antiqueПетрПутенихинИсследование переменных параметров ХабблаrusПетрПутенихинcalibre 0.8.3817.5.20211d6d7e39-76cd-42c9-8c91-1012245695281.0

Путенихин П.В.

Исследование переменных параметров Хаббла

Researching Hubble's variable parameters

Оглавление

Введение

Исследование переменных параметров Хаббла

     1. Наблюдаемые диаграммы Хаббла

     2. Параметр Хаббла с изгибом

     3. Проблема сверхсветового движения

          Диаграммы для действительных источников

     4. Спорные выводы об ускоренном расширении

Выводы

О возрасте Вселенной

Аннотация

It is believed that one of the main arguments from which the conclusion about the accelerated expansion of the Universe follows is the reduced brightness of distant supernovae. A mathematical study of this phenomenon unexpectedly led to the exact opposite conclusion. If distant supernovae are seen dimmer, then the universe is expanding decelerate.

Считается, что одним из главных аргументов, из которого следует вывод об ускоренном расширении Вселенной, является пониженная яркость дальних сверхновых. Математическое исследование этого явления неожиданно привело к прямо противоположному выводу. Если дальние сверхновые видны более тусклыми, то Вселенная расширяется с замедлением.

Ключевые слова

закон Хаббла, яркость, тусклая сверхновая, скорость расширения Вселенной, диаграмма Хаббла, красное смещение, фотон, стационарная Вселенная

Введение

Существует предположение, что Вселенная расширяется в наши дни с ускорением, а в более далёком прошлом, напротив, она расширялась замедленно. Какими наблюдениями можно подтвердить это? Считается, что одним из главных аргументов, обосновывающих ускоренное расширение Вселенной, является пониженная яркость дальних сверхновых. Однако это правило выглядит довольно необычно. В самом деле, если они более тусклые, то они находятся дальше, что соответствует скорее их более быстрому удалению в прошлом. Хотя приведённое правило "тусклый-дальний" в целом корректно, правильнее всё-таки говорить, что при ускоренном расширении сверхновые имеют меньшую скорость, меньшее красное смещение, чем этого требует стандартный закон Хаббла. Хотя удалились они на меньшее расстояние, находятся они ближе, чем в равномерно расширяющейся Вселенной. Действительно, несложный анализ показывает, что при ускоренном расширении Вселенной и на самом деле любой объект оказывается на более близком удалении, проходит меньший путь, нежели при замедленном или равномерном расширении.

Давно замечено одно интересное и важное свойство математики – делать верные описания нашего мира, предсказания, на первых порах кажущиеся буквально выдумками:

"... почему математика столь эффективна при описании нашего мира и столь хорошо описывает его эволюцию? ... Почему эти правила так хорошо работают?" (Линде)

Вряд ли следует слишком уж сильно этому удивляться. Эта математика так хорошо работает просто потому, что мы и вывели её из прямых наблюдений за окружающей действительностью. Эффективно работает, значит, верно подсмотрели. Более того, в науке и в физике, в частности, уже давно замечена еще одна интересная закономерность: кажущиеся поначалу абстрактными математические выражения, уравнения вдруг оказываются описанием какого-нибудь вполне реального явления:

"... физики обнаруживают, что математические построения, необходимые им для описания нового класса явления, уже исследованы математиками по причинам, не имеющим ничего общего с обсуждаемыми явлениями" (Виленкин).

Однако даже при таком явно полезном подходе следует все-таки быть предельно осторожным при формулировке выводов и следствий из этих математических построений. Можно привести ряд примеров, когда такие выкладки приводят не просто к противоречиям со здравым смыслом, но к довольно заметным противоречиям с логикой, содержат логические ошибки.

В предлагаемой работе мы покажем, что математические выводы в физике во многом зависят как от их последующей трактовки, так и от предположений, оснований, исходных положений, использованных в процессе получения этих выводов. Вообще говоря, это очевидно: если изменить исходные положения, заявить истинными другие, выводы также будут иными, а то и противоположными. Но какие из этих взаимоисключающих выводов верны?

Нас интересуют логические построения, послужившие основой для утверждения об ускоренном расширении Вселенной, опирающиеся на факт пониженной яркости дальних сверхновых. Поскольку детали этой логики нам неизвестны, мы проведём собственные построения, пытаясь прийти к такому же выводу, утверждению.

В дальнейшем мы используем следующую систему обозначений. Все дистанции мы измеряем в миллиардах световых лет, а время – в миллиардах лет. В этом случае скорость света равна единице. Обозначение шкалы скоростей v/c и означает, что значения представлена в долях от скорости света.

У переменных, параметров и графиков нижние индексы состоят из букв и их комбинаций: a – ускорение (accelerate); d – замедление (decelerate); s – условная стационарность, на отдельном участке; о – неизменный, обычный, традиционный. Например, ad – обозначает ускорение – замедление и наоборот. Возможны и более длинные индексы, поскольку некоторые параметры на всём протяжении состоят из нескольких участков: ada – ускорение, замедление, ускорение. Последний индекс - o, по сути, является эквивалентом нуля, используемый просто для удобства записи. То есть, Ho и H0 – это один и тот же параметр Хаббла в нашей Вселенной. Некоторые параметры имеют отдельные обозначения: Rф – график движения фотонов в системе отсчёта сверхновой; Rco – путь, пройденный фотонами по их измерениям; Rc и Vc – графики движения условного источника фотонов, пройденный им путь и его скорость.

Большинство параметров на приводимых диаграммах зависят от времени, однако эту зависимость в их обозначениях для краткости мы чаще всего опускаем.

Все рассмотренные Вселенные и параметры их расширения являются вымышленными, условными и имеют к настоящей, нашей Вселенной лишь качественное, демонстрационное отношение. При этом на одной диаграмме для сопоставления будут изображены параметры движения сразу нескольких Вселенный, никак не связанных друг с другом.

Исследование переменных параметров Хаббла

В основу всех построений, в качестве базового, фундаментального принципа мы положим закон изменения во времени параметра Хаббла. Именно он определяет скорость расширения Вселенной. Скорость его изменения во времени, собственно, и является показателем, эквивалентом ускоренности расширения пространства. Параметр Хаббла неявно, но всегда присутствует на диаграммах Хаббла, зависимости R(v), удалённости некого космологического объекта от скорости его удаления от наблюдателя, Земли. Мы не используем диаграммы Хаббла вида R(z), в частности, по причине их двусмысленности. Связь скорости и красного смещения линейна лишь при малых значениях скорости, однако в литературе очень часто встречаются записи, отождествляющие их даже при больших значениях скоростей: v = cz = z.

Закон Хаббла и соответствующие ему диаграммы можно представить в трёх вариантах. Это: а) теоретический, действительный закон, в реальности – экстраполяционный, б) наблюдаемый и в) условный, для начальных удалённостей объектов. Экстраполяционный характер теоретического закона заключается в том, что эта диаграмма строится на основе наблюдений в ближайшей области пространства, после чего неограниченно продляется.

Определить значение параметра Хаббла в прошлом на основе теоретического закона, видимо, невозможно, поскольку все значения параметров на соответствующей ему (теоретической, действительной) диаграмме относятся к текущему моменту времени, в который параметр Хаббла H0 един для всех объектов во Вселенной. При любом прошлом законе изменения параметра Хаббла теоретическая диаграмма для наших дней всегда имеет один и тот же вид.

Напротив, наблюдаемый закон и наблюдаемая диаграмма Хаббла, вероятно, несут в себе некоторую информацию о величине параметра Хаббла в разные эпохи. Вместе с тем, по этой наблюдаемой величине точное определение его прошлого значение представляет достаточно неочевидную задачу. Рассмотрим, например, следующие три условные варианта изменения параметра Хаббла во времени и соответствующие им три разные Вселенные:

Рис.1. Графики изменения во времени параметров Хаббла.

Первый из них, параметр Hd в своей Вселенной убывает от произвольно выбранного значения порядка 2H0 до современного значения H0. Сразу же отметим, что при рассмотрении любого закона изменения параметра Хаббла, его конечное значение обязательно должно быть равно современному значению, поскольку именно это значение мы наблюдаем сегодня в любой из Вселенных. Следовательно, при уменьшении параметра Хаббла во времени, все его прошлые значения определённо могут быть только больше современного. И, наоборот, для возрастающего во времени параметра Хаббла его начальное и все прошлые значения обязательно должны быть меньше современного.

Построим диаграммы Хаббла для расширения Вселенной с каждым из этих параметров, используя выведенный в предыдущих разделах алгоритм определения удалённости объектов с учётом времени в пути света от них до наблюдателей на Земле. Для наглядности изобразим диаграммы с приведёнными на рис.1 параметрами на одном рисунке: Ro для равномерно расширяющейся Вселенной с параметром H0; Rd для Вселенной, расширяющейся замедленно с убывающим параметром Хаббла Hd; диаграмму Ra для Вселенной с параметром Ha, возрастающим от некой произвольной малой величины до современного значения, и теоретическую диаграмму Хаббла R(v):

Рис.2. Наблюдаемые диаграммы Хаббла для Вселенных с параметрами H0, Hd и Ha.

На рисунке чёрный график R(v) – это слившиеся в один графики теоретического закона Хаббла для Вселенной с H0 и теоретических графиков законов Хаббла для Вселенной с убывающим, Hd и с возрастающим, Ha параметрами Хаббла. Это слияние индивидуальных теоретических графиков Хаббла, собственно говоря, достаточно очевидно, поскольку является следствием того, что в конечной точке, в наши дни все параметры равны H0, и диаграммы соответствуют именно этому моменту времени, когда во всей Вселенной объекты удаляются друг от друга именно с этим параметром Хаббла. При этом отметим некоторую условность: мы принимаем, что скорость источника фотонов равна скорости удаления сверхновой на момент наблюдения фотонов.

Как видим, взаимное положение графиков, диаграмм Хаббла сохранилось, но теперь ни один из них не является прямой линией, как на рис.1 и как на стандартной диаграмме. При этом замечаем, что графически при некоторой произвольной скорости сверхновой параметр Хаббла H0 для красного графика больше, чем для зелёного Ha, и меньше, чем для синего Hd. Говоря иначе, график с H0 на рисунке расположен ниже графика с Ha и выше графика Hd:

Эти неравенства соответствуют изначально принятым нами условиям. Отмеченные обстоятельства, соотношения мы определённо можем трактовать так, будто при замедленном расширении Вселенной объекты находятся к наблюдателю ближе и, соответственно, видны более яркими. Напротив, при ускоренном расширении Вселенной далёкие галактики, сверхновые видны менее яркими, чем при обычном, равномерном расширении Вселенной, и, тем более, при её замедленном расширении. Иначе говоря, пониженная яркость дальних сверхновых при таком подходе позволяет однозначно идентифицировать, определить ускоренное расширение Вселенной. На рис.2 для трёх разных Вселенных представлены только диаграммы Хаббла, поэтому для более детального анализа нанесём на этот рисунок дополнительные графики: времени вспышки и начальной удалённости сверхновых. Штриховые вертикальные линии соответствующего цвета помечают время вспышки самой дальней, самой старой сверхновой для каждой из рассматриваемых Вселенных – 14 млрд. лет назад.

Рис.3. Наблюдаемые диаграммы Хаббла, графики времени и начальных удалённостей сверхновых для Вселенных с параметрами H0, Hd и Ha.

Теперь на рисунке можно заметить отмеченную выше небольшую неточность в интерпретации пары параметров яркость-скорость (удалённость), в формулировке "тусклая, поэтому более далёкая". Правильнее всё-таки говорить "яркая, но медленная", то есть, хотя сверхновая более яркая, но движется она медленнее, чем это следует из стандартного закона Хаббла, его диаграммы. Из этого сразу же следует, что с момента вспышки в ускоренно расширяющейся Вселенной сверхновая удалилась на меньшее расстояние и, соответственно, видна более яркой, чем такая же сверхновая в равномерно или замедленно расширяющейся Вселенной. Пониженная яркость ускоренно удаляющейся сверхновой является кажущейся, поскольку вызвана тем, что она изначально находилась на большем удалении от наблюдателя. Малая скорость её удаления и кажущееся большее расстояние до наблюдателя согласно стандартному закону Хаббла вызвана её действительно меньшей скоростью за всю историю движения. А пониженная яркость вызвана её изначально большему удалению.

Действительно, сравним две сверхновые: в ускоренной и равномерной Вселенных, вспыхнувшие в один и тот же момент времени Ta= To= T12= 12 млрд. лет назад. По графикам на рис.3 видим, что ускоренная сверхновая в момент вспышки находилась на удалении r12 ~ 10,5 млрд. световых лет, а в момент наблюдения – на удалении R12 = 12 млрд. световых лет. Следовательно, за 12 млрд. лет сверхновая "прошла путь", равный  R12 – r12 ~ 1,5 млрд. световых лет. Соответственно, находим для сверхновой в равномерно расширяющейся Вселенной: r12=6, R12=10,5, откуда R12 – r12 ~ 4,5 млрд. световых лет. В ускоренно расширяющейся Вселенной сверхновая оказалась на большем удалении, поскольку она изначально находилась дальше, хотя и "прошла" меньший путь.

1. Наблюдаемые диаграммы Хаббла

Наблюдаемыми диаграммами мы называем диаграммы, построенные на основе наблюдаемых яркостей и скоростей сверхновых, которые явно зависят от времени в пути света от них. Отметим, что наблюдаемые параметры Хаббла, согласно этим диаграммам, будут отличаться от исходных, действительных параметров, изображённых на рис.1. На рис.3 каждой точке диаграмм R соответствуют значения скорости и времени, следовательно, мы можем по точкам этих графиков построить и графики H(t)=v(t)/R(t). Но можно использовать и те же таблицы данных, по которым построены эти диаграммы R(v). Правильным способом построения параметров Хаббла в этом случае является дифференциальный, согласно уравнениям:

(1)

Для построения воспользуемся линиями трендов, позволяющими заменить графические построения аналитическим, на основе уравнений этих линий, что заметно упрощает процедуру. Поскольку у нас есть уравнения R(v), используем первое выражение (1), найдя производную R'(v), после чего производим параметрическое построение наблюдаемых параметров H(t), используя v как параметр и учитывая, что H(t)=1/R'(t).

Рис.4. Наблюдаемые параметры Хаббла в трёх разных Вселенных.

Обращаем внимание: хотя исходные законы изменени параметров Хаббла разнонаправленны (возрастание, убывание, рис.1), наблюдаемые законы их изменения на рис.4 все убывающие. Формально это означает замедленное расширение Вселенной, просто с возрастающим параметром Хаббла мы наблюдаем менее быстрое замедление расширения.

2. Параметр Хаббла с изгибом

Проведённые выше исследования, таким образом, согласуются с известным утверждением, что при монотонном изменении параметров Хаббла пониженная яркость дальних сверхновых соответствует более медленному расширению Вселенной в прошлом, то есть, её ускоренному расширению в наши дни. Иначе говоря, монотонное уменьшение или рост параметра Хаббла за время существования Вселенной приводит к соответствующим наблюдательным данным об его уменьшении или росте. Напомним, что рассмотренные монотонные параметры Хаббла условные.

Вместе с тем, пока неясно, что покажут наблюдения при "ломаном" изменении параметра Хаббла: его замедлении с последующим ростом и наоборот.

Для того чтобы выяснить, какими могут быть наблюдения Вселенной с таким ломаным параметром Хаббла, расширявшейся сначала замедленно, а затем ускоренно, вновь спроектируем ещё одну функцию изменения во времени соответствующего параметра Хаббла для некоторой условной Вселенной.

Требуемый параметр Хаббла в простейшем, условном виде скомбинируем из линейных отрезков, исходя из трёх контрольных точек. Первая – это начальное значение параметра Хаббла, вторая – точка изменения направления роста и третья – современное значение параметра.

Точку излома установим в 8 млрд. лет от начала расширения, поскольку считается, что именно тогда началось ускоренное расширение нашей Вселенной, примерно 6 млрд. лет назад. Один из вариантов такого параметра Хаббла с изломом может иметь, например, вид, представленный на врезке на следующем рисунке. На рисунке приведены диаграммы Хаббла – графики движения R(v) сверхновых в двух разных Вселенных: условной, расширяющейся с указанным параметром Хаббла Hda (замедление – ускорение), и нашей с современным параметром H0. Повторим, что все представленные на рисунке параметры – условные, подобраны по смыслу и не являются реальными данными наблюдений:

Рис.5. Диаграммы Хаббла для Вселенных с H0 (штриховые) и Hda. На врезке приведён параметр Хаббла Hda для условной Вселенной, расширяющейся сначала замедленно, затем ускоренно.

Скорость, обозначенная как Vinv ~ 1,45с – это скорость, при которой яркость дальних сверхновых изменяется с повышенной (раньше) на пониженную (позже от начала расширения). Инверсия произошла через 1,5 млрд. лет после начала расширения, что видно по графикам Tо, Tda. Точка ΔRmax – это точка, когда условное расстояние между ускоренной и равномерно удаляющейся сверхновой – максимально. Здесь это ~ 5 млрд. лет от начала расширения, скорость удаления сверхновой ~ 0,9с.

На удалённостях ближе 12,5 млрд. световых лет сверхновые видны более тусклыми, они дальше. На ещё большей удалённости – более яркими. Этот момент времени, 12,5 млрд. лет назад для наблюдателя, на графиках является ничем не примечательной точкой, хотя в этот момент замедленное расширение сменилось ускоренным. На удалении 9 млрд. световых лет разница яркости сверхновых максимальна.

Используя рассмотренный выше алгоритм, построим по линиям трендов наблюдаемые графики изменения во времени параметров Хаббла:

Рис.6. Зависимости от времени параметров Хаббла в ускоренной и равномерной Вселенной

Хотя излом, перегиб на рассмотренном параметре Хаббла выглядит несколько неестественно, рассмотренный вариант с его плавным изменением, тем не менее, привёл к графикам, не имеющим принципиальных отличий от рассмотренных.

Отметим это ещё раз явно: рассмотренные параметры Хаббла условно наблюдаемые, то есть, построены на основе вымышленных наблюдений без каких-либо космологических корректировок, без явного учёта времени в пути света от вспышек сверхновых.

На приведённых диаграммах мы видим, что в прошлом ускоренно двигавшиеся дальние сверхновые в наши дни видны более тусклыми. Действительно, их скорости соответствуют большей удалённости. Напротив, ещё более далёкие сверхновые видны более яркими, чем это следовало бы из стандартного закона Хаббла. Однако, на что следует обратить внимание, ближние сверхновые и галактики видны практически неразличимо одинаково яркими, независимо от их скорости в прошлом. Вывод об ускоренном расширении сделан на сопоставлении параметров движения дальних сверхновых с параметрами ближайших к нам. Бесспорно, что вплоть до наших дней Вселенная расширялась ускоренно. Но можно ли утверждать, что и ныне эта тенденция сохранена?

Мы умышленно сформировали такой закон изменения параметра Хаббла рис.5, на врезке, что с ним в наши дни Вселенная расширяется ускоренно. Несмотря на это, на рис.6 мы видим, что наблюдаемые параметры Хаббла для обоих Вселенных – ниспадающие, отличные от диаграмм на вставке рис.5, что формально следовало бы трактовать как замедленное расширение в обоих случаях.

Строго говоря, утверждать ускоренность или замедленность расширения Вселенной следует на основе данных именно по начальному участку диаграмм, по изменению скорости расширения в наши дни. Вместе с тем, на основе полученных достаточно логичных и убедительных результатов мы определённо имеем право вновь повторить сделанный ранее вывод:

Пониженная яркость дальней сверхновой по сравнению с её яркостью в равномерно расширяющейся Вселенной является свидетельством более быстрого расширения Вселенной в настоящее время, начиная со времени порядка 6 млрд. лет назад.

Вместе с тем наблюдаемые законы изменения параметров Хаббла в обоих случаях – ниспадающие. Это выглядит как, наоборот, замедление расширения Вселенной. Кроме того эти ускоренные сверхновые видны на самом деле более яркими, чем они были бы видны во Вселенной с неизменным параметром Хаббла, они к наблюдателю ближе.

3. Проблема сверхсветового движения

Графики движения сверхновых со сформированным выше параметром Хаббла ускоренной Вселенной можно представить в следующем виде:

Рис.7. Графики движения сверхновой и света от неё в ускоренно расширяющейся Вселенной

На рисунке R(t) – это удалённость по времени сверхновой от Земли (или наоборот). Скорость разбегания сверхновой и Земли – V(t), на рисунке для лучшей видности она увеличена в 10 раз. Графики Rф и Rco – это графики движения фотонов от сверхновой – реальная удалённость и наблюдаемая. На табличке в центре показано, что отношение скорости удаления сверхновой Vda=V(t) к её наблюдаемой удалённости Rco, определяемой по яркости, даёт значение параметра Хаббла Hda = 0,13, что почти в 2 раза превышает значение Ho и явно не соответствует астрономическим наблюдениям.

Кроме этого замечаем, что на рис.2, рис.3, рис.5 и данном, рис.7 наблюдаемая скорость удаления галактик превышает скорость света. Однако уравнения для связи скорости и красного смещения такой скорости не допускают. В чём же дело? Мы видим, что фотоны явно достигли наблюдателя – траектории Земли и фотонов, графики R(t) и Rф пересеклись в наши дни, но при этом их физический источник, сверхновая удаляется со скоростью V(t), превышающей скорость света, уравнения для эффекта Доплера неприменимы.

Проблема возникла вследствие того, что мы молчаливо приняли, будто скорость источника фотонов вспышки в момент наблюдения равна скорости удаления наблюдаемой сверхновой в этот же момент.

Однако скорость области пространства, соответствующего удалённости реального источника, соответствующей его наблюдаемой яркости, определённо не равна удалённости галактики. Мы выяснили, что галактика в момент наблюдения фотонов находится существенно дальше, чем путь, пройденный фотонами от неё до наблюдателя. Следовательно, нам следует учитывать не скорость удаляющейся в этот момент галактики, а скорость точки пространства, находящейся на удалении, равном пути, пройденному фотонами.

Согласно разработанным нами уравнениям движения графики движения самой дальней наблюдаемой сверхновой с параметром Хаббла Hda имеют вид рис.7. Учитывая указанное обстоятельство, на этот рисунок нам следует добавить ещё два графика "дистанция-скорость", как показано на рис.8.

Согласно этим графикам "исходная" удалённость "действительного" источника фотонов меньше, чем исходная удалённость наблюдаемой сверхновой, а его скорость даже от самой дальней сверхновой теперь уже не превышает скорости света. В этом случае может возникнуть желание определять скорость такого действительного источника фотонов как разность скоростей. Действительно, Земля удаляется от сверхновой со скоростью V(t), а источник фотонов – со скоростью Vc. Следовательно, Земля от источника удаляется вроде бы с разностной скоростью: V(t) – Vc.

Рис.8. Графики движения действительного "источника" фотонов.

Однако, во-первых, наблюдаемой величиной является только Vc, а скорость V(t) для наблюдения недоступна, она сверхсветовая. Во-вторых, такой разностный подход сразу же приводит к труднообъяснимой, даже абсурдной ситуации. При вычислениях наблюдаемый параметр Хаббла Hda в этом случае стремительно уменьшается по мере приближения точки вспышки сверхновой к Земле с 0,06 (самая дальняя сверхновая) до 0,0007 (самая ближняя сверхновая). Как видно на рис.8:

Но является ли в этом случае отношение наблюдаемых величин Vc/Rco реальным значением параметра Хаббла? Вернее, можно ли принять, что мы наблюдаем яркость именно по действительной удалённости источника фотонов Re(t)=Rco и его скорости Vc? Да, дело обстоит именно так.

Рис.9. Обоснование графиков движения действительного "источника" фотонов.

Момент этот довольно тонкий, поэтому рассмотрим его как можно более детально. Следует признать, что вопрос этот довольно сложный как для восприятия, так и для доказательного изложения. Для его более наглядного рассмотрения нанесём на рис.8а несколько дополнительных декоративных элементов: схематичные изображения Земли, сверхновой и фотонов, рис.9.

На рисунке показано, что в начальный момент времени, в момент начала расширения Вселенной t = 0 звезда, готовая стать сверхновой, и Земля находятся в точках с координатами 0 и r0 ~ 5,6 млрд. световых лет. В этом положении звезда изображена приглушённым серым цветом – она ещё не вспыхнула. Поскольку параметр Хаббла Hda в этот момент не равен нулю, Земля имеет начальную скорость относительно звезды (и, соответственно, наоборот), уже превышающую скорость света: v0 ~ 1,1c. Синяя линия, график R(t) показывает текущую удалённость Земли от сверхновой, а линия V(t) горчичного цвета – скорость этого удаления в каждый момент времени. Как видим, в наши дни с этим условным параметром Хаббла Hda расстояние между Землёй и рассматриваемой сверхновой равно почти 22 млрд. световых лет, а их относительная скорость превышает скорость света в 1,5 раза.

В рассматриваемом примере через 1 млрд. лет после начала расширения Вселенной звезда вспыхнула, превратившись в сверхновую. На рисунке в этой точке пространства звезда обозначена теперь уже красно-жёлтым всполохом. Красные волнистые короткие линии со стрелками изображают в каждый момент времени положение одной и той же группы фотонов, испущенных сверхновой. График их движения изображён красной линией Rф удалённости от сверхновой. Отметим, что реальная скорость удаления фотонов от сверхновой превышает скорость света, и в момент прихода к Земле составляет почти 2,5 скорости света. Это не противоречит положениям специальной теории относительности, поскольку, как это традиционно трактуется, "разбегание" Земли и сверхновой не является движением физическим, это кажущееся движение, вызванное расширением пространства между ними.

Противоречие заключается в другом. В момент получения фотонов на Земле они удаляются от своего физического источника, от сверхновой со сверхсветовой скоростью v > c = 1. Из этого следует, что уравнения для эффекта Доплера применить мы не можем. Невозможно подставить в уравнение скорость, превышающую скорость света, поскольку под корнем появляется отрицательная величина:

Не существует, наоборот, также и значений красного смещения z, при котором могла быть получена скорость v, превышающая скорость света, поскольку при любом z величина дроби меньше единицы:

Иначе говоря, мы не имеем права использовать реальную скорость удаления сверхновой от Земли в качестве скорости источника излучения в эффекте Доплера. Поэтому мы не имеем права использовать эту скорость и для определения наблюдаемого параметра Хаббла Hda, поскольку определить скорость v(t) по красному смещению мы не можем физически, следовательно:

Однако здесь вступает в силу выявленная ранее особенность движения фотонов в расширяющемся пространстве. Поскольку пространство расширяется не только перед фотонами, но и позади них, реальный пройденный фотонами путь от сверхновой до Земли меньше, чем конечное расстояние между последними. Следовательно, красный график движения фотонов Rф(t) не является реальным графиком их движения, он не отражается действительной, физической протяжённости, длины пути, пройденного фотонами.

Как мы определили в предыдущих разделах, действительным графиком пути является светоподобная линия, график Rco. Следовательно, в момент получения на Земле фотонов от сверхновой они прошли меньший путь. В случае рис.9 он составил не 21,5 млрд., а только 13 млрд. световых лет, как указывает конечная точка графика Rco.

С другой стороны, это означает также, что за 13 млрд. лет, времени расширения пространства с момента вспышки сверхновой, в точку встречи "прибыл" и получатель фотонов – Земля. Другими словами, до точки встречи с Землёй фотоны прошли путь 13 млрд. световых лет. Но это также означает, что фактический, реальный источник фотонов в момент излучения находился теперь уже не в точке нахождения сверхновой, а на таком же удалении от Земли.

Если трактовать ситуацию буквально, то с момента этой реальной "вспышки", начала движения фотонов, Земля двигалась 13 млрд. лет и прошла от источника фотонов этот самый путь – 13 млрд. световых лет. Ввиду изотропности Вселенной, все эти 13 млрд. световых лет Земля двигалась с параметром Хаббла Hda, точно так же, как и реальный, фактический источник фотонов удалялся от неё с этим же параметром. Развернув течение времени в обратную сторону, мы можем по конечному значению взаимной удалённости определить и начальную точку ri в пространстве этого источника фотонов в момент времени t = 0. Нужно всё-таки понимать, что на самом деле это некий фиктивный, не физический источник, он даже не совпадает в пространстве с местоположением сверхновой. Его появление и проявление связано не с физическим движением участников, а с движением вследствие расширения пространства.

Итог такой инверсии времени показан на рис.9. Обращаем внимание, что фиктивный источник в момент времени t = 0 фотонов не испускал. Как видно на рисунке, на момент начала расширения пространства источник находится не в точке вспышки ri ~ 3,5, а в точке, удалённой от сверхновой на 4 млрд. световых лет. Именно от этой точки фотоны и начали свой путь к Земле длительностью 13 млрд. лет.

Выглядит, конечно, весьма странно: источник фотонов не совпадает со сверхновой в момент её вспышки. Ещё раз подчеркнём это важное обстоятельство. В момент начала пути от указанного источника фотон "видит вдали" силуэт Земли. Земля непрерывно удаляется от фотона, но при встрече с нею одометр (измеритель пути) фотона покажет именно дистанцию Rco = 13, пройденную им за эти годы. Инверсия времени в пути (13 млрд. лет) даст нам начальную удалённость источника от Земли, а инверсия всего времени от начала расширения Вселенной (14 млрд. лет) даст нам удалённость этого же фиктивного источника от сверхновой ri, что хорошо видно на рис.9. Но и эта точка, и точка испускания фотонов лежат на одной и той же кривой – на Rc(t). Следовательно, первая из них за 14 млрд. лет, а вторая за 13 млрд. лет придут в одну и ту же конечную точку пространства: на рис.9 это 13 млрд. световых лет. Ещё раз, выглядит это, несомненно, довольно странно: к моменту встречи с Землёй фотоны прошли путь до неё такой же, на какой они удалились от источника - сверхновой, но при этом Земля оказалась от сверхновой на существенно большем реальном удалении – 21,5 млрд. св. лет.

Из этого странного, но строго логичного обстоятельства и следует главный вывод. Поскольку нам известна начальная удалённость ri источника от сверхновой, источника фиктивного, но ставшего впоследствии, через 1 млрд. лет реальным, мы в изотропной Вселенной можем вычислить также и vi – скорость удаления источника от сверхновой. Но фотоны, в конце концов, достигли Земли, следовательно, в этот момент эта скорость vi является также и скоростью Земли относительно источника фотонов – сверхновой, поскольку их движение описывается теми же уравнениями расширения пространства, но за более короткий период.

Здесь следует признаться, что приведённые рассуждения кажутся нам всё-таки довольно сложными и выглядящими недостаточно полными, убедительными, поэтому приведём дополнительные пояснения.

Наши вычисления о пройденном фотонами пути относятся к системе отсчёта, которую можно называть фиктивной системой отсчёта источника. Можно сказать иначе: к системе отсчёта фиктивного источника. В этой системе учитывается путь, пространство, удлиняющиеся только перед движущимся фотоном. Следовательно, в момент встречи фотона с Землёй оба они находятся на реальном удалении от источника, равном пути, пройденному фотоном, в данном случае – 13 млрд. световых лет.

Исходя из этого, эквивалентно получается, что это именно Земля удалилась от источника в результате расширения пространства с соответствующим параметром Хаббла – H(t). Следовательно, в момент получения фотона на Земле она приобретёт скорость, отвечающую этому движению, то есть скорость приёмника относительно источника, используемую в расчётах эффекта Доплера. Особо отметим, что это не приращение скорости Земли, приёмника, а её абсолютное значение относительно источника. Действительно, в момент "отправки" фотона от источника их начальные скорости равны, поэтому их можно принять равными нулю. В момент "встречи" с Землёй область пространства с фотоном, приёмник имеет отмеченную абсолютную скорость относительно её положения в момент излучения.

Исходя из этого, мы имеем все основания произвести ретроспективу времени и найти условное начальное положение Земли относительно источника, как эквивалент начального положения движущейся области с фотоном. Для удобства, удобства графического изображения находим положение области в момент начала расширения Вселенной, то есть, не 13, а 14 млрд. лет назад - ri. Это допустимо, поскольку обе эти точки находятся на одном графике. Теперь мы можем найти и скорость Земли в момент наибольшего её удаления от источника, в наши дни – Vc(14), в момент получения фотона. Ещё раз отметим, что на самом деле от источника удалялся "доплеровский приёмник", а не Земля, которая в конце просто оказалась в нужной точке.

Диаграммы для действительных источников

Исходя из этих данных, наблюдаемые диаграммы Хаббла рис.3 и рис.5 также приобретут иной вид, вид, который соответствует действительным источникам фотонов – вид прямолинейной теоретической диаграммы Хаббла R(v) на рис.3.

В этом случае получается, что даже на основе наблюдаемых параметров сверхновых мы будет получать одну и ту же связку скорость-яркость любой сверхновой, одну и ту же, как для равномерно расширяющейся Вселенной, так и расширяющейся ускоренно или замедленно.

Как видно на рис.8абв, для сверхновой, вспыхнувшей 14, 10 или 6 млрд. лет назад мы получаем одно и то же значение параметра Хаббла:

Буквально это означает, что при таком подходе никакие реальные наблюдения не позволяют нам определить предысторию расширения Вселенной, как она расширялась в прошлом. Однако эти же тождества сразу же поднимают серьёзный вопрос. Они никак не проясняют, почему дальние сверхновые видны более тусклыми, чем это следует из стандартной линейной диаграммы Хаббла, в том числе и из полученных здесь новых диаграмм. На каждой из них, для соответствующего значения параметра Хаббла, точки каждого единичного наблюдения неизбежно должны ложиться на стандартную прямолинейную диаграмму. Никакой пониженной яркости при этом возникнуть не может.

Однако астрономические наблюдения показывают, что дальние сверхновые всё-таки менее яркие. Для объяснения этой пониженной яркости можно сделать достаточно разумное предположение. В процессе движения потока фотонов пространство испытывает помимо продольного расширения также и поперечное. Совершенно очевидно, что удлиняется не только волна излучения, но и расстояние между соседними волнами в пучке света.

Как известно, угловое расширение светового потока позволяет определить удалённость источника света. Действительно, о реальной удалённости объекта, стандартной свечи мы судим по её яркости, которая в неподвижном пространстве уменьшается обратно пропорционально квадрату расстояния.

В неподвижном пространстве отношение площадей фотонных потоков прямо пропорционально квадратам их удалённостей. Но в расширяющемся пространстве эта квадратичная пропорция нарушается, поскольку площадь сечения потока в процессе удаления увеличивается за счёт расширения пространства. Яркость же обратно пропорциональна площади потока, поэтому в расширяющемся пространстве источник виден тем более тусклым по сравнению с квадратичной зависимостью, чем дальше он находится.

Допустим, мы наблюдаем только один фотон, который никакого поперечного хаббловского расширения не имеет. Траекторию такого фотона можно представить как очень длинный пучок прижатых друг к другу струн, трос. Их толщина одна и та же, как в исходной, так и в конечной точках. Определить яркость источника по одному единственному фотону, или по этому пучку "фотонных струн", тросу мы не сможем.

Но если трос расплести, если принять пучок струн-фотонов разбегающимся в стороны, то в наш телескоп попадут тем меньше фотонов, струн или нитей троса, чем дальше от нас их начало. И здесь уже важно, по какому закону эти фотоны удаляются от осевой линии, как резко разлетаются в стороны струны троса. Если траектории фотонов и натянутые струны расходятся, оставаясь прямолинейными, натянутыми, то плотность принятого потока будет одна, обратно квадратичная. Но если по мере удаления расталкивать их в разные стороны принудительно и очень сильно, то плотность принимаемого потока будет существенно ниже. То есть, помимо обычного углового расширения фотонного потока требуется учитывать и его принудительное хаббловское расширение.

Можно эту картину описать и в виде традиционной раздувающейся сферы. В момент вспышки сверхновой от неё начинает удаляться фотонная сфера изначального диаметра. Пусть это будет условный 1 метр. Эта фотонная сфера вблизи сверхновой полностью попадёт в наш телескоп с диаметром входной трубы тоже 1 метр. То есть вблизи от сверхновой мы увидим все эти фотоны полностью, получив максимальную яркость сигнала. Если же телескоп находится на удалении, скажем 100 метров от точки взрыва, то фотонная сфера увеличится в диаметре в соответствии с характером расширения пространства. Следовательно, во входную трубу нашего телескопа попадёт уже меньшее количество фотонов из этой сферы. И этих фотонов будет тем меньше, чем дальше телескоп находится от точки вспышки и чем быстрее сфера раздувается.

Легко догадаться, что диаметр фотонной сферы будет увеличиваться строго по закону расширения пространства, то есть будет иметь, например, значение R(t) рис.9. Но мы на диаграммах показываем именно значение R(t), а не площадь принятой части фотонной сферы, которая имеет значение πr2, где r – диаметр входной трубы телескопа. В предыдущих рассуждениях это обстоятельство мы не учитывали. С учётом же этого фактора вывод о невозможности наблюдения скорости расширения или наблюдения ускорения, сделанные выше, выглядят теперь уже не таким убедительными, их следует серьёзно пересмотреть.

Действительно, в случае учёта поперечного хаббловского расширения пространства, даже при линейном законе расширения H0 = const дальние сверхновые будут видны более тусклыми, что приведёт к искривлению даже прямолинейной диаграммы Хаббла, что даже в этом случае может трактоваться как космологическое ускорение.

Более того, независимо от характера расширения Вселенной: равномерного, ускоренного или замедленного, полученные в последних выкладках диаграммы Хаббла R(v) сразу же приобретают вид, отличный от прямолинейного, они искривляются кверху.

Как следствие, такое отличие реально наблюдаемой криволинейной с изгибом вверх диаграммы Хаббла R(v) от классической прямолинейной не может считаться однозначным свидетельством ускоренного расширения Вселенной. То, что наблюдаемая диаграмма Хаббла R(v) изгибается вверх от прямолинейной теоретической диаграммы, может быть следствием поперечного хаббловского расширения светового потока по мере его удаления от источника.

Можно предположить, что при замедленном расширении пространства эта тенденция также будет наблюдаться, но в меньшей степени. То есть, буквально, наблюдаемая диаграмма Хаббла для замедленно расширяющейся Вселенной будет выглядеть так, будто она расширяется ускоренно.

Иначе говоря, по сравнению со Вселенной с параметром H0 все сверхновые будут видны менее яркими. Вероятно, при этом дальняя сверхновая в ускоренной Вселенной будет видна более тусклой, чем в равномерной и замедленной. Однако мы имеем только один реальный набор измерений, сравнить который не с чем, вследствие чего остаётся лишь один определённый вывод: независимо от характера расширения Вселенной дальние сверхновые видны более тусклыми, чем в пространстве, описываемом прямолинейной диаграммой Хаббла. Любые наблюдения должны показать ускоренное расширение, отклонение диаграммы Хаббла R(v) вверх от теоретической, прямолинейной.

Рис.10. Аналитически-графическое представление связи удалённости и яркости светящегося объекта в разных Вселенных

Вместе с тем в этом случае открытым остаётся ещё один вопрос: почему сверхдальние сверхновые имеют, наоборот, тенденцию к увеличению яркости. Возможная причина – характер расширения Вселенной в этот сверхдальний период резко, инфляционно ускоренный.

Чтобы немного прояснить эти странности, попробуем определить степень влияния на диаграммы поперечного хаббловского расширения пространства.

Обратимся к рис.10, на котором представлен график ускоренного расширения. За 14 млрд. лет фрагмент пространства расширился с 10 до 16 млрд. св. лет, то есть, в 1,6 раза. Это значит, что самая дальняя сверхновая должны быть видна более удалённой, то есть, бледнее, а именно – в 1,6 раза. То есть, самая дальняя сверхновая на рисунке должна находиться не в конечной точке штрихового графика R14, а выше, в конечной точке графика Ra(t), то есть в 1,6 раза дальше.

Напротив, ближняя сверхновая должна точно совпадать с конечной точкой штрихового графика. Видимо, коэффициент умножения можно выявить из выражения:

Здесь, r0(t) и r14 – удалённость от сверхновой в момент вспышки t и в момент наблюдения вспышки в наши дни, а R14 – кажущаяся наблюдаемая удалённость сверхновой, вспыхнувшей в момент t. По сути, этот коэффициент и является действительной удалённостью Rk(t) = Ra(t). Проверим по рисунку уравнения на осмысленность. Для дальней сверхновой:

Для ближайшей сверхновой находим:

И, наконец, для самой дальней:

Получается, что самая дальняя сверхновая видна на большем удалении, чем даже она находится на самом деле. Рассмотрев сверхновые с начальной удалённостью 10, но в момент вспышки находящиеся на разных удалённостях от Земли, получаем для них ориентировочно, "на глазок" следующий график значений действительной, реально наблюдаемой удалённости:

Рис.11. График реальной удалённости сверхновой с учётом времени в пути света от вспышки

Чем дальше в прошлом произошла вспышка сверхновой (вправо, время вспышки от наших дней), тем она бледнее. Этот график построен на основе графиков рис.10. Из использованного для построений выражения:

можно определить и величину коэффициента умножения, коэффициента поперечного хаббловского растяжения пространства:

Максимально возможное значение коэффициента, являющегося коэффициентом увеличения удалённости kR = r14/r0 в зависимости от r0 и r14 удобно определить по рисункам рис.9 и рис.10.

Замечаем, что максимальное значение этого коэффициента зависит только от параметра Хаббла. Коэффициент изменяется от 1 (в наши дни) до максимального значения (в прошлом, 14 млрд. лет назад). Для разных параметров Хаббла, рассмотренных в наших вычислениях, коэффициенты kR имеют максимальные значения:

Для сверхновых, вспыхнувших в более позднее время, этот коэффициент уменьшается до единицы в соответствии с законом изменения параметра Хаббла. Обнаруживаем интересную закономерность. Сильнее всего свет дальних сверхновых ослабляется во Вселенной, расширяющейся замедленно, а слабее всего – в ускоряющейся Вселенной.

Параметры H(t) представлены у нас графически и аналитически, поэтому мы можем вычислить максимальные значения kR для Ho, Hd и Ha также и интегрированием соответствующих функций. Например, для Hd:

Ожидаем, что обнаруженное соотношение коэффициентов приведёт к интересным следствиям. Для следующих построений используем параметры Хаббла, представленные на врезке к следующему рисунку. Используя их, строим диаграммы Хаббла, учитывающие коэффициент kR для поперечного хаббловского расширения светового потока:

Рис.12. Наблюдаемые диаграммы Хаббла с учётом времени в пути света от вспышки сверхновой

В первую очередь на рисунке нас интересует параметр Хаббла Hda, соответствующий принятой модели расширения Вселенной – замедленно, затем ускоренно. При его функциональном проектировании мы исходили из достаточно разумного предположения, что в начальный момент времени этот параметр был равен нулю, после чего он резко увеличился до своего максимального значения, условно через 1 млрд. лет. Далее он уменьшался до момента времени t = 8 млрд. лет, после чего начал расти вплоть до наших дней. Максимальное и минимальное значение параметра – условные.

Обнаруживаем, что, действительно, результат оказался, по меньшей мере, странным. С таким параметром Hda мы не смогли получить диаграмму Хаббла, соответствующую ускоренному расширению. На рис.12 видим, что с параметром Hda дальние сверхновые видны более яркими, а сверхдальние, наоборот, менее яркие. Сам параметр Hda определённо соответствует замедленно - ускоренно расширяющейся Вселенной, но производная от него диаграмма Хаббла явно демонстрирует замедленное расширение, если использовать модель "дальний – тусклый". Буквально это означает, что такая модель не является показателем ускоренного расширения.

Но, возможно, другая форма параметра Hda подтвердит эту модель? Этот параметр Хаббла мы спроектировали параметрическим, то есть, его форму мы можем легко изменить, меняя его 4 настроечных параметра: пик и его время и минимум и его время. Простым подбором этих параметров, наугад пытаемся получить такую его зависимость от времени, чтобы характер Hda изменился на противоположный, на ускоренно - замедленный параметр Had, приводящий к известной диаграмме Хаббла, когда дальние сверхновые выглядят более тусклыми, а сверхдальние – более яркими, чем с параметром Хаббла H0.

Такая корректировка прошла успешно и мы получили требуемый инверсный параметр Hda. Строим диаграмму Хаббла на его основе и обнаруживаем, что с такой зависимостью мы получаем известное соотношение: дальние сверхновые теперь видны более тусклыми, а сверхдальние, напротив, со временем замедляют скорость своего разбегания, рис.13.

Рис.13. Только при ускоренно-замедленном расширении Вселенной с параметром Had дальние сверхновые видны более тусклыми, а сверхдальние – более яркими

Падение параметра Had происходит по выпуклой параболе, что приводит к его резкому уменьшению в наши дни. Предполагаем, что более естественным было бы его падение по экспоненте или гиперболе. Наш регулируемый параметр Hda ~ Had не позволяет скорректировать этот участок, поскольку он функционально задан параболой.

Для простоты сформируем этот параметр Had из прямолинейных отрезков, выбрав 14 интервалов. Такой способ формирования параметра позволяет подогнать его форму под любой вид диаграммы Хаббла. В результате удалось получить диаграмму, соответствующую утверждению об ускоренном расширении Вселенной: тусклые дальние сверхновые и более яркие – сверхдальние. Однако и в этот раз мы обнаруживаем, что сам параметр Хаббла соответствует противоположному характеру расширения Вселенной – замедленному. Вселенная сначала расширяется ускоренно, затем замедленно. Буквально это означает, что в наши дни согласно наблюдениям за тусклыми сверхновыми мы получаем замедленное расширение Вселенной.

Рис.14. При ускоренно-замедленном расширении Вселенной с параметром Had дальние сверхновые видны более тусклыми, а сверхдальние – более яркими. Параметр представлен в виде ломаной линии

Для справки и наглядности приводим полный набор графиков движения самой дальней сверхновой в такой Вселенной, который имеет вид:

Рис.15. Графики движения самой дальней видимой сверхновой в замедленной Вселенной с параметром Had

Определённым недостатком такого построения параметра Хаббла является его статичность. Изменить что-либо в нём непросто – нужно менять все его точки, причём соблюдая результирующую плавность ломаной кривой. Кроме того его ломаный вид и острый пик вызывают некоторое недоверие, сможем ли мы получить похожий результат при других, гладких экстремумах. Поэтому проектируем такой же вид параметра, состоящий из гладких кривых.

В результате мы получили параметрическую кривую с изменяемыми координатами пика и кривизной роста - спада. Кривая сформирована из трёх функций. Начальный участок – парабола переменной степени n = 1,5...5; средний участок – перевёрнутая квадратичная парабола; третий – гипербола переменной степени n = 2...12.

Перебирая регулировочные коэффициенты, формируем требуемый параметр Хаббла Hda – вкладка к рис.16, максимально соответствующий модели тусклых дальних сверхновых, и более ярких – сверхдальних.

Рис.16. При ускоренно-замедленном расширении Вселенной с параметром Had дальние сверхновые видны более тусклыми, а сверхдальние – более яркими. Параметр представлен в виде гладкой кривой

Как видим на рисунке, параметр Хаббла Had соответствует замедленно расширяющейся Вселенной, но диаграмма Хаббла, согласно модели "тусклый – дальний" явно трактуется как ускоренно расширяющаяся Вселенная. Это несоответствие вновь говорит о том, что пониженная яркость дальних сверхновых не может служить основанием для заключения об ускоренном расширении Вселенной.

Рис.17. Графики движения самой дальней видимой сверхновой в замедленной Вселенной с параметром Had в виде гладкой кривой

Этот рисунок без комментариев мы приводим также для справки и наглядности: полный набор графиков движения самой дальней наблюдаемой сверхновой во Вселенной со сформированным параметром Хаббла Had.

4. Спорные выводы об ускоренном расширении

Поскольку параметр Хаббла Hd также имеет настроечные величины для пика и минимума, мы проделали такую же процедуру подбора и с ним. Оказалось, что его регулировкой наугад можно получить вид диаграммы Хаббла, довольно близко приближающийся к виду диаграммы ускоренного расширения Вселенной, рис.18. Отметим, что правильнее было бы параметр Хаббла в этом случае именовать Hada.

Рис.18. При замедленно-ускоренном расширении Вселенной с параметром Had дальние сверхновые видны более тусклыми, а сверхдальние – более яркими

Единственным, неуверенно слабым местом этой диаграммы является участок ближних сверхновых. Хотя и очень слабо, но можно заметить их некоторую повышенную яркость по отношению к стандартной диаграмме Хаббла. Лучше всего это заметно на увеличенном фрагменте диаграммы:

Рис.19. Увеличенный фрагмент диаграммы для замедленно-ускоренного расширения Вселенной с параметром Hda. Ближние сверхновые видны с повышенной яркостью.

Хотя и на пределе точности, но все-таки заметно, что ближние сверхновые при действительно ускоренном расширении Вселенной видны более яркими, чем в стандартной Вселенной, равномерно расширяющейся с параметром Хаббла H0. Отметим, что на диаграмме Вселенной с параметром Хаббла Had такой эффект определённо отсутствует:

Рис.20. Увеличенный фрагмент диаграммы для ускоренно - замедленного расширения Вселенной с гладким параметром Had. Яркость ближних сверхновых не превышает классических значений для Вселенной с параметром H0.

Как видно на рисунке, ближние сверхновые в такой Вселенной всегда не ярче, они заметно дальше сверхновых в стандартной Вселенной с H0.

В заключение рассмотрим ещё один вариант. Каким должен быть параметр Хаббла, чтобы наблюдаемая диаграмма выглядела классически – прямолинейной, с наклоном, соответствующим параметру Хаббла H0. Как оказалось, этому условию отвечает стационарная Вселенная, расширение в которой началось буквально вчера (штриховая линия Hs) – рис.21.

На рисунке показано, что рассмотренная диаграмма Хаббла "замедленно - ускоренной" Вселенной также будет получена по наблюдениям во Вселенной с параметрами Хаббла, имеющими вид, подобный показанным на врезке рисунка. На всём протяжении существования такой Вселенной значения параметров были не только близки к нулю, но были даже отрицательными. То есть, большее время, почти 11 млрд. лет Вселенная не расширялась, а сжималась. И только в последние 1,5 млрд. лет началось её стремительное расширение.

Рис.21. Диаграммы Хаббла, наблюдаемые во Вселенных с параметрами H0, Ha и Hs

Следует отметить, что приведённая на рисунке прямолинейная диаграмма с параметром Хаббла Hs (индекс s обозначает стационарность) выглядит как классическая диаграмма, так, будто Вселенная все 14 млрд. лет расширялась с неизменным параметром Хаббла H0. Но на врезке видно, что это не так, лишь через 13 млрд. лет после рождения Вселенной параметр Hs испытал резкий скачок, всплеск. Простым логическим анализом не удалось выявить с определённостью причину такого расхождения. Действительно, свет от самой дальней наблюдаемой сверхновой прошёл путь в 14 млрд. св. лет, но до последнего момента скорость источника, звезды была нулевой. В момент регистрации света, всего за 1 млрд. лет скорость получателя, Земли возросла почти до скорости света. Но тогда на этом интервале времени свет и от других, ближних сверхновых должен достичь Земли. Как получается, что относительно дальней сверхновой скорость удаления Земли выше, чем относительно ближних? Достаточно разумным объяснением является, вероятно, следующее. На "набор" нужной относительной скорости как раз и повлиял этот последний интервал ускорения расширения пространства. Каждая из сверхновых – ближняя и далёкая – в момент скачка параметра Хаббла определённо находились на разных расстояниях от Земли, иначе они не достигли бы её одновременно. Поэтому ближние двигались на поздних этапах расширения и набрали, соответственно, меньшую скорость. Самая дальняя же "захватила" период скачка полностью и, следовательно, набрала более высокую скорость.

Но особенность этой модели заключается и в том, что известный вид диаграммы "тусклый - яркий" присущ также и Вселенной, расширяющийся в наши дни замедленно:

Рис.22. Диаграммы Хаббла, наблюдаемые во Вселенных с параметрами H0, Hd и Hs

Вновь заметим, что в этом случае параметр Хаббла следовало бы обозначить как Had. На рисунках рис.21 и рис.22 видно, что за всё время существования Вселенной она расширялась фактически с ускоренным параметром Хаббла, кроме последнего периода, на рис.22. В последний миллиард лет расширения произошла инверсия – Вселенная расширялась с замедлением. Тем не менее, в обоих случаях диаграммы Хаббла демонстрируют пониженную яркость дальних сверхновых и повышенную – сверхдальних. Участок диаграммы для дальних сверхновых находится выше диаграммы для Вселенной, условно расширяющейся равномерно с параметром Hs. Участок для ещё более далёких сверхновых смещён ниже этой диаграммы.

Для рассмотренного на рис.22 варианта Вселенной приведём полный набор графиков движения самой далёкой наблюдаемой сверхновой:

Рис.23. Графики движения самой дальней видимой сверхновой в замедляющейся Вселенной с параметром Hd

Обращаем внимание, что в соответствии с отрицательным знаком параметра Хаббла такая условная Вселенная на начальном этапе не расширялась, а сжималась, график R(t). Иначе говоря, изначально она обладала отрицательной скоростью v(t), которая в наши дни изменила направление и достигла скорости света. Свет от вспышки сверхновой распространялся практически так же, как в стационарной Вселенной. По графикам мы видим, что пути фотонов – фактический Rco и хаббловский Rф – практически не различаются, поэтому влияние поперечного хаббловского расширения светового потока сводится к обычному закону обратных квадратов.

Наблюдаемые удалённость Rco и скорость Vc также практически совпадают с теоретическими значениями R(t) и v(t), соответственно.

Лишь спустя 8 млрд. лет расширение Вселенной стало заметным, а спустя 12 млрд. лет скорость расширения резко возросла. В последний миллиард лет наша условная Вселенная вновь стала расширяться с замедлением вплоть до наших дней.

Хотя Вселенная выглядит отчасти как стационарная, диаграммы Хаббла отчётливо показывают зависимость красного смещения, скорости сверхновых от их удалённости. Причём осталась в силе метафора "дальние сверхновые более тусклые". И, соответственно: сверхдальние, наоборот, более яркие.

Выходит, что в классическом, прямолинейном виде диаграмма Хаббла R(v) = Rs(v) будет в нашей реальной Вселенной наблюдаема только в случае параметра Хаббла H0(t) = Hs, имеющего форму латинской буквы L, лежащей "на спине". Именно с таким параметром Хаббла диаграмма будет иметь классический вид рис.24:

Рис.24. Наблюдаемые диаграммы Хаббла для Вселенных с параметрами H0 и Hs. Диаграмма Rs соответствует Вселенной, которая большую часть времени была стационарной

На рисунке представлены две диаграммы Хаббла, а на врезке – соответствующие им параметры Хаббла. Обе диаграммы являются наблюдаемыми, результаты измерения скорости и яркости каждой сверхновой в своей Вселенной в точности лягут на соответствующую линию. Однако на самом деле их реальные, действительные удалённости и скорости удаления будут иными. Отметим, что Вселенная с параметром Hs основную часть времени существования является стационарной. Графики движения самой дальней наблюдаемой сверхновой в этой Вселенной показаны на рис.25. Графики соответствуют частично стационарному параметру Хаббла Hs.

Рис.25. Графики движения самой дальней сверхновой во Вселенной с параметром Hs. Основную часть времени Вселенная стационарна

На рисунке видно практически полное совпадение пар параметров: R(t) и Rc (путь фиктивного источника фотонов), Rф (хаббловский путь фотонов) и Rco (измеренный, реальный путь фотонов), v(t) и Vc (скорость фиктивного источника фотонов). Это означает, что наблюдаемые данные (вторые в парах) практически совпадают с теоретическими параметрами, которые соответствуют постоянной Хаббла H0. Буквально это следует трактовать, что классические диаграммы Хаббла соответствуют параметру Hs, а не параметру H0. Именно во Вселенной с параметром Hs будет наблюдаться классическая, прямолинейная диаграмма Хаббла с углом наклона, соответствующим величине H0. Хотя на предыдущей диаграмме видна действительная L-образная форма параметра Хаббла.

Напротив, наблюдаемые скорости и удалённости сверхновых во Вселенной с действительным параметром Хаббла H0 попадут точно на изогнутую вверх диаграмму R0 рисунка рис.24. В этом случае наблюдения покажут иное значение параметра Хаббла, явно отличающееся от реального, от H0.

Выводы

Вывод об ускоренном расширении Вселенной по признаку пониженной яркости дальних сверхновых одинаково справедлив как при ускоренном, так и при замедленном реальном, фактическом расширении Вселенной. О действительной скорости расширения Вселенной нельзя судить только лишь по разнице наблюдаемых параметров Хаббла в далёком прошлом и в наши дни, это ускорение – замедление является кажущимся.

Такая же неопределённая картина наблюдается и в случае первично стационарной Вселенной с L-образным параметром Хаббла Hs. Такая условная Вселенная была стационарной 13 млрд. лет, после чего начала стремительно расширяться до современного значения параметра Хаббла – H0. Иначе говоря, форма параметра Hs напоминает латинскую букву L, лежащую "на спине". При этом по наблюдаемым данным, яркости и красному смещению будет формироваться традиционная, прямолинейная диаграмма Хаббла. Производный от этой диаграммы параметр Хаббла будет неизменным и равным современному значению параметра Хаббла H0 в нашей Вселенной. В действительности же такая Вселенная большее время была стационарной.

И при ускоренном и при замедленном расширении в такой условно стационарной Вселенной отличие диаграмм от "базовой", с L-образным параметром Hs, являющимся эквивалентом параметра H0 нашей Вселенной, точно так же будет наблюдаться эффект "дальняя – более тусклая" и "сверхдальняя – более яркая". Правда, модель ускоренного расширения пространства в такой Вселенной имеет "дефект", пусть и слабый, но всё-таки понижающий её достоверность – это повышенная яркость ближних сверхновых.

Во всех случаях приоритетной моделью, видимо, следует считать модель замедленного расширения Вселенной.

О возрасте Вселенной

Принято считать, что, согласно ретроспективному анализу расширения Вселенной, все галактики сливаются в одну точку в начальный момент времени. Однако это не совсем верно. Если рассмотреть расширение Вселенной в ретроспективе, согласно известному экспоненциальному уравнению, то в начальный момент времени t = 0 галактики (область, в которой эта галактика оказалась впоследствии) не окажутся в некоторой общей нулевой точке. Каждая галактика окажется в этот момент в некоторой своей собственной начальной точке r0:

Это несоответствие должна была устранить гипотеза об инфляции. Согласно ей, сразу после Большого Взрыва Вселенная расширилась до некоторого начального объёма, имеющего по разным мнениям размеры от зёрнышка до апельсина, то есть, все точки пространства тем самым заняли своё "исходное положение" с собственными начальными значениями r0. В частности, по одной из гипотез после Взрыва Вселенная расширилась в 10 в степени 1012 раз, что ничего не говорит явно об её начальном диаметре после инфляции.

Можно обнаружить, что одним из следствий инфляционной гипотезы является наличие конкретной физической, вещественной границы расширяющейся Вселенной, что выглядит весьма и весьма спорно. Действительно, и зёрнышко и апельсин определённо имеют внешнюю границу.

Вместе с тем, рассматриваемые нами диаграммы наталкивают на мысль, что эти диаграммы, графики можно продлить влево от нулевой точки, в отрицательные значения времени. Отрицательное время, конечно, условность, связанная с принудительным заданием нулевой точки, являющейся такой же условностью. Буквально это означает, что до Большого Взрыва все точки бесконечного пространства Вселенной уже существовали. Формально, вопрос "что было до большого Взрыва?" превращается в риторический. Все галактики и звёзды, вернее области пространства, где они возникнут в будущем, в общую ретроспективную точку "сольются" лишь в бесконечно удалённом прошлом. Но бесконечное прошлое – это математическая абстракция, из которой сразу же вытекает отсутствие границы пустого пространства, на которое при этом не требуется налагать никаких условий замкнутости, многомерности.

Согласно такому подходу изначально расширялось вечно существующее материальное пространство пустой Вселенной, а формирование вещественных объектов – звёзд, галактик, туманностей – началось в некоторый момент времени, принятый за нулевой, начальный. Даже если считать, что прошлое простирается в бесконечность, то в некоторый очень отдалённый момент времени t ~ ∞ все точки пространства "пустой" Вселенной также находятся каждая на своём собственном месте, ничтожно малом удалении от некоторого условного наблюдателя.

В самом общем виде уравнение движения некоторого объекта в такой расширяющейся Вселенной можно представить в следующем виде:

До момента времени t = 0 Вселенная постепенно заполняется первичным достаточно холодным "бульоном" из кварков и/или каких-либо других первичных субстанций. Пространство такой Вселенной расширяется за счёт веществолизации материи, непрерывного возникновения всё новых и новых вещественных "атомов" пространства, "атомов" физического вакуума. Лишь спустя длительное время вакуум становится настолько плотным, что формируется известная ныне "горячая" Вселенная. Конечно, вполне допустима и одномоментная, взрывная веществолизация материи. В дальнейшем все процессы в овеществлённой области Материи могут быть описаны традиционной моделью Большого Взрыва, но без инфляции. На следующем рисунке представлен график движения некоторой точки пространства, которая в нулевой момент времени будет иметь координату r0, и в которой в этот момент возникнет некий объект. Излом графика в нулевой точке связан с тем, что левая сторона рисунка имеет обратную степенную шкалу времени, некое подобие логарифмической шкалы, а правая – обычную, линейную. На самом деле при равномерной шкале времени график представляет собой плавную кривую.

Рис.12.01. Расширение Вселенной из бесконечного прошлого

Данная гипотеза не нуждается в мистическом акте Творения, возникновения чего-то из Ничего и последующей инфляции. Данная модель решает ещё одну проблему. Согласно одной из гипотез, за известное время существования Вселенной без тёмной материи в ней не могли возникнуть никакие материальные объекты: газы, звёзды, вещество. В представленной модели для возникновения вещества времени более чем достаточно.

27.07 – 10.12.2020

Сконвертировано и опубликовано на https://SamoLit.com/

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