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Champions League GW3 Preview – Wednesday
Posted on October 18, 2017 October 28, 2017 by FightingLikeBeavers
Welcome to another Champions League GW3 preview article for the Wednesday slate. Hopefully things went well for you on Tuesday; unfortunately I moved off Liverpool in cash games on DK once the team news came out (possibly a big mistake) and went heavy on Aubameyang in all formats. I rarely have high exposure to the chalk strikers, so it was frustrating that he bombed yesterday, especially when I had some nice low owned picks in yesterday's article with Promes and Augustin. I had the Yarmolenko + Ronaldo combo higher than Coutinho + Salah and didn't think I could upgrade much with the extra funds, so I went with the former instead in a lot of spots after being on the latter for most of the day.
At first glance the Wednesday slate looks much tougher; Barcelona, Bayern Munich and PSG are all huge favourites and it will be tough to decide between them. It doesn't look like there are enough good value plays to get exposure to each team. It goes without saying that the likes of Mbappe, Dybala, Muller, Kurzawa, Sandro, Alaba etc are all in play, but I will try to focus on some cheaper players here at midfield and defence.
Joshua Kimmich – Bayern Munich, $6700 DK, £19 YH (M)
Kimmich is the top option on the slate, but it's going to be very hard to fit him in with so many expensive forwards in good spots. Kimmich is still taking some set pieces for Bayern and he is averaging 10.1 Crs90 in the Bundesliga this season, which is ridiculous for a defender. If some value opens up once team news is released then it makes sense to try and get him into your lineups.
Dani Alves – PSG, $6200 DK, £17 YH
As with Kimmich it will be tough to fit Alves in on Wednesday, but he is one of the best options on the slate if you can make it work. Alves has strong assist potential, so it's a good idea to add him to PSG stacks in GPPs.
Aleksander Kolarov – Roma, $4900 DK, £14 YH
Kolarov has been taking a lot of the set pieces for Roma this season, with 5 Crs90 and 1.6 Sh90 in Serie A. Chelsea are the favourites here, but I can easily see Roma keeping this one closer than expected. We know that Kolarov will get forward regardless, so even if his clean sheet chances are lower than normal he could still do enough to be worth using.
Georgy Schennikov – CSKA Moscow, $4300 DK, £13 YH
Unfortunately it looks like a lot of the value plays will be in this game, which is never what you want to hear. CSKA play with wingbacks in a 3-5-2 system and are at home to Basel, which is the best matchup Schennikov and Fernandes will get in the competition. I am leaning towards Schennikov, who is slightly cheaper and has crossed more frequently than his teammate in the RPL this season. CSKA also have a couple of CBs that are minimum salary on Yahoo if you need to save even more money (Aleksei Berezutski and Kirill Nababkin).
Alejandro Grimaldo – Benfica, $4000 DK, £14 YH
I'm not in love with Grimaldo here, but he is at least cheap enough to consider on DraftKings. Benfica need to win this game and we could see Jose deploy a more conservative style here away from home, with a point from this fixture likely enough to see United into the knockout phase given the overall strength of the group. That said, Benfica did get destroyed 5-0 by Basel in the last game, so tread lightly here.
Jonathan Silva – Sporting Lisbon, $3500 DK, £10 YH
Taking a fullback on the road in Turin doesn't sound remotely appealing, but Silva is cheap on both sites at least! Silva has averaged 4.1 Crs90, 2.4 TW90 and 1.9 Int90 in his career, so hopefully he can do enough to be worth using at that price tag…
Fade: Davide Zappacosta – Chelsea, $5500 DK, £15 YH
With Moses out I can see people being excited to use Zappacosta, but he is still priced up from last season despite the fact that he hasn't taken any set pieces for Chelsea. I would rather use Kolarov if you are looking for a fullback from this game.
Arjen Robben/James Rodriguez – Bayern Munich, $8500/8400 DK, £22/21 YH
Bayern were soundly beaten in the last game against PSG, but a home fixture against Celtic is just what the doctor ordered, and they should hopefully be able to score multiple goals here. Bayern have a lot of options in their squad so it's not clear who will start, but I would be fine with either of these two in all formats on Wednesday. If both play I would lean towards James in cash (more crosses) and Robben in GPPs (more shots), but it's a close decision either way and I think you can use them both together.
Arturo Vidal – Bayern Munich, $7000 DK, £16 YH
If Vidal starts, he is a great option on Yahoo at just £16. The Chilean midfielder has a safe floor from defensive stats and passes, and he also gets forward and shoots frequently. Vidal was disappointing in the last game, but that was a tough test against PSG and he should bounce back nicely in a much easier matchup. If Tolisso starts instead then he is worth a strong look on DraftKings at $5300. Sebastian Rudy is far less interesting for fantasy purposes as he will sit deep the entire game, but he is minimum salary on DK and £13 on YH.
Alessandro Florenzi – Roma, $5100 DK, £15 YH
The $5k price range on DraftKings is full of OK players in suboptimal situations, so I have gone with Florenzi here given that he is cheaper than most of these guys on Yahoo. Florenzi missed the bulk of last season with a serious knee injury, but he is back fit and started on the wing last weekend against Napoli. Florenzi sometimes takes a couple of set pieces, and he will shoot and cross from open play as well. If Cengiz Under starts instead I am a lot more interested; he is similar to Florenzi for fantasy purposes but is much cheaper at just $3300 on DraftKings and £14 on Yahoo.
Mohamed Elyounoussi – FC Basel – $4700 DK, £14 YH
I do prefer the home side in this one, but Basel did manage their biggest ever win in European competition last time out, whereas CSKA suffered their worst ever home defeat in Europe. Unfortunately for Basel their top scorer Ricky van Wolfswinkel is injured, and whilst Demitri Oberlin did well in the last game I am more interested in Mohamed Elyounoussi. The Norwegian likes to shoot and has 2 goals and 7 assists in 9 league games this season. His playing time is a potential issue, but if Basel can score Elyounoussi is a strong bet to be involved somehow when he is on the field. You have to go back to 2006 for Akinfeev's last clean sheet in the competition, which weirdly came in an away game against Henry and van Persie (two of the current CBs were also playing that day). He has no clean sheets at home in 25 UCL appearances.
Bibras Natcho – CSKA Moscow – $4300 DK, £13 YH
On the other side of the ball is Bibras Natcho, who is a cheap CM that should take a few of the set pieces for CSKA. Natcho looks like a high floor/low ceiling play better suited to cash games than GPPs, but he does chip in with goals and assists every so often (2 goals this season in 12 appearances). Also give a look to teammates Aleksandr Golovin and Georgi Milanov, who are even cheaper than Natcho but are more suited for GPPs.
Fade – Sporting Lisbon Midfielders
Marcos Acuna, Bruno Fernandes and Gelson Martins are all decent fantasy options in some matchups and they are cheap on this slate too, especially on DraftKings. I expect people to be a bit interested in them for that reason, but this is a horrible matchup for Sporting. If you are desperate for more value plays then I would look at the Benfica players instead, although I'm not excited about either team.
Lionel Messi – Barcelona, $12300 DK, £29 YH
Messi just keeps getting better; he has just under 7 Sh90 this season, which is ahead of everyone in Europe. Just play him!
Neymar – PSG, $11500 DK, £26 YH
Ideally we would play Messi and Neymar in the same lineup, but unfortunately there is a salary cap. Neymar is obviously a fantastic play against Anderlecht, but if choosing between the two I would do my best to find the extra salary for Messi. If you can fit both without damaging the rest of your lineup too much, then it makes sense to do so.
Robert Lewandowski – Bayern Munich, $10400 DK, £27 YH
I expect people to use Lewandowski, but he is probably going to end up being a bit underowned here even in an amazing matchup. There are so many good options at forward and people will probably look to pair some of the Bayern midfielders with the likes of Messi and Neymar instead, given that Barca and PSG don't have many players we are interested in outside of the forwards. That said, Messi and Neymar's floors are so high that one goal from them will probably be enough to cancel out 2 goals from Lewandowski, so he is the 3rd choice here in all formats if you aren't worried about ownership.
Edinson Cavani – PSG, $10100 DK, £25 YH
If PSG can get goals against Anderlecht then Cavani is more likely to be the one scoring them instead of Neymar. He isn't in play in cash games for me, but it's always worth giving him a look in GPPs. As with Lewandowski, he will need 2+ goals to be worth using on this slate, so don't roster him unless you agree with the bookies that PSG will score multiple goals here.
Luis Suarez – Barcelona, $9400 DK, £24 YH
Suarez is always slightly overlooked in DFS as a result of being on the same team as Messi. I like pairing both together in GPPs and hoping for a big win from Barca, which is something that people might be reluctant to do because it means you have to fade players from PSG and Bayern.
Vitinho – CSKA Moscow – $6300 DK, £19 YH
Vitinho is still shooting 5.6 times per 90 and he takes some of the set pieces for CSKA as well. He is a great mid-range option with a high floor and nice ceiling in this matchup, although it's hard to use a forward spot on him on Yahoo, where Messi and Neymar don't have multi-position eligibility.
Henry Onyekuru – Anderlecht, $4200 DK, £13 YH
I think the correct move is to spend up at forward on this slate, but if you are desperate for a punt then Onyekuru is my favourite of the bunch despite a nightmare matchup against PSG. Everton need to recall this guy ASAP; he leads his team with 6 goals in 9 appearances this season as a winger and has excellent stats to back it up.
I wasn't sure about Henry Onyekuru this summer due to lack of data, but this is stompy.
Looks like Everton nailed this one. pic.twitter.com/8fFtU61P3H
— Ted Knutson (@mixedknuts) October 17, 2017
Fade: Eden Hazard – Chelsea, $8600 DK, £23 YH
This is probably an obvious one, but I saw a lot of people using Hazard last weekend so I thought I would mention that he is overrated for fantasy purposes and has a tough matchup on Wednesday. It's probably safe to fade him on most slates, especially when he is priced above $8k on DK and £20 on YH.
Thanks for reading! Please share the article if you enjoyed it, and look out for the EPL preview later in the week. Good luck in your contests today!
Champions League GW3 Preview – Tuesday
EPL Gameweek 9 Preview


A child dedication is a public statement by parents who want to thank God for their child, entrust them to him, and commit themselves to raise their child to follow God. Parents who dedicate their child to God are acknowledging their role as a loving steward of His gift, and are asking for God's wisdom and direction in parenting. Dedication is a biblical parenting principle found in both the Old & New Testaments.
to celebrate the gift they are to their families and to the church.
to give parents the opportunity to publicly commit to raise their child to be a follower of Christ.
to give family members, friends and church family the opportunity to express their commitment to support the parents.
During the dedication ceremony, you will stand before the church with your child. The ceremony includes a charge for parents, a commitment from the church to be a source of support for your family, and a prayer for your family. Parents may invite as many friends and family members as they like to celebrate the occasion. A church photographer takes photos during each dedication. We'll share a link to a private gallery so you can download pictures to cherish and share.
If you are interested in dedicating your child, fill out the form below, and we'll get in touch with you soon.


Cuban collars were one of the defining trends to emerge from the SS18 runway, and this Bottega Veneta navy-blue iteration is a refined take. It's crafted in Italy from lightweight cotton and shaped for a slim fit with button-fastening cuffs, jet pockets at the chest, and side hem slits to ease the waist. Runs true to size.


How can I initiate a Panorama Scan?
The App will initiate Panorama scan at the beginning of every search, provided the signal quality is good enough.
Panorama scan is done once per search, as triggered by the App. If you want to repeat the Panorama Scan, cancel the search by tapping on the < back arrow in the upper left corner of the screen. This will exit the current search and you can tap the item in your list to start a new Panorama Scan.


As the Economist points out, and as I personally can confirm (since I am constantly having to alter and update my excel sheets), Greek government statistics are notoriously unreliable - indeed, I would say that along with the Bulgarians the Greek statistical agencies are the joint worst in the entire EU. But rarely can the numbers have seemed more erratic and subject to such sharp revisions as they have been in recent months. Following the election of the new government in October (who obviously decided to get as much of the bad news out of the way as quickly as possible) we suddenly learnt that far from having been being "spared" the worst of the global economic contraction, the Greek economy in fact entered a period of negative growth in the first quarter of 2009 (shrinking by 0.5% on the quarter instead of growing by 0.3% as the stats office had previously "estimated") wherein it is has subsequently remained. And of course given the size of the correction the Greek economy is now entering it is likely the economy will stay in this mode for some considerable time to come. And as if that wasn't shocking enough, the forecast for this year's budget deficit more than doubled overnight, from 6% to 12.7% of GDP.
But this problem has a history, and quite a long one. The 2008 general government deficit as notified by the Greek authorities on 21 October to the EU Commission was put at 7.75% of GDP (up by 2.75% percentage points from the earlier estimate of 5% sent in April). But even this version was found to be deficient, and Eurostat specifically expressed reservations about the figures, largely due to the inclusion of a one-off capital expenditure item of around 1.25 percentage points of GDP mainly connected with public hospitals arrears to paramedical and pharmaceutical suppliers.
Moving forward, the official public deficit estimate for 2009 stands at 12.75% of GDP, as compared with the agreed budgetary target of 3.7% of GDP included in the January 2009 update of the EU stability programme. The EU Commission also estimate (in their November forecast) that under a no-policy-change scenario assumption and on the back of the discontinuation of the one-off expenditure item which has in fact been repeated in 2009, the headline deficit is set to remain above 12% of GDP in 2010 and then to continue to increase, exceeding 12.75% of GDP by 2011.
Both falling revenue-to-GDP and rising expenditure-to-GDP ratios will contribute to the anticipated fiscal deterioration. The economic downturn, coupled with high budget deficits, is expected to push debt higher, from 113% of GDP in 2009 to over 135% of GDP by 2011, thus weakening the already fragile sustainability of Greek public finances in the long term.
i) On the basis of the data notified by the Greek authorities to the EU Commission in October 2008 the Commission, on 18 February 2009, adopted an Article 104(3) report for Greece on 18 February 2009. Subsequently, and in accordance with Article 104(4), the Economic and Financial Committee formulated an opinion on the Commission report on 27 February 2009.
ii) On 24 March 2009 the Commission, having taken into account its report under Article 104(3) and the opinion of the Economic and Financial Committee under Article 104(4), addressed to the Council, in accordance with Article 104(5), its opinion that an excessive deficit existed in Greece.
iii) On 27 April 2009, following a recommendation by the Commission, the Council decided, in accordance with Article 104(6) of the Treaty, that an excessive deficit existed in Greece in 2007 and 2008. At the same time, and in accordance with Article 104(7) the Council, addressed recommendations to Greece to put an end to the present excessive deficit situation by 2010, by bringing the general government deficit below 3% of GDP in a credible and sustainable manner.
(iv) continue efforts to improve the collection and processing of statistical data and in particular general government data".
The Council established a deadline of 27 October 2009 for Greece to take effective action and to specify the measures deemed sufficient to ensure adequate progress towards the correction of the excessive deficit by 2010.
"is also crucial for recovering competitiveness losses and addressing the existing external imbalances. To this end, the Greek authorities should implement permanent measures, including the recently announced measures to control current primary expenditure, including public wages; and urgently implement structural reforms. In particular, the Greek authorities should ensure that fiscal consolidation measures are also geared towards enhancing the quality of public finances within the framework of a comprehensive reform programme, while strengthening the binding nature of its budgetary framework, improving monitoring of the budget execution and swiftly implementing policies to further reforming the tax administration".
That is to say, in depth reforms, and not simply cosmetics are called for.
Following their 11 November ruling the European Commission sent a large delegation of experts to Athens to assess the situation for themselves on the spot, amid strong rumours circulating Athens that the EU was not willing to approve the government's 2010 9.4% of GDP deficit first draft. In reponse to the evident EU reservations the Greek finance ministry circulated on November 20 what they called their revised "final" 2010 budget, with a projected deficit of 9.1% of GDP. But the patience of the EU had by that time become exhausted, and on November 30 2009 the Council of the European Union resolved Greece has not taken effective action in response to the Council Recommendation of 27 April 2009 within the period laid down in that Recommendation.
So we now move on to the next stage of the process, within which the Greek government will again be given a period of time (two months) to advance convincing measures, and another period of time (a minimum of four months and a maximum 16 months) to start to take the required measures (in 4 months) and give unequivocal evidence that they are credibly implementing them (within a maximum of 16, but my reading is that this process could be drawn to a close at any point before then if there were sufficient effidence that agreed steps were not being taken).
If a participating Member State fails to act in compliance with the successive decisions of the Council in accordance with Article 104(7) and (9) of the Treaty, the decision of the Council to impose sanctions, in accordance with Article 104(11), shall be taken as a rule within sixteen months of the reporting dates established in Article 4(2) and (3) of Regulation (EC) No 3605/93. In case Article 3(5) or 5(2) of this Regulation is applied, the sixteen-month deadline is amended accordingly. An expedited procedure shall be used in the case of a deliberately planned deficit which the Council decides is excessive.
Evidently Greek politicians have come under continuous pressure in recent weeks, and Finance Minister George Papaconstantinou has been doing his best to convince the outside world that he means business. In a statement of intent issued at the end of last week he declared that "the fiscal priority for 2010 is a significant reduction of the deficit as a percentage of GDP and containing the increase in public debt which undermine the immediate and long term future of our country."
Fine words these, although the actual details available leave us far from clear about the precise nature of the deep seated reforms the Greek government is actually contemplating, or the extent to which they realise that significant wage and price deflation will be necessary if the are to restore competitiveness.
The decision to bring forward by several weeks the announcement of the government economic strategy underlines the new sense of urgency which now surrounds policymaking, and comes in response to a 10-day battering of Greek bonds and falling share prices on the Athens bourse. Yet it seems the investing community - the key players in the Greek democratic system now - remain unconvinced, since the yield on the Greek 10-year bond climbed on Tuesday morning 24 basis points to hit 5.70 percent, while the 30-year yield rose 15 basis points to 6.23 percent. Thus the extra yield, or spread, investors demand to hold 10-year Greek bonds instead of German bund equivalents widened 24 basis point to reach 253 basis points, the highest level since April 2.
Mr Papaconstantinou, has effectively been given the objective of reducing the deficit by 3.3% of GDP next year. The reduction is likely to come mainly from increasing revenues (asking businesses, for example, to pay a one-off extra 10% tax on last year's profits) rather than from spending cuts, although this approach is unlikely to satisfy the Commission, who want long term structural shifts.
In an attempt to head off just this kind of pressure from the European Commission Mr Papaconstantinou has stated he will immediately tackle the structural reforms issue. Among ideas being currently touted are steadily raising the retirement age to 65 for women, opening up a range of "closed" services (from notaries to taxi drivers) to improve competitiveness and introducing five-year rolling budgets for ministries (to curb pre-election spending).
Yet despite the fact that Greek Prime Minister Papandreou promised on Monday that his government will, over the next three months, take all those decisions which "have not been taken for decades", plenty of grounds for scepticism still remain. Mr Papandreou's said many choices will be "painful," though he pledged to protect poorer and middle-income Greeks. Unfortunately the speech was short on real detail, while the little we have is not impressive.
Talk about obtaining capital to reduce debt via securitization of income from the state's tax on major property holdings sounds incredibly like simply taking some of the debt off the balance sheet to me, but I'm sure Eurostat will have their say when the time comes.
Whether the announced measures will be enough to satisfy the EU Commission remains unclear. In particular Mr Papandreou's reluctance to follow the example of Ireland, which announced a 4 billion euro correction of its 2010 budget with public sector salary cuts of five to 15%, has had a lukewarm reception from analysts and commentators alike. Greece faces mounting pressure from markets and its European partners to follow Ireland and adopt stronger fiscal measures such as a public sector pay freeze, a ban on civil service hiring and hikes in indirect taxes, to restore competitiveness as well as bring the deficit under control.
I doubt the sort of measures we are hearing about will do that much to convince either Standard & Poor's or Moody's Investors Service, both of whom are currently considering cutting the Greek Sovereign rating. Moody's rates Greece at A1, well above the S&P level of A- (different classificatory systems) and seems willing to give the government time to address the situation.
But a cut by S&P, which is worried both about the amount of deficit reduction and how it is being achieved, would take their rating down into the triple-B category, where Fitch currently have it. ECB board member Lorenzo Bini Smaghi issued a scarcely veiled warning on Monday when he pointed out Greece had just one year to ensure it still had a single-A rating if it wanted its debt to qualify as collateral for central bank lending operations.
The whole discourse from the Greek political leadership gives us an indication of how far one discourse has moved away from the other. While Athens fiddles, the EU Commission continues to demand serious and sustained structural reform. I fear even the most recent of Greek governments only sees a short term problem of deficit reduction, and herein lies the danger, not only for the citizens of Greece themselves, but for all of us who live in the eurozone. This time the situation is for real, and I only wish I could believe that Mr Papaconstantinou and Mr Papandreou were aware of that fact. From this point on there is nowhere left to hide.


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Guatemala Congress Opens Door for Prosecution of President
September 1, 2015 10:00 PM
Demonstrators wave Guatemalan flags as they celebrate that Congress voted to withdraw President Otto Perez Molina's immunity from prosecution, in Guatemala City, Tuesday, Sep. 1, 2015.
GUATEMALA CITY - Guatemala's Congress lifted President Otto Perez Molina's immunity of office Tuesday, opening him up to possible prosecution in a widening customs corruption scandal that has rocked his administration and the country's political system.
With 132 of 158 lawmakers approving the measure, prosecutors now can file criminal charges against Perez Molina just like any other citizen, and a judge can order his detention.
About 200 people outside the capitol cheered and set off firecrackers as news of the vote reached them, an echo of earlier massive street protests calling for his removal from office.
"Excellent! It is a step forward for Guatemala," said Gerardo Corzo, a 71-year-old retiree.
Perez Molina, 64, has said he is innocent of corruption and has vowed to face the legal process against him.
"The president is aware of the new scenario, which was not the most desirable but was very probable," his spokesman, Jorge Ortega, told The Associated Press. "He has said he will be very respectful and submit himself to the rule of law."
Those voting against Perez Molina included members of his own ruling party.
"The party gave us permission to vote and withdraw the president's immunity," lawmaker Luis Fernandez Chenal said. "He who owes nothing, fears nothing."
The customs scandal already has claimed the job of Perez Molina's former Vice President Roxana Baldetti, who is in jail awaiting trial on accusations that she accepted millions in bribes in return for letting others avoid paying import duties. A number of Cabinet officials also have left office.
Protesters are demanding not only that Perez Molina step down but that Sunday's presidential elections be postponed. He has refused to do so.
Earlier in the day, civilians formed a wall of bodies to let lawmakers into Congress, protecting them from dozens of presidential loyalists who had blocked access to the building since the morning in a bid to delay the proceedings.
Interior Department Vice Minister Elmer Sosa had arrived with riot police to "guarantee the safety of protesters and Congress," and lawmakers were finally able to go inside and begin the session.
"It was impressive that the people themselves came and created a human chain and a path so we could enter," said opposition legislator Leonel Lira.
Government agents have had to step in on several occasions to keep backers and critics of the president away from each other.
In the wake of the fraud scandal, major government offices and businesses, the power base in Guatemala, have urged him to reconsider his resolve to stay, including the government comptrollers' office. Even Guatemala's national Council of Bishops issued a statement urging him to resign.
"This decision demonstrates that the people and their collective actions can get results, but this is just the beginning," said activist Byron Garon. "Now we want him and his vice president to be tried and convicted, and for them to give back to Guatemala all that they stole."
Guatemalan Prosecutors Urge President to Resign Amid Scandal
Pressure grew Thursday on Guatemalan President Otto Perez Molina to resign as business and government offices closed, protesters marched by the thousands and the attorney general's office urged him to step down "to prevent ungovernability that could destabilize the nation."The government comptrollers' office also issued a statement saying Perez Molina, whose government has been shaken by corruption scandals, should resign "to avoid greater social unrest that could have…
Guatemala Corruption Scandal Shows Positive Signs for Change
The decisive steps taken against politicians caught in Guatemala's growing corruption scandal are signs that positive change could be coming to a poor country long pillaged by its elite, analysts said. Former Vice President Roxana Baldetti was ordered to stand trial on Tuesday, while Guatemala's Supreme Court took the first step in allowing impeachment proceedings against President Otto Perez Molina in a fraud case that has pushed the country into political crisis…

redpajama_set_name: RedPajamaCommonCrawl


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\section{Introduction}

High-redshift galaxy formation and evolution at $z \geq 6$ have been amongst the primary foci of extragalactic astrophysics for the past two decades. In particular, the Cosmic Dawn era, beginning just a few hundred million years after the Big Bang, is a crucial time period that saw the birth of the first stars. The Cosmic Dawn not only ended the Dark Ages, but these early populations of stars were the progenitors of all the structure that we see today and would later become the drivers of reionization, the last global phase transition of the Universe.

Despite the challenge of probing such a distant Universe, in recent years there has been substantial improvement in our census of luminous objects at $z \sim 6$--$8$ \citep{mclure_new_2013, schenker_uv_2013, schmidt_luminosity_2014, bouwens_uv_2015, steven_l_finkelstein_evolution_2015, atek_are_2015, bowler_unveiling_2017, livermore_directly_2017}, which has allowed us to develop a good understanding of bright galaxies at these times.
Higher redshifts are just now beginning to be explored by JWST, but the situation is not yet clear \citep{oesch_remarkably_2016, Donnan2022, Castellano2022,  Harikane2022}.

Nonetheless, there have been numerous recent efforts to model various aspects of galaxy evolution in this era (\citealt{furlanetto_minimalist_2017}; \citealt{furlanetto_bursty_2021}; \citealt{moster_emerge_2018}; \citealt{popping_dust_2017}; also see \citet{dayal_early_2018} for an overview of the physics of early galaxy formation and the types of theoretical tools used to model it). In general, these models show that the properties of observed galaxies at $6 \la z \la 8$ can be understood in a similar fashion to galaxies at later times. For example, models in which stellar feedback regulates star formation can quite successfully explain the observed luminosity function \citep{mason_galaxy_2015, furlanetto_minimalist_2017, mirocha_global_2017}.

One particularly critical aspect of this feedback is the presence of ``winds" driven by supernovae. Examples of cosmological simulations incorporating galactic winds include \citet{scannapieco_highredshift_2001, springel_cosmological_2003, theuns_galactic_2002, suresh_impact_2015, fierlinger_stellar_2016, nelson_first_2019}, while those using analytic or semi-analytic methods include \citet{nath_enrichment_1997, ferrara_mixing_2000, madau_early_2001, hayward_how_2017}). Many of these studies show how such supernova explosions can help regulate star formation inside galaxies. But beyond the host galaxy, these winds can also influence the surrounding intergalactic medium (IGM) by ejecting material into it. A significant consequence of this in the early Universe is metal enrichment over large volumes of the Universe -- including gas that will later be accreted onto growing galaxies. Though there are many other processes that may have contributed to widespread enrichment (e.g. quasar winds, radiation pressure-driven dust outflows), winds from star-forming galaxies are the clearest culprit for polluting the large volumes of the Universe that we see today \citep{ferrara_mixing_2000,madau_early_2001, furlanetto_metal_2003}.

Existing observations have shown that metals are relatively widespread by $z \sim 3$ \citep{crane_ionization_1995, cowie_heavy-element_1998, songaila_metal_1996, ellison_enrichment_2000, adelberger_galaxies_2003, schaye_detection_2000}, at least outside of underdense voids in the galaxy distribution (where the observations cannot yet probe). More recent observations revealed metal absorbers existing by $z \sim 5-6$. For example,  \citet{ryan-weber_intergalactic_2006} and \citet{becker_discovery_2006} reported absorption of [CIV] in $z \sim 5$ and [OI] in $z \sim 6$ QSO systems respectively, while  \citet{sparre_metallicity_2014} discovered Fe II fine line structures in the afterglow of GRBs at $z \sim 5$. However, only a handful of absorbers are known, and it is difficult to know how widespread the metal enrichment was during this era.

Meanwhile, it is clear that high-$z$ galaxies are themselves significantly enriched: \citet{capak_galaxies_2015} detected [CII] gas emission in star forming galaxies at $z \sim 5-6$, and  \citet{faisst_rest-uv_2016} found that the average galaxy population at $z \sim 5$ has a metallicity comparable to those at $z \sim 3.5$. Therefore, it is becoming increasingly evident that metal enrichment started in the very early stages of the bright Universe, though its precise timeline remain unclear. A crucial uncertainty lies in the phase of galaxy evolution at which metals are ejected -- whether it begins with the birth and death of Population III stars or as part of ``normal" feedback regulation at later times.

Interpretation of the handful of metal-line measurements presents an important challenge. While the overall metal production rate depends mostly on the star formation rate (but also on stellar models), the extent of metal enrichment
is necessary to understand the impact of these metals. There have been many attempts to estimate this factor in the past, but early calculations were not based on realistic galaxy populations, which likely resulted in an overestimate of the filling factor (or the fraction of the Universe enriched by metals at a given time; e.g.,  \citealt{madau_early_2001, furlanetto_metal_2003}). More recent calculations have typically used numerical simulations, so it is difficult to assess the uncertainties associated with them (e.g., \citealt{jaacks_baseline_2018}).  The recent launch of the James Webb Space Telescope, which will make extraordinary progress in understanding metal lines during this era \citep{mason_galaxy_2015, yung_semi-analytic_2019-1, vogelsberger_high-redshift_2020}, makes modeling the process very timely.

The extent of metal enrichment is important both for understanding galaxy evolution (as the IGM gas provides the fuel for later star formation) and also for probes of reionization and the radiation fields at early times. For galaxy evolution, the extent of the winds can be used to constrain models of stellar feedback. The  evolution of the radiation background determines the ionization state of the metals, so in principle quasar metal lines are powerful probes of reionization \citep{oh_probing_2002, hennawi_probing_2021}, but only if the distribution of metals themselves is known as well (or at least can be jointly measured).
In this paper, we address this question by pairing a simple galaxy evolution model with a simple wind model. In particular, we use the minimalist model for feedback regulation described in \citet{furlanetto_minimalist_2017} to calibrate the galaxy parameters to reproduce observed luminosity functions.  We then follow the resulting winds using the model of \citet{furlanetto_metal_2003} (which is in turn based upon \citealt{tegmark_late_1993}). Although such a simplistic model cannot follow star formation or wind propagation in as detailed a manner as numerical simulations, its flexibility allows for a broader study of the many uncertainties in early galaxy formation. We explore how the overall volume filling factor and absorption line statistics at $z \gtrsim 4$ depend upon assumptions about star-forming galaxies during this era. As it remains difficult to detect metals at $z \geq 6$, we also consider distortions to the cosmic microwave background (CMB) through the Sunyaev–Zel'dovich effect as an alternative observable for such winds \citep{oh_sunyaev-zeldovich_2003}.

This paper is organized as follows. In Section \ref{sec:galaxy_model}, we introduce our galaxy evolution model, including a description of the underlying dark matter halo population and our prescription for feedback regulation. In Section \ref{sec:wind_model}, we present our wind model and show its solutions for individual galactic winds. We then discuss the implications of our model for the extent of metal enrichment, as well as potential probes of it, in Section \ref{sec:results}. Finally, we conclude in Section \ref{sec:discussion}.

In accordance with the Planck Collaboration XIII results \citep{planck_collaboration_planck_2016}, we use the following cosmological parameters: $\Omega_{m} = 0.308$, $\Omega_{\Lambda} = 0.692$, $\Omega_{b} = 0.0484$, and $h = 0.678$. For astrophysical constants, we use those found in Section~2 of \citet{beringer_review_2012}.

\section{A model for star-forming galaxies} \label{sec:galaxy_model}

In this section, we introduce our simple galaxy model. We refer the reader to \citet{furlanetto_minimalist_2017} for more details.

\subsection{Dark matter haloes} \label{ssec: darkmatterhalos}

We define the co-moving number density of dark matter haloes in the mass range
$\left(M_h, M_h+\textup{d}M_h\right)$ at redshift $z$ as $n_h\left(M_h,z\right)\textup{d} M_h$. By convention, we write
\begin{equation}
    n_h\left(M_h,z\right) = f\left(\sigma\right)\frac{\bar{\rho}}{M_h}\frac{d \ln\left(1/\sigma\right)}{\textup{d}M_h}
\end{equation}
where $\bar{\rho}$ is the comoving average matter density, $\sigma\left(M\right)$ is the linear rms fluctuation of the matter density field smoothed on a scale $M$,
and $f\left(\sigma\right)$ is a dimensionless function taken from a fit to high-$z$ cosmological simulations \citep{trac_scorch_2015}:
\begin{equation}
    f\left(\sigma\right) = 0.150\left[1+\left(\frac{\sigma}{2.54}\right)^{-1.36}\right]e^{-{1.14}/{\sigma^2}}.
\end{equation}

Our galaxy model requires the accretion rate onto dark matter haloes. We use the method described in \citet{furlanetto_minimalist_2017}, which assumes that accretion proceeds smoothly without mergers. In analogy with abundance matching \citep{vale_linking_2004}, we demand that the overall number density of the haloes remains constant, with each halo growing in order to maintain the underlying mass function. We assume that the haloes remain in the same relative ordering in mass as well. In other words,  at any two redshifts $z_1$ and $z_2$,
\begin{equation}
    \int_{M_1}^{\infty}\textup{d}M\ n_h\left(M\vert z_1\right) = \int_{M_2}^{\infty}\textup{d}M\ n_h\left(M\vert z_2\right),
\end{equation}
where $M_1$ and $M_2$ are the respective masses of a given halo at the two redshifts. The accretion rate, $\dot{M}_h$, is then obtained by requiring that this relation be satisfied at all redshifts and all halo masses.

We must also specify the range of halo masses allowed to form stars (and hence drive winds).  This threshold is determined by two conditions: (1) the halo must exceed the ``filter mass" \citep{gnedin_probing_1998} above which baryons can accrete and (2) the halo gas must cool efficiently after virialization. Motivated by the second condition, for most of our models we take this minimum mass to correspond to a virial temperature of $10^4$~K, above which atomic cooling becomes efficient. We consider winds from haloes below this mass threshold in Section~\ref{ssec: burst}.

\subsection{Star formation and feedback}

Next, we describe star formation and feedback in the minimalist galaxy formation model (referred to as the ``normal" model hereafter) as well as a ``bursty" model for lower mass halos.

The fundamental idea of this model is the assumption that stellar feedback controls the star formation rate (SFR) of each galaxy. First, we balance the rate at which baryons are accreted onto the halo, $\dot{m}_b$, with the SFR, $\dot{m}_{\star}$, and the rate of at which baryons are expelled through processes such as radiation pressure and supernovae, $\dot{m}_w$:
\begin{equation}
    \dot{m}_b = \dot{m}_{\star} + \dot{m}_w.
\end{equation}
Because the feedback is driven by star formation, we then assume $\dot{m}_w = \eta\dot{m}_{\star}$. The constant of proportionality, $\eta$, is in general a function of halo mass and redshift and represents the strength of stellar feedback.

The star formation efficiency (SFE) is defined as the fraction of the accreted baryons which form stars, $f_{\star} = \dot{m}_{\star}/\dot{m}_b$. Thus,
\begin{equation}
\label{eqn:fstar}
    f_{\star} = \frac{1}{1+\eta(M_h,z)}.
\end{equation}

However, to match the observed star formation efficiencies at large halo masses, quenching processes which suppress accretion are often invoked. Although such effects do not significantly affect our results, we will incorporate virial shock heating as one example, suppressing accretion by a factor $f_{\rm shock}$ following \citet{faucher-giguere_baryonic_2011}. We also impose an upper limit, $f_{\star,\rm max}$, on the star formation efficiency. Incorporating both these effects in such a way as to maintain continuity, we obtain
\begin{equation}
    f_{\star} = \frac{f_{\rm shock}}{f_{\star,\rm max}^{-1} + \eta(M_h,z)}. \label{fstar}
\end{equation}

We are interested in considering a wide range of metal enrichment scenarios so include two distinct models of feedback regulation. Firstly, the ``energy-regulated'' model is derived by balancing a fixed fraction of the supernova energy (the portion in a kinetic form) with the kinetic energy of the gas that is lifted out as a wind:
\begin{equation}
    \frac{1}{2}\dot{m}_w v_{\rm esc}^2 = \dot{m}_{\star}\epsilon_k\omega_{\rm SN},
\end{equation}
where $v_{\rm esc}$ is the halo escape velocity and $\omega_{\rm SN} = 10^{49}$erg/$M_{\odot}$ is the supernova energy produced per unit mass of star formation. This results in
\begin{equation} \label{eqn:eta}
    \eta = C \left(\frac{10^{11.5}M_{\odot}}{M_h}\right)^{\xi}\left(\frac{9}{1+z}\right)^{\sigma},
\end{equation}
with exponents $\xi = \frac{2}{3}$ and $\sigma = 1$. However, due to radiative cooling or other processes that occur in high density and temperature environments, feedback can in practice be much less efficient. Therefore, we also consider a  ``momentum-regulated'' case, balancing the momentum of the supernova blastwaves with the gas escaping the halo, done in a similar way to energy conservation above. This yields $\xi = \frac{1}{3}$ and $\sigma = \frac{1}{2}$. These scenarios provide helpful intuition for the parameters in equation~(\ref{eqn:eta}); given the difficulty of modeling the interaction of stellar feedback with the interstellar medium, we do not try to do better here (though see \citealt{hayward_how_2017, furlanetto_quasi-equilibrium_2021}). The normalization constant $C$ and the parameter $f_{\star,\rm max}$ are set by comparison to observations. For completeness, we also consider a model with a constant star formation efficiency in all halos ($\xi=0$ and $\sigma=0$), which matches the assumptions in many early treatments of metal enrichment.
The parameters used for each model are summarized in Table~\ref{tab:Table 1}. All of these feedback-regulated models have an SFE that increases with halo mass in faint galaxies, aside from the constant model.

We also consider a ``bursty'' model, motivated by the results of \citet{furlanetto_bursty_2021}, who showed that incorporating a time delay between star formation and supernova feedback results in oscillations (i.e. repeated ``bursts") of the SFR, particularly in lower mass halos. This moderates the dependence of the average SFE on halo mass, boosting the star formation rate of small galaxies. It has little effect on massive galaxies, because eventually feedback is no longer able to eject all of the halo gas, at which point the SFR transitions to a smooth function. To mimic this case, the bursty model imposes a minimum SFE value $f_{\star,\rm min}$ = 0.03.

We emphasize that all of these parameter choices  provide reasonable fits to the observed luminosity functions (LFs) at $z \sim 6$--8, (see Fig.~3 of \citealt{furlanetto_minimalist_2017} and Fig.~8 of \citealt{furlanetto_bursty_2021}).  Importantly, the minimum SFE in the bursty model only affects faint galaxies that have not yet been observed. We emphasize that the model with constant SFE, $f_\star = 0.1$ does not produce realistic galaxy populations, at least at $z \sim 6$--8, but we include it anyway for comparison with earlier results.

 \begin{table}
     \centering
     \begin{tabular}{|c c c c c|}
          \hline \hline
          Model & $C$ & $\xi$ & $\sigma$ & $f*_{\rm max}^{-1}$ \\
          \hline
          Energy-reg & $10\epsilon_k$ & $\frac{2}{3}$ & $1$ & $0.1$ \\
          Momentum-reg & $\epsilon_p$ & $\frac{1}{3}$ & $\frac{1}{2}$ & $0.2$ \\
          \hline
     \end{tabular}
     \caption{SFE parameters for normal galaxy models. We set $\epsilon_k = 0.1$ and $\epsilon_p = 5$ which reproduces observed luminosity functions.}
     \label{tab:Table 1}
 \end{table}

\subsection{Metal production} \label{ssec:MP}

We follow \citet{furlanetto_metal_2003} to estimate the metal production rate of our galaxies. The mass of a metal $i$ produced by a galaxy is
\begin{equation} \label{eqn:M_metal}
    M_{i} = Y_i n_{\rm SN}\frac{\Omega_b}{\Omega_m}X_\star M_h
\end{equation}
where $Y_i$ is the average yield of the element $i$ produced per Type II supernova, $n_{\rm SN} \approx 10^{-2} M_{\odot}^{-1}$ is the number of supernovae per unit mass of star formation, and $X_\star$ is the time-averaged SFE for the galaxy in question, which is slightly smaller than the instantaneous SFE $f_\star$ in our models (see \citealt{furlanetto_minimalist_2017}).

We then estimate the total column density of the metal along a line of sight through the wind bubble via
\begin{equation}
    N_i \approx \frac{M_{i}}{m_i}\frac{1}{4\pi R^2}
\end{equation}
where $m_i = A_i u$ is the mass of a metal atom. Note that this is simply an estimate of the characteristic column density; even if the metals are distributed uniformly, the column density would vary with the impact parameter. We emphasize that this equation assumes a roughly uniform distribution of metals; if the metals were clumped, fewer lines of sight would encounter metals, but those that did would have stronger absorption.

For an unsaturated line, the equivalent width is given by
\begin{multline} \label{eqn:EW}
    W \approx 0.8 \mbox{\AA} \, \left(\frac{f_{\rm osc}}{0.05}\frac{Y_x}{0.5 M_{\odot}}\frac{16}{A_x}\right)\left(\frac{\lambda_m}{1300\mathring{A}}\right)^2\\\left(\frac{X*}{0.1}\frac{\omega_{\rm SN}}{10^{51}\rm{ergs}/126M_{\odot}}\frac{\Omega_b/\Omega_m}{0.05/0.3}\frac{M_h}{10^9 M_{\odot}}\right)\left(\frac{0.02 \rm Mpc}{R}\right)^2\left(\frac{1+z}{10}\right)
\end{multline}
where $\lambda_{m}$ is the wavelength of the transition and $f_{\rm osc}$ is its oscillator strength.

In Section \ref{ssec: dndx}, we compare predictions for the OI and CIV transitions to observations. We choose these because they are amongst the very few which have been observed at $z \sim 6$ and because OI is a strong line in mostly neutral gas while CIV is a strong line in mostly ionized gas. For OI, we take $Y = 0.5 \ M_{\odot}$,  $f_{\rm osc} = 0.04887$,  $\lambda_{m} = 1302.2$ \AA; while for CIV we take $Y = 0.1 \  M_{\odot}$,  $f_{\rm osc} = 0.1908$,  $\lambda_{m} = 1548.2$ \AA \space (CIV is a doublet, so these values correspond to the stronger transition, as reported in \citet{becker_high-redshift_2009} to which we compare in Section~\ref{ssec: dndx}). These average yields are those produced per Type II supernova from \citet{woosley_evolution_1995} which depends on the energy of the supernova. Following \citet{furlanetto_metal_2003}, we take the values from the lower energy model and neglect chemical enrichment within the galaxies. These approximations are reasonable as uncertainties arising from them are likely less significant than those from other simplifications in our model.

\subsection{Pop~III star formation} \label{ssec: burst}

So far, we have focused on ``normal'' galaxies during the Cosmic Dawn. But many models expect another set of star-forming haloes hosting Population III stars. We thus also consider potential contributions from smaller haloes hosting these exotic stars.

The criteria to form Pop~III stars are complex: haloes must be massive enough to accrete baryons and then form H$_2$, which is the primary coolant in this regime. But H$_2$ is fragile and is destroyed by ultraviolet photons -- which are themselves produced by Pop~III stars. Models of this era show that the minimum halo mass can therefore have a complex evolutionary history (e.g., \citealt{jaacks_baseline_2018, mebane_persistence_2018}). We therefore consider two simple models. For one, we take a minimum virial temperature of $1000$~K for this population (``constant $T_{\min}$ model''). This is a few times smaller than the ``plateau'' found in the minimum halo in \citet{mebane_persistence_2018}, so it provides an optimistic estimate of the enrichment. In addition, we also consider a prescription for $T_{\rm min}$ with a stronger time dependence inspired by an updated version of the \citet{mebane_persistence_2018} model (S.~Hegde, private communication). This includes improved estimates of halo self-shielding and dark matter--baryon streaming (as studied recently in \citealt{kulkarni_critical_2021}).  We find the following simple fit roughly matches the new model:
\begin{equation}
    M_{\rm h0} = 0.916 e^{-0.568\left(z-38.3\right)} + 1.21 \times 10^6 \ M_\odot
\end{equation}

Such low mass halos likely have brief periods of rapid star formation that then shut off, until they are able to accrete enough mass to retain gas and begin normal star formation (e.g,  \citealt{abel_formation_2002, bromm_formation_2002}).
Although each such halo will form only a small mass of stars, the haloes are so numerous that their contribution to the overall filling factor of winds can nevertheless be substantial (especially at very early times). We model these sources as hosting a \emph{single} burst of  star formation, producing a specified total stellar mass $M_* \sim 100$--$1000 \ M_{\odot}$, after which star formation halts completely. This is in contrast to our fiducial model, in which star formation continues over long time periods. We also assume that Pop~III stars have more energetic supernovae than those in normal haloes, so we take $\omega_{\rm SN} = 10^{50}$~erg/$M_\odot$, about an order of magnitude larger.
We assume the  metal yields are the same as our fiducial model, although these could be underestimates if pair-instability supernovae dominate. Our treatment of Population~III star-forming haloes is thus very approximate, but it should provide some intuition for the contribution of these kinds of haloes.

\section{Wind-driven bubbles}\label{sec:wind_model}

\subsection{Wind physics}

In this section, we provide an overview of our wind expansion model, which is a simplified version of \citet{furlanetto_metal_2003} (which is itself an implementation of \citealt{tegmark_late_1993}). In the model, winds are driven by supernovae inside galaxies and commence as soon as star formation does. Once a wind forms, it expands and sweeps up the ambient IGM. While a fraction of the material forms the shell, the remaining gas enters the hot, rarefied interior, whose thermal pressure helps expand the shell. Using the thin-shell approximation and assuming spherical symmetry, the following system of equations can be used to describe the evolution of an individual wind bubble:
\begin{align}
    M_d &= M_h + \frac{4\pi}{3}\Bar{\rho}_d^0\left(1+z\right)^3R^3 \\
    \ddot{R} &= \frac{4\pi R^2}{M_s}P-\frac{G}{R^2}\left(M_d+\frac{M_s}{2}\right)-\frac{\dot{M}_s}{M_s}\left(\dot{R}-HR\right)\\
    \dot{P} &= \frac{L}{2\pi R^3}-5P\frac{\dot{R}}{R} \\
    \dot{M}_s &= \left.
    \begin{cases} 0 ,& \text{for } \dot{R} \leq HR \\
    4 \pi R^2 \Bar{\rho}_b^0\left(1+z\right)^3\left(\dot{R}-HR\right), & \text{for } \dot{R} > HR
    \end{cases}
    \right.
\end{align}
where $M_h$ is the mass of the halo in which the source galaxy resides, $M_d$ is the dark matter mass enclosed by the wind shell (with mass $M_s$), $R$ is the radius of the wind bubble, $P$ is the pressure of the bubble interior, and $L$ is the rate energy is injected to drive the wind. $\Bar{\rho}_d^0$ and $\Bar{\rho}_b^0$ are the average dark matter and baryon densities at $z = 0$, respectively. Furthermore, the accretion rate, $\dot{M}_h$, is obtained from abundance matching as described in Section \ref{ssec: darkmatterhalos}.

The luminosity term in the pressure equation is made up of two components:
\begin{equation}
    L = L_w\left(r\right) + L_{\rm comp}.
\end{equation}
Here $L_{w}$ is the energy injected by stellar feedback and $L_{\rm comp}$ is the rate at which the thermal energy is lost to Compton cooling. For the first,
\begin{equation}
    L_{w}\left(r\right) = f_{\star}\frac{\Omega_b}{\Omega_m}\dot{M}_h\epsilon_k\omega_{\rm SN} \label{eqn:Lw}
\end{equation}
where $f_\star$ is determined for normal galaxies as described in Section~\ref{sec:galaxy_model} and $\epsilon_k$ is the fraction of this energy released in the wind. As described in Section \ref{ssec: burst}, in the Pop III model, this component is set to zero after the initial burst of star formation.

$L_{\rm comp}$ is the energy lost through inverse Compton cooling, where energy is transferred from a hot charged particle in the wind to a cosmic microwave background (CMB) photon through scattering; while this process is very slow in the present Universe, the increased CMB energy density at $z > 6$ can make it important (and potentially observable, as we will explore in Section~\ref{ssec:CMB_effects}).  The rate energy is lost through this process is
\begin{equation}
    L_{\rm comp} = - \frac{3}{2} \frac{PV}{t_{\rm comp}},
\end{equation}
where $V$ is the volume enclosed by the wind and the cooling time is
\begin{equation}
    t_{\rm comp} = \left(\frac{8\sigma_Ta_{\rm rad}T_{\gamma}^4}{3m_e c}\right)^{-1} = 1.2 \times 10^8 \left(\frac{1+z}{10}\right)^{-4} \ {\rm yr}
\end{equation}

\begin{figure*}
    \centering
    \includegraphics[width=\columnwidth]{Figure_1_a.png}
    \includegraphics[width=\columnwidth]{Figure_1_b_i.png}
    \includegraphics[width=\columnwidth]{Figure_1_c.png}
    \includegraphics[width=\columnwidth]{Figure_1_d.png}
    \caption{Evolution of the radius of an individual bubble starting at different initial redshifts for different SFE cases of the normal and bursty galaxy models: \emph{Top left:}  Constant $f_{\star}$ model. (0.1) \emph{Bottom left:} Momentum-regulated model. \emph{Top right:} Energy-regulated model (for this case, we also plot curves with energy losses from Compton cooling switched off). \emph{Bottom right:} Corresponding plots for several initial stellar masses for the Pop III model. We show the results for both a constant minimum virial temperature of 1000~K and a varying temperature which approximates the evolution of initial halo mass in a more complex history (see Section~\ref{ssec: burst}).}
    \label{fig:single_normal_bursty}
\end{figure*}

In comparison to \citet{tegmark_late_1993} and \citet{furlanetto_metal_2003}, we ignore a term that accounts for the uncertain fate of energy dissipated when particles are swept up. We find that, in the regime relevant to these galaxies, including this term
increases the wind radii by $\la 10$\%; ignoring it thus makes our filling factor estimates somewhat conservative.

Furthermore, if the velocity of the wind slows down to the Universe's expansion speed at $z=z_f$ and corresponding radius $R=R_f$, it will simply expand with the usual Hubble flow. If this condition occurs, we therefore set the comoving size to a constant.

We initialize each wind at the redshift $z_{\rm init}$ the halo passes the relevant threshold to form stars. We assume that it instantly transforms a fraction $f_\star$ (evaluated at $z_{\rm init}$ and this threshold mass) into stars. We assume for simplicity that the wind begins at the halo's virial radius. We then distribute the energy released by the initial wave of supernovae equally between the hot interior (whose pressure drives the expansion) and the kinetic energy of the shell itself. Finally, we set the shell properties by demanding that it begins at the halo's escape velocity, unless the corresponding mass would exceed the halo's non-stellar baryonic mass. We have verified that these assumptions do not affect the resulting radii of the winds by more than $\sim 25\%$.

\subsection{Example wind bubbles} \label{sec:examples}

Three panels in Figure \ref{fig:single_normal_bursty} show the evolution of the wind bubble radius $R$ for a single halo in the normal and bursty galaxy models starting at initial redshifts $z_{\rm{init}} = 8, 12,$ and $20$ for three SFE cases ($f_{\star} = 0.1$, energy-regulated and momentum-regulated). The parameters for the latter two models are summarized in Table \ref{tab:Table 1}.
In almost all cases, the bubbles initially expand rapidly before slowing, thanks to the sweeping up of ambient material. Energy loss from Compton cooling also contributes to the slowdown (and suppresses the final bubble radius), but it is not a large effect, as shown explicitly for the energy-regulated case in the upper right panel of Figure~\ref{fig:single_normal_bursty}.

Overall, we find that winds that begin at earlier times grow larger than those launched at later times (even though they begin at smaller radii -- which simply reflects the smaller virial radius at those times, $r_{\rm vir} \propto M_{h}[1+z]^{-1}$]). Even though winds that begin at earlier times must propagate through higher-density material, their source haloes grow steadily, so the haloes forming at $z=20$ have much more mass (and stellar mass) than those at $z=8$. Moreover, even if these early winds ``stall," they continue to expand thanks to the Hubble flow.

\begin{figure*}
    \centering
    \includegraphics[width=\columnwidth]{Figure_2.png}
    \includegraphics[width=\columnwidth]{Figure_2_b.png}
    \caption{\emph{Left:} Radii of the wind bubble as a function of halo mass at various redshifts for the normal energy-regulated (ER) and momentum-regulated (MR) models as well as the bursty energy-regulated model. \emph{Right:} A power law fit of the form $R = A {M_h}^n$ to the normal momentum-regulated ($n = 4/15$) models at z = 8, 12, and 20.}
    \label{fig:finalradii}
\end{figure*}

It is worth noting that initial expansion phase is much stronger for the bursty model, and for $z_{\rm init} = 8$ and 12, they reach significantly larger radii by $z = 6$ than the normal cases. This demonstrates the impact of increasing the star formation efficiency of the smallest haloes, when the wind expansion begins (here $f_{\star,\rm min} = 0.03$). On the other hand, while the winds starting at $z_{\rm init} = 20$ are initially larger in the bursty models, they  approach the normal ones at later redshifts, because these haloes have grown far above the low-mass limit in which the burstiness criterion matters. In addition, the difference is generally less significant for the momentum-regulated case because that model is less effective at suppressing star formation in small haloes anyway, so the haloes have a larger $f_\star$ throughout.

Similar trends can be identified for the Pop~III model in the lower right panel of Figure \ref{fig:single_normal_bursty}. The key difference is that these wind bubbles are much smaller (note the different vertical axis in this panel). This is simply because they form many fewer stars, in contrast to the normal and bursty models, which continue to form stars throughout. Note as well that many of the Pop III model haloes which form at later times actually grow larger than those that begin at earlier times -- opposite to the normal galaxies. This is because the surrounding medium has a higher density at early times, which makes it harder for the shell to expand. This effect is more significant for the Pop III model because all of the energy is injected at once. Thus, a wind that formed earlier gets stalled and those that form later, where the density is lower, quickly overtake them even though they have not had as much time to grow. This effect is less important for the normal galaxies because they continuously inject energy over a long period of time.

The left panel of Figure \ref{fig:finalradii} plots the final radii of the wind bubbles as a function of source halo mass at three different redshifts. We show three different models at $z_{\rm{final}} = 8, 12,$ and $20$. These curves can be compared with Fig. 1. of \citet{furlanetto_metal_2003}. The qualitative agreement between our results and these older ones is reasonable (with one exception; see below). Note, however, that our models generally have smaller star formation efficiencies than assumed in that earlier work, especially in the normal energy-regulated model. The general trend of the radii rising sharply at low masses $\sim 10^8 M_{\odot}$ and then quickly slowing down to a relatively shallow slope is present in both figures. In addition, the radii increase significantly as the redshift decreases because the wind bubbles have had a longer time to expand. (In the newer models, the star formation efficiency also increases at lower redshifts.)

But there is one notable difference with earlier work -- our results do not show the rapid decline in radii at high halo masses present in \citeauthor{furlanetto_metal_2003}. This can likely be attributed to the simplifying assumptions  in our initial conditions: the more detailed, merger-tree construction of \citet{furlanetto_metal_2003} found that winds from high-mass haloes would quickly be ``trapped" by the halo. This is generically expected because the halo binding energy increases more rapidly with halo mass than the energy supplied by supernovae. We do not attempt to model this regime in detail because our galaxy formation model is likely not accurate at high masses (because it ignores feedback from active galactic nuclei) and because haloes at these high masses are sufficiently rare that winds from massive galaxies do not significantly affect our final results, which focus on the cumulative impact of the entire wind bubble population (see Section \ref{ssec:Q}).

The right panel of Figure \ref{fig:finalradii} compares a power law of the form $R \propto M_h^n$, to our results. The power law index $n$ is set to $4/15$ here, which is appropriate for the  momentum-regulated case. This expectation follows by applying energy conservation at the asymptotic comoving radius of the wind. In detail, we assume that all the energy from supernova blastwaves goes into accelerating the swept-up IGM material to the Hubble flow velocity:
\begin{equation}
    E_{\rm SN} = \frac{1}{2} M_h \left(HR\right)^2 = \frac{1}{2}\frac{4\pi}{3} \rho\left(z\right) R^3 \left(HR\right)^2 \propto R^5.
    \label{eq:esn-fit}
\end{equation}
We then have $E_{\rm SN} \propto f_{\star} M_h$ (eq. \ref{eqn:Lw}) and $ f_{\star} \propto \left(M_h\right)^{\xi}$ (eq. \ref{eqn:eta}), we expect $R \propto M_h^n$ where $n = \left(\xi + 1\right)/5$, which match the choices in Figure~\ref{fig:finalradii}. The estimate matches our momentum-regulated case relatively well, except at the low mass end, where the winds have only recently turned on so have not yet reached their maximum extent. It is clear from the success of this simple estimate that energy losses are not significant for the dynamics of these winds (though, again, we have not included the gravitational potential well at high masses accurately). Clearly this power law fit can be a useful approximation to the overall extent of wind bubbles. The energy-regulated case is not as close a match, but the estimate is still reasonable.

\section{Results} \label{sec:results}

\subsection{Widespread enrichment of metal-carrying winds} \label{ssec:Q}

We next turn to our main result, the fraction of space filled by wind-driven metals. The volume filling factor of bubbles, $Q$, is
\begin{equation} \label{eqn:Q}
    Q = \int_{M_{\rm min}}^{\infty}dM_h\frac{dn}{dM_h}\frac{4\pi}{3}R_c^3,
\end{equation}
with $R_c = R(1+z)$ obtained for each halo mass as in Figure \ref{fig:finalradii}.

\begin{figure}
    \centering
    \includegraphics[width=0.5\textwidth]{Figure_3.png}
    \caption{The integrand of equation~(\ref{eqn:Q}), or the contribution to the total volume filling factor from haloes in each logarithmic mass range. The different line styles denote different star formation models: the minimalist energy-regulated model (solid curves), the bursty energy-regulated model (black dashed curves), and the bursty momentum-regulated model (magenta dashed curves); the two bursty cases are so close that they are hard to distinguish. In all cases, the integral is dominated by relatively low-mass halos.}
   \label{fig:Q_int}
\end{figure}

Figure~\ref{fig:Q_int} shows the integrand of this equation as a function of halo mass, thus providing the contribution of each mass range to $Q$. As mentioned briefly in Section~\ref{sec:examples}, we  see that halos with $M_h \lesssim 10^{11} M_{\odot}$ dominate over those at higher masses, especially at higher redshifts. Therefore, ignoring gravity in the highest mass haloes does not significantly affect our results. We also see that the smallest haloes are most important in the bursty models, because their star formation rate is strongly suppressed in our ``normal" energy-regulated model.

\begin{figure*}
    \centering
    \includegraphics[width=\columnwidth]{Figure_4_a.png}
    \includegraphics[width=\columnwidth]{Figure_4_b.png}
    \caption{Evolution of the volume filling factor $Q$ in our models. \emph{Left:} ``Normal" galaxies, in which star formation proceeds over long timescales. From top to bottom, the solid lines take $f_\star=0.1$, momentum-regulated feedback, and energy-regulated feedback. The dashed curves impose a minimum $f_\star=0.03$, mimicking bursts in very small galaxies. \emph{Right:} Population~III models, in which star formation only occurs in a single burst. The curves assume different stellar masses in this burst and take two prescriptions for the mass at which this star formation occurs.}
    \label{fig:Q}
\end{figure*}

The left panel of Figure \ref{fig:Q} shows the evolution of $Q$ for several models of the star formation efficiency. From top to bottom, the solid curves take a constant $f_\star=0.1$, momentum-regulated feedback, and energy regulated feedback. The dashed curves show the bursty models, which closely overlap in the case of the momentum-regulated models. We remind the reader that the constant $f_\star=0.1$ model will not fit observed luminosity functions, but it is included for comparison with earlier results. We see that, regardless of the feedback prescription, $Q\sim0.1-1\%$ at $z\sim10$ and between $\sim1-10\%$ at $z\sim6$.

The right panel of Figure \ref{fig:Q} shows the evolution of $Q$ for several Pop III models. Here, we show conservative and optimistic cases, corresponding to stellar masses of $100 \ M_\odot$ and $1000 \ M_\odot$, for both a constant threshold virial temperature and the fit to semi-analytic results. The values of $Q$ lie roughly between $\sim0.1$--$10\%$ (depending on $M*$) for all $z \lesssim 10$; by that time, relatively few new Pop~III stars are forming in these models. Note that the structure in the time-varying $T_{\rm min}$ curves comes from that threshold mass evolution.

In both sets of models, metals can be \emph{relatively} widespread by $z=6$, but they are very unlikely to be pervasive, at least if they are spread by winds from star-forming galaxies. Only if the star formation efficiency of very small galaxies is increased dramatically can $Q \sim 1$ at $z=6$; even then it is difficult to imagine that it will be $\ga 10\%$ at $z \sim 10$.

Interestingly, we see that the overall shape of the curves for normal galaxies are quite similar, regardless of the particular star formation model.
At all redshifts, $Q$ in the $f_\star=0.1$ model is about an order magnitude larger than that in the energy and momentum-regulated models. The bursty model, with enhanced star formation in small haloes, does have significantly more widespread enrichment. This underlines the important role of the initial phases of star formation in early galaxies to metal enrichment and motivates further studies of it in greater detail.

Comparing the two panels of Figure~\ref{fig:Q}, we find that the Pop~III haloes, even with the conservative $100 M_{\odot}$ estimate for the stellar mass,  contribute significantly to the total enriched volume, dominating at high redshifts. Although the stellar mass in each halo is very small in these cases, they have several advantages: they get very early starts (and so have time to reach their maximum size) and they form in abundant, low-mass haloes. The filling factor from these sources is particularly large at early times, although the gap closes in at later redshifts because the continuous star formation in normal galaxies helps their bubbles continue to grow over time. However, we will see in the next section that the amount of enrichment in these regions is very small.

There are relatively few estimates of metal enrichment during this era, but we can compare our model to a few earlier calculations.
\citet{furlanetto_metal_2003} used a more complex wind model but made similar assumptions to our $f_{\star}=0.1$ model (see their Fig.~5). We find  good agreement between the two calculations,
\citet{scannapieco_early_2002} also considered a very similar case (see their Fig.~1), finding $Q \sim 0.2$ at $z \sim 6$.
We conclude our simplifications to galaxy growth and the wind model are reasonable. However, we emphasize that a constant-$f_{\star}$ model is no longer viable in light of the observed luminosity function at $z \sim 6$--8, so the true filling factor is likely to be many times smaller than suggested by these earlier papers -- in the range $Q \sim 0.01$--0.05 rather than near unity.

\citet{furlanetto_metal_2003} also considered enrichment from Pop~III haloes, but they made far more optimistic assumptions about star formation in them, so it is difficult to compare directly, although the relative shapes of the curves are similar. However, \citet{jaacks_baseline_2018} used a simulation-based method to estimate the filling fraction. Their star formation model is much more sophisticated, but their results are roughly similar to our models with a stellar mass of $100\ M_\odot$ per halo, although our volume filling fraction is larger at late times. This may be because we assume that all the Pop~III wind bubbles remain independent; in reality, their source haloes likely merge over time. In that case, we would overestimate their contribution at late times.

\begin{figure*}
    \centering
    \includegraphics[width=\columnwidth]{Figure_5_a.png}
    \includegraphics[width=\columnwidth]{Figure_5_b.png}
    \caption{Evolution of the average metallicity of the Universe (\emph{left panel}) and of the wind-driven bubbles (\emph{right panel}) for several normal and bursty galaxy models, as in Fig.~\ref{fig:Q}. }
    \label{fig:Z}
\end{figure*}

\subsection{Average Metallicity} \label{ssec:Z}

Next, we consider how the average metallicity of the Universe $\left<Z\right>$, as well as that of the bubbles $\left<Z\right>_{b}$, evolve with time. The comoving mass density of metals produced is
\begin{equation}
   \rho_{Z} = \int^{\infty}_{M_{\rm min}} M_{\rm metal}\left(M_h\right) \frac{dn}{dM_h} dM_h,
\end{equation}
where $M_{\rm metal}$ is the mass of a metal produced by a halo of mass $M_h$ (eq.~\ref{eqn:M_metal} with the fraction of stellar mass turned into metals, $y_Z = 0.01$, in place of $Y_i n_{\rm SN}$). The mean metallicity of the Universe is then $\left<Z\right> = \rho_{Z}/\bar{\rho}_{b0}$, and the mean metallicity of the enriched material inside bubbles is then $\left<Z\right>_{b} = \left<Z\right>/Q$. We report our results in units of solar metallicity, which we take to be $Z_{\odot} = 0.0196$ \citep{von_steiger_solar_2015}.

The left panel of Figure \ref{fig:Z} shows the evolution of $\left<Z\right>$ with redshift. As expected, we see that it increases with time as the total number of stars rises, reaching $\sim 10^{-4}$ to $10^{-3}$ by $z = 6$. This can be compared to Figure 10 in \citet{furlanetto_minimalist_2017}. The two figures are in good agreement, sharing similar shapes of the curves as well as very comparable $\left<Z\right>$ values.

Meanwhile, we see from the right panel of Figure \ref{fig:Z} that the average metallicity inside the bubbles $\left<Z\right>_b$ falls with time. This is because, in most models, the wind bubbles expand fast enough that the metals are diluted more rapidly than new star formation produces them. This is natural for any wind. For example, the simple Sedov-Taylor blastwave has $R \propto E^{3/5}$. Because the input energy is proportional to the stellar mass -- and hence also the metal mass -- we would expect $\left< Z \right>_b \propto E/R^3 \propto M_*^{-4/5}$. Our winds are more complex than the simple Sedov-Taylor solution (with continuous energy injection and an expanding medium) but follow the same qualitative trend.

For the Pop~III models, the metallicities are typically much smaller, with $\left<Z\right>$ ranging between $\sim 10^{-7} - 10^{-8}$: these winds are efficient at spreading metals around the Universe, but only at extremely low levels. This would remain true even if Pop~III supernovae had larger metal yields.

\subsection{Metal-line absorption systems} \label{ssec: dndx}

At present, the only way to observe metal enrichment in the early Universe is through metal line absorption systems seen against luminous background quasars. This does not directly probe the filling factor, but rather the incidence of (strong) enrichment along each line of sight. To compare to these observations, in this section we therefore estimate the incidence of such absorbers in our models.

\begin{figure}
    \centering
    \includegraphics[width=0.5\textwidth]{Figure_6.png}
    \caption{Equivalent width of the OI line produced by wind bubbles as a function of halo masses at several redshifts and for different galaxy models. The dotted line at 0.05\AA \ represents the sensitivity limit of \citet{becker_evolution_2019}. The other dotted lines show the equivalent width of the CIV line instead.}
    \label{fig:EW}
\end{figure}

To do so, we must specify how metals are distributed within the wind bubble. We assume for simplicity that every line of sight through a wind bubble has a metal absorber and that all have equal column density. In reality, there may be a non-uniform distribution of metals, which would generally lead to fewer (but stronger) lines. Furthermore, we assume that all metals are in the ionization state of the observed line. Therefore, our model should only be taken as an upper limit to the true incidence of absorbers.

\begin{figure*}
    \centering
    \includegraphics[width=\columnwidth]{Figure_7_O.png}
    \includegraphics[width=\columnwidth]{Figure_7_C.png}
    \caption{Comparison of predicted line incidence for OI (\emph{left panel}) and CIV (\emph{right panel}) to observations at $z\sim 6$ . In each case, the theoretical curves include only those systems that exceed the limiting equivalent width of the observations. We show several different model predictions. For OI, the data point is from Table 2 in \citet{becker_evolution_2019}, for $z = 5.7-6.5$.  For CIV, the data points are a combination of the upper limits of the number of absorbers (95\% and 99\% confidence intervals) from four sightlines in \citet{becker_high-redshift_2009} and one absorber detected in \citet{ryan-weber_intergalactic_2006}. The Pop III model shown here is for constant $T_{\rm min}$.}
   \label{fig:dNdX}
\end{figure*}

We can then use the metal yields from Section \ref{ssec:MP} to estimate the equivalent width of each wind bubble, shown in Figure \ref{fig:EW} for the OI line. We show the equivalent width as a function of halo mass for several galaxy models at $z = 6$, $12$, and $20$.  We see that $W$ is relatively insensitive to the star formation law in the normal and bursty galaxy models: equation~(\ref{eqn:EW}) shows that $W \propto X_\star M_h/R^2 \propto 1/(\xi + 1)$ using the fits in equation~(\ref{eq:esn-fit}), which does not depend strongly on the star formation law. The Pop~III models are all much lower, however, because they have much smaller stellar masses driving them (which are also independent of the halo mass).

The horizontal dotted line in Figure~\ref{fig:EW} at 0.05~\AA \ corresponds to the observational cutoff below which the absorption cannot be detected in \citet{becker_evolution_2019}. We see that nearly all the wind bubbles around ``normal'' galaxies are likely detectable, except in the Pop~III models.

Under these simplifying assumptions, the number density of absorption  lines per unit path length, $dN/dX$, is an integral over the cross-sectional area of the bubbles:
\begin{equation}
    \frac{dN}{dX} = 90 \ {\rm Mpc} \  \left(\frac{7}{1+z}\right)^2\int dM_h\frac{dn}{dM_h}\pi R_c^2.
\end{equation}
The prefactor  comes from $X$, the absorption path length interval, originally defined in equation~(4) of \citet{bahcall_statistical_1969}, where
\begin{equation}
    \frac{dX}{dz} = \left(1+z\right)^2\frac{H_0}{H(z)}.
\end{equation}
We set the limits of the integral to include only those haloes whose wind bubbles exceed the threshold for line detection (e.g., the horizontal line in Fig.~\ref{fig:EW}).

The evolution with redshift of $dN/dX$ for OI for several different galaxy models is shown in the left panel of Figure \ref{fig:dNdX}. For the normal galaxy population, the incidence of absorbers rises rapidly with redshift, so that their relative abundance more or less follows that of $Q$ in Figure \ref{fig:Q}. However, the Pop III model -- even taking an extreme case in which each halo forms $1000 \ M_{\odot}$ of stars -- remains relatively constant and sharply falls off at $z \sim 10$. This follows from the discussion of $W$ -- although the volume occupied by the wind bubbles, and thus $Q$, from these galaxies are significant, the metal yields are low and thus only a small fraction of the lowest mass halos have detectable absorption. By $z \sim 9$, they completely fall below the observational limit. The right panel of Figure~\ref{fig:dNdX} shows analogous results for CIV; given our extremely simplified assumptions about the metal distribution, the curves are very similar, with the only differences due to the different atomic yields and line properties. (Recall that we assume these species dominate the ionized fractions, which is not realistic!)

The extent of metal enrichment can be studied through quasar spectra and recent observations have pushed back the redshift limits to $z \sim 5$--$6$. As one example, \citet{becker_evolution_2019} presents a survey of 74 systems between $z = 3.2 - 6.5$ with O I $\lambda1302$ absorption with equivalent width $W > 0.05$ \AA.  \citeauthor{becker_evolution_2019} present constraints on $dN/dX$ at $z = 5.7-6.5$, with $dN/dX = 0.421$, shown  on the left panel of Figure \ref{fig:EW}. On the right panel, we compare to the combined detections of CIV absorbers along sightlines to quasars in \citet{becker_high-redshift_2009} and \citet{ryan-weber_intergalactic_2006}.

We see that all of our normal galaxy models fill a sufficient volume to explain the current observations of OI and CIV line number densities at $z \sim 5-6$. Typically we overproduce the lines by an order of magnitude or so, which is not surprising in light of our very optimistic assumptions of a uniform metal distribution and that all atoms are in the appropriate ionic state. We conclude that our estimates are consistent with the current observations, but future data and better models of the connection between wind bubbles and absorbers are needed to make any stronger conclusions.

We also find that absorbers from the Pop III model are rare except at the highest redshifts. Thus, although they may contribute to a low-level enriched background, they cannot explain the observed systems at $z \sim 5-6$. This would remain true even if we treated them as more energetic pair instability supernovae: the metal yields can be larger, but the increased explosion energies dilute them significantly.

\subsection{Effects on the CMB} \label{ssec:CMB_effects}

While metal lines surveys provide one probe of the extent of metal enrichment, they are challenging to perform and can only probe a small number of lines of sight at the highest redshift. It is therefore worth considering alternative observational constraints on high-$z$ winds. In this subsection,  we consider the Sunyaev-Zeldovich (SZ) effect as a possible alternative approach to detect the effects of galactic winds. This occurs when CMB photons scatter off of hot electrons along the line of sight, imprinting a fluctuation (with a characteristic spectral distortion) on the CMB \citep{rashid_alievich_sunyaev_small-scale_1970}. It has proved to be a powerful tool in observational cosmology, especially for searching for galaxy clusters (e.g., \citealt{planck_collaboration_planck_2014, bleem_galaxy_2015}). But any source of hot electrons can imprint SZ fluctuations, including high-redshift star formation \citep{oh_sunyaev-zeldovich_2003} and winds from quasars and galaxies (e.g, \citealt{majumdar_sunyaev--zeldovich_2001, platania_sunyaev-zeldovich_2002, white_simulating_2002}).

In this section, we estimate the amount of CMB heating due to the Compton cooling of the hot gas inside the wind bubbles. Following \citet{oh_sunyaev-zeldovich_2003}, we focus on estimating the Compton-$y$ parameter, which characterizes the total average spectral distortion along a line of sight, to look at how the magnitude of the SZ effect depends on some of our input parameters. In our model, the energy density injected by supernovae is
\begin{equation}
    \epsilon_{\rm SN} = \omega_{\rm SN}f_{\rm comp} \tilde{f}_{\star}f_{\rm coll}\Omega_b {\rho}_{\rm crit},
\end{equation}
where $f_{\rm coll} = \rho_{\rm halo}/\bar{\rho}_m$ is the fraction of matter contained in star-forming haloes, $\tilde{f}_{\star}$ is the mass-averaged star formation efficiency, and a fraction $f_{\rm comp}$ of the energy is injected into the CMB. Thus, the energy input per baryon can be written
\begin{equation}
    \frac{\epsilon_{\rm SN}}{\bar{n}_b}
    \approx0.25 \ { \rm eV} \left(\frac{\omega_{\rm SN}}{10^{49} {\rm erg}/M_{\odot}}\right)\left(\frac{f_{\rm comp}}{0.1}\right)\left(\frac{f_{\star}}{0.01}\right)\left(\frac{f_{\rm coll}}{0.05}\right)\left(\frac{f_{\star}}{0.01}\right)
\end{equation}
where $\bar{n}_b = \Omega_b \rho_{\rm crit} / m_p$ is the average number density of baryons.

As a simple estimate of the amplitude of the SZ effect, we compute the Componization parameter, or the average $y$-distortion \citep{as_kompaneets_establishment_1957}, which is the dimensionless timescale for the collision of a photon in an electron field. It equals the integrated electron pressure along a line of sight and can be approximated as the ratio of energy density injected through Compton cooling of the wind bubbles to the energy density of the CMB \citep{zeldovich_interaction_1969}, with $y = -1/2*\Delta T / T$. We find  (see Section 3.3 of \citealt{oh_sunyaev-zeldovich_2003})
\begin{eqnarray}
    y & \approx &  10^{-8}\left(\frac{7}{1+z}\right)\left(\frac{f_{\rm comp}}{0.1}\right) \left(\frac{f_{\star}}{0.01}\right)  \nonumber
    \\ & & \times \left(\frac{f_{\rm coll}}{0.05}\right)\left(\frac{\omega_{\rm SN}}{10^{49} \rm erg/M_{\odot}}\right)
    \label{eqn:y-est}
\end{eqnarray}
Note that the amplitude is proportional to $f_{\star}$ as well as to $f_{\rm comp}$, the fraction of SN energy which is lost through Compton cooling. The latter can be approximated as $u_{\rm comp}/u_{\rm SN}$ where $u_{\rm SN}$ is the total energy density injected into the winds by supernovae, and $u_{\rm comp}$ is the energy density transferred from the winds to the CMB.

Equation~(\ref{eqn:y-est}) suggests that the CMB distortion due to high-$z$ winds is quite small. The COBE FIRAS instrument constrained $y < 1.5 \times 10^{-5}$ \citep{fixsen_cosmic_1996}, for example. Our result is also well below the predictions of \citet{oh_sunyaev-zeldovich_2003}, because they made much  more optimistic assumptions about the efficiency of Pop~III star formation.

For a more detailed estimate of the distortion, we trace the energy lost by our wind bubbles to the CMB, $E_{\rm comp}$, in a process similar to that used to obtain Figure \ref{fig:finalradii}. The resulting energy density is
\begin{equation}
    u_{\rm comp} = \int_{M_{\rm min}}^{\infty}dM_h\frac{dn}{dM_h}E_{\rm comp}\left(M_h\right).
\end{equation}
This expression allows us to obtain the redshift evolution of the Compton cooling rate from the wind bubbles for all galaxy models. The total energy injected increases with cosmic time, because bubbles continue to grow (and lose energy) as galaxies form more stars.

The total energy density $u$ of the CMB evolves as
\begin{align}
    \frac{du}{dz} = - 4 H\left(z\right)\frac{dt}{dz}u\left(z\right) + \frac{du_{\rm comp}}{dz} \left(1+z\right)^3.
\end{align}
The first term simply represents the decreasing density due to the expanding Universe and the second term accounts for the energy injected from the wind bubbles (in proper units). Using $u(z) = aT(z)^4$, the temperature distortion $\Delta T$ caused by the inclusion of the second term can be calculated.

\begin{figure}
    \centering
    \includegraphics[width=0.5\textwidth]{Figure_8.png}
    \caption{Evolution of the Compton-$y$ parameter for several of our galaxy models.}
    \label{fig:y}
\end{figure}

Figure \ref{fig:y} shows the resulting $y$-distortion to the CMB. The Compton $y$-parameter increases from  $\la 10^{-11}$ to $\sim 10^{-8}$ in our models that are calibrated to the luminosity function, as expected from our simple estimate.
The $y$-distortions produced by the Pop III models are even smaller, ranging from $10^{-13}$ to $10^{-10}$, although these could increase by at least an order of magnitude if the Pop~III supernovae are particularly energetic (as with pair instability supernovae).

The smallness of the induced $y$-distortion will make it difficult to measure on average, because it cannot be separated from other sources (like the hot IGM at lower redshifts). \citet{oh_sunyaev-zeldovich_2003} pointed out, however, that the distortion will not be inhomogeneous, thanks to the clustering of the wind sources: a line of sight passing through an overdense area of high-$z$ galaxies will observe a large distortion. This is captured in the SZ power spectrum $C_{l}\left(y\right)$ (see eq.~13 of \citealt{oh_sunyaev-zeldovich_2003}). Unfortuantely, the clustering signal is proportional to $y^2$, which is so small in our models that angular fluctuations in the SZE as a result of wind bubbles are unlikely to be detected. (However, their angular structure should be similar to those in \citealt{oh_sunyaev-zeldovich_2003}, as our source models are not dramatically different.)

\section{Discussion}  \label{sec:discussion}

In this paper, we studied the extent of metal enrichment at $z \ga 6$ from winds driven by star-forming galaxies. We combined a simple model of star formation that matches observations at $6 \la z \la 8$ \citep{furlanetto_minimalist_2017} with a simple model of wind expansion \citep{tegmark_late_1993, furlanetto_metal_2003} that allowed us to study several star formation scenarios.

We find that, when the galaxy model is calibrated to existing observations of high-$z$ galaxies, the volume filling factor $Q$, representing the fraction of the Universe enriched by metals through winds, only reaches at most $\sim 1 - 10 \%$ at $z \sim 6-8$. This is true even assuming relatively efficient star formation in small haloes. Therefore, although it is easy to imagine galactic winds permeating a substantial fraction of space, it is challenging to get a majority of the Universe enriched. This implies that the process of chemical enrichment is highly nonuniform, so that one can imagine metal-free star formation taking place at late times, at least in principle. Our predictions for the filling factor are somewhat smaller than many earlier works (e.g., \citealt{furlanetto_metal_2003}, because those works made optimistic assumptions about the overall star formation efficiency of high-$z$ galaxies (and so do not reproduce the observed luminosity function).

Despite this relatively low level of enrichment, we find that normal galaxies can easily account for the few observations of $dN/dX$ currently available at $z \sim 5-6$ \citep{becker_high-redshift_2009, becker_evolution_2019, ryan-weber_intergalactic_2006}, although we did not attempt to model the distribution of metals or ionic species within each wind bubble. Nevertheless, the inhomogeneity implies that metal lines cannot be used ``out of the box'' to study reionization but will require simultaneous modeling of the metal and ionization distributions \citep{oh_probing_2002, hennawi_probing_2021}.

In addition to the ``normal'' galaxy population, we also considered enrichment from Pop~III stars forming in minihaloes. While each such halo only forms a few stars, we find that they are sufficiently numerous to provide a comparable filling factor to normal galaxies at later times, with $Q \sim 1\%$ in reasonable models. At early times ($z \ga 10$), enrichment from these sources can even dominate over normal galaxies, because they are so widespread. However, the star formation driving these episodes is at a very low level, and the resulting enriched regions have a very low average metallicity, well below the limits of metal-line systems.

Finally, we considered the Sunyaev-Zel'dovich distortions produced by Compton cooling of the hot wind bubbles by calculating the $y$ distortion. We found that, for all models, the $y$ values are significantly smaller than the upper limit from COBE FIRAS \citep{fixsen_cosmic_1996} as well as those predicted by \citet{oh_sunyaev-zeldovich_2003}, because we made much less optimistic assumptions about the efficiency of early star formation. Even if this signal could be detected, it would require separation from low-$z$ sources of SZ distortions.

\section*{Data Availability}

No new data were obtained as part of this work. Results used to generate the figures are available from the authors upon reasonable request.

\section*{Acknowledgments}

This work was supported by the National Science Foundation through award AST-1812458. In addition, this work was directly supported by the NASA Solar System Exploration Research Virtual Institute cooperative agreement number 80ARC017M0006. We also acknowledge a NASA contract supporting the ``WFIRST Extragalactic Potential Observations (EXPO) Science Investigation Team" (15-WFIRST15-0004), administered by GSFC. N.~Y. thanks the UCLA Department of Physics \& Astronomy for support during its 2021 Undergraduate Summer Research Program.

\bibliographystyle{mnras}

redpajama_set_name: RedPajamaArXiv


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SciBar is a regional event, which sees informal discussions around scientific topics take place in a bar. The Chester SciBar is part of the new North West Science Alliance, and organised by staff and students at the University of Chester.
Professor Nigel John leads the Medical Graphics Team in the Faculty of Science and Engineering at Thornton Science Park. The team was recently named by Universities UK as one of the Nation's Lifesavers – the top 100 individuals or groups based in universities, whose work is saving lives and making a life-changing difference to our health and wellbeing.
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In early December 2015 Rick Cruz, Coordinator for the Toys for Tots Foundation of The Greater Bridgeport Area, received word of a school wide donation occurring at Elizabeth Shelton School. Students in grades K-4 participated by donating unwrapped and unopened toys for those in need. Marissa Fusco, Senior at Shelton High School, had been working on her Capstone graduation requirement at Elizabeth Shelton School. In that time, she had decided, along with fourth grade students, to include the entire school in a donation for Toys for Tots. On December 10, 2015 Rick Cruz collected the toys donated by students as well as a generous $500 contribution from Elizabeth Shelton PTO. All donations were greatly appreciated and put towards a great cause this holiday season! Marissa was joined in her efforts by Amanda Wandishion and Laura Ferraro (not pictured) and the fourth grade student leaders.