7,000,000,000 People

The world is approaching 7 billion people. According to Wikipedia,

The "Day of 7 Billion" has been targeted by the United States Census Bureau to be in July 2012,^[13] while the Population Division of the United Nations suggests some time in late 2011.^[5]

Take this table from Wikipedia:

World population estimates milestones

Population (in billions)	1		2		3		4		5		6		7		8		9
Year	1804		1927		1960		1974		1987		1999		2011		2025		2045–50
Years elapsed		123		33		14		13		12		12		14		20–25

Notice how it took until the year 1804 for the world to reach 1 billion people.  That's 5564 years since creation.  Quite a long time.
For the population to reach 2 billion, it took another 123 years.  Pretty long, but miniscule compared to the first billion.
After that, the year interval between billions has gotten considerably shorter.

Where is the source to this in the Torah?

Perhaps, we can say that the 7 billion people in the world are compared to the first 7 people in the world.

Pirkei DeRibbi Eliezer (among others) says:

ושנים עשר שעות היום בשעה ראשונה הוצבר עפרו של אדם בשנייה גבלו בשלישית רקמו ברביעית נזרקה בו נשמה בחמשית קם על רגליו בששית קרא שמות בשביעית נזדמנה לו חוה בשמינית נצטוו על פירות בתשיעית עלו למטה שנים וירדו ארבעה בעשירית עבר על צוויו באחד עשר נדון בשנים עשר נתגרש שנא' ויגרש את האדם

Adam HaRishon represents the first billion people.  He was only created on day 6 and the latter half of the 24-hour day.  5564 is the latter half of the 6th millennium.

Until he was Kam Al Raglav (got up on his feet), it took a few hours.  Then it took another couple hours - or perhaps a bit less - until Havah was created. Havah was created in the 7th daylight hour, so from the time he got up on his feet until Havah was created - in cosmic years - is close to the 123-year period.  (I'll admit that this is a bit of a stretch since we are suspending the years of Adam's formation.  However, the main point isn't the exact calculation.  Rather, it's the relations of the intervals.)

Next, we must look at a few Pesukim in Bereishit.  The verses in Bereishit say:

א וְהָאָדָם, יָדַע אֶת-חַוָּה אִשְׁתּוֹ; וַתַּהַר, וַתֵּלֶד אֶת-קַיִן, וַתֹּאמֶר, קָנִיתִי אִישׁ אֶת-יְהוָה.

1 And the man knew Eve his wife; and she conceived and bore Cain, and said: 'I have gotten a man with the help of the LORD.'

ב וַתֹּסֶף לָלֶדֶת, אֶת-אָחִיו אֶת-הָבֶל; וַיְהִי-הֶבֶל, רֹעֵה צֹאן, וְקַיִן, הָיָה עֹבֵד אֲדָמָה.

2 And again she bore his brother Abel. And Abel was a keeper of sheep, but Cain was a tiller of the ground.

ג וַיְהִי, מִקֵּץ יָמִים; וַיָּבֵא קַיִן מִפְּרִי הָאֲדָמָה, מִנְחָה--לַיהוָה.

3 And in process of time it came to pass, that Cain brought of the fruit of the ground an offering unto the LORD.

As Midrash Rabba says:

 אר' אלעזר בן עזריה: שלושה פלאים נעשו באותו היום.
בו ביום נבראו,
בו ביום שמשו, בו ביום הוציאו תולדות.

אמר לו רבי יהושע בן קרחה: עלו למיטה שנים, וירדו שבעה.
קין ותאומתו,
והבל ושתי תאומותיו.

My translation:

Ribbi Elazar ben Azarya said: 3 wonders were done on that day: On that day they were created, on that day they had relations, and on that day they brought forth offspring.
Ribbi Yehoshua ben Korha said to him: 2 went up on the bed, and 7 came down - Kayin and his twin sister - and Hevel and his 2 twin sisters.

The subsequent jump in billions in population were relatively instantaneous - just like Kayin, Hevel, and their sisters. And we will soon be at 7 billion.

Rashi in Humash learns the fact that the twin sisters were born from the extra words "את", before each son's name. Then, the next Pasuk says "ויהי מקץ ימים" - and it was at the end of days. Perhaps, it's telling us that at the end of days, we will be 7 billion in population corresponding to the 7 people who existed in Gan Eden before Adam's sin.

Are we ready to reach a level of Adam HaRishon before the Heit?