that may be available with themany exoplanet telescopes that are upcoming or are cur-rently being proposed. As per the local search, if a suf-ficiently cold exoplanet is discovered, this would allow aconstraint to be set on the DM mass and scattering crosssection.We calculated the DM parameter space sensitivity tobrown dwarfs and Jupiters that may have their temper-ature measured in the near future. We determined thatDM with masses above about an MeV can be probedwith exoplanets, with DM-proton and DM-electron scat-tering cross sections down to about 10 − cm , strongerthan some existing limits by up to six orders of magni-tude. We pointed out that this DM mass sensitivity islighter than many other celestial body searches for DMheat flow. This is because brown dwarfs and Jupitershave large integrated column densities, and given the ra-tio between the gravitational potential and the core tem-perature, it is more difficult for light DM to evaporatein these systems. Interestingly, we found that new pro-cesses also become relevant in this previously unprobedsub-GeV DM-heating regime. In particular, we foundthat certain 3 → ACKNOWLEDGMENTS
We thank John Beacom, Joe Bramante, Chris Cap-piello, Rouven Essig, Katie Freese, Savannah Jacklin,Shirley Li, Nirmal Raj, Pat Scott, Sara Seager, AaronVincent, and Ji Wang for helpful comments and discus-sions. RKL was supported by the Office of High EnergyPhysics of the U.S. Department of Energy under GrantNo. DE-SC00012567 and DE-SC0013999, as well as theNASA Fermi Guest Investigator Program under GrantNo. 80NSSC19K1515, and later at SLAC under Con-tract DE-AC02-76SF00515. J.S. is largely supported bya Feodor Lynen Fellowship from the Alexander von Hum-boldt foundation.
Appendix A: Impact of Atmospheric Emissivity
Exoplanet atmospheric emissivity can have an impacton JWST searches. This is because emissivity can trapsome exoplanet heat flow, leading to higher temperatures.We now briefly demonstrate how this can improve JWSTsensitivities.Figure 8 (left) shows the impact of varied exoplanetemissivity on the spectral flux density. As the inter-nal heat and the power output of a planet is a con-served quantity, a higher temperature can be obtainedfor smaller emissivity values, at the cost of a drop inthe normalization of the spectral flux. As energy is con-served, this leads to the same total integrated flux forall emissivities, but the temperature peaks at a shorterwavelength. The main benefit is therefore being able toexploit the more powerful filters available on JWST’s in-struments; longer wavelengths generally have worse fluxsensitivity than the shorter wavelengths. The examplescenario shown here is for a DM-heated Jupiter at 109 m ]012345 Sp e c t r a l F l u x D e n s i t y [ m J y ] = 1, T = 116= 0.1, T = 207= 0.01, T = 369= 0.001, T = 657Emissivity Impact m ]051015202530 Sp e c t r a l F l u x D e n s i t y [ m J y ] T = 116, T = 207, blackbodyWavelength Dependent Emissivity T = , . M e a s u r e d T e m p Figure 8. Impact of atmospheric emissivity on heating signals.
Left:
Temperature peak shifts for smaller emissivities.
Right:
Schematic example of wavelength-dependent emissivity values, which can substantially boost signal intensity. This example hasemissivity of one above about 0.08 µm − , and very suppressed emissivities below this frequency. This leads to a non-suppressedpeak at higher frequencies, shown as the orange “measured temp” curve. The non-suppressed version of the emissivity equalto 0.1 curve is shown in solid blue. The dashed blue is the correctly rescaled version of the emissivity 0.1 case (i.e., the 0.1penalty is applied to the blackbody temperature). The emissivity equal to one case is shown as solid magenta. pc, which is a slice at d = 10 pc through Fig. 4. Whilean increased emissivity can lead to a larger effect alsofor the longer-distance searches, the effect is generallynot as pronounced, as the higher temperature filters arenot substantially more powerful than the already high-temperature filters used at large distances in the (cid:15) = 1case shown in Fig. 3.Figure 8 (right) shows a schematic wavelength-dependent emissivity scenario. Indeed, in reality, an ex-oplanet may have different emissivities at different wave-lengths, due to some wavelengths being better reflectedby the atmosphere. For example, one could imaginean atmosphere leaving optical wavelengths mostly un-affected, while internally reflecting infrared wavelengths.This would lead to an extreme departure from the usualblackbody spectrum, similar to what is shown in the rightfigure. This also can allow both a boost in flux densitycompared to an emissivity value that is constant at allwavelengths, as well as applicability of better filters atshorter wavelengths. A planet could look, for example,truly like a higher temperature planet if only observing the edge of the spectrum (e.g. if a telescope was wave-length limited), without decreased normalization, whilein other wavelengths, the normalization could be greatlysuppressed due to the emissivity factor. In such a sce-nario, the area under the curves would still be conserved(e.g. the area under all of the smaller three fluxes inFig. 8 is conserved). This variance of temperature peaks,at different emissivities, when the planet could not oth-erwise reach such high temperatures without DM, wouldbe a smoking gun signal of a DM-heated planet. To beconservative, we do not use wavelength-dependent emis-sivities in our main results; we only point out this canpotentially considerably boost sensitivities.Lastly, note that at small emissivity values, the exo-planet surface temperature might become completely un-accessible, and DM-heating may instead only impact thetemperature of the atmosphere in an energy exchangeprocess. The details of such an effect will however de-pend on the exoplanet in question, and is an interestingpossibility to study in a dedicated simulation, which isoutside the scope of this work. [1] I. Goldman and S. Nussinov, “Weakly Interacting Mas-sive Particles and Neutron Stars,” Phys. Rev. D40 ,3221–3230 (1989). [2] Andrew Gould, Bruce T. Draine, Roger W. Romani,and Shmuel Nussinov, “Neutron Stars: Graveyard ofCharged Dark Matter,” Phys. Lett.
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