A Small Contribution to Reionization from Accretion Shocks

The Shocking Truth: The small contribution to hydrogen reionization from gravitational infall

Abstract

It is commonly thought that stars are responsible for reionizing the Universe. However, deep searches for star-forming galaxies during the epoch of reionization have not yet found sufficient galaxies to provide the necessary ionizing flux. Fast accretion shocks associated with gravitational infall of baryons during the formation of galaxies have recently been proposed as an alternative method of generating the required ionising photons. In this Letter we assess the contribution to hydrogen reionization from shocked gas associated with gravitational infall. We find that shocks can ionize no more than a few percent of the cosmic hydrogen by . However, the small fraction of ionising radiation produced by fast accretion shocks would be significantly more biased than that associated with stars, leading to a modification of the luminosity weighted source clustering by . This modification of the bias may be measurable with future precision experiments utilising the redshifted 21cm line to study the distribution of hydrogen during the reionization era.

Subject headings:
cosmology: theory, reionization, diffuse radiation
6

1. Introduction

Star-bursting galaxies and quasars have been the leading candidates for the sources of the UV radiation required to reionize the hydrogen gas in the inter-galactic medium (IGM) (Loeb, 2010). The quasar population is observed to decline quickly at (e.g., Fan et al., 2002) and so it is believed that galaxies contributed the bulk of UV photons that drove reionization (Madau et al., 1999; Srbinovsky & Wyithe, 2007; Bolton & Haehnelt, 2007). The observed number counts of high-redshift galaxy candidates (Bouwens et al., 2011; Yan et al., 2010) discovered with the Hubble Space Telescope Ultra Deep Field (HUDF) have been used to build up a statistical description of star-forming activity at redshift . At , 8.6 and 10.6 the flux limits correspond to absolute magnitudes , and  mag, respectively. While impressively faint, these observations do not reach the levels corresponding to the faintest galaxies thought to exist at these early epochs (e.g., Barkana & Loeb, 2000), and the observed stars in the HUDF are insufficient to reionize the Universe.

Trenti et al. (2010) have constructed an empirical model for the luminosity function and find that the observed population could have reionized the Universe if it extends to luminosities fainter than observed. In addition to galaxies, the discovery of high redshift gamma ray bursts (Salvaterra et al., 2009; Tanvir et al., 2009) can be used to probe the star formation rate up to (Kistler et al., 2009), based on which Wyithe et al. (2010) found sufficient star-formation to achieve reionization by .

Thus, there is evidence that stars are capable of reionizing the Universe. However, until the sources are identified directly, it is prudent to study alternatives. One such alternative is provided by cooling radiation associated with shock-heated gas, which can process gravitational energy associated with structure formation into an ionising radiation background. For example, Furlanetto & Loeb (2004) have considered the effect of large scale structure shocks on the IGM during reionization, finding that shocks heat the early IGM and produce a radiation background that effects molecular hydrogen formation and the spin temperature of neutral hydrogen. In a complementary study, Miniati et al. (2004) calculated the UV background from cooling radiation associated with virialised gas that is shock heated during halo formation. They evaluated the ionising rate in the optically thin IGM at , and determined that the ionising background at the hydrogen ionisation edge produced following virialisation shocks was likely to be much smaller than for stars, unless supernova feedback could efficiently reheat the galactic gas (in which case the galaxies would likely produce a significant contribution to the ionising flux from star light). Miniati et al. (2004) concluded that the hard spectrum produced may doubly reionize helium by , and also described the resulting effects on the thermal history of the IGM.

Recently, Dopita et al. (2011) presented high resolution simulations of the shocks associated with the supersonic gravitational infall of gas onto a nascent galactic disk that arises when the cooling rate is much shorter than the free-fall time. They argued that this provided a source of photons that could augment the ionising flux from galaxies, and indeed might dominate the reionization of the Universe. If true this finding would have significant implications for studies of reionization. In particular, the direct link between star-formation and the ionisation structure of the IGM would be removed, so that future redshifted 21cm experiments would not provide a fruitful route to study the first stars.

In this Letter we assess the contribution to hydrogen reionization from ionising photons produced in fast radiative accretion shocks. We note that virialisation (Miniati et al., 2004) and fast accretion shocks (Dopita et al., 2011) should lead to similar ionizing luminosities since the cooling radiation following the virialization shock contains energy comparable to the gravitational potential energy available to drive the fast accretion shock. In our numerical examples, we adopt the standard set of cosmological parameters (Komatsu et al., 2011), with values of , and for the matter, baryon, and dark energy fractional density respectively, , for the dimensionless Hubble constant, and .

2. Ionizing photons

We begin by computing the number of ionizations per baryon processed through shocks. We also discuss the ionisation history due to stars for comparison.

2.1. Ionizing photons based on collapsed fraction

Figure 1.— Plots of the number of photons produced per hydrogen atom in the Universe. Upper Left: The number produced when all gas that enters a halo shocks once when the halo forms. Upper Right: The number produced when gas re-shocks following each merger (based on the merger rate of halos). Lower Left: The number produced assuming all gravitational energy of collapsed gas was released as photons with 13.6eV. Lower Right: The number number of ionizations obtained for stars, assuming . Results are shown as a function of redshift, assuming minimum halo masses corresponding to , and . The grey strips represent uncertainty in .

Dopita et al. (2011) presented a fitting formula for the number of ionising photons that enter the IGM per baryon processed through a shock, which in our terminology is

(1)

where is the velocity of the shocked gas. Dopita et al. (2011) argue that where is the virial velocity of the halo, and we utilise this throughout the current work. The virial velocity is estimated from the halo mass using

(2)

where is close to unity and defined as , , , and (see equations 22–25 in Barkana & Loeb 2001 for more details).

Under the assumption that gas shocks once when a halo forms, the number of photons produced per hydrogen atom in the Universe can be estimated from the Press & Schechter (1974) mass function (with modifications due to Sheth & Tormen, 1999)

(3)

where and , are the baryonic mass inside a dark matter halo and baryonic mass density in the IGM respectively. Results are shown as a function of redshift in the upper left panel of Figure 1, assuming minimum halo masses corresponding to (the cooling threshold for hydrogen), and (the Jeans threshold in an ionized IGM, Dijkstra et al., 2004). The results are independent of this choice owing to the dominance of massive halos in producing radiation from fast accretion shocks (Dopita et al., 2011). We find that only a few percent of the IGM is reionized by , and at , indicating that there are 1-2 orders of magnitude too few ionising photons produced in fast accretion shocks to reionize the Universe.

We note that virialisation shocks do not exist in low mass halos due to the presence of cold flows. Given the halo mass dependent fraction of cold flow accretion where no shock is produced Kereš et al. (2009); Faucher-Giguère et al. (2010), one should exclude cold mode accretion material from the calculation of ionising luminosity associated with virialisation shocks (Miniati et al., 2004). This would reduce the predictions in Miniati et al. (2004) by a factor of , where is weighted over the mass and number of halos. However Dopita et al. (2011) argue that with sufficient resolution the cold flow material is found to shock at the intersection with the nascent galactic disk. As a result no correction for cold flow accretion should be applied to estimates of fast accretion shock produced ionising photons in equation (3) or later in this Letter.

2.2. ionising photons from stars

We compare the above result to the number of ionizations obtained for stars (lower right panel of Figure 1). Here we utilise equation (3) but assume , appropriate for a Salpeter IMF with a fiducial value of , where and are the star-formation efficiency and escape fraction of ionising photons respectively. We again show minimum halo masses corresponding to and , leading to significant variation in the total number of ionising photons produced. In difference to shock produced photons, these fiducial stellar populations are easily able to reionize the Universe by (as is well known). This result is plausible since the nuclear efficiency of stars is larger by many orders of magnitude than the efficiency of converting rest mass to radiation by a shock, , for the shock speeds of interest ().

2.3. Ionizing photons based on merger rates

The calculation in § 2.1 utilises each baryon only once, whereas, unlike the case for stars, a baryon may be processed through shocks several times during the hierarchical formation of a galaxy. We therefore recalculate the contribution to reionization from fast accretion shocks based on the merger rate of halos. Specifically, when a halo of mass merges with a halo of mass , we assume that the baryons contained within halo 1 are shocked at . We assume that the star-formation efficiency is negligible so that all baryons are available to shock during each merger. The resulting expression for the rate of production of all baryons in galaxies is

(4)

yielding

(5)

Here is the number of mergers of a halo with mass between and that merge with a halo of mass in a redshift interval .

The assumption that baryons shock during each merger is optimistic. While mergers will certainly increase the virial energy per baryon, it is not clear how the baryons would be recycled through virialization shocks on multiple occasions if they cool efficiently to produce the ionizing radiation. If the baryons cool from the halo into a disk, they will not be heated to the virial temperature of the halo (unless they have been expelled through some feedback process) and so should not participate in the halo shock of the next merger. Thus, our results from equation (2.3) represent the maximum ionisation flux possible from the fast accretion shock mechanism summarised in equation (2.1).

Results are shown as a function of redshift in the upper right panel of Figure 1, assuming minimum halo masses corresponding to and (again with no discernible difference owing to the dominance of massive halos). We find that the re-processing of baryons through shocks in multiple mergers increases the ionising photon output by a factor of a few relative to the collapsed fraction calculation. However, we still find that only percent of the IGM is reionized by , and at , indicating that shocks provide insufficient ionising luminosity to reionize the IGM.

The difference between our findings and the results presented in Dopita et al. (2011) originates partly from the adoption of a large value of in that work, and partly from an error in the calculation method (Lawrence Krauss, private communication). Dopita et al. (2011) calculate the accretion rate and corresponding ionising luminosity as a function of halo mass (see their Figure 2). They then integrate over the mass-function and redshift. This procedure effectively sets the rate at which gas in the halo doubles to be equal to the inverse of dynamical time at the virial radius (which is shorter by an order of magnitude than the Hubble time), and so does not account for the duty-cycle of the shocks (which should be only ). As a result, the calculation in Dopita et al. (2011) accretes an order of magnitude more gas than available per halo.

2.4. A maximum of ionizing photons based on binding energy

In the previous subsections we have estimated the number of ionising photons available per hydrogen in the IGM. We next estimate the maximum number of photons that equal the binding energy of all baryons in the halo. This should provide an upper limit to the number of ionising photons produced by shocks (in the absence of significant feedback, Miniati et al., 2004), and hence an upper limit on the contribution of shocks to reionization. For this calculation, we again appeal to equation (3), setting to be

(6)

Results are shown in the lower-left panel of Figure 1 assuming minimum halo masses corresponding to and . We find that the total gravitational energy available for ionisation of hydrogen corresponds to less than 1 ionising photon per 3 hydrogens by and less than 1 ionising photon per 10 hydrogens by .

3. Reionization Histories

Next we use the estimate of flux based on our merger calculation of ionising radiation from shocks as the source term in a calculation of the reionization history. Miralda-Escudé et al. (2000) presented a model which allows the calculation of an effective recombination rate in an inhomogeneous universe by assuming a maximum overdensity () penetrated by ionizing photons within HII regions. The model assumes that reionization progresses rapidly through islands of lower density prior to the overlap of individual cosmological ionized regions. Following the overlap epoch, the remaining regions of high density are gradually ionized. Wyithe & Loeb (2003) employed this prescription within a semi-analytic model of reionization, and we refer the reader to that paper for details of the model. Within this formalism, the epoch of overlap is precisely defined as the time when the volume fraction of the universe ionized up to an overdensity , reaches unity. After the overlap epoch, ionizing photons will experience attenuation due to residual overdense pockets of HI gas. The model also follows the mass averaged ionized fraction ().

Figure 2 shows the resulting model for the reionization of the IGM and the subsequent post-overlap evolution due to ionising sources from fast accretion shocks (solid lines). Here we have used an ionising photon rate based on equation (2.3). The case shown corresponds to a value for the critical overdensity prior to the overlap epoch of , and both the volume averaged (dark lines) and mass-averaged (grey lines) ionisation fractions are shown. We find that shocks can reionize less than 10% of the IGM (by volume or mass) prior to , and cannot complete reionization until . For comparison we compute the reionization history for stars (dashed lines), where the ionising photon production rate is based on , with based on equation (3) with . In this model, stars complete reionization by , at which time the relative contributions from stars and fast accretion shocks differ by a factor in excess of 100.

Figure 2.— Plots of the reionization history of the IGM and the subsequent post-overlap evolution due to ionising sources from merger induced shocks (solid lines), and stars with (dashed lines). The case shown corresponds to a value for the critical overdensity prior to the overlap epoch of , and both the volume averaged (thick lines) and mass-averaged (thin lines) ionisation fractions are shown.

4. implications for the bias of ionising sources and 21cm studies

In this Letter we have demonstrated that recent estimates of the ionising luminosity from fast accretion shocks associated with galaxy formation are not sufficient to drive reionization. However the ionising photons produced by shocks are dominated by massive halos (Miniati et al., 2004; Dopita et al., 2011). This is in contrast to the ionising radiation from stars, which is both predicted and observed to be dominated by low mass galaxies. As a result, the ionising radiation produced in shocks is significantly more biased relative to the underlying large scale density of the IGM than are ionising photons produced in galaxies. It is easy to see the physics of the dominance of massive halos by noting that the collapse energy available in equation (6) is proportional to (or ), whereas the stellar mass (assuming a constant mass-to-light ratio) is proportional to (or ).

The ionisation structure of the IGM, particularly the scale of HII regions produced is a sensitive function of the bias of ionising sources (McQuinn et al., 2007). It is this relation between the bias of ionising sources and the resulting ionisation structure during reionization that motivates redshifted 21cm experiments with the ultimate aim of connecting galaxy properties to the power-spectrum of 21cm fluctuations (Barkana, 2009). Here we quantify the effect of fast accretion shocks on the bias of ionising sources. The halo bias for a halo mass at redshift may be approximated using the Press & Schechter (1974) formalism, modified to include non-spherical collapse (Sheth et al., 2001). The power-spectrum of the space distribution of sources is proportional to squared. The luminosity weighted bias of ionising radiation produced by shocks arising in mergers can be evaluated using the expression

(7)

where the bias is evaluated at a mass . The resulting bias is plotted as a function of redshift in the upper panel of Figure 3. Prior to reionization, fast accretion shock powered ionising sources have a luminosity weighted bias of .

For comparison, we calculate the luminosity weighted bias for stellar sources () based on the derivative of equation (3), and in analogy with equation (2.3). The result is also plotted in the upper panel of Figure 3. Prior to reionization, stellar ionising sources have a smaller luminosity weighted bias of . Finally, we evaluate the luminosity weighted bias obtained when fast accretion shock powered ionisation sources are added to the stellar sources needed for reionization. We then calculate the fractional change in observed clustering of ionising radiation relative to the stars-only reionization history []. This fractional change is plotted in the lower panel of Figure 3, which shows that the power-spectrum of ionising sources is increased by owing to ionising radiation produced in shocks. This change in the clustering of ionising sources will lead to comparable changes in the amplitude of redshifted 21cm fluctuations (Wyithe & Morales, 2007), that will be detectable by planned low frequency radio telescopes (Wyithe et al., 2009).

Figure 3.— Plots of the luminosity weighted galaxy bias. Upper Panel: the bias is plotted as a function of redshift due to ionising sources from merger induced shocks (solid lines), and stars with (dashed lines). Lower Panel: the fractional change in observed clustering of ionising radiation when compared with the stars-only history.

5. summary

Based on recent high resolution simulations (Dopita et al., 2011), we have quantified the contribution that gravitationally powered fast accretion shocks during galaxy formation can make to the reionzation of hydrogen. We find that ionising radiation from fast accretion shocks represents a negligible contribution to the overall reionization history of hydrogen, leaving the dominant contribution to be provided by stars. This conclusion is consistent with expectations based on observations of cosmic background radiation. The energy released by star formation at high redshift is stored in the cosmic infrared background with nW/m/Str (Hauser & Dwek, 2001). On the other hand, any energy surplus from gravitational shocks is stored in the cosmic soft x-ray background at nW/m/Str. This three orders of magnitude difference is suggestive of the relative efficiency of shocks and star formation in illuminating the Universe at high redshift.

Despite their small contribution to hydrogen reionization, shocks may have observable consequences for studies of the reionization era. As discussed by Miniati et al. (2004), the harder spectrum associated with shocks will lead to a modification of the thermal history. In particular, the reionization of hydrogen by shocks would be accompanied by reionization of singly ionized Helium, thus heating the IGM to levels above those observed at Becker et al. (2011). While this likely rules out reionization by shocks, independently from the hydrogen ionisation photon budget, heating by shocks may still have important consequences for star-formation at high redshift (e.g. Dijkstra et al., 2004).

In addition, we find that because the small contribution from fast accretion shocks is produced in highly biased galaxies, their presence modifies the mean clustering bias of the combined ionising radiation. This modification will likely lead to observable changes in the redshifted 21cm fluctuations from neutral hydrogen during reionization, and so will need to be considered in analyses which aim to use precision measurements of 21cm fluctuations to study the properties of very high redshift galaxies.

Acknowledgments We thank James Bolton, Jonathan Bittner and Lawrence Krauss for helpful discussions during this work. The Centre for All-sky Astrophysics is an Australian Research Council Centre of Excellence, funded by grant CE11E0090. AL was supported in part by NSF grant AST-0907890 and NASA grants NNX08AL43G and NNA09DB30A.

Footnotes

  1. affiliation: School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia
  2. affiliation: ARC Centre of Excellence for All-sky Astrophysics (CAASTRO)
  3. affiliation: Center for Astrophysics and Super Computing, Swinburne University, Hawthorn Vic 3122, Australia
  4. affiliation: ARC Centre of Excellence for All-sky Astrophysics (CAASTRO)
  5. affiliation: Astronomy Department, Harvard University, 60 Garden Street, Cambridge, MA 02138, USA
  6. slugcomment: Submitted to ApJ. Lett.

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